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Theorem pythagtriplem12 15738
Description: Lemma for pythagtrip 15746. Calculate the square of 𝑀. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Hypothesis
Ref Expression
pythagtriplem11.1 𝑀 = (((√‘(𝐶 + 𝐵)) + (√‘(𝐶𝐵))) / 2)
Assertion
Ref Expression
pythagtriplem12 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (𝑀↑2) = ((𝐶 + 𝐴) / 2))

Proof of Theorem pythagtriplem12
StepHypRef Expression
1 pythagtriplem11.1 . . 3 𝑀 = (((√‘(𝐶 + 𝐵)) + (√‘(𝐶𝐵))) / 2)
21oveq1i 6803 . 2 (𝑀↑2) = ((((√‘(𝐶 + 𝐵)) + (√‘(𝐶𝐵))) / 2)↑2)
3 nncn 11230 . . . . . . . . 9 (𝐶 ∈ ℕ → 𝐶 ∈ ℂ)
4 nncn 11230 . . . . . . . . 9 (𝐵 ∈ ℕ → 𝐵 ∈ ℂ)
5 addcl 10220 . . . . . . . . 9 ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐶 + 𝐵) ∈ ℂ)
63, 4, 5syl2anr 584 . . . . . . . 8 ((𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐶 + 𝐵) ∈ ℂ)
763adant1 1124 . . . . . . 7 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐶 + 𝐵) ∈ ℂ)
87sqrtcld 14384 . . . . . 6 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (√‘(𝐶 + 𝐵)) ∈ ℂ)
9 subcl 10482 . . . . . . . . 9 ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐶𝐵) ∈ ℂ)
103, 4, 9syl2anr 584 . . . . . . . 8 ((𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐶𝐵) ∈ ℂ)
11103adant1 1124 . . . . . . 7 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐶𝐵) ∈ ℂ)
1211sqrtcld 14384 . . . . . 6 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (√‘(𝐶𝐵)) ∈ ℂ)
138, 12addcld 10261 . . . . 5 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → ((√‘(𝐶 + 𝐵)) + (√‘(𝐶𝐵))) ∈ ℂ)
14133ad2ant1 1127 . . . 4 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((√‘(𝐶 + 𝐵)) + (√‘(𝐶𝐵))) ∈ ℂ)
15 2cn 11293 . . . . . 6 2 ∈ ℂ
16 2ne0 11315 . . . . . 6 2 ≠ 0
17 sqdiv 13135 . . . . . 6 ((((√‘(𝐶 + 𝐵)) + (√‘(𝐶𝐵))) ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0) → ((((√‘(𝐶 + 𝐵)) + (√‘(𝐶𝐵))) / 2)↑2) = ((((√‘(𝐶 + 𝐵)) + (√‘(𝐶𝐵)))↑2) / (2↑2)))
1815, 16, 17mp3an23 1564 . . . . 5 (((√‘(𝐶 + 𝐵)) + (√‘(𝐶𝐵))) ∈ ℂ → ((((√‘(𝐶 + 𝐵)) + (√‘(𝐶𝐵))) / 2)↑2) = ((((√‘(𝐶 + 𝐵)) + (√‘(𝐶𝐵)))↑2) / (2↑2)))
1915sqvali 13150 . . . . . 6 (2↑2) = (2 · 2)
2019oveq2i 6804 . . . . 5 ((((√‘(𝐶 + 𝐵)) + (√‘(𝐶𝐵)))↑2) / (2↑2)) = ((((√‘(𝐶 + 𝐵)) + (√‘(𝐶𝐵)))↑2) / (2 · 2))
2118, 20syl6eq 2821 . . . 4 (((√‘(𝐶 + 𝐵)) + (√‘(𝐶𝐵))) ∈ ℂ → ((((√‘(𝐶 + 𝐵)) + (√‘(𝐶𝐵))) / 2)↑2) = ((((√‘(𝐶 + 𝐵)) + (√‘(𝐶𝐵)))↑2) / (2 · 2)))
2214, 21syl 17 . . 3 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((((√‘(𝐶 + 𝐵)) + (√‘(𝐶𝐵))) / 2)↑2) = ((((√‘(𝐶 + 𝐵)) + (√‘(𝐶𝐵)))↑2) / (2 · 2)))
2383ad2ant1 1127 . . . . . . 7 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (√‘(𝐶 + 𝐵)) ∈ ℂ)
24123ad2ant1 1127 . . . . . . 7 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (√‘(𝐶𝐵)) ∈ ℂ)
25 binom2 13186 . . . . . . 7 (((√‘(𝐶 + 𝐵)) ∈ ℂ ∧ (√‘(𝐶𝐵)) ∈ ℂ) → (((√‘(𝐶 + 𝐵)) + (√‘(𝐶𝐵)))↑2) = ((((√‘(𝐶 + 𝐵))↑2) + (2 · ((√‘(𝐶 + 𝐵)) · (√‘(𝐶𝐵))))) + ((√‘(𝐶𝐵))↑2)))
2623, 24, 25syl2anc 573 . . . . . 6 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (((√‘(𝐶 + 𝐵)) + (√‘(𝐶𝐵)))↑2) = ((((√‘(𝐶 + 𝐵))↑2) + (2 · ((√‘(𝐶 + 𝐵)) · (√‘(𝐶𝐵))))) + ((√‘(𝐶𝐵))↑2)))
