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Theorem pythagtrip 12395

Description: Parameterize the Pythagorean triples. If 𝐴, 𝐵, and 𝐶 are naturals, then they obey the Pythagorean triple formula iff they are parameterized by three naturals. This proof follows the Isabelle proof at http://afp.sourceforge.net/entries/Fermat3_4.shtml. This is Metamath 100 proof #23. (Contributed by Scott Fenton, 19-Apr-2014.)

**Assertion**
Ref	Expression
pythagtrip	⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ↔ ∃𝑛 ∈ ℕ ∃𝑚 ∈ ℕ ∃𝑘 ∈ ℕ ({𝐴, 𝐵} = {(𝑘 · ((𝑚↑2) − (𝑛↑2))), (𝑘 · (2 · (𝑚 · 𝑛)))} ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2))))))

Distinct variable groups: 𝐴,𝑘,𝑚,𝑛 𝐵,𝑘,𝑚,𝑛 𝐶,𝑘,𝑚,𝑛

**Proof of Theorem pythagtrip**
Step	Hyp	Ref	Expression
1		divgcdodd 12255	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (¬ 2 ∥ (𝐴 / (𝐴 gcd 𝐵)) ∨ ¬ 2 ∥ (𝐵 / (𝐴 gcd 𝐵))))
2	1	3adant3 1019	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (¬ 2 ∥ (𝐴 / (𝐴 gcd 𝐵)) ∨ ¬ 2 ∥ (𝐵 / (𝐴 gcd 𝐵))))
3	2	adantr 276	. . . . 5 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2)) → (¬ 2 ∥ (𝐴 / (𝐴 gcd 𝐵)) ∨ ¬ 2 ∥ (𝐵 / (𝐴 gcd 𝐵))))
4		pythagtriplem19 12394	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ¬ 2 ∥ (𝐴 / (𝐴 gcd 𝐵))) → ∃𝑛 ∈ ℕ ∃𝑚 ∈ ℕ ∃𝑘 ∈ ℕ (𝐴 = (𝑘 · ((𝑚↑2) − (𝑛↑2))) ∧ 𝐵 = (𝑘 · (2 · (𝑚 · 𝑛))) ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2)))))
5	4	3expia 1207	. . . . . 6 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2)) → (¬ 2 ∥ (𝐴 / (𝐴 gcd 𝐵)) → ∃𝑛 ∈ ℕ ∃𝑚 ∈ ℕ ∃𝑘 ∈ ℕ (𝐴 = (𝑘 · ((𝑚↑2) − (𝑛↑2))) ∧ 𝐵 = (𝑘 · (2 · (𝑚 · 𝑛))) ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2))))))
6		simp12 1030	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ¬ 2 ∥ (𝐵 / (𝐴 gcd 𝐵))) → 𝐵 ∈ ℕ)
7		simp11 1029	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ¬ 2 ∥ (𝐵 / (𝐴 gcd 𝐵))) → 𝐴 ∈ ℕ)
8		simp13 1031	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ¬ 2 ∥ (𝐵 / (𝐴 gcd 𝐵))) → 𝐶 ∈ ℕ)
9		nnsqcl 10654	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℕ → (𝐴↑2) ∈ ℕ)
10	9	nncnd 8982	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℕ → (𝐴↑2) ∈ ℂ)
11	10	3ad2ant1 1020	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐴↑2) ∈ ℂ)
12		nnsqcl 10654	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ ℕ → (𝐵↑2) ∈ ℕ)
13	12	nncnd 8982	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ ℕ → (𝐵↑2) ∈ ℂ)
14	13	3ad2ant2 1021	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐵↑2) ∈ ℂ)
15	11, 14	addcomd 8156	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → ((𝐴↑2) + (𝐵↑2)) = ((𝐵↑2) + (𝐴↑2)))
16	15	eqeq1d 2198	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ↔ ((𝐵↑2) + (𝐴↑2)) = (𝐶↑2)))
17	16	biimpa 296	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2)) → ((𝐵↑2) + (𝐴↑2)) = (𝐶↑2))
