Theorem: BoxIntegral.Box.le_TFAE
File: Mathlib/Analysis/BoxIntegral/Box/Basic.lean

Proof steps:
state_before: ι : Type u_1
I J : Box ι
x y : ι → ℝ
⊢ [I ≤ J, ↑I ⊆ ↑J, Icc I.lower I.upper ⊆ Icc J.lower J.upper, J.lower ≤ I.lower ∧ I.upper ≤ J.upper].TFAE
state_after: case tfae_1_iff_2
ι : Type u_1
I J : Box ι
x y : ι → ℝ
⊢ I ≤ J ↔ ↑I ⊆ ↑J

ι : Type u_1
I J : Box ι
x y : ι → ℝ
tfae_1_iff_2 : I ≤ J ↔ ↑I ⊆ ↑J
⊢ [I ≤ J, ↑I ⊆ ↑J, Icc I.lower I.upper ⊆ Icc J.lower J.upper, J.lower ≤ I.lower ∧ I.upper ≤ J.upper].TFAE
tactic: tfae_have 1 ↔ 2
state_before: ι : Type u_1
I J : Box ι
x y : ι → ℝ
tfae_1_iff_2 : I ≤ J ↔ ↑I ⊆ ↑J
⊢ [I ≤ J, ↑I ⊆ ↑J, Icc I.lower I.upper ⊆ Icc J.lower J.upper, J.lower ≤ I.lower ∧ I.upper ≤ J.upper].TFAE
state_after: case tfae_2_to_3
ι : Type u_1
I J : Box ι
x y : ι → ℝ
tfae_1_iff_2 : I ≤ J ↔ ↑I ⊆ ↑J
⊢ ↑I ⊆ ↑J → Icc I.lower I.upper ⊆ Icc J.lower J.upper

ι : Type u_1
I J : Box ι
x y : ι → ℝ
tfae_1_iff_2 : I ≤ J ↔ ↑I ⊆ ↑J
tfae_2_to_3 : ↑I ⊆ ↑J → Icc I.lower I.upper ⊆ Icc J.lower J.upper
⊢ [I ≤ J, ↑I ⊆ ↑J, Icc I.lower I.upper ⊆ Icc J.lower J.upper, J.lower ≤ I.lower ∧ I.upper ≤ J.upper].TFAE
tactic: tfae_have 2 → 3
state_before: ι : Type u_1
I J : Box ι
x y : ι → ℝ
tfae_1_iff_2 : I ≤ J ↔ ↑I ⊆ ↑J
tfae_2_to_3 : ↑I ⊆ ↑J → Icc I.lower I.upper ⊆ Icc J.lower J.upper
⊢ [I ≤ J, ↑I ⊆ ↑J, Icc I.lower I.upper ⊆ Icc J.lower J.upper, J.lower ≤ I.lower ∧ I.upper ≤ J.upper].TFAE
state_after: case tfae_3_iff_4
ι : Type u_1
I J : Box ι
x y : ι → ℝ
tfae_1_iff_2 : I ≤ J ↔ ↑I ⊆ ↑J
tfae_2_to_3 : ↑I ⊆ ↑J → Icc I.lower I.upper ⊆ Icc J.lower J.upper
⊢ Icc I.lower I.upper ⊆ Icc J.lower J.upper ↔ J.lower ≤ I.lower ∧ I.upper ≤ J.upper

ι : Type u_1
I J : Box ι
x y : ι → ℝ
tfae_1_iff_2 : I ≤ J ↔ ↑I ⊆ ↑J
tfae_2_to_3 : ↑I ⊆ ↑J → Icc I.lower I.upper ⊆ Icc J.lower J.upper
tfae_3_iff_4 : Icc I.lower I.upper ⊆ Icc J.lower J.upper ↔ J.lower ≤ I.lower ∧ I.upper ≤ J.upper
⊢ [I ≤ J, ↑I ⊆ ↑J, Icc I.lower I.upper ⊆ Icc J.lower J.upper, J.lower ≤ I.lower ∧ I.upper ≤ J.upper].TFAE
tactic: tfae_have 3 ↔ 4
state_before: ι : Type u_1
I J : Box ι
x y : ι → ℝ
tfae_1_iff_2 : I ≤ J ↔ ↑I ⊆ ↑J
tfae_2_to_3 : ↑I ⊆ ↑J → Icc I.lower I.upper ⊆ Icc J.lower J.upper
tfae_3_iff_4 : Icc I.lower I.upper ⊆ Icc J.lower J.upper ↔ J.lower ≤ I.lower ∧ I.upper ≤ J.upper
⊢ [I ≤ J, ↑I ⊆ ↑J, Icc I.lower I.upper ⊆ Icc J.lower J.upper, J.lower ≤ I.lower ∧ I.upper ≤ J.upper].TFAE
state_after: case tfae_4_to_2
ι : Type u_1
I J : Box ι
x y : ι → ℝ
tfae_1_iff_2 : I ≤ J ↔ ↑I ⊆ ↑J
tfae_2_to_3 : ↑I ⊆ ↑J → Icc I.lower I.upper ⊆ Icc J.lower J.upper
tfae_3_iff_4 : Icc I.lower I.upper ⊆ Icc J.lower J.upper ↔ J.lower ≤ I.lower ∧ I.upper ≤ J.upper
⊢ J.lower ≤ I.lower ∧ I.upper ≤ J.upper → ↑I ⊆ ↑J

ι : Type u_1
I J : Box ι
x y : ι → ℝ
tfae_1_iff_2 : I ≤ J ↔ ↑I ⊆ ↑J
tfae_2_to_3 : ↑I ⊆ ↑J → Icc I.lower I.upper ⊆ Icc J.lower J.upper
tfae_3_iff_4 : Icc I.lower I.upper ⊆ Icc J.lower J.upper ↔ J.lower ≤ I.lower ∧ I.upper ≤ J.upper
tfae_4_to_2 : J.lower ≤ I.lower ∧ I.upper ≤ J.upper → ↑I ⊆ ↑J
⊢ [I ≤ J, ↑I ⊆ ↑J, Icc I.lower I.upper ⊆ Icc J.lower J.upper, J.lower ≤ I.lower ∧ I.upper ≤ J.upper].TFAE
tactic: tfae_have 4 → 2
state_before: ι : Type u_1
I J : Box ι
x y : ι → ℝ
tfae_1_iff_2 : I ≤ J ↔ ↑I ⊆ ↑J
tfae_2_to_3 : ↑I ⊆ ↑J → Icc I.lower I.upper ⊆ Icc J.lower J.upper
tfae_3_iff_4 : Icc I.lower I.upper ⊆ Icc J.lower J.upper ↔ J.lower ≤ I.lower ∧ I.upper ≤ J.upper
tfae_4_to_2 : J.lower ≤ I.lower ∧ I.upper ≤ J.upper → ↑I ⊆ ↑J
⊢ [I ≤ J, ↑I ⊆ ↑J, Icc I.lower I.upper ⊆ Icc J.lower J.upper, J.lower ≤ I.lower ∧ I.upper ≤ J.upper].TFAE
state_after: no goals
tactic: tfae_finish