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Theorem pythagtriplem12 16828
Description: Lemma for pythagtrip 16836. 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 7434 . 2 (𝑀↑2) = ((((√‘(𝐶 + 𝐵)) + (√‘(𝐶𝐵))) / 2)↑2)
3 nncn 12272 . . . . . . . . 9 (𝐶 ∈ ℕ → 𝐶 ∈ ℂ)
4 nncn 12272 . . . . . . . . 9 (𝐵 ∈ ℕ → 𝐵 ∈ ℂ)
5 addcl 11240 . . . . . . . . 9 ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐶 + 𝐵) ∈ ℂ)
63, 4, 5syl2anr 595 . . . . . . . 8 ((𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐶 + 𝐵) ∈ ℂ)
763adant1 1127 . . . . . . 7 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐶 + 𝐵) ∈ ℂ)
87sqrtcld 15442 . . . . . 6 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (√‘(𝐶 + 𝐵)) ∈ ℂ)
9 subcl 11509 . . . . . . . . 9 ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐶𝐵) ∈ ℂ)
103, 4, 9syl2anr 595 . . . . . . . 8 ((𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐶𝐵) ∈ ℂ)
11103adant1 1127 . . . . . . 7 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐶𝐵) ∈ ℂ)
1211sqrtcld 15442 . . . . . 6 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (√‘(𝐶𝐵)) ∈ ℂ)
138, 12addcld 11283 . . . . 5 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → ((√‘(𝐶 + 𝐵)) + (√‘(𝐶𝐵))) ∈ ℂ)
14133ad2ant1 1130 . . . 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 12339 . . . . . 6 2 ∈ ℂ
16 2ne0 12368 . . . . . 6 2 ≠ 0
17 sqdiv 14140 . . . . . 6 ((((√‘(𝐶 + 𝐵)) + (√‘(𝐶𝐵))) ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0) → ((((√‘(𝐶 + 𝐵)) + (√‘(𝐶𝐵))) / 2)↑2) = ((((√‘(𝐶 + 𝐵)) + (√‘(𝐶𝐵)))↑2) / (2↑2)))
1815, 16, 17mp3an23 1450 . . . . 5 (((√‘(𝐶 + 𝐵)) + (√‘(𝐶𝐵))) ∈ ℂ → ((((√‘(𝐶 + 𝐵)) + (√‘(𝐶𝐵))) / 2)↑2) = ((((√‘(𝐶 + 𝐵)) + (√‘(𝐶𝐵)))↑2) / (2↑2)))
1915sqvali 14198 . . . . . 6 (2↑2) = (2 · 2)
2019oveq2i 7435 . . . . 5 ((((√‘(𝐶 + 𝐵)) + (√‘(𝐶𝐵)))↑2) / (2↑2)) = ((((√‘(𝐶 + 𝐵)) + (√‘(𝐶𝐵)))↑2) / (2 · 2))
2118, 20eqtrdi 2782 . . . 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 1130 . . . . . . 7 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (√‘(𝐶 + 𝐵)) ∈ ℂ)
24123ad2ant1 1130 . . . . . . 7 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (√‘(𝐶𝐵)) ∈ ℂ)
25 binom2 14235 . . . . . . 7 (((√‘(𝐶 + 𝐵)) ∈ ℂ ∧ (√‘(𝐶𝐵)) ∈ ℂ) → (((√‘(𝐶 + 𝐵)) + (√‘(𝐶𝐵)))↑2) = ((((√‘(𝐶 + 𝐵))↑2) + (2 · ((√‘(𝐶 + 𝐵)) · (√‘(𝐶𝐵))))) + ((√‘(𝐶𝐵))↑2)))
2623, 24, 25syl2anc 582 . . . . . 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 12271 . . . . . . . . . . . 12 (𝐶 ∈ ℕ → 𝐶 ∈ ℝ)
28 nnre 12271 . . . . . . . . . . . 12 (𝐵 ∈ ℕ → 𝐵 ∈ ℝ)
29 readdcl 11241 . . . . . . . . . . . 12 ((𝐶 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐶 + 𝐵) ∈ ℝ)
3027, 28, 29syl2anr 595 . . . . . . . . . . 11 ((𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐶 + 𝐵) ∈ ℝ)
31303adant1 1127 . . . . . . . . . 10 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐶 + 𝐵) ∈ ℝ)
32313ad2ant1 1130 . . . . . . . . 9 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (𝐶 + 𝐵) ∈ ℝ)
33273ad2ant3 1132 . . . . . . . . . . . 12 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝐶 ∈ ℝ)
34283ad2ant2 1131 . . . . . . . . . . . 12 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝐵 ∈ ℝ)
35 nngt0 12295 . . . . . . . . . . . . 13 (𝐶 ∈ ℕ → 0 < 𝐶)