27 nnre 11229 . . . . . . . . . . . 12 (𝐶 ∈ ℕ → 𝐶 ∈ ℝ)
28 nnre 11229 . . . . . . . . . . . 12 (𝐵 ∈ ℕ → 𝐵 ∈ ℝ)
29 readdcl 10221 . . . . . . . . . . . 12 ((𝐶 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐶 + 𝐵) ∈ ℝ)
3027, 28, 29syl2anr 584 . . . . . . . . . . 11 ((𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐶 + 𝐵) ∈ ℝ)
31303adant1 1124 . . . . . . . . . 10 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐶 + 𝐵) ∈ ℝ)
32313ad2ant1 1127 . . . . . . . . 9 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (𝐶 + 𝐵) ∈ ℝ)
33273ad2ant3 1129 . . . . . . . . . . . 12 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝐶 ∈ ℝ)
34283ad2ant2 1128 . . . . . . . . . . . 12 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝐵 ∈ ℝ)
35 nngt0 11251 . . . . . . . . . . . . 13 (𝐶 ∈ ℕ → 0 < 𝐶)
36353ad2ant3 1129 . . . . . . . . . . . 12 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 0 < 𝐶)
37 nngt0 11251 . . . . . . . . . . . . 13 (𝐵 ∈ ℕ → 0 < 𝐵)
38373ad2ant2 1128 . . . . . . . . . . . 12 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 0 < 𝐵)
3933, 34, 36, 38addgt0d 10804 . . . . . . . . . . 11 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 0 < (𝐶 + 𝐵))
40393ad2ant1 1127 . . . . . . . . . 10 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 0 < (𝐶 + 𝐵))
41 0re 10242 . . . . . . . . . . 11 0 ∈ ℝ
42 ltle 10328 . . . . . . . . . . 11 ((0 ∈ ℝ ∧ (𝐶 + 𝐵) ∈ ℝ) → (0 < (𝐶 + 𝐵) → 0 ≤ (𝐶 + 𝐵)))
4341, 42mpan 670 . . . . . . . . . 10 ((𝐶 + 𝐵) ∈ ℝ → (0 < (𝐶 + 𝐵) → 0 ≤ (𝐶 + 𝐵)))
4432, 40, 43sylc 65 . . . . . . . . 9 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 0 ≤ (𝐶 + 𝐵))
45 resqrtth 14204 . . . . . . . . 9 (((𝐶 + 𝐵) ∈ ℝ ∧ 0 ≤ (𝐶 + 𝐵)) → ((√‘(𝐶 + 𝐵))↑2) = (𝐶 + 𝐵))
4632, 44, 45syl2anc 573 . . . . . . . 8 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((√‘(𝐶 + 𝐵))↑2) = (𝐶 + 𝐵))
4746oveq1d 6808 . . . . . . 7 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (((√‘(𝐶 + 𝐵))↑2) + (2 · ((√‘(𝐶 + 𝐵)) · (√‘(𝐶𝐵))))) = ((𝐶 + 𝐵) + (2 · ((√‘(𝐶 + 𝐵)) · (√‘(𝐶𝐵))))))
48 resubcl 10547 . . . . . . . . . . 11 ((𝐶 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐶𝐵) ∈ ℝ)
4927, 28, 48syl2anr 584 . . . . . . . . . 10 ((𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐶𝐵) ∈ ℝ)
50493adant1 1124 . . . . . . . . 9 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐶𝐵) ∈ ℝ)
51503ad2ant1 1127 . . . . . . . 8 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (𝐶𝐵) ∈ ℝ)
52 pythagtriplem10 15732 . . . . . . . . . 10 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2)) → 0 < (𝐶𝐵))
53523adant3 1126 . . . . . . . . 9 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 0 < (𝐶𝐵))
54 ltle 10328 . . . . . . . . . 10 ((0 ∈ ℝ ∧ (𝐶𝐵) ∈ ℝ) → (0 < (𝐶𝐵) → 0 ≤ (𝐶𝐵)))
5541, 54mpan 670 . . . . . . . . 9 ((𝐶𝐵) ∈ ℝ → (0 < (𝐶𝐵) → 0 ≤ (𝐶𝐵)))
5651, 53, 55sylc 65 . . . . . . . 8 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 0 ≤ (𝐶𝐵))
