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Theorem pythagtriplem12 16868
Description: Lemma for pythagtrip 16876. 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 7455 . 2 (𝑀↑2) = ((((√‘(𝐶 + 𝐵)) + (√‘(𝐶𝐵))) / 2)↑2)
3 nncn 12297 . . . . . . . . 9 (𝐶 ∈ ℕ → 𝐶 ∈ ℂ)
4 nncn 12297 . . . . . . . . 9 (𝐵 ∈ ℕ → 𝐵 ∈ ℂ)
5 addcl 11262 . . . . . . . . 9 ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐶 + 𝐵) ∈ ℂ)
63, 4, 5syl2anr 596 . . . . . . . 8 ((𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐶 + 𝐵) ∈ ℂ)
763adant1 1130 . . . . . . 7 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐶 + 𝐵) ∈ ℂ)
87sqrtcld 15482 . . . . . 6 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (√‘(𝐶 + 𝐵)) ∈ ℂ)
9 subcl 11531 . . . . . . . . 9 ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐶𝐵) ∈ ℂ)
103, 4, 9syl2anr 596 . . . . . . . 8 ((𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐶𝐵) ∈ ℂ)
11103adant1 1130 . . . . . . 7 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐶𝐵) ∈ ℂ)
1211sqrtcld 15482 . . . . . 6 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (√‘(𝐶𝐵)) ∈ ℂ)
138, 12addcld 11305 . . . . 5 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → ((√‘(𝐶 + 𝐵)) + (√‘(𝐶𝐵))) ∈ ℂ)
14133ad2ant1 1133 . . . 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 12364 . . . . . 6 2 ∈ ℂ
16 2ne0 12393 . . . . . 6 2 ≠ 0
17 sqdiv 14167 . . . . . 6 ((((√‘(𝐶 + 𝐵)) + (√‘(𝐶𝐵))) ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0) → ((((√‘(𝐶 + 𝐵)) + (√‘(𝐶𝐵))) / 2)↑2) = ((((√‘(𝐶 + 𝐵)) + (√‘(𝐶𝐵)))↑2) / (2↑2)))
1815, 16, 17mp3an23 1453 . . . . 5 (((√‘(𝐶 + 𝐵)) + (√‘(𝐶𝐵))) ∈ ℂ → ((((√‘(𝐶 + 𝐵)) + (√‘(𝐶𝐵))) / 2)↑2) = ((((√‘(𝐶 + 𝐵)) + (√‘(𝐶𝐵)))↑2) / (2↑2)))
1915sqvali 14225 . . . . . 6 (2↑2) = (2 · 2)
2019oveq2i 7456 . . . . 5 ((((√‘(𝐶 + 𝐵)) + (√‘(𝐶𝐵)))↑2) / (2↑2)) = ((((√‘(𝐶 + 𝐵)) + (√‘(𝐶𝐵)))↑2) / (2 · 2))
2118, 20eqtrdi 2790 . . . 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 1133 . . . . . . 7 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (√‘(𝐶 + 𝐵)) ∈ ℂ)
24123ad2ant1 1133 . . . . . . 7 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (√‘(𝐶𝐵)) ∈ ℂ)
25 binom2 14262 . . . . . . 7 (((√‘(𝐶 + 𝐵)) ∈ ℂ ∧ (√‘(𝐶𝐵)) ∈ ℂ) → (((√‘(𝐶 + 𝐵)) + (√‘(𝐶𝐵)))↑2) = ((((√‘(𝐶 + 𝐵))↑2) + (2 · ((√‘(𝐶 + 𝐵)) · (√‘(𝐶𝐵))))) + ((√‘(𝐶𝐵))↑2)))
2623, 24, 25syl2anc 583 . . . . . 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 12296 . . . . . . . . . . . 12 (𝐶 ∈ ℕ → 𝐶 ∈ ℝ)
28 nnre 12296 . . . . . . . . . . . 12 (𝐵 ∈ ℕ → 𝐵 ∈ ℝ)
29 readdcl 11263 . . . . . . . . . . . 12 ((𝐶 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐶 + 𝐵) ∈ ℝ)
3027, 28, 29syl2anr 596 . . . . . . . . . . 11 ((𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐶 + 𝐵) ∈ ℝ)
31303adant1 1130 . . . . . . . . . 10 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐶 + 𝐵) ∈ ℝ)
32313ad2ant1 1133 . . . . . . . . 9 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (𝐶 + 𝐵) ∈ ℝ)
33273ad2ant3 1135 . . . . . . . . . . . 12 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝐶 ∈ ℝ)
34283ad2ant2 1134 . . . . . . . . . . . 12 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝐵 ∈ ℝ)
35 nngt0 12320 . . . . . . . . . . . . 13 (𝐶 ∈ ℕ → 0 < 𝐶)