18	17	3adant3 1019	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ¬ 2 ∥ (𝐵 / (𝐴 gcd 𝐵))) → ((𝐵↑2) + (𝐴↑2)) = (𝐶↑2))
19		nnz 9322	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℤ)
20	19	3ad2ant1 1020	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝐴 ∈ ℤ)
21		nnz 9322	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℤ)
22	21	3ad2ant2 1021	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝐵 ∈ ℤ)
23	22	adantr 276	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2)) → 𝐵 ∈ ℤ)
24		gcdcom 12084	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 gcd 𝐵) = (𝐵 gcd 𝐴))
25	20, 23, 24	syl2an2r 595	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2)) → (𝐴 gcd 𝐵) = (𝐵 gcd 𝐴))
26	25	oveq2d 5922	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2)) → (𝐵 / (𝐴 gcd 𝐵)) = (𝐵 / (𝐵 gcd 𝐴)))
27	26	breq2d 4037	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2)) → (2 ∥ (𝐵 / (𝐴 gcd 𝐵)) ↔ 2 ∥ (𝐵 / (𝐵 gcd 𝐴))))
28	27	notbid 668	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2)) → (¬ 2 ∥ (𝐵 / (𝐴 gcd 𝐵)) ↔ ¬ 2 ∥ (𝐵 / (𝐵 gcd 𝐴))))
29	28	biimp3a 1356	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ¬ 2 ∥ (𝐵 / (𝐴 gcd 𝐵))) → ¬ 2 ∥ (𝐵 / (𝐵 gcd 𝐴)))
30		pythagtriplem19 12394	. . . . . . . 8 ⊢ (((𝐵 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐵↑2) + (𝐴↑2)) = (𝐶↑2) ∧ ¬ 2 ∥ (𝐵 / (𝐵 gcd 𝐴))) → ∃𝑛 ∈ ℕ ∃𝑚 ∈ ℕ ∃𝑘 ∈ ℕ (𝐵 = (𝑘 · ((𝑚↑2) − (𝑛↑2))) ∧ 𝐴 = (𝑘 · (2 · (𝑚 · 𝑛))) ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2)))))
31	6, 7, 8, 18, 29, 30	syl311anc 1263	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ¬ 2 ∥ (𝐵 / (𝐴 gcd 𝐵))) → ∃𝑛 ∈ ℕ ∃𝑚 ∈ ℕ ∃𝑘 ∈ ℕ (𝐵 = (𝑘 · ((𝑚↑2) − (𝑛↑2))) ∧ 𝐴 = (𝑘 · (2 · (𝑚 · 𝑛))) ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2)))))
32	31	3expia 1207	. . . . . 6 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2)) → (¬ 2 ∥ (𝐵 / (𝐴 gcd 𝐵)) → ∃𝑛 ∈ ℕ ∃𝑚 ∈ ℕ ∃𝑘 ∈ ℕ (𝐵 = (𝑘 · ((𝑚↑2) − (𝑛↑2))) ∧ 𝐴 = (𝑘 · (2 · (𝑚 · 𝑛))) ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2))))))
33	5, 32	orim12d 787	. . . . 5 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2)) → ((¬ 2 ∥ (𝐴 / (𝐴 gcd 𝐵)) ∨ ¬ 2 ∥ (𝐵 / (𝐴 gcd 𝐵))) → (∃𝑛 ∈ ℕ ∃𝑚 ∈ ℕ ∃𝑘 ∈ ℕ (𝐴 = (𝑘 · ((𝑚↑2) − (𝑛↑2))) ∧ 𝐵 = (𝑘 · (2 · (𝑚 · 𝑛))) ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2)))) ∨ ∃𝑛 ∈ ℕ ∃𝑚 ∈ ℕ ∃𝑘 ∈ ℕ (𝐵 = (𝑘 · ((𝑚↑2) − (𝑛↑2))) ∧ 𝐴 = (𝑘 · (2 · (𝑚 · 𝑛))) ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2)))))))
34	3, 33	mpd 13	. . . 4 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2)) → (∃𝑛 ∈ ℕ ∃𝑚 ∈ ℕ ∃𝑘 ∈ ℕ (𝐴 = (𝑘 · ((𝑚↑2) − (𝑛↑2))) ∧ 𝐵 = (𝑘 · (2 · (𝑚 · 𝑛))) ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2)))) ∨ ∃𝑛 ∈ ℕ ∃𝑚 ∈ ℕ ∃𝑘 ∈ ℕ (𝐵 = (𝑘 · ((𝑚↑2) − (𝑛↑2))) ∧ 𝐴 = (𝑘 · (2 · (𝑚 · 𝑛))) ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2))))))
