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Theorem pythagtriplem4 16011
Description: Lemma for pythagtrip 16026. Show that 𝐶𝐵 and 𝐶 + 𝐵 are relatively prime. (Contributed by Scott Fenton, 12-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
pythagtriplem4 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((𝐶𝐵) gcd (𝐶 + 𝐵)) = 1)

Proof of Theorem pythagtriplem4
StepHypRef Expression
1 simp3r 1183 . . 3 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ¬ 2 ∥ 𝐴)
2 nnz 11816 . . . . . . . . . . . . 13 (𝐶 ∈ ℕ → 𝐶 ∈ ℤ)
3 nnz 11816 . . . . . . . . . . . . 13 (𝐵 ∈ ℕ → 𝐵 ∈ ℤ)
4 zsubcl 11836 . . . . . . . . . . . . 13 ((𝐶 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐶𝐵) ∈ ℤ)
52, 3, 4syl2anr 588 . . . . . . . . . . . 12 ((𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐶𝐵) ∈ ℤ)
653adant1 1111 . . . . . . . . . . 11 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐶𝐵) ∈ ℤ)
763ad2ant1 1114 . . . . . . . . . 10 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (𝐶𝐵) ∈ ℤ)
8 simp13 1186 . . . . . . . . . . . 12 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 𝐶 ∈ ℕ)
9 simp12 1185 . . . . . . . . . . . 12 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 𝐵 ∈ ℕ)
108, 9nnaddcld 11491 . . . . . . . . . . 11 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (𝐶 + 𝐵) ∈ ℕ)
1110nnzd 11898 . . . . . . . . . 10 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (𝐶 + 𝐵) ∈ ℤ)
12 gcddvds 15711 . . . . . . . . . 10 (((𝐶𝐵) ∈ ℤ ∧ (𝐶 + 𝐵) ∈ ℤ) → (((𝐶𝐵) gcd (𝐶 + 𝐵)) ∥ (𝐶𝐵) ∧ ((𝐶𝐵) gcd (𝐶 + 𝐵)) ∥ (𝐶 + 𝐵)))
137, 11, 12syl2anc 576 . . . . . . . . 9 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (((𝐶𝐵) gcd (𝐶 + 𝐵)) ∥ (𝐶𝐵) ∧ ((𝐶𝐵) gcd (𝐶 + 𝐵)) ∥ (𝐶 + 𝐵)))
1413simprd 488 . . . . . . . 8 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((𝐶𝐵) gcd (𝐶 + 𝐵)) ∥ (𝐶 + 𝐵))
15 breq1 4929 . . . . . . . . 9 (((𝐶𝐵) gcd (𝐶 + 𝐵)) = 2 → (((𝐶𝐵) gcd (𝐶 + 𝐵)) ∥ (𝐶 + 𝐵) ↔ 2 ∥ (𝐶 + 𝐵)))
1615biimpd 221 . . . . . . . 8 (((𝐶𝐵) gcd (𝐶 + 𝐵)) = 2 → (((𝐶𝐵) gcd (𝐶 + 𝐵)) ∥ (𝐶 + 𝐵) → 2 ∥ (𝐶 + 𝐵)))
1714, 16mpan9 499 . . . . . . 7 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) ∧ ((𝐶𝐵) gcd (𝐶 + 𝐵)) = 2) → 2 ∥ (𝐶 + 𝐵))
18 2z 11826 . . . . . . . 8 2 ∈ ℤ
19 simpl13 1231 . . . . . . . . . 10 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) ∧ ((𝐶𝐵) gcd (𝐶 + 𝐵)) = 2) → 𝐶 ∈ ℕ)
2019nnzd 11898 . . . . . . . . 9 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) ∧ ((𝐶𝐵) gcd (𝐶 + 𝐵)) = 2) → 𝐶 ∈ ℤ)
21 simpl12 1230 . . . . . . . . . 10 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) ∧ ((𝐶𝐵) gcd (𝐶 + 𝐵)) = 2) → 𝐵 ∈ ℕ)
2221nnzd 11898 . . . . . . . . 9 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) ∧ ((𝐶𝐵) gcd (𝐶 + 𝐵)) = 2) → 𝐵 ∈ ℤ)
2320, 22zaddcld 11903 . . . . . . . 8 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) ∧ ((𝐶𝐵) gcd (𝐶 + 𝐵)) = 2) → (𝐶 + 𝐵) ∈ ℤ)
2420, 22zsubcld 11904 . . . . . . . 8 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) ∧ ((𝐶𝐵) gcd (𝐶 + 𝐵)) = 2) → (𝐶𝐵) ∈ ℤ)
25 dvdsmultr1 15506 . . . . . . . 8 ((2 ∈ ℤ ∧ (𝐶 + 𝐵) ∈ ℤ ∧ (𝐶𝐵) ∈ ℤ) → (2 ∥ (𝐶 + 𝐵) → 2 ∥ ((𝐶 + 𝐵) · (𝐶𝐵))))
2618, 23, 24, 25mp3an2i 1446 . . . . . . 7 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) ∧ ((𝐶𝐵) gcd (𝐶 + 𝐵)) = 2) → (2 ∥ (𝐶 + 𝐵) → 2 ∥ ((𝐶 + 𝐵) · (𝐶𝐵))))