57 resqrtth 14204 . . . . . . . 8 (((𝐶𝐵) ∈ ℝ ∧ 0 ≤ (𝐶𝐵)) → ((√‘(𝐶𝐵))↑2) = (𝐶𝐵))
5851, 56, 57syl2anc 573 . . . . . . 7 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((√‘(𝐶𝐵))↑2) = (𝐶𝐵))
5947, 58oveq12d 6811 . . . . . 6 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((((√‘(𝐶 + 𝐵))↑2) + (2 · ((√‘(𝐶 + 𝐵)) · (√‘(𝐶𝐵))))) + ((√‘(𝐶𝐵))↑2)) = (((𝐶 + 𝐵) + (2 · ((√‘(𝐶 + 𝐵)) · (√‘(𝐶𝐵))))) + (𝐶𝐵)))
6073ad2ant1 1127 . . . . . . . 8 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (𝐶 + 𝐵) ∈ ℂ)
618, 12mulcld 10262 . . . . . . . . . 10 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → ((√‘(𝐶 + 𝐵)) · (√‘(𝐶𝐵))) ∈ ℂ)
62 mulcl 10222 . . . . . . . . . 10 ((2 ∈ ℂ ∧ ((√‘(𝐶 + 𝐵)) · (√‘(𝐶𝐵))) ∈ ℂ) → (2 · ((√‘(𝐶 + 𝐵)) · (√‘(𝐶𝐵)))) ∈ ℂ)
6315, 61, 62sylancr 575 . . . . . . . . 9 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (2 · ((√‘(𝐶 + 𝐵)) · (√‘(𝐶𝐵)))) ∈ ℂ)
64633ad2ant1 1127 . . . . . . . 8 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (2 · ((√‘(𝐶 + 𝐵)) · (√‘(𝐶𝐵)))) ∈ ℂ)
65113ad2ant1 1127 . . . . . . . 8 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (𝐶𝐵) ∈ ℂ)
6660, 64, 65add32d 10465 . . . . . . 7 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (((𝐶 + 𝐵) + (2 · ((√‘(𝐶 + 𝐵)) · (√‘(𝐶𝐵))))) + (𝐶𝐵)) = (((𝐶 + 𝐵) + (𝐶𝐵)) + (2 · ((√‘(𝐶 + 𝐵)) · (√‘(𝐶𝐵))))))
6733ad2ant3 1129 . . . . . . . . . 10 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝐶 ∈ ℂ)
68673ad2ant1 1127 . . . . . . . . 9 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 𝐶 ∈ ℂ)
69 nncn 11230 . . . . . . . . . . 11 (𝐴 ∈ ℕ → 𝐴 ∈ ℂ)
70693ad2ant1 1127 . . . . . . . . . 10 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝐴 ∈ ℂ)
71703ad2ant1 1127 . . . . . . . . 9 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 𝐴 ∈ ℂ)
72 adddi 10227 . . . . . . . . . 10 ((2 ∈ ℂ ∧ 𝐶 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (2 · (𝐶 + 𝐴)) = ((2 · 𝐶) + (2 · 𝐴)))
7315, 72mp3an1 1559 . . . . . . . . 9 ((𝐶 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (2 · (𝐶 + 𝐴)) = ((2 · 𝐶) + (2 · 𝐴)))
7468, 71, 73syl2anc 573 . . . . . . . 8 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (2 · (𝐶 + 𝐴)) = ((2 · 𝐶) + (2 · 𝐴)))
7543ad2ant2 1128 . . . . . . . . . . . 12 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝐵 ∈ ℂ)
76753ad2ant1 1127 . . . . . . . . . . 11 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 𝐵 ∈ ℂ)
7768, 76, 68ppncand 10634 . . . . . . . . . 10 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((𝐶 + 𝐵) + (𝐶𝐵)) = (𝐶 + 𝐶))
78682timesd 11477 . . . . . . . . . 10 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (2 · 𝐶) = (𝐶 + 𝐶))
7977, 78eqtr4d 2808 . . . . . . . . 9 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((𝐶 + 𝐵) + (𝐶𝐵)) = (2 · 𝐶))
80 oveq1 6800 . . . . . . . . . . . . . 14 (((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) → (((𝐴↑2) + (𝐵↑2)) − (𝐵↑2)) = ((𝐶↑2) − (𝐵↑2)))
81803ad2ant2 1128 . . . . . . . . . . . . 13 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (((𝐴↑2) + (𝐵↑2)) − (𝐵↑2)) = ((𝐶↑2) − (𝐵↑2)))
