Theorem: LO.FirstOrder.Arith.Model.PPow2.sq_sqrt_eq (Arithmetization/ISigmaZero/Exponential/PPow2.lean)

State before:
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐈𝚫₀
i : M
ppi : PPow2 i
ne2 : i ≠ 2
⊢ (√i) ^ 2 = i
Tactic:
rcases ppi with ⟨pi, m, _, sppm, lb⟩
State after:
case intro.intro.intro.intro
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐈𝚫₀
i : M
ne2 : i ≠ 2
pi : Pow2 i
m : M
left✝ : m < 2 * i
sppm : SPPow2 m
lb : LenBit i m
⊢ (√i) ^ 2 = i

State before:
case intro.intro.intro.intro
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐈𝚫₀
i : M
ne2 : i ≠ 2
pi : Pow2 i
m : M
left✝ : m < 2 * i
sppm : SPPow2 m
lb : LenBit i m
⊢ (√i) ^ 2 = i
Tactic:
exact ((sppm.lenbit_iff lb.le pi (lt_of_le_of_ne (one_lt_iff_two_le.mp $ sppm.one_lt lb) (Ne.symm ne2))).mp lb).1
State after:
no goals