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Theorem pythagtriplem12 15935

Description: Lemma for pythagtrip 15943. 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)) + (√‘(𝐶 − 𝐵))) / 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**
Step	Hyp	Ref	Expression
1		pythagtriplem11.1	. . 3 ⊢ 𝑀 = (((√‘(𝐶 + 𝐵)) + (√‘(𝐶 − 𝐵))) / 2)
2	1	oveq1i 6932	. 2 ⊢ (𝑀↑2) = ((((√‘(𝐶 + 𝐵)) + (√‘(𝐶 − 𝐵))) / 2)↑2)
3		nncn 11383	. . . . . . . . 9 ⊢ (𝐶 ∈ ℕ → 𝐶 ∈ ℂ)
4		nncn 11383	. . . . . . . . 9 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℂ)
5		addcl 10354	. . . . . . . . 9 ⊢ ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐶 + 𝐵) ∈ ℂ)
6	3, 4, 5	syl2anr 590	. . . . . . . 8 ⊢ ((𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐶 + 𝐵) ∈ ℂ)
7	6	3adant1 1121	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐶 + 𝐵) ∈ ℂ)
8	7	sqrtcld 14584	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (√‘(𝐶 + 𝐵)) ∈ ℂ)
9		subcl 10621	. . . . . . . . 9 ⊢ ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐶 − 𝐵) ∈ ℂ)
10	3, 4, 9	syl2anr 590	. . . . . . . 8 ⊢ ((𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐶 − 𝐵) ∈ ℂ)
11	10	3adant1 1121	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐶 − 𝐵) ∈ ℂ)
12	11	sqrtcld 14584	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (√‘(𝐶 − 𝐵)) ∈ ℂ)
13	8, 12	addcld 10396	. . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → ((√‘(𝐶 + 𝐵)) + (√‘(𝐶 − 𝐵))) ∈ ℂ)
14	13	3ad2ant1 1124	. . . 4 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((√‘(𝐶 + 𝐵)) + (√‘(𝐶 − 𝐵))) ∈ ℂ)
15		2cn 11450	. . . . . 6 ⊢ 2 ∈ ℂ
16		2ne0 11486	. . . . . 6 ⊢ 2 ≠ 0
17		sqdiv 13246	. . . . . 6 ⊢ ((((√‘(𝐶 + 𝐵)) + (√‘(𝐶 − 𝐵))) ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0) → ((((√‘(𝐶 + 𝐵)) + (√‘(𝐶 − 𝐵))) / 2)↑2) = ((((√‘(𝐶 + 𝐵)) + (√‘(𝐶 − 𝐵)))↑2) / (2↑2)))
18	15, 16, 17	mp3an23 1526	. . . . 5 ⊢ (((√‘(𝐶 + 𝐵)) + (√‘(𝐶 − 𝐵))) ∈ ℂ → ((((√‘(𝐶 + 𝐵)) + (√‘(𝐶 − 𝐵))) / 2)↑2) = ((((√‘(𝐶 + 𝐵)) + (√‘(𝐶 − 𝐵)))↑2) / (2↑2)))
19	15	sqvali 13262	. . . . . 6 ⊢ (2↑2) = (2 · 2)
20	19	oveq2i 6933	. . . . 5 ⊢ ((((√‘(𝐶 + 𝐵)) + (√‘(𝐶 − 𝐵)))↑2) / (2↑2)) = ((((√‘(𝐶 + 𝐵)) + (√‘(𝐶 − 𝐵)))↑2) / (2 · 2))
21	18, 20	syl6eq 2830	. . . 4 ⊢ (((√‘(𝐶 + 𝐵)) + (√‘(𝐶 − 𝐵))) ∈ ℂ → ((((√‘(𝐶 + 𝐵)) + (√‘(𝐶 − 𝐵))) / 2)↑2) = ((((√‘(𝐶 + 𝐵)) + (√‘(𝐶 − 𝐵)))↑2) / (2 · 2)))
22	14, 21	syl 17	. . 3 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((((√‘(𝐶 + 𝐵)) + (√‘(𝐶 − 𝐵))) / 2)↑2) = ((((√‘(𝐶 + 𝐵)) + (√‘(𝐶 − 𝐵)))↑2) / (2 · 2)))
23	8	3ad2ant1 1124	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (√‘(𝐶 + 𝐵)) ∈ ℂ)
24	12	3ad2ant1 1124	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (√‘(𝐶 − 𝐵)) ∈ ℂ)