35		simplll 533	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ)) ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ ℕ)
36		simpllr 534	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ)) ∧ 𝑘 ∈ ℕ) → 𝐵 ∈ ℕ)
37		nnz 9322	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℤ)
38	37	adantl 277	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ)) ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℤ)
39		simplrr 536	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ)) ∧ 𝑘 ∈ ℕ) → 𝑚 ∈ ℕ)
40	39	nnzd 9424	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ)) ∧ 𝑘 ∈ ℕ) → 𝑚 ∈ ℤ)
41		zsqcl 10655	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ ℤ → (𝑚↑2) ∈ ℤ)
42	40, 41	syl 14	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ)) ∧ 𝑘 ∈ ℕ) → (𝑚↑2) ∈ ℤ)
43		simplrl 535	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ)) ∧ 𝑘 ∈ ℕ) → 𝑛 ∈ ℕ)
44	43	nnzd 9424	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ)) ∧ 𝑘 ∈ ℕ) → 𝑛 ∈ ℤ)
45		zsqcl 10655	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℤ → (𝑛↑2) ∈ ℤ)
46	44, 45	syl 14	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ)) ∧ 𝑘 ∈ ℕ) → (𝑛↑2) ∈ ℤ)
47	42, 46	zsubcld 9430	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ)) ∧ 𝑘 ∈ ℕ) → ((𝑚↑2) − (𝑛↑2)) ∈ ℤ)
48	38, 47	zmulcld 9431	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ)) ∧ 𝑘 ∈ ℕ) → (𝑘 · ((𝑚↑2) − (𝑛↑2))) ∈ ℤ)
49		2z 9331	. . . . . . . . . . . . . 14 ⊢ 2 ∈ ℤ
50	49	a1i 9	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ)) ∧ 𝑘 ∈ ℕ) → 2 ∈ ℤ)
51	40, 44	zmulcld 9431	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ)) ∧ 𝑘 ∈ ℕ) → (𝑚 · 𝑛) ∈ ℤ)
52	50, 51	zmulcld 9431	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ)) ∧ 𝑘 ∈ ℕ) → (2 · (𝑚 · 𝑛)) ∈ ℤ)
53	38, 52	zmulcld 9431	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ)) ∧ 𝑘 ∈ ℕ) → (𝑘 · (2 · (𝑚 · 𝑛))) ∈ ℤ)
54		preq12bg 3795	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ ((𝑘 · ((𝑚↑2) − (𝑛↑2))) ∈ ℤ ∧ (𝑘 · (2 · (𝑚 · 𝑛))) ∈ ℤ)) → ({𝐴, 𝐵} = {(𝑘 · ((𝑚↑2) − (𝑛↑2))), (𝑘 · (2 · (𝑚 · 𝑛)))} ↔ ((𝐴 = (𝑘 · ((𝑚↑2) − (𝑛↑2))) ∧ 𝐵 = (𝑘 · (2 · (𝑚 · 𝑛)))) ∨ (𝐴 = (𝑘 · (2 · (𝑚 · 𝑛))) ∧ 𝐵 = (𝑘 · ((𝑚↑2) − (𝑛↑2)))))))
55	35, 36, 48, 53, 54	syl22anc 1250	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ)) ∧ 𝑘 ∈ ℕ) → ({𝐴, 𝐵} = {(𝑘 · ((𝑚↑2) − (𝑛↑2))), (𝑘 · (2 · (𝑚 · 𝑛)))} ↔ ((𝐴 = (𝑘 · ((𝑚↑2) − (𝑛↑2))) ∧ 𝐵 = (𝑘 · (2 · (𝑚 · 𝑛)))) ∨ (𝐴 = (𝑘 · (2 · (𝑚 · 𝑛))) ∧ 𝐵 = (𝑘 · ((𝑚↑2) − (𝑛↑2)))))))
56	55	anbi1d 465	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ)) ∧ 𝑘 ∈ ℕ) → (({𝐴, 𝐵} = {(𝑘 · ((𝑚↑2) − (𝑛↑2))), (𝑘 · (2 · (𝑚 · 𝑛)))} ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2)))) ↔ (((𝐴 = (𝑘 · ((𝑚↑2) − (𝑛↑2))) ∧ 𝐵 = (𝑘 · (2 · (𝑚 · 𝑛)))) ∨ (𝐴 = (𝑘 · (2 · (𝑚 · 𝑛))) ∧ 𝐵 = (𝑘 · ((𝑚↑2) − (𝑛↑2))))) ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2))))))