2717, 26mpd 15 . . . . . 6 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) ∧ ((𝐶𝐵) gcd (𝐶 + 𝐵)) = 2) → 2 ∥ ((𝐶 + 𝐵) · (𝐶𝐵)))
2819nncnd 11456 . . . . . . 7 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) ∧ ((𝐶𝐵) gcd (𝐶 + 𝐵)) = 2) → 𝐶 ∈ ℂ)
2921nncnd 11456 . . . . . . 7 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) ∧ ((𝐶𝐵) gcd (𝐶 + 𝐵)) = 2) → 𝐵 ∈ ℂ)
30 subsq 13386 . . . . . . 7 ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐶↑2) − (𝐵↑2)) = ((𝐶 + 𝐵) · (𝐶𝐵)))
3128, 29, 30syl2anc 576 . . . . . 6 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) ∧ ((𝐶𝐵) gcd (𝐶 + 𝐵)) = 2) → ((𝐶↑2) − (𝐵↑2)) = ((𝐶 + 𝐵) · (𝐶𝐵)))
3227, 31breqtrrd 4954 . . . . 5 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) ∧ ((𝐶𝐵) gcd (𝐶 + 𝐵)) = 2) → 2 ∥ ((𝐶↑2) − (𝐵↑2)))
33 simpl2 1173 . . . . . . 7 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) ∧ ((𝐶𝐵) gcd (𝐶 + 𝐵)) = 2) → ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2))
3433oveq1d 6990 . . . . . 6 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) ∧ ((𝐶𝐵) gcd (𝐶 + 𝐵)) = 2) → (((𝐴↑2) + (𝐵↑2)) − (𝐵↑2)) = ((𝐶↑2) − (𝐵↑2)))
35 simpl11 1229 . . . . . . . . 9 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) ∧ ((𝐶𝐵) gcd (𝐶 + 𝐵)) = 2) → 𝐴 ∈ ℕ)
3635nnsqcld 13419 . . . . . . . 8 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) ∧ ((𝐶𝐵) gcd (𝐶 + 𝐵)) = 2) → (𝐴↑2) ∈ ℕ)
3736nncnd 11456 . . . . . . 7 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) ∧ ((𝐶𝐵) gcd (𝐶 + 𝐵)) = 2) → (𝐴↑2) ∈ ℂ)
3821nnsqcld 13419 . . . . . . . 8 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) ∧ ((𝐶𝐵) gcd (𝐶 + 𝐵)) = 2) → (𝐵↑2) ∈ ℕ)
3938nncnd 11456 . . . . . . 7 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) ∧ ((𝐶𝐵) gcd (𝐶 + 𝐵)) = 2) → (𝐵↑2) ∈ ℂ)
4037, 39pncand 10798 . . . . . 6 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) ∧ ((𝐶𝐵) gcd (𝐶 + 𝐵)) = 2) → (((𝐴↑2) + (𝐵↑2)) − (𝐵↑2)) = (𝐴↑2))
4134, 40eqtr3d 2811 . . . . 5 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) ∧ ((𝐶𝐵) gcd (𝐶 + 𝐵)) = 2) → ((𝐶↑2) − (𝐵↑2)) = (𝐴↑2))
4232, 41breqtrd 4952 . . . 4 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) ∧ ((𝐶𝐵) gcd (𝐶 + 𝐵)) = 2) → 2 ∥ (𝐴↑2))
43 nnz 11816 . . . . . . . 8 (𝐴 ∈ ℕ → 𝐴 ∈ ℤ)
44433ad2ant1 1114 . . . . . . 7 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝐴 ∈ ℤ)
45443ad2ant1 1114 . . . . . 6 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 𝐴 ∈ ℤ)
4645adantr 473 . . . . 5 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) ∧ ((𝐶𝐵) gcd (𝐶 + 𝐵)) = 2) → 𝐴 ∈ ℤ)
47 2prm 15891 . . . . . 6 2 ∈ ℙ
48 2nn 11512 . . . . . 6 2 ∈ ℕ
49 prmdvdsexp 15914 . . . . . 6 ((2 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 2 ∈ ℕ) → (2 ∥ (𝐴↑2) ↔ 2 ∥ 𝐴))
5047, 48, 49mp3an13 1432 . . . . 5 (𝐴 ∈ ℤ → (2 ∥ (𝐴↑2) ↔ 2 ∥ 𝐴))
5146, 50syl 17 . . . 4 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) ∧ ((𝐶𝐵) gcd (𝐶 + 𝐵)) = 2) → (2 ∥ (𝐴↑2) ↔ 2 ∥ 𝐴))
5242, 51mpbid 224 . . 3 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) ∧ ((𝐶𝐵) gcd (𝐶 + 𝐵)) = 2) → 2 ∥ 𝐴)
531, 52mtand 804 . 2 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ¬ ((𝐶𝐵) gcd (𝐶 + 𝐵)) = 2)