8271sqcld 13213 . . . . . . . . . . . . . 14 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (𝐴↑2) ∈ ℂ)
8376sqcld 13213 . . . . . . . . . . . . . 14 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (𝐵↑2) ∈ ℂ)
8482, 83pncand 10595 . . . . . . . . . . . . 13 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (((𝐴↑2) + (𝐵↑2)) − (𝐵↑2)) = (𝐴↑2))
85 subsq 13179 . . . . . . . . . . . . . 14 ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐶↑2) − (𝐵↑2)) = ((𝐶 + 𝐵) · (𝐶𝐵)))
8668, 76, 85syl2anc 573 . . . . . . . . . . . . 13 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((𝐶↑2) − (𝐵↑2)) = ((𝐶 + 𝐵) · (𝐶𝐵)))
8781, 84, 863eqtr3rd 2814 . . . . . . . . . . . 12 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((𝐶 + 𝐵) · (𝐶𝐵)) = (𝐴↑2))
8887fveq2d 6336 . . . . . . . . . . 11 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (√‘((𝐶 + 𝐵) · (𝐶𝐵))) = (√‘(𝐴↑2)))
8932, 44, 51, 56sqrtmuld 14371 . . . . . . . . . . 11 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (√‘((𝐶 + 𝐵) · (𝐶𝐵))) = ((√‘(𝐶 + 𝐵)) · (√‘(𝐶𝐵))))
90 nnre 11229 . . . . . . . . . . . . . 14 (𝐴 ∈ ℕ → 𝐴 ∈ ℝ)
91903ad2ant1 1127 . . . . . . . . . . . . 13 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝐴 ∈ ℝ)
92913ad2ant1 1127 . . . . . . . . . . . 12 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 𝐴 ∈ ℝ)
93 nnnn0 11501 . . . . . . . . . . . . . . 15 (𝐴 ∈ ℕ → 𝐴 ∈ ℕ0)
9493nn0ge0d 11556 . . . . . . . . . . . . . 14 (𝐴 ∈ ℕ → 0 ≤ 𝐴)
95943ad2ant1 1127 . . . . . . . . . . . . 13 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 0 ≤ 𝐴)
96953ad2ant1 1127 . . . . . . . . . . . 12 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 0 ≤ 𝐴)
9792, 96sqrtsqd 14366 . . . . . . . . . . 11 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (√‘(𝐴↑2)) = 𝐴)
9888, 89, 973eqtr3d 2813 . . . . . . . . . 10 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((√‘(𝐶 + 𝐵)) · (√‘(𝐶𝐵))) = 𝐴)
9998oveq2d 6809 . . . . . . . . 9 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (2 · ((√‘(𝐶 + 𝐵)) · (√‘(𝐶𝐵)))) = (2 · 𝐴))
10079, 99oveq12d 6811 . . . . . . . 8 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (((𝐶 + 𝐵) + (𝐶𝐵)) + (2 · ((√‘(𝐶 + 𝐵)) · (√‘(𝐶𝐵))))) = ((2 · 𝐶) + (2 · 𝐴)))
10174, 100eqtr4d 2808 . . . . . . 7 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (2 · (𝐶 + 𝐴)) = (((𝐶 + 𝐵) + (𝐶𝐵)) + (2 · ((√‘(𝐶 + 𝐵)) · (√‘(𝐶𝐵))))))
10266, 101eqtr4d 2808 . . . . . 6 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (((𝐶 + 𝐵) + (2 · ((√‘(𝐶 + 𝐵)) · (√‘(𝐶𝐵))))) + (𝐶𝐵)) = (2 · (𝐶 + 𝐴)))
10326, 59, 1023eqtrd 2809 . . . . 5 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (((√‘(𝐶 + 𝐵)) + (√‘(𝐶𝐵)))↑2) = (2 · (𝐶 + 𝐴)))
104103oveq1d 6808 . . . 4 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((((√‘(𝐶 + 𝐵)) + (√‘(𝐶𝐵)))↑2) / (2 · 2)) = ((2 · (𝐶 + 𝐴)) / (2 · 2)))
105 addcl 10220 . . . . . . . . 9 ((𝐶 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (𝐶 + 𝐴) ∈ ℂ)
1063, 69, 105syl2anr 584 . . . . . . . 8 ((𝐴 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐶 + 𝐴) ∈ ℂ)