57	56	rexbidva 2487	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ)) → (∃𝑘 ∈ ℕ ({𝐴, 𝐵} = {(𝑘 · ((𝑚↑2) − (𝑛↑2))), (𝑘 · (2 · (𝑚 · 𝑛)))} ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2)))) ↔ ∃𝑘 ∈ ℕ (((𝐴 = (𝑘 · ((𝑚↑2) − (𝑛↑2))) ∧ 𝐵 = (𝑘 · (2 · (𝑚 · 𝑛)))) ∨ (𝐴 = (𝑘 · (2 · (𝑚 · 𝑛))) ∧ 𝐵 = (𝑘 · ((𝑚↑2) − (𝑛↑2))))) ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2))))))
58	57	2rexbidva 2513	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (∃𝑛 ∈ ℕ ∃𝑚 ∈ ℕ ∃𝑘 ∈ ℕ ({𝐴, 𝐵} = {(𝑘 · ((𝑚↑2) − (𝑛↑2))), (𝑘 · (2 · (𝑚 · 𝑛)))} ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2)))) ↔ ∃𝑛 ∈ ℕ ∃𝑚 ∈ ℕ ∃𝑘 ∈ ℕ (((𝐴 = (𝑘 · ((𝑚↑2) − (𝑛↑2))) ∧ 𝐵 = (𝑘 · (2 · (𝑚 · 𝑛)))) ∨ (𝐴 = (𝑘 · (2 · (𝑚 · 𝑛))) ∧ 𝐵 = (𝑘 · ((𝑚↑2) − (𝑛↑2))))) ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2))))))
59		andir 820	. . . . . . . . . . 11 ⊢ ((((𝐴 = (𝑘 · ((𝑚↑2) − (𝑛↑2))) ∧ 𝐵 = (𝑘 · (2 · (𝑚 · 𝑛)))) ∨ (𝐴 = (𝑘 · (2 · (𝑚 · 𝑛))) ∧ 𝐵 = (𝑘 · ((𝑚↑2) − (𝑛↑2))))) ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2)))) ↔ (((𝐴 = (𝑘 · ((𝑚↑2) − (𝑛↑2))) ∧ 𝐵 = (𝑘 · (2 · (𝑚 · 𝑛)))) ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2)))) ∨ ((𝐴 = (𝑘 · (2 · (𝑚 · 𝑛))) ∧ 𝐵 = (𝑘 · ((𝑚↑2) − (𝑛↑2)))) ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2))))))
60		df-3an 982	. . . . . . . . . . . 12 ⊢ ((𝐴 = (𝑘 · ((𝑚↑2) − (𝑛↑2))) ∧ 𝐵 = (𝑘 · (2 · (𝑚 · 𝑛))) ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2)))) ↔ ((𝐴 = (𝑘 · ((𝑚↑2) − (𝑛↑2))) ∧ 𝐵 = (𝑘 · (2 · (𝑚 · 𝑛)))) ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2)))))
61		df-3an 982	. . . . . . . . . . . 12 ⊢ ((𝐴 = (𝑘 · (2 · (𝑚 · 𝑛))) ∧ 𝐵 = (𝑘 · ((𝑚↑2) − (𝑛↑2))) ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2)))) ↔ ((𝐴 = (𝑘 · (2 · (𝑚 · 𝑛))) ∧ 𝐵 = (𝑘 · ((𝑚↑2) − (𝑛↑2)))) ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2)))))
62	60, 61	orbi12i 765	. . . . . . . . . . 11 ⊢ (((𝐴 = (𝑘 · ((𝑚↑2) − (𝑛↑2))) ∧ 𝐵 = (𝑘 · (2 · (𝑚 · 𝑛))) ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2)))) ∨ (𝐴 = (𝑘 · (2 · (𝑚 · 𝑛))) ∧ 𝐵 = (𝑘 · ((𝑚↑2) − (𝑛↑2))) ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2))))) ↔ (((𝐴 = (𝑘 · ((𝑚↑2) − (𝑛↑2))) ∧ 𝐵 = (𝑘 · (2 · (𝑚 · 𝑛)))) ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2)))) ∨ ((𝐴 = (𝑘 · (2 · (𝑚 · 𝑛))) ∧ 𝐵 = (𝑘 · ((𝑚↑2) − (𝑛↑2)))) ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2))))))
63		3ancoma 987	. . . . . . . . . . . 12 ⊢ ((𝐴 = (𝑘 · (2 · (𝑚 · 𝑛))) ∧ 𝐵 = (𝑘 · ((𝑚↑2) − (𝑛↑2))) ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2)))) ↔ (𝐵 = (𝑘 · ((𝑚↑2) − (𝑛↑2))) ∧ 𝐴 = (𝑘 · (2 · (𝑚 · 𝑛))) ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2)))))