36353ad2ant3 1132 . . . . . . . . . . . 12 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 0 < 𝐶)
37 nngt0 12295 . . . . . . . . . . . . 13 (𝐵 ∈ ℕ → 0 < 𝐵)
38373ad2ant2 1131 . . . . . . . . . . . 12 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 0 < 𝐵)
3933, 34, 36, 38addgt0d 11839 . . . . . . . . . . 11 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 0 < (𝐶 + 𝐵))
40393ad2ant1 1130 . . . . . . . . . 10 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 0 < (𝐶 + 𝐵))
41 0re 11266 . . . . . . . . . . 11 0 ∈ ℝ
42 ltle 11352 . . . . . . . . . . 11 ((0 ∈ ℝ ∧ (𝐶 + 𝐵) ∈ ℝ) → (0 < (𝐶 + 𝐵) → 0 ≤ (𝐶 + 𝐵)))
4341, 42mpan 688 . . . . . . . . . 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 15260 . . . . . . . . 9 (((𝐶 + 𝐵) ∈ ℝ ∧ 0 ≤ (𝐶 + 𝐵)) → ((√‘(𝐶 + 𝐵))↑2) = (𝐶 + 𝐵))
4632, 44, 45syl2anc 582 . . . . . . . 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 7439 . . . . . . 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 11574 . . . . . . . . . . 11 ((𝐶 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐶𝐵) ∈ ℝ)
4927, 28, 48syl2anr 595 . . . . . . . . . 10 ((𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐶𝐵) ∈ ℝ)
50493adant1 1127 . . . . . . . . 9 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐶𝐵) ∈ ℝ)
51503ad2ant1 1130 . . . . . . . 8 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (𝐶𝐵) ∈ ℝ)
52 pythagtriplem10 16822 . . . . . . . . . 10 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2)) → 0 < (𝐶𝐵))
53523adant3 1129 . . . . . . . . 9 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 0 < (𝐶𝐵))
54 ltle 11352 . . . . . . . . . 10 ((0 ∈ ℝ ∧ (𝐶𝐵) ∈ ℝ) → (0 < (𝐶𝐵) → 0 ≤ (𝐶𝐵)))
5541, 54mpan 688 . . . . . . . . 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 15260 . . . . . . . 8 (((𝐶𝐵) ∈ ℝ ∧ 0 ≤ (𝐶𝐵)) → ((√‘(𝐶𝐵))↑2) = (𝐶𝐵))
5851, 56, 57syl2anc 582 . . . . . . 7 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((√‘(𝐶𝐵))↑2) = (𝐶𝐵))
5947, 58oveq12d 7442 . . . . . 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 1130 . . . . . . . 8 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (𝐶 + 𝐵) ∈ ℂ)
618, 12mulcld 11284 . . . . . . . . . 10 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → ((√‘(𝐶 + 𝐵)) · (√‘(𝐶𝐵))) ∈ ℂ)
62 mulcl 11242 . . . . . . . . . 10 ((2 ∈ ℂ ∧ ((√‘(𝐶 + 𝐵)) · (√‘(𝐶𝐵))) ∈ ℂ) → (2 · ((√‘(𝐶 + 𝐵)) · (√‘(𝐶𝐵)))) ∈ ℂ)
6315, 61, 62sylancr 585 . . . . . . . . 9 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (2 · ((√‘(𝐶 + 𝐵)) · (√‘(𝐶𝐵)))) ∈ ℂ)
64633ad2ant1 1130 . . . . . . . 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 1130 . . . . . . . 8 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (𝐶𝐵) ∈ ℂ)
6660, 64, 65add32d 11491 . . . . . . 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 1132 . . . . . . . . . 10 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝐶 ∈ ℂ)
68673ad2ant1 1130 . . . . . . . . 9 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 𝐶 ∈ ℂ)
69 nncn 12272 . . . . . . . . . . 11 (𝐴 ∈ ℕ → 𝐴 ∈ ℂ)
70693ad2ant1 1130 . . . . . . . . . 10 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝐴 ∈ ℂ)
71703ad2ant1 1130 . . . . . . . . 9 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 𝐴 ∈ ℂ)
72 adddi 11247 . . . . . . . . 9 ((2 ∈ ℂ ∧ 𝐶 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (2 · (𝐶 + 𝐴)) = ((2 · 𝐶) + (2 · 𝐴)))