25		binom2 13298	. . . . . . 7 ⊢ (((√‘(𝐶 + 𝐵)) ∈ ℂ ∧ (√‘(𝐶 − 𝐵)) ∈ ℂ) → (((√‘(𝐶 + 𝐵)) + (√‘(𝐶 − 𝐵)))↑2) = ((((√‘(𝐶 + 𝐵))↑2) + (2 · ((√‘(𝐶 + 𝐵)) · (√‘(𝐶 − 𝐵))))) + ((√‘(𝐶 − 𝐵))↑2)))
26	23, 24, 25	syl2anc 579	. . . . . 6 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (((√‘(𝐶 + 𝐵)) + (√‘(𝐶 − 𝐵)))↑2) = ((((√‘(𝐶 + 𝐵))↑2) + (2 · ((√‘(𝐶 + 𝐵)) · (√‘(𝐶 − 𝐵))))) + ((√‘(𝐶 − 𝐵))↑2)))
27		nnre 11382	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ ℕ → 𝐶 ∈ ℝ)
28		nnre 11382	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℝ)
29		readdcl 10355	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐶 + 𝐵) ∈ ℝ)
30	27, 28, 29	syl2anr 590	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐶 + 𝐵) ∈ ℝ)
31	30	3adant1 1121	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐶 + 𝐵) ∈ ℝ)
32	31	3ad2ant1 1124	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (𝐶 + 𝐵) ∈ ℝ)
33	27	3ad2ant3 1126	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝐶 ∈ ℝ)
34	28	3ad2ant2 1125	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝐵 ∈ ℝ)
35		nngt0 11407	. . . . . . . . . . . . 13 ⊢ (𝐶 ∈ ℕ → 0 < 𝐶)
36	35	3ad2ant3 1126	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 0 < 𝐶)
37		nngt0 11407	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ ℕ → 0 < 𝐵)
38	37	3ad2ant2 1125	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 0 < 𝐵)
39	33, 34, 36, 38	addgt0d 10950	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 0 < (𝐶 + 𝐵))
40	39	3ad2ant1 1124	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 0 < (𝐶 + 𝐵))
41		0re 10378	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ
42		ltle 10465	. . . . . . . . . . 11 ⊢ ((0 ∈ ℝ ∧ (𝐶 + 𝐵) ∈ ℝ) → (0 < (𝐶 + 𝐵) → 0 ≤ (𝐶 + 𝐵)))
43	41, 42	mpan 680	. . . . . . . . . 10 ⊢ ((𝐶 + 𝐵) ∈ ℝ → (0 < (𝐶 + 𝐵) → 0 ≤ (𝐶 + 𝐵)))
44	32, 40, 43	sylc 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 14403	. . . . . . . . 9 ⊢ (((𝐶 + 𝐵) ∈ ℝ ∧ 0 ≤ (𝐶 + 𝐵)) → ((√‘(𝐶 + 𝐵))↑2) = (𝐶 + 𝐵))
46	32, 44, 45	syl2anc 579	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((√‘(𝐶 + 𝐵))↑2) = (𝐶 + 𝐵))
47	46	oveq1d 6937	. . . . . . 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)) · (√‘(𝐶 − 𝐵))))) = ((𝐶 + 𝐵) + (2 · ((√‘(𝐶 + 𝐵)) · (√‘(𝐶 − 𝐵))))))
48		resubcl 10687	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐶 − 𝐵) ∈ ℝ)
49	27, 28, 48	syl2anr 590	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐶 − 𝐵) ∈ ℝ)
50	49	3adant1 1121	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐶 − 𝐵) ∈ ℝ)
51	50	3ad2ant1 1124	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (𝐶 − 𝐵) ∈ ℝ)
52		pythagtriplem10 15929	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2)) → 0 < (𝐶 − 𝐵))