64	63	orbi2i 763	. . . . . . . . . . 11 ⊢ (((𝐴 = (𝑘 · ((𝑚↑2) − (𝑛↑2))) ∧ 𝐵 = (𝑘 · (2 · (𝑚 · 𝑛))) ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2)))) ∨ (𝐴 = (𝑘 · (2 · (𝑚 · 𝑛))) ∧ 𝐵 = (𝑘 · ((𝑚↑2) − (𝑛↑2))) ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2))))) ↔ ((𝐴 = (𝑘 · ((𝑚↑2) − (𝑛↑2))) ∧ 𝐵 = (𝑘 · (2 · (𝑚 · 𝑛))) ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2)))) ∨ (𝐵 = (𝑘 · ((𝑚↑2) − (𝑛↑2))) ∧ 𝐴 = (𝑘 · (2 · (𝑚 · 𝑛))) ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2))))))
65	59, 62, 64	3bitr2i 208	. . . . . . . . . 10 ⊢ ((((𝐴 = (𝑘 · ((𝑚↑2) − (𝑛↑2))) ∧ 𝐵 = (𝑘 · (2 · (𝑚 · 𝑛)))) ∨ (𝐴 = (𝑘 · (2 · (𝑚 · 𝑛))) ∧ 𝐵 = (𝑘 · ((𝑚↑2) − (𝑛↑2))))) ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2)))) ↔ ((𝐴 = (𝑘 · ((𝑚↑2) − (𝑛↑2))) ∧ 𝐵 = (𝑘 · (2 · (𝑚 · 𝑛))) ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2)))) ∨ (𝐵 = (𝑘 · ((𝑚↑2) − (𝑛↑2))) ∧ 𝐴 = (𝑘 · (2 · (𝑚 · 𝑛))) ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2))))))
66	65	rexbii 2497	. . . . . . . . 9 ⊢ (∃𝑘 ∈ ℕ (((𝐴 = (𝑘 · ((𝑚↑2) − (𝑛↑2))) ∧ 𝐵 = (𝑘 · (2 · (𝑚 · 𝑛)))) ∨ (𝐴 = (𝑘 · (2 · (𝑚 · 𝑛))) ∧ 𝐵 = (𝑘 · ((𝑚↑2) − (𝑛↑2))))) ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2)))) ↔ ∃𝑘 ∈ ℕ ((𝐴 = (𝑘 · ((𝑚↑2) − (𝑛↑2))) ∧ 𝐵 = (𝑘 · (2 · (𝑚 · 𝑛))) ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2)))) ∨ (𝐵 = (𝑘 · ((𝑚↑2) − (𝑛↑2))) ∧ 𝐴 = (𝑘 · (2 · (𝑚 · 𝑛))) ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2))))))
67	66	2rexbii 2499	. . . . . . . 8 ⊢ (∃𝑛 ∈ ℕ ∃𝑚 ∈ ℕ ∃𝑘 ∈ ℕ (((𝐴 = (𝑘 · ((𝑚↑2) − (𝑛↑2))) ∧ 𝐵 = (𝑘 · (2 · (𝑚 · 𝑛)))) ∨ (𝐴 = (𝑘 · (2 · (𝑚 · 𝑛))) ∧ 𝐵 = (𝑘 · ((𝑚↑2) − (𝑛↑2))))) ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2)))) ↔ ∃𝑛 ∈ ℕ ∃𝑚 ∈ ℕ ∃𝑘 ∈ ℕ ((𝐴 = (𝑘 · ((𝑚↑2) − (𝑛↑2))) ∧ 𝐵 = (𝑘 · (2 · (𝑚 · 𝑛))) ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2)))) ∨ (𝐵 = (𝑘 · ((𝑚↑2) − (𝑛↑2))) ∧ 𝐴 = (𝑘 · (2 · (𝑚 · 𝑛))) ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2))))))
68		r19.43 2648	. . . . . . . . . 10 ⊢ (∃𝑘 ∈ ℕ ((𝐴 = (𝑘 · ((𝑚↑2) − (𝑛↑2))) ∧ 𝐵 = (𝑘 · (2 · (𝑚 · 𝑛))) ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2)))) ∨ (𝐵 = (𝑘 · ((𝑚↑2) − (𝑛↑2))) ∧ 𝐴 = (𝑘 · (2 · (𝑚 · 𝑛))) ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2))))) ↔ (∃𝑘 ∈ ℕ (𝐴 = (𝑘 · ((𝑚↑2) − (𝑛↑2))) ∧ 𝐵 = (𝑘 · (2 · (𝑚 · 𝑛))) ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2)))) ∨ ∃𝑘 ∈ ℕ (𝐵 = (𝑘 · ((𝑚↑2) − (𝑛↑2))) ∧ 𝐴 = (𝑘 · (2 · (𝑚 · 𝑛))) ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2))))))
69	68	2rexbii 2499	. . . . . . . . 9 ⊢ (∃𝑛 ∈ ℕ ∃𝑚 ∈ ℕ ∃𝑘 ∈ ℕ ((𝐴 = (𝑘 · ((𝑚↑2) − (𝑛↑2))) ∧ 𝐵 = (𝑘 · (2 · (𝑚 · 𝑛))) ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2)))) ∨ (𝐵 = (𝑘 · ((𝑚↑2) − (𝑛↑2))) ∧ 𝐴 = (𝑘 · (2 · (𝑚 · 𝑛))) ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2))))) ↔ ∃𝑛 ∈ ℕ ∃𝑚 ∈ ℕ (∃𝑘 ∈ ℕ (𝐴 = (𝑘 · ((𝑚↑2) − (𝑛↑2))) ∧ 𝐵 = (𝑘 · (2 · (𝑚 · 𝑛))) ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2)))) ∨ ∃𝑘 ∈ ℕ (𝐵 = (𝑘 · ((𝑚↑2) − (𝑛↑2))) ∧ 𝐴 = (𝑘 · (2 · (𝑚 · 𝑛))) ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2))))))
