Theorem: LinearOrderedField.exists_mem_cutMap_mul_self_of_lt_inducedMap_mul_self
File: Mathlib/Algebra/Order/CompleteField.lean

Proof steps:
state_before: F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝³ : LinearOrderedField α
inst✝² : ConditionallyCompleteLinearOrderedField β
inst✝¹ : ConditionallyCompleteLinearOrderedField γ
inst✝ : Archimedean α
a : α
b✝ : β
q : ℚ
ha : 0 < a
b : β
hba : b < inducedMap α β a * inducedMap α β a
⊢ ∃ c ∈ cutMap β (a * a), b < c
state_after: case inl
F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝³ : LinearOrderedField α
inst✝² : ConditionallyCompleteLinearOrderedField β
inst✝¹ : ConditionallyCompleteLinearOrderedField γ
inst✝ : Archimedean α
a : α
b✝ : β
q : ℚ
ha : 0 < a
b : β
hba : b < inducedMap α β a * inducedMap α β a
hb : b < 0
⊢ ∃ c ∈ cutMap β (a * a), b < c

case inr
F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝³ : LinearOrderedField α
inst✝² : ConditionallyCompleteLinearOrderedField β
inst✝¹ : ConditionallyCompleteLinearOrderedField γ
inst✝ : Archimedean α
a : α
b✝ : β
q : ℚ
ha : 0 < a
b : β
hba : b < inducedMap α β a * inducedMap α β a
hb : 0 ≤ b
⊢ ∃ c ∈ cutMap β (a * a), b < c
tactic: obtain hb | hb := lt_or_le b 0
state_before: case inr
F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝³ : LinearOrderedField α
inst✝² : ConditionallyCompleteLinearOrderedField β
inst✝¹ : ConditionallyCompleteLinearOrderedField γ
inst✝ : Archimedean α
a : α
b✝ : β
q : ℚ
ha : 0 < a
b : β
hba : b < inducedMap α β a * inducedMap α β a
hb : 0 ≤ b
⊢ ∃ c ∈ cutMap β (a * a), b < c
state_after: case inr.intro.intro.intro
F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝³ : LinearOrderedField α
inst✝² : ConditionallyCompleteLinearOrderedField β
inst✝¹ : ConditionallyCompleteLinearOrderedField γ
inst✝ : Archimedean α
a : α
b✝ : β
q✝ : ℚ
ha : 0 < a
b : β
hba : b < inducedMap α β a * inducedMap α β a
hb : 0 ≤ b
q : ℚ
hq : 0 < q
hbq : b < ↑q ^ 2
hqa : ↑q ^ 2 < inducedMap α β a * inducedMap α β a
⊢ ∃ c ∈ cutMap β (a * a), b < c
tactic: obtain ⟨q, hq, hbq, hqa⟩ := exists_rat_pow_btwn two_ne_zero hba (hb.trans_lt hba)
state_before: case inr.intro.intro.intro
F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝³ : LinearOrderedField α
inst✝² : ConditionallyCompleteLinearOrderedField β
inst✝¹ : ConditionallyCompleteLinearOrderedField γ
inst✝ : Archimedean α
a : α
b✝ : β
q✝ : ℚ
ha : 0 < a
b : β
hba : b < inducedMap α β a * inducedMap α β a
hb : 0 ≤ b
q : ℚ
hq : 0 < q
hbq : b < ↑q ^ 2
hqa : ↑q ^ 2 < inducedMap α β a * inducedMap α β a
⊢ ∃ c ∈ cutMap β (a * a), b < c
state_after: case inr.intro.intro.intro
F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝³ : LinearOrderedField α
inst✝² : ConditionallyCompleteLinearOrderedField β
inst✝¹ : ConditionallyCompleteLinearOrderedField γ
inst✝ : Archimedean α
a : α
b✝ : β
q✝ : ℚ
ha : 0 < a
b : β
hba : b < inducedMap α β a * inducedMap α β a
hb : 0 ≤ b
q : ℚ
hq : 0 < q
hbq : b < ↑(q ^ 2)
hqa : ↑q ^ 2 < inducedMap α β a * inducedMap α β a
⊢ ∃ c ∈ cutMap β (a * a), b < c
tactic: rw [← cast_pow] at hbq
state_before: case inr.intro.intro.intro
F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝³ : LinearOrderedField α
inst✝² : ConditionallyCompleteLinearOrderedField β
inst✝¹ : ConditionallyCompleteLinearOrderedField γ
inst✝ : Archimedean α
a : α
b✝ : β
q✝ : ℚ
ha : 0 < a
b : β
hba : b < inducedMap α β a * inducedMap α β a
hb : 0 ≤ b
q : ℚ
hq : 0 < q
hbq : b < ↑(q ^ 2)
hqa : ↑q ^ 2 < inducedMap α β a * inducedMap α β a
⊢ ∃ c ∈ cutMap β (a * a), b < c
state_after: case inr.intro.intro.intro
F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝³ : LinearOrderedField α
inst✝² : ConditionallyCompleteLinearOrderedField β
inst✝¹ : ConditionallyCompleteLinearOrderedField γ
inst✝ : Archimedean α