36353ad2ant3 1135 . . . . . . . . . . . 12 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 0 < 𝐶)
37 nngt0 12320 . . . . . . . . . . . . 13 (𝐵 ∈ ℕ → 0 < 𝐵)
38373ad2ant2 1134 . . . . . . . . . . . 12 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 0 < 𝐵)
3933, 34, 36, 38addgt0d 11861 . . . . . . . . . . 11 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 0 < (𝐶 + 𝐵))
40393ad2ant1 1133 . . . . . . . . . 10 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 0 < (𝐶 + 𝐵))
41 0re 11288 . . . . . . . . . . 11 0 ∈ ℝ
42 ltle 11374 . . . . . . . . . . 11 ((0 ∈ ℝ ∧ (𝐶 + 𝐵) ∈ ℝ) → (0 < (𝐶 + 𝐵) → 0 ≤ (𝐶 + 𝐵)))
4341, 42mpan 689 . . . . . . . . . 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 15300 . . . . . . . . 9 (((𝐶 + 𝐵) ∈ ℝ ∧ 0 ≤ (𝐶 + 𝐵)) → ((√‘(𝐶 + 𝐵))↑2) = (𝐶 + 𝐵))
4632, 44, 45syl2anc 583 . . . . . . . 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 7460 . . . . . . 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 11596 . . . . . . . . . . 11 ((𝐶 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐶𝐵) ∈ ℝ)
4927, 28, 48syl2anr 596 . . . . . . . . . 10 ((𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐶𝐵) ∈ ℝ)
50493adant1 1130 . . . . . . . . 9 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐶𝐵) ∈ ℝ)
51503ad2ant1 1133 . . . . . . . 8 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (𝐶𝐵) ∈ ℝ)
52 pythagtriplem10 16862 . . . . . . . . . 10 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2)) → 0 < (𝐶𝐵))
53523adant3 1132 . . . . . . . . 9 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 0 < (𝐶𝐵))
54 ltle 11374 . . . . . . . . . 10 ((0 ∈ ℝ ∧ (𝐶𝐵) ∈ ℝ) → (0 < (𝐶𝐵) → 0 ≤ (𝐶𝐵)))
5541, 54mpan 689 . . . . . . . . 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 15300 . . . . . . . 8 (((𝐶𝐵) ∈ ℝ ∧ 0 ≤ (𝐶𝐵)) → ((√‘(𝐶𝐵))↑2) = (𝐶𝐵))
5851, 56, 57syl2anc 583 . . . . . . 7 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((√‘(𝐶𝐵))↑2) = (𝐶𝐵))
5947, 58oveq12d 7463 . . . . . 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 1133 . . . . . . . 8 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (𝐶 + 𝐵) ∈ ℂ)
618, 12mulcld 11306 . . . . . . . . . 10 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → ((√‘(𝐶 + 𝐵)) · (√‘(𝐶𝐵))) ∈ ℂ)
62 mulcl 11264 . . . . . . . . . 10 ((2 ∈ ℂ ∧ ((√‘(𝐶 + 𝐵)) · (√‘(𝐶𝐵))) ∈ ℂ) → (2 · ((√‘(𝐶 + 𝐵)) · (√‘(𝐶𝐵)))) ∈ ℂ)
6315, 61, 62sylancr 586 . . . . . . . . 9 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (2 · ((√‘(𝐶 + 𝐵)) · (√‘(𝐶𝐵)))) ∈ ℂ)
64633ad2ant1 1133 . . . . . . . 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 1133 . . . . . . . 8 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (𝐶𝐵) ∈ ℂ)
6660, 64, 65add32d 11513 . . . . . . 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 1135 . . . . . . . . . 10 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝐶 ∈ ℂ)
68673ad2ant1 1133 . . . . . . . . 9 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 𝐶 ∈ ℂ)
69 nncn 12297 . . . . . . . . . . 11 (𝐴 ∈ ℕ → 𝐴 ∈ ℂ)
70693ad2ant1 1133 . . . . . . . . . 10 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝐴 ∈ ℂ)
71703ad2ant1 1133 . . . . . . . . 9 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 𝐴 ∈ ℂ)