70		r19.43 2648	. . . . . . . . . . 11 ⊢ (∃𝑚 ∈ ℕ (∃𝑘 ∈ ℕ (𝐴 = (𝑘 · ((𝑚↑2) − (𝑛↑2))) ∧ 𝐵 = (𝑘 · (2 · (𝑚 · 𝑛))) ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2)))) ∨ ∃𝑘 ∈ ℕ (𝐵 = (𝑘 · ((𝑚↑2) − (𝑛↑2))) ∧ 𝐴 = (𝑘 · (2 · (𝑚 · 𝑛))) ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2))))) ↔ (∃𝑚 ∈ ℕ ∃𝑘 ∈ ℕ (𝐴 = (𝑘 · ((𝑚↑2) − (𝑛↑2))) ∧ 𝐵 = (𝑘 · (2 · (𝑚 · 𝑛))) ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2)))) ∨ ∃𝑚 ∈ ℕ ∃𝑘 ∈ ℕ (𝐵 = (𝑘 · ((𝑚↑2) − (𝑛↑2))) ∧ 𝐴 = (𝑘 · (2 · (𝑚 · 𝑛))) ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2))))))
71	70	rexbii 2497	. . . . . . . . . 10 ⊢ (∃𝑛 ∈ ℕ ∃𝑚 ∈ ℕ (∃𝑘 ∈ ℕ (𝐴 = (𝑘 · ((𝑚↑2) − (𝑛↑2))) ∧ 𝐵 = (𝑘 · (2 · (𝑚 · 𝑛))) ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2)))) ∨ ∃𝑘 ∈ ℕ (𝐵 = (𝑘 · ((𝑚↑2) − (𝑛↑2))) ∧ 𝐴 = (𝑘 · (2 · (𝑚 · 𝑛))) ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2))))) ↔ ∃𝑛 ∈ ℕ (∃𝑚 ∈ ℕ ∃𝑘 ∈ ℕ (𝐴 = (𝑘 · ((𝑚↑2) − (𝑛↑2))) ∧ 𝐵 = (𝑘 · (2 · (𝑚 · 𝑛))) ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2)))) ∨ ∃𝑚 ∈ ℕ ∃𝑘 ∈ ℕ (𝐵 = (𝑘 · ((𝑚↑2) − (𝑛↑2))) ∧ 𝐴 = (𝑘 · (2 · (𝑚 · 𝑛))) ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2))))))
72		r19.43 2648	. . . . . . . . . 10 ⊢ (∃𝑛 ∈ ℕ (∃𝑚 ∈ ℕ ∃𝑘 ∈ ℕ (𝐴 = (𝑘 · ((𝑚↑2) − (𝑛↑2))) ∧ 𝐵 = (𝑘 · (2 · (𝑚 · 𝑛))) ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2)))) ∨ ∃𝑚 ∈ ℕ ∃𝑘 ∈ ℕ (𝐵 = (𝑘 · ((𝑚↑2) − (𝑛↑2))) ∧ 𝐴 = (𝑘 · (2 · (𝑚 · 𝑛))) ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2))))) ↔ (∃𝑛 ∈ ℕ ∃𝑚 ∈ ℕ ∃𝑘 ∈ ℕ (𝐴 = (𝑘 · ((𝑚↑2) − (𝑛↑2))) ∧ 𝐵 = (𝑘 · (2 · (𝑚 · 𝑛))) ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2)))) ∨ ∃𝑛 ∈ ℕ ∃𝑚 ∈ ℕ ∃𝑘 ∈ ℕ (𝐵 = (𝑘 · ((𝑚↑2) − (𝑛↑2))) ∧ 𝐴 = (𝑘 · (2 · (𝑚 · 𝑛))) ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2))))))
73	71, 72	bitri 184	. . . . . . . . 9 ⊢ (∃𝑛 ∈ ℕ ∃𝑚 ∈ ℕ (∃𝑘 ∈ ℕ (𝐴 = (𝑘 · ((𝑚↑2) − (𝑛↑2))) ∧ 𝐵 = (𝑘 · (2 · (𝑚 · 𝑛))) ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2)))) ∨ ∃𝑘 ∈ ℕ (𝐵 = (𝑘 · ((𝑚↑2) − (𝑛↑2))) ∧ 𝐴 = (𝑘 · (2 · (𝑚 · 𝑛))) ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2))))) ↔ (∃𝑛 ∈ ℕ ∃𝑚 ∈ ℕ ∃𝑘 ∈ ℕ (𝐴 = (𝑘 · ((𝑚↑2) − (𝑛↑2))) ∧ 𝐵 = (𝑘 · (2 · (𝑚 · 𝑛))) ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2)))) ∨ ∃𝑛 ∈ ℕ ∃𝑚 ∈ ℕ ∃𝑘 ∈ ℕ (𝐵 = (𝑘 · ((𝑚↑2) − (𝑛↑2))) ∧ 𝐴 = (𝑘 · (2 · (𝑚 · 𝑛))) ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2))))))
74	69, 73	bitri 184	. . . . . . . 8 ⊢ (∃𝑛 ∈ ℕ ∃𝑚 ∈ ℕ ∃𝑘 ∈ ℕ ((𝐴 = (𝑘 · ((𝑚↑2) − (𝑛↑2))) ∧ 𝐵 = (𝑘 · (2 · (𝑚 · 𝑛))) ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2)))) ∨ (𝐵 = (𝑘 · ((𝑚↑2) − (𝑛↑2))) ∧ 𝐴 = (𝑘 · (2 · (𝑚 · 𝑛))) ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2))))) ↔ (∃𝑛 ∈ ℕ ∃𝑚 ∈ ℕ ∃𝑘 ∈ ℕ (𝐴 = (𝑘 · ((𝑚↑2) − (𝑛↑2))) ∧ 𝐵 = (𝑘 · (2 · (𝑚 · 𝑛))) ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2)))) ∨ ∃𝑛 ∈ ℕ ∃𝑚 ∈ ℕ ∃𝑘 ∈ ℕ (𝐵 = (𝑘 · ((𝑚↑2) − (𝑛↑2))) ∧ 𝐴 = (𝑘 · (2 · (𝑚 · 𝑛))) ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2))))))