53	52	3adant3 1123	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 0 < (𝐶 − 𝐵))
54		ltle 10465	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ ∧ (𝐶 − 𝐵) ∈ ℝ) → (0 < (𝐶 − 𝐵) → 0 ≤ (𝐶 − 𝐵)))
55	41, 54	mpan 680	. . . . . . . . 9 ⊢ ((𝐶 − 𝐵) ∈ ℝ → (0 < (𝐶 − 𝐵) → 0 ≤ (𝐶 − 𝐵)))
56	51, 53, 55	sylc 65	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 0 ≤ (𝐶 − 𝐵))
57		resqrtth 14403	. . . . . . . 8 ⊢ (((𝐶 − 𝐵) ∈ ℝ ∧ 0 ≤ (𝐶 − 𝐵)) → ((√‘(𝐶 − 𝐵))↑2) = (𝐶 − 𝐵))
58	51, 56, 57	syl2anc 579	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((√‘(𝐶 − 𝐵))↑2) = (𝐶 − 𝐵))
59	47, 58	oveq12d 6940	. . . . . 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)) · (√‘(𝐶 − 𝐵))))) + ((√‘(𝐶 − 𝐵))↑2)) = (((𝐶 + 𝐵) + (2 · ((√‘(𝐶 + 𝐵)) · (√‘(𝐶 − 𝐵))))) + (𝐶 − 𝐵)))
60	7	3ad2ant1 1124	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (𝐶 + 𝐵) ∈ ℂ)
61	8, 12	mulcld 10397	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → ((√‘(𝐶 + 𝐵)) · (√‘(𝐶 − 𝐵))) ∈ ℂ)
62		mulcl 10356	. . . . . . . . . 10 ⊢ ((2 ∈ ℂ ∧ ((√‘(𝐶 + 𝐵)) · (√‘(𝐶 − 𝐵))) ∈ ℂ) → (2 · ((√‘(𝐶 + 𝐵)) · (√‘(𝐶 − 𝐵)))) ∈ ℂ)
63	15, 61, 62	sylancr 581	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (2 · ((√‘(𝐶 + 𝐵)) · (√‘(𝐶 − 𝐵)))) ∈ ℂ)
64	63	3ad2ant1 1124	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (2 · ((√‘(𝐶 + 𝐵)) · (√‘(𝐶 − 𝐵)))) ∈ ℂ)
65	11	3ad2ant1 1124	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (𝐶 − 𝐵) ∈ ℂ)
66	60, 64, 65	add32d 10603	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (((𝐶 + 𝐵) + (2 · ((√‘(𝐶 + 𝐵)) · (√‘(𝐶 − 𝐵))))) + (𝐶 − 𝐵)) = (((𝐶 + 𝐵) + (𝐶 − 𝐵)) + (2 · ((√‘(𝐶 + 𝐵)) · (√‘(𝐶 − 𝐵))))))
67	3	3ad2ant3 1126	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝐶 ∈ ℂ)
68	67	3ad2ant1 1124	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 𝐶 ∈ ℂ)
69		nncn 11383	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℂ)
70	69	3ad2ant1 1124	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝐴 ∈ ℂ)
71	70	3ad2ant1 1124	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 𝐴 ∈ ℂ)
72		adddi 10361	. . . . . . . . 9 ⊢ ((2 ∈ ℂ ∧ 𝐶 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (2 · (𝐶 + 𝐴)) = ((2 · 𝐶) + (2 · 𝐴)))
73	15, 68, 71, 72	mp3an2i 1539	. . . . . . . 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)))
74	4	3ad2ant2 1125	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝐵 ∈ ℂ)
75	74	3ad2ant1 1124	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 𝐵 ∈ ℂ)
76	68, 75, 68	ppncand 10774	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((𝐶 + 𝐵) + (𝐶 − 𝐵)) = (𝐶 + 𝐶))
77	68	2timesd 11625	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (2 · 𝐶) = (𝐶 + 𝐶))
78	76, 77	eqtr4d 2817	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((𝐶 + 𝐵) + (𝐶 − 𝐵)) = (2 · 𝐶))