72 adddi 11269 . . . . . . . . 9 ((2 ∈ ℂ ∧ 𝐶 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (2 · (𝐶 + 𝐴)) = ((2 · 𝐶) + (2 · 𝐴)))
7315, 68, 71, 72mp3an2i 1466 . . . . . . . 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 1134 . . . . . . . . . . . 12 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝐵 ∈ ℂ)
75743ad2ant1 1133 . . . . . . . . . . 11 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 𝐵 ∈ ℂ)
7668, 75, 68ppncand 11683 . . . . . . . . . 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 12532 . . . . . . . . . 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 2777 . . . . . . . . 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 7452 . . . . . . . . . . . . . 14 (((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) → (((𝐴↑2) + (𝐵↑2)) − (𝐵↑2)) = ((𝐶↑2) − (𝐵↑2)))
80793ad2ant2 1134 . . . . . . . . . . . . 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 14190 . . . . . . . . . . . . . 14 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (𝐴↑2) ∈ ℂ)
8275sqcld 14190 . . . . . . . . . . . . . 14 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (𝐵↑2) ∈ ℂ)
8381, 82pncand 11644 . . . . . . . . . . . . 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 14255 . . . . . . . . . . . . . 14 ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐶↑2) − (𝐵↑2)) = ((𝐶 + 𝐵) · (𝐶𝐵)))
8568, 75, 84syl2anc 583 . . . . . . . . . . . . 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 2783 . . . . . . . . . . . 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 6923 . . . . . . . . . . 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 15469 . . . . . . . . . . 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 12296 . . . . . . . . . . . . . 14 (𝐴 ∈ ℕ → 𝐴 ∈ ℝ)
90893ad2ant1 1133 . . . . . . . . . . . . 13 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝐴 ∈ ℝ)
91903ad2ant1 1133 . . . . . . . . . . . 12 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 𝐴 ∈ ℝ)
92 nnnn0 12556 . . . . . . . . . . . . . . 15 (𝐴 ∈ ℕ → 𝐴 ∈ ℕ0)
9392nn0ge0d 12612 . . . . . . . . . . . . . 14 (𝐴 ∈ ℕ → 0 ≤ 𝐴)
94933ad2ant1 1133 . . . . . . . . . . . . 13 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 0 ≤ 𝐴)
95943ad2ant1 1133 . . . . . . . . . . . 12 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 0 ≤ 𝐴)
9691, 95sqrtsqd 15464 . . . . . . . . . . 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 2782 . . . . . . . . . 10 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((√‘(𝐶 + 𝐵)) · (√‘(𝐶𝐵))) = 𝐴)
9897oveq2d 7461 . . . . . . . . 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 7463 . . . . . . . 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 2777 . . . . . . 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 2777 . . . . . 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 2778 . . . . 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 7460 . . . 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 11262 . . . . . . . . 9 ((𝐶 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (𝐶 + 𝐴) ∈ ℂ)
1053, 69, 104syl2anr 596 . . . . . . . 8 ((𝐴 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐶 + 𝐴) ∈ ℂ)