75	67, 74	bitri 184	. . . . . . 7 ⊢ (∃𝑛 ∈ ℕ ∃𝑚 ∈ ℕ ∃𝑘 ∈ ℕ (((𝐴 = (𝑘 · ((𝑚↑2) − (𝑛↑2))) ∧ 𝐵 = (𝑘 · (2 · (𝑚 · 𝑛)))) ∨ (𝐴 = (𝑘 · (2 · (𝑚 · 𝑛))) ∧ 𝐵 = (𝑘 · ((𝑚↑2) − (𝑛↑2))))) ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2)))) ↔ (∃𝑛 ∈ ℕ ∃𝑚 ∈ ℕ ∃𝑘 ∈ ℕ (𝐴 = (𝑘 · ((𝑚↑2) − (𝑛↑2))) ∧ 𝐵 = (𝑘 · (2 · (𝑚 · 𝑛))) ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2)))) ∨ ∃𝑛 ∈ ℕ ∃𝑚 ∈ ℕ ∃𝑘 ∈ ℕ (𝐵 = (𝑘 · ((𝑚↑2) − (𝑛↑2))) ∧ 𝐴 = (𝑘 · (2 · (𝑚 · 𝑛))) ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2))))))
76	58, 75	bitrdi 196	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (∃𝑛 ∈ ℕ ∃𝑚 ∈ ℕ ∃𝑘 ∈ ℕ ({𝐴, 𝐵} = {(𝑘 · ((𝑚↑2) − (𝑛↑2))), (𝑘 · (2 · (𝑚 · 𝑛)))} ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2)))) ↔ (∃𝑛 ∈ ℕ ∃𝑚 ∈ ℕ ∃𝑘 ∈ ℕ (𝐴 = (𝑘 · ((𝑚↑2) − (𝑛↑2))) ∧ 𝐵 = (𝑘 · (2 · (𝑚 · 𝑛))) ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2)))) ∨ ∃𝑛 ∈ ℕ ∃𝑚 ∈ ℕ ∃𝑘 ∈ ℕ (𝐵 = (𝑘 · ((𝑚↑2) − (𝑛↑2))) ∧ 𝐴 = (𝑘 · (2 · (𝑚 · 𝑛))) ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2)))))))
77	76	3adant3 1019	. . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (∃𝑛 ∈ ℕ ∃𝑚 ∈ ℕ ∃𝑘 ∈ ℕ ({𝐴, 𝐵} = {(𝑘 · ((𝑚↑2) − (𝑛↑2))), (𝑘 · (2 · (𝑚 · 𝑛)))} ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2)))) ↔ (∃𝑛 ∈ ℕ ∃𝑚 ∈ ℕ ∃𝑘 ∈ ℕ (𝐴 = (𝑘 · ((𝑚↑2) − (𝑛↑2))) ∧ 𝐵 = (𝑘 · (2 · (𝑚 · 𝑛))) ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2)))) ∨ ∃𝑛 ∈ ℕ ∃𝑚 ∈ ℕ ∃𝑘 ∈ ℕ (𝐵 = (𝑘 · ((𝑚↑2) − (𝑛↑2))) ∧ 𝐴 = (𝑘 · (2 · (𝑚 · 𝑛))) ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2)))))))
78	77	adantr 276	. . . 4 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2)) → (∃𝑛 ∈ ℕ ∃𝑚 ∈ ℕ ∃𝑘 ∈ ℕ ({𝐴, 𝐵} = {(𝑘 · ((𝑚↑2) − (𝑛↑2))), (𝑘 · (2 · (𝑚 · 𝑛)))} ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2)))) ↔ (∃𝑛 ∈ ℕ ∃𝑚 ∈ ℕ ∃𝑘 ∈ ℕ (𝐴 = (𝑘 · ((𝑚↑2) − (𝑛↑2))) ∧ 𝐵 = (𝑘 · (2 · (𝑚 · 𝑛))) ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2)))) ∨ ∃𝑛 ∈ ℕ ∃𝑚 ∈ ℕ ∃𝑘 ∈ ℕ (𝐵 = (𝑘 · ((𝑚↑2) − (𝑛↑2))) ∧ 𝐴 = (𝑘 · (2 · (𝑚 · 𝑛))) ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2)))))))
79	34, 78	mpbird 167	. . 3 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2)) → ∃𝑛 ∈ ℕ ∃𝑚 ∈ ℕ ∃𝑘 ∈ ℕ ({𝐴, 𝐵} = {(𝑘 · ((𝑚↑2) − (𝑛↑2))), (𝑘 · (2 · (𝑚 · 𝑛)))} ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2)))))
80	79	ex 115	. 2 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) → ∃𝑛 ∈ ℕ ∃𝑚 ∈ ℕ ∃𝑘 ∈ ℕ ({𝐴, 𝐵} = {(𝑘 · ((𝑚↑2) − (𝑛↑2))), (𝑘 · (2 · (𝑚 · 𝑛)))} ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2))))))
81		pythagtriplem2 12378	. . 3 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (∃𝑛 ∈ ℕ ∃𝑚 ∈ ℕ ∃𝑘 ∈ ℕ ({𝐴, 𝐵} = {(𝑘 · ((𝑚↑2) − (𝑛↑2))), (𝑘 · (2 · (𝑚 · 𝑛)))} ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2)))) → ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2)))