79		oveq1 6929	. . . . . . . . . . . . . 14 ⊢ (((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) → (((𝐴↑2) + (𝐵↑2)) − (𝐵↑2)) = ((𝐶↑2) − (𝐵↑2)))
80	79	3ad2ant2 1125	. . . . . . . . . . . . 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)))
81	71	sqcld 13325	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (𝐴↑2) ∈ ℂ)
82	75	sqcld 13325	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (𝐵↑2) ∈ ℂ)
83	81, 82	pncand 10735	. . . . . . . . . . . . 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 13291	. . . . . . . . . . . . . 14 ⊢ ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐶↑2) − (𝐵↑2)) = ((𝐶 + 𝐵) · (𝐶 − 𝐵)))
85	68, 75, 84	syl2anc 579	. . . . . . . . . . . . 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) · (𝐶 − 𝐵)))
86	80, 83, 85	3eqtr3rd 2823	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((𝐶 + 𝐵) · (𝐶 − 𝐵)) = (𝐴↑2))
87	86	fveq2d 6450	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (√‘((𝐶 + 𝐵) · (𝐶 − 𝐵))) = (√‘(𝐴↑2)))
88	32, 44, 51, 56	sqrtmuld 14571	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (√‘((𝐶 + 𝐵) · (𝐶 − 𝐵))) = ((√‘(𝐶 + 𝐵)) · (√‘(𝐶 − 𝐵))))
89		nnre 11382	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℝ)
90	89	3ad2ant1 1124	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝐴 ∈ ℝ)
91	90	3ad2ant1 1124	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 𝐴 ∈ ℝ)
92		nnnn0 11650	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℕ₀)
93	92	nn0ge0d 11705	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℕ → 0 ≤ 𝐴)
94	93	3ad2ant1 1124	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 0 ≤ 𝐴)
95	94	3ad2ant1 1124	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 0 ≤ 𝐴)
96	91, 95	sqrtsqd 14566	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (√‘(𝐴↑2)) = 𝐴)
97	87, 88, 96	3eqtr3d 2822	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((√‘(𝐶 + 𝐵)) · (√‘(𝐶 − 𝐵))) = 𝐴)
98	97	oveq2d 6938	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (2 · ((√‘(𝐶 + 𝐵)) · (√‘(𝐶 − 𝐵)))) = (2 · 𝐴))
99	78, 98	oveq12d 6940	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (((𝐶 + 𝐵) + (𝐶 − 𝐵)) + (2 · ((√‘(𝐶 + 𝐵)) · (√‘(𝐶 − 𝐵))))) = ((2 · 𝐶) + (2 · 𝐴)))
100	73, 99	eqtr4d 2817	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (2 · (𝐶 + 𝐴)) = (((𝐶 + 𝐵) + (𝐶 − 𝐵)) + (2 · ((√‘(𝐶 + 𝐵)) · (√‘(𝐶 − 𝐵))))))
101	66, 100	eqtr4d 2817	. . . . . 6 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (((𝐶 + 𝐵) + (2 · ((√‘(𝐶 + 𝐵)) · (√‘(𝐶 − 𝐵))))) + (𝐶 − 𝐵)) = (2 · (𝐶 + 𝐴)))
102	26, 59, 101	3eqtrd 2818	. . . . 5 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (((√‘(𝐶 + 𝐵)) + (√‘(𝐶 − 𝐵)))↑2) = (2 · (𝐶 + 𝐴)))