a : α
b✝ : β
q✝ : ℚ
ha : 0 < a
b : β
hba : b < inducedMap α β a * inducedMap α β a
hb : 0 ≤ b
q : ℚ
hq : 0 < q
hbq : b < ↑(q ^ 2)
hqa : ↑q ^ 2 < inducedMap α β a * inducedMap α β a
⊢ ↑(q ^ 2) < a * a
tactic: refine ⟨(q ^ 2 : ℚ), coe_mem_cutMap_iff.2 ?_, hbq⟩
state_before: case inr.intro.intro.intro
F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝³ : LinearOrderedField α
inst✝² : ConditionallyCompleteLinearOrderedField β
inst✝¹ : ConditionallyCompleteLinearOrderedField γ
inst✝ : Archimedean α
a : α
b✝ : β
q✝ : ℚ
ha : 0 < a
b : β
hba : b < inducedMap α β a * inducedMap α β a
hb : 0 ≤ b
q : ℚ
hq : 0 < q
hbq : b < ↑(q ^ 2)
hqa : ↑q ^ 2 < inducedMap α β a * inducedMap α β a
⊢ ↑(q ^ 2) < a * a
state_after: case inr.intro.intro.intro
F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝³ : LinearOrderedField α
inst✝² : ConditionallyCompleteLinearOrderedField β
inst✝¹ : ConditionallyCompleteLinearOrderedField γ
inst✝ : Archimedean α
a : α
b✝ : β
q✝ : ℚ
ha : 0 < a
b : β
hba : b < inducedMap α β a * inducedMap α β a
hb : 0 ≤ b
q : ℚ
hq : 0 < q
hbq : b < ↑(q ^ 2)
hqa : ↑q * ↑q < inducedMap α β a * inducedMap α β a
⊢ ↑(q * q) < a * a
tactic: rw [pow_two] at hqa ⊢
state_before: case inr.intro.intro.intro
F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝³ : LinearOrderedField α
inst✝² : ConditionallyCompleteLinearOrderedField β
inst✝¹ : ConditionallyCompleteLinearOrderedField γ
inst✝ : Archimedean α
a : α
b✝ : β
q✝ : ℚ
ha : 0 < a
b : β
hba : b < inducedMap α β a * inducedMap α β a
hb : 0 ≤ b
q : ℚ
hq : 0 < q
hbq : b < ↑(q ^ 2)
hqa : ↑q * ↑q < inducedMap α β a * inducedMap α β a
⊢ ↑(q * q) < a * a
state_after: case inr.intro.intro.intro
F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝³ : LinearOrderedField α
inst✝² : ConditionallyCompleteLinearOrderedField β
inst✝¹ : ConditionallyCompleteLinearOrderedField γ
inst✝ : Archimedean α
a : α
b✝ : β
q✝ : ℚ
ha : 0 < a
b : β
hba : b < inducedMap α β a * inducedMap α β a
hb : 0 ≤ b
q : ℚ
hq : 0 < q
hbq : b < ↑(q ^ 2)
hqa : ↑q * ↑q < inducedMap α β a * inducedMap α β a
⊢ ↑q * ↑q < a * a
tactic: push_cast
state_before: case inr.intro.intro.intro
F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝³ : LinearOrderedField α
inst✝² : ConditionallyCompleteLinearOrderedField β
inst✝¹ : ConditionallyCompleteLinearOrderedField γ
inst✝ : Archimedean α
a : α
b✝ : β
q✝ : ℚ
ha : 0 < a
b : β
hba : b < inducedMap α β a * inducedMap α β a
hb : 0 ≤ b
q : ℚ
hq : 0 < q
hbq : b < ↑(q ^ 2)
hqa : ↑q * ↑q < inducedMap α β a * inducedMap α β a
⊢ ↑q * ↑q < a * a
state_after: case inr.intro.intro.intro.intro.intro
F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝³ : LinearOrderedField α
inst✝² : ConditionallyCompleteLinearOrderedField β
inst✝¹ : ConditionallyCompleteLinearOrderedField γ
inst✝ : Archimedean α
a : α
b✝ : β
q✝ : ℚ
ha : 0 < a
b : β
hba : b < inducedMap α β a * inducedMap α β a
hb : 0 ≤ b
q : ℚ
hq : 0 < q
hbq : b < ↑(q ^ 2)
hqa : ↑q * ↑q < inducedMap α β a * inducedMap α β a
q' : ℚ
hq' : ↑q < ↑q'
hqa' : ↑q' < a
⊢ ↑q * ↑q < a * a
tactic: obtain ⟨q', hq', hqa'⟩ := lt_inducedMap_iff.1 (lt_of_mul_self_lt_mul_self
    (inducedMap_nonneg ha.le) hqa)
state_before: case inr.intro.intro.intro.intro.intro
F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝³ : LinearOrderedField α
inst✝² : ConditionallyCompleteLinearOrderedField β
inst✝¹ : ConditionallyCompleteLinearOrderedField γ
inst✝ : Archimedean α
a : α
b✝ : β
q✝ : ℚ
ha : 0 < a
b : β
hba : b < inducedMap α β a * inducedMap α β a
hb : 0 ≤ b
q : ℚ
hq : 0 < q
hbq : b < ↑(q ^ 2)
hqa : ↑q * ↑q < inducedMap α β a * inducedMap α β a
q' : ℚ
hq' : ↑q < ↑q'
hqa' : ↑q' < a
⊢ ↑q * ↑q < a * a
state_after: no goals
tactic: exact mul_self_lt_mul_self (mod_cast hq.le) (hqa'.trans' <| by assumption_mod_cast)