1061053adant2 1131 . . . . . . 7 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐶 + 𝐴) ∈ ℂ)
1071063ad2ant1 1133 . . . . . 6 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (𝐶 + 𝐴) ∈ ℂ)
108 mulcl 11264 . . . . . 6 ((2 ∈ ℂ ∧ (𝐶 + 𝐴) ∈ ℂ) → (2 · (𝐶 + 𝐴)) ∈ ℂ)
10915, 107, 108sylancr 586 . . . . 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 12499 . . . . . 6 (2 ∈ ℂ ∧ 2 ≠ 0)
111 divdiv1 12001 . . . . . 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 1453 . . . . 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 2777 . . 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 11971 . . . . . 6 (((𝐶 + 𝐴) ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0) → ((2 · (𝐶 + 𝐴)) / 2) = (𝐶 + 𝐴))
11615, 16, 115mp3an23 1453 . . . . 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 7460 . . 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 2778 . 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 2786 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 395  w3a 1087   = wceq 1537  wcel 2103  wne 2942   class class class wbr 5169  cfv 6572  (class class class)co 7445  cc 11178  cr 11179  0cc0 11180  1c1 11181   + caddc 11183   · cmul 11185   < clt 11320  cle 11321  cmin 11516   / cdiv 11943  cn 12289  2c2 12344  cexp 14108  csqrt 15278  cdvds 16296   gcd cgcd 16534
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705  ax-sep 5320  ax-nul 5327  ax-pow 5386  ax-pr 5450  ax-un 7766  ax-cnex 11236  ax-resscn 11237  ax-1cn 11238  ax-icn 11239  ax-addcl 11240  ax-addrcl 11241  ax-mulcl 11242  ax-mulrcl 11243  ax-mulcom 11244  ax-addass 11245  ax-mulass 11246  ax-distr 11247  ax-i2m1 11248  ax-1ne0 11249  ax-1rid 11250  ax-rnegex 11251  ax-rrecex 11252  ax-cnre 11253  ax-pre-lttri 11254  ax-pre-lttrn 11255  ax-pre-ltadd 11256  ax-pre-mulgt0 11257  ax-pre-sup 11258
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2890  df-ne 2943  df-nel 3049  df-ral 3064  df-rex 3073  df-rmo 3383  df-reu 3384  df-rab 3439  df-v 3484  df-sbc 3799  df-csb 3916  df-dif 3973  df-un 3975  df-in 3977  df-ss 3987  df-pss 3990  df-nul 4348  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5021  df-br 5170  df-opab 5232  df-mpt 5253  df-tr 5287  df-id 5597  df-eprel 5603  df-po 5611  df-so 5612  df-fr 5654  df-we 5656  df-xp 5705  df-rel 5706  df-cnv 5707  df-co 5708  df-dm 5709  df-rn 5710  df-res 5711  df-ima 5712  df-pred 6331  df-ord 6397  df-on 6398  df-lim 6399  df-suc 6400  df-iota 6524  df-fun 6574  df-fn 6575  df-f 6576  df-f1 6577  df-fo 6578  df-f1o 6579  df-fv 6580  df-riota 7401  df-ov 7448  df-oprab 7449  df-mpo 7450  df-om 7900  df-2nd 8027  df-frecs 8318  df-wrecs 8349  df-recs 8423  df-rdg 8462  df-er 8759  df-en 9000  df-dom 9001  df-sdom 9002  df-sup 9507  df-pnf 11322  df-mnf 11323  df-xr 11324  df-ltxr 11325  df-le 11326  df-sub 11518  df-neg 11519  df-div 11944  df-nn 12290  df-2 12352  df-3 12353  df-n0 12550  df-z 12636  df-uz 12900  df-rp 13054  df-seq 14049  df-exp 14109  df-cj 15144  df-re 15145  df-im 15146  df-sqrt 15280  df-abs 15281
This theorem is referenced by:  pythagtriplem15  16871  pythagtriplem17  16873
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