103	102	oveq1d 6937	. . . 4 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((((√‘(𝐶 + 𝐵)) + (√‘(𝐶 − 𝐵)))↑2) / (2 · 2)) = ((2 · (𝐶 + 𝐴)) / (2 · 2)))
104		addcl 10354	. . . . . . . . 9 ⊢ ((𝐶 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (𝐶 + 𝐴) ∈ ℂ)
105	3, 69, 104	syl2anr 590	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐶 + 𝐴) ∈ ℂ)
106	105	3adant2 1122	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐶 + 𝐴) ∈ ℂ)
107	106	3ad2ant1 1124	. . . . . 6 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (𝐶 + 𝐴) ∈ ℂ)
108		mulcl 10356	. . . . . 6 ⊢ ((2 ∈ ℂ ∧ (𝐶 + 𝐴) ∈ ℂ) → (2 · (𝐶 + 𝐴)) ∈ ℂ)
109	15, 107, 108	sylancr 581	. . . . 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 11592	. . . . . 6 ⊢ (2 ∈ ℂ ∧ 2 ≠ 0)
111		divdiv1 11086	. . . . . 6 ⊢ (((2 · (𝐶 + 𝐴)) ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 ≠ 0) ∧ (2 ∈ ℂ ∧ 2 ≠ 0)) → (((2 · (𝐶 + 𝐴)) / 2) / 2) = ((2 · (𝐶 + 𝐴)) / (2 · 2)))
112	110, 110, 111	mp3an23 1526	. . . . 5 ⊢ ((2 · (𝐶 + 𝐴)) ∈ ℂ → (((2 · (𝐶 + 𝐴)) / 2) / 2) = ((2 · (𝐶 + 𝐴)) / (2 · 2)))
113	109, 112	syl 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)))
114	103, 113	eqtr4d 2817	. . 3 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((((√‘(𝐶 + 𝐵)) + (√‘(𝐶 − 𝐵)))↑2) / (2 · 2)) = (((2 · (𝐶 + 𝐴)) / 2) / 2))
115		divcan3 11059	. . . . . 6 ⊢ (((𝐶 + 𝐴) ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0) → ((2 · (𝐶 + 𝐴)) / 2) = (𝐶 + 𝐴))
116	15, 16, 115	mp3an23 1526	. . . . 5 ⊢ ((𝐶 + 𝐴) ∈ ℂ → ((2 · (𝐶 + 𝐴)) / 2) = (𝐶 + 𝐴))
117	107, 116	syl 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))
118	117	oveq1d 6937	. . 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))
119	22, 114, 118	3eqtrd 2818	. 2 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((((√‘(𝐶 + 𝐵)) + (√‘(𝐶 − 𝐵))) / 2)↑2) = ((𝐶 + 𝐴) / 2))
120	2, 119	syl5eq 2826	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 386 ∧ w3a 1071 = wceq 1601 ∈ wcel 2107 ≠ wne 2969 class class class wbr 4886 ‘cfv 6135 (class class class)co 6922 ℂcc 10270 ℝcr 10271 0cc0 10272 1c1 10273 + caddc 10275 · cmul 10277 < clt 10411 ≤ cle 10412 − cmin 10606 / cdiv 11032 ℕcn 11374 2c2 11430 ↑cexp 13178 √csqrt 14380 ∥ cdvds 15387 gcd cgcd 15622

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 ax-pre-sup 10350

This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-sup 8636 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-div 11033 df-nn 11375 df-2 11438 df-3 11439 df-n0 11643 df-z 11729 df-uz 11993 df-rp 12138 df-seq 13120 df-exp 13179 df-cj 14246 df-re 14247 df-im 14248 df-sqrt 14382 df-abs 14383

This theorem is referenced by: pythagtriplem15 15938 pythagtriplem17 15940

W3C validator