82	81	3adant3 1019	. 2 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (∃𝑛 ∈ ℕ ∃𝑚 ∈ ℕ ∃𝑘 ∈ ℕ ({𝐴, 𝐵} = {(𝑘 · ((𝑚↑2) − (𝑛↑2))), (𝑘 · (2 · (𝑚 · 𝑛)))} ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2)))) → ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2)))
83	80, 82	impbid 129	1 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ↔ ∃𝑛 ∈ ℕ ∃𝑚 ∈ ℕ ∃𝑘 ∈ ℕ ({𝐴, 𝐵} = {(𝑘 · ((𝑚↑2) − (𝑛↑2))), (𝑘 · (2 · (𝑚 · 𝑛)))} ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2))))))

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 709 ∧ w3a 980 = wceq 1364 ∈ wcel 2160 ∃wrex 2469 {cpr 3615 class class class wbr 4025 (class class class)co 5906 ℂcc 7856 + caddc 7861 · cmul 7863 − cmin 8176 / cdiv 8677 ℕcn 8968 2c2 9019 ℤcz 9303 ↑cexp 10583 ∥ cdvds 11904 gcd cgcd 12053

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4140 ax-sep 4143 ax-nul 4151 ax-pow 4199 ax-pr 4234 ax-un 4458 ax-setind 4561 ax-iinf 4612 ax-cnex 7949 ax-resscn 7950 ax-1cn 7951 ax-1re 7952 ax-icn 7953 ax-addcl 7954 ax-addrcl 7955 ax-mulcl 7956 ax-mulrcl 7957 ax-addcom 7958 ax-mulcom 7959 ax-addass 7960 ax-mulass 7961 ax-distr 7962 ax-i2m1 7963 ax-0lt1 7964 ax-1rid 7965 ax-0id 7966 ax-rnegex 7967 ax-precex 7968 ax-cnre 7969 ax-pre-ltirr 7970 ax-pre-ltwlin 7971 ax-pre-lttrn 7972 ax-pre-apti 7973 ax-pre-ltadd 7974 ax-pre-mulgt0 7975 ax-pre-mulext 7976 ax-arch 7977 ax-caucvg 7978

This theorem depends on definitions: df-bi 117 df-stab 832 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-xor 1387 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2758 df-sbc 2982 df-csb 3077 df-dif 3151 df-un 3153 df-in 3155 df-ss 3162 df-nul 3443 df-if 3554 df-pw 3599 df-sn 3620 df-pr 3621 df-op 3623 df-uni 3832 df-int 3867 df-iun 3910 df-br 4026 df-opab 4087 df-mpt 4088 df-tr 4124 df-id 4318 df-po 4321 df-iso 4322 df-iord 4391 df-on 4393 df-ilim 4394 df-suc 4396 df-iom 4615 df-xp 4657 df-rel 4658 df-cnv 4659 df-co 4660 df-dm 4661 df-rn 4662 df-res 4663 df-ima 4664 df-iota 5203 df-fun 5244 df-fn 5245 df-f 5246 df-f1 5247 df-fo 5248 df-f1o 5249 df-fv 5250 df-riota 5861 df-ov 5909 df-oprab 5910 df-mpo 5911 df-1st 6180 df-2nd 6181 df-recs 6345 df-frec 6431 df-1o 6456 df-2o 6457 df-er 6574 df-en 6782 df-sup 7029 df-pnf 8042 df-mnf 8043 df-xr 8044 df-ltxr 8045 df-le 8046 df-sub 8178 df-neg 8179 df-reap 8580 df-ap 8587 df-div 8678 df-inn 8969 df-2 9027 df-3 9028 df-4 9029 df-n0 9227 df-z 9304 df-uz 9579 df-q 9671 df-rp 9706 df-fz 10061 df-fzo 10195 df-fl 10325 df-mod 10380 df-seqfrec 10505 df-exp 10584 df-cj 10960 df-re 10961 df-im 10962 df-rsqrt 11116 df-abs 11117 df-dvds 11905 df-gcd 12054 df-prm 12220

This theorem is referenced by: (None)

W3C validator