54 neg1z 11830 . . . . . . . 8 -1 ∈ ℤ
55 gcdaddm 15732 . . . . . . . 8 ((-1 ∈ ℤ ∧ (𝐶𝐵) ∈ ℤ ∧ (𝐶 + 𝐵) ∈ ℤ) → ((𝐶𝐵) gcd (𝐶 + 𝐵)) = ((𝐶𝐵) gcd ((𝐶 + 𝐵) + (-1 · (𝐶𝐵)))))
5654, 7, 11, 55mp3an2i 1446 . . . . . . 7 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((𝐶𝐵) gcd (𝐶 + 𝐵)) = ((𝐶𝐵) gcd ((𝐶 + 𝐵) + (-1 · (𝐶𝐵)))))
578nncnd 11456 . . . . . . . 8 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 𝐶 ∈ ℂ)
589nncnd 11456 . . . . . . . 8 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 𝐵 ∈ ℂ)
59 pnncan 10727 . . . . . . . . . . 11 ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐶 + 𝐵) − (𝐶𝐵)) = (𝐵 + 𝐵))
60593anidm23 1402 . . . . . . . . . 10 ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐶 + 𝐵) − (𝐶𝐵)) = (𝐵 + 𝐵))
61 subcl 10684 . . . . . . . . . . . . 13 ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐶𝐵) ∈ ℂ)
6261mulm1d 10892 . . . . . . . . . . . 12 ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (-1 · (𝐶𝐵)) = -(𝐶𝐵))
6362oveq2d 6991 . . . . . . . . . . 11 ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐶 + 𝐵) + (-1 · (𝐶𝐵))) = ((𝐶 + 𝐵) + -(𝐶𝐵)))
64 addcl 10416 . . . . . . . . . . . 12 ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐶 + 𝐵) ∈ ℂ)
6564, 61negsubd 10803 . . . . . . . . . . 11 ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐶 + 𝐵) + -(𝐶𝐵)) = ((𝐶 + 𝐵) − (𝐶𝐵)))
6663, 65eqtrd 2809 . . . . . . . . . 10 ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐶 + 𝐵) + (-1 · (𝐶𝐵))) = ((𝐶 + 𝐵) − (𝐶𝐵)))
67 2times 11582 . . . . . . . . . . 11 (𝐵 ∈ ℂ → (2 · 𝐵) = (𝐵 + 𝐵))
6867adantl 474 . . . . . . . . . 10 ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (2 · 𝐵) = (𝐵 + 𝐵))
6960, 66, 683eqtr4d 2819 . . . . . . . . 9 ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐶 + 𝐵) + (-1 · (𝐶𝐵))) = (2 · 𝐵))
7069oveq2d 6991 . . . . . . . 8 ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐶𝐵) gcd ((𝐶 + 𝐵) + (-1 · (𝐶𝐵)))) = ((𝐶𝐵) gcd (2 · 𝐵)))
7157, 58, 70syl2anc 576 . . . . . . 7 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((𝐶𝐵) gcd ((𝐶 + 𝐵) + (-1 · (𝐶𝐵)))) = ((𝐶𝐵) gcd (2 · 𝐵)))
7256, 71eqtrd 2809 . . . . . 6 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((𝐶𝐵) gcd (𝐶 + 𝐵)) = ((𝐶𝐵) gcd (2 · 𝐵)))
739nnzd 11898 . . . . . . . . 9 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 𝐵 ∈ ℤ)
74 zmulcl 11843 . . . . . . . . 9 ((2 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (2 · 𝐵) ∈ ℤ)
7518, 73, 74sylancr 579 . . . . . . . 8 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (2 · 𝐵) ∈ ℤ)
76 gcddvds 15711 . . . . . . . 8 (((𝐶𝐵) ∈ ℤ ∧ (2 · 𝐵) ∈ ℤ) → (((𝐶𝐵) gcd (2 · 𝐵)) ∥ (𝐶𝐵) ∧ ((𝐶𝐵) gcd (2 · 𝐵)) ∥ (2 · 𝐵)))
777, 75, 76syl2anc 576 . . . . . . 7 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (((𝐶𝐵) gcd (2 · 𝐵)) ∥ (𝐶𝐵) ∧ ((𝐶𝐵) gcd (2 · 𝐵)) ∥ (2 · 𝐵)))
7877simprd 488 . . . . . 6 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((𝐶𝐵) gcd (2 · 𝐵)) ∥ (2 · 𝐵))
7972, 78eqbrtrd 4948 . . . . 5 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((𝐶𝐵) gcd (𝐶 + 𝐵)) ∥ (2 · 𝐵))
80 1z 11824 . . . . . . . 8 1 ∈ ℤ
81 gcdaddm 15732 . . . . . . . 8 ((1 ∈ ℤ ∧ (𝐶𝐵) ∈ ℤ ∧ (𝐶 + 𝐵) ∈ ℤ) → ((𝐶𝐵) gcd (𝐶 + 𝐵)) = ((𝐶𝐵) gcd ((𝐶 + 𝐵) + (1 · (𝐶𝐵)))))
8280, 7, 11, 81mp3an2i 