7315, 68, 71, 72mp3an2i 1463 . . . . . . . 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)))
7443ad2ant2 1131 . . . . . . . . . . . 12 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝐵 ∈ ℂ)
75743ad2ant1 1130 . . . . . . . . . . 11 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 𝐵 ∈ ℂ)
7668, 75, 68ppncand 11661 . . . . . . . . . 10 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((𝐶 + 𝐵) + (𝐶𝐵)) = (𝐶 + 𝐶))
77682timesd 12507 . . . . . . . . . 10 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (2 · 𝐶) = (𝐶 + 𝐶))
7876, 77eqtr4d 2769 . . . . . . . . 9 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((𝐶 + 𝐵) + (𝐶𝐵)) = (2 · 𝐶))
79 oveq1 7431 . . . . . . . . . . . . . 14 (((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) → (((𝐴↑2) + (𝐵↑2)) − (𝐵↑2)) = ((𝐶↑2) − (𝐵↑2)))
80793ad2ant2 1131 . . . . . . . . . . . . 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)))
8171sqcld 14163 . . . . . . . . . . . . . 14 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (𝐴↑2) ∈ ℂ)
8275sqcld 14163 . . . . . . . . . . . . . 14 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (𝐵↑2) ∈ ℂ)
8381, 82pncand 11622 . . . . . . . . . . . . 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))
84 subsq 14228 . . . . . . . . . . . . . 14 ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐶↑2) − (𝐵↑2)) = ((𝐶 + 𝐵) · (𝐶𝐵)))
8568, 75, 84syl2anc 582 . . . . . . . . . . . . 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)))
8680, 83, 853eqtr3rd 2775 . . . . . . . . . . . 12 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((𝐶 + 𝐵) · (𝐶𝐵)) = (𝐴↑2))
8786fveq2d 6905 . . . . . . . . . . 11 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (√‘((𝐶 + 𝐵) · (𝐶𝐵))) = (√‘(𝐴↑2)))
8832, 44, 51, 56sqrtmuld 15429 . . . . . . . . . . 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))))
89 nnre 12271 . . . . . . . . . . . . . 14 (𝐴 ∈ ℕ → 𝐴 ∈ ℝ)
90893ad2ant1 1130 . . . . . . . . . . . . 13 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝐴 ∈ ℝ)
91903ad2ant1 1130 . . . . . . . . . . . 12 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 𝐴 ∈ ℝ)
92 nnnn0 12531 . . . . . . . . . . . . . . 15 (𝐴 ∈ ℕ → 𝐴 ∈ ℕ0)
9392nn0ge0d 12587 . . . . . . . . . . . . . 14 (𝐴 ∈ ℕ → 0 ≤ 𝐴)
94933ad2ant1 1130 . . . . . . . . . . . . 13 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 0 ≤ 𝐴)
95943ad2ant1 1130 . . . . . . . . . . . 12 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 0 ≤ 𝐴)
9691, 95sqrtsqd 15424 . . . . . . . . . . 11 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (√‘(𝐴↑2)) = 𝐴)
9787, 88, 963eqtr3d 2774 . . . . . . . . . 10 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((√‘(𝐶 + 𝐵)) · (√‘(𝐶𝐵))) = 𝐴)
9897oveq2d 7440 . . . . . . . . 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))
9978, 98oveq12d 7442 . . . . . . . 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)))
10073, 99eqtr4d 2769 . . . . . . 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))))))
10166, 100eqtr4d 2769 . . . . . 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)))
10226, 59, 1013eqtrd 2770 . . . . 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)))