1071063adant2 1125 . . . . . . 7 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐶 + 𝐴) ∈ ℂ)
1081073ad2ant1 1127 . . . . . 6 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (𝐶 + 𝐴) ∈ ℂ)
109 mulcl 10222 . . . . . 6 ((2 ∈ ℂ ∧ (𝐶 + 𝐴) ∈ ℂ) → (2 · (𝐶 + 𝐴)) ∈ ℂ)
11015, 108, 109sylancr 575 . . . . 5 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (2 · (𝐶 + 𝐴)) ∈ ℂ)
111 2cnne0 11444 . . . . . 6 (2 ∈ ℂ ∧ 2 ≠ 0)
112 divdiv1 10938 . . . . . 6 (((2 · (𝐶 + 𝐴)) ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 ≠ 0) ∧ (2 ∈ ℂ ∧ 2 ≠ 0)) → (((2 · (𝐶 + 𝐴)) / 2) / 2) = ((2 · (𝐶 + 𝐴)) / (2 · 2)))
113111, 111, 112mp3an23 1564 . . . . 5 ((2 · (𝐶 + 𝐴)) ∈ ℂ → (((2 · (𝐶 + 𝐴)) / 2) / 2) = ((2 · (𝐶 + 𝐴)) / (2 · 2)))
114110, 113syl 17 . . . 4 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (((2 · (𝐶 + 𝐴)) / 2) / 2) = ((2 · (𝐶 + 𝐴)) / (2 · 2)))
115104, 114eqtr4d 2808 . . 3 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((((√‘(𝐶 + 𝐵)) + (√‘(𝐶𝐵)))↑2) / (2 · 2)) = (((2 · (𝐶 + 𝐴)) / 2) / 2))
116 divcan3 10913 . . . . . 6 (((𝐶 + 𝐴) ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0) → ((2 · (𝐶 + 𝐴)) / 2) = (𝐶 + 𝐴))
11715, 16, 116mp3an23 1564 . . . . 5 ((𝐶 + 𝐴) ∈ ℂ → ((2 · (𝐶 + 𝐴)) / 2) = (𝐶 + 𝐴))
118108, 117syl 17 . . . 4 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((2 · (𝐶 + 𝐴)) / 2) = (𝐶 + 𝐴))
119118oveq1d 6808 . . 3 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (((2 · (𝐶 + 𝐴)) / 2) / 2) = ((𝐶 + 𝐴) / 2))
12022, 115, 1193eqtrd 2809 . 2 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((((√‘(𝐶 + 𝐵)) + (√‘(𝐶𝐵))) / 2)↑2) = ((𝐶 + 𝐴) / 2))
1212, 120syl5eq 2817 1 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (𝑀↑2) = ((𝐶 + 𝐴) / 2))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382  w3a 1071   = wceq 1631  wcel 2145  wne 2943   class class class wbr 4786  cfv 6031  (class class class)co 6793  cc 10136  cr 10137  0cc0 10138  1c1 10139   + caddc 10141   · cmul 10143   < clt 10276  cle 10277  cmin 10468   / cdiv 10886  cn 11222  2c2 11272  cexp 13067  csqrt 14181  cdvds 15189   gcd cgcd 15424
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-cnex 10194  ax-resscn 10195  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-mulcom 10202  ax-addass 10203  ax-mulass 10204  ax-distr 10205  ax-i2m1 10206  ax-1ne0 10207  ax-1rid 10208  ax-rnegex 10209  ax-rrecex 10210  ax-cnre 10211  ax-pre-lttri 10212  ax-pre-lttrn 10213  ax-pre-ltadd 10214  ax-pre-mulgt0 10215  ax-pre-sup 10216
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-er 7896  df-en 8110  df-dom 8111  df-sdom 8112  df-sup 8504  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-sub 10470  df-neg 10471  df-div 10887  df-nn 11223  df-2 11281  df-3 11282  df-n0 11495  df-z 11580  df-uz 11889  df-rp 12036  df-seq 13009  df-exp 13068  df-cj 14047  df-re 14048  df-im 14049  df-sqrt 14183  df-abs 14184
This theorem is referenced by:  pythagtriplem15  15741  pythagtriplem17  15743
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