103102oveq1d 7439 . . . 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)))
104 addcl 11240 . . . . . . . . 9 ((𝐶 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (𝐶 + 𝐴) ∈ ℂ)
1053, 69, 104syl2anr 595 . . . . . . . 8 ((𝐴 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐶 + 𝐴) ∈ ℂ)
1061053adant2 1128 . . . . . . 7 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐶 + 𝐴) ∈ ℂ)
1071063ad2ant1 1130 . . . . . 6 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (𝐶 + 𝐴) ∈ ℂ)
108 mulcl 11242 . . . . . 6 ((2 ∈ ℂ ∧ (𝐶 + 𝐴) ∈ ℂ) → (2 · (𝐶 + 𝐴)) ∈ ℂ)
10915, 107, 108sylancr 585 . . . . 5 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (2 · (𝐶 + 𝐴)) ∈ ℂ)
110 2cnne0 12474 . . . . . 6 (2 ∈ ℂ ∧ 2 ≠ 0)
111 divdiv1 11976 . . . . . 6 (((2 · (𝐶 + 𝐴)) ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 ≠ 0) ∧ (2 ∈ ℂ ∧ 2 ≠ 0)) → (((2 · (𝐶 + 𝐴)) / 2) / 2) = ((2 · (𝐶 + 𝐴)) / (2 · 2)))
112110, 110, 111mp3an23 1450 . . . . 5 ((2 · (𝐶 + 𝐴)) ∈ ℂ → (((2 · (𝐶 + 𝐴)) / 2) / 2) = ((2 · (𝐶 + 𝐴)) / (2 · 2)))
113109, 112syl 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)))
114103, 113eqtr4d 2769 . . 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))
115 divcan3 11949 . . . . . 6 (((𝐶 + 𝐴) ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0) → ((2 · (𝐶 + 𝐴)) / 2) = (𝐶 + 𝐴))
11615, 16, 115mp3an23 1450 . . . . 5 ((𝐶 + 𝐴) ∈ ℂ → ((2 · (𝐶 + 𝐴)) / 2) = (𝐶 + 𝐴))
117107, 116syl 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))
118117oveq1d 7439 . . 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))
11922, 114, 1183eqtrd 2770 . 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))
1202, 119eqtrid 2778 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 394  w3a 1084   = wceq 1534  wcel 2099  wne 2930   class class class wbr 5153  cfv 6554  (class class class)co 7424  cc 11156  cr 11157  0cc0 11158  1c1 11159   + caddc 11161   · cmul 11163   < clt 11298  cle 11299  cmin 11494   / cdiv 11921  cn 12264  2c2 12319  cexp 14081  csqrt 15238  cdvds 16256   gcd cgcd 16494
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11214  ax-resscn 11215  ax-1cn 11216  ax-icn 11217  ax-addcl 11218  ax-addrcl 11219  ax-mulcl 11220  ax-mulrcl 11221  ax-mulcom 11222  ax-addass 11223  ax-mulass 11224  ax-distr 11225  ax-i2m1 11226  ax-1ne0 11227  ax-1rid 11228  ax-rnegex 11229  ax-rrecex 11230  ax-cnre 11231  ax-pre-lttri 11232  ax-pre-lttrn 11233  ax-pre-ltadd 11234  ax-pre-mulgt0 11235  ax-pre-sup 11236
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6312  df-ord 6379  df-on 6380  df-lim 6381  df-suc 6382  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-riota 7380  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7877  df-2nd 8004  df-frecs 8296  df-wrecs 8327  df-recs 8401  df-rdg 8440  df-er 8734  df-en 8975  df-dom 8976  df-sdom 8977  df-sup 9485  df-pnf 11300  df-mnf 11301  df-xr 11302  df-ltxr 11303  df-le 11304  df-sub 11496  df-neg 11497  df-div 11922  df-nn 12265  df-2 12327  df-3 12328  df-n0 12525  df-z 12611  df-uz 12875  df-rp 13029  df-seq 14022  df-exp 14082  df-cj 15104  df-re 15105  df-im 15106  df-sqrt 15240  df-abs 15241
This theorem is referenced by:  pythagtriplem15  16831  pythagtriplem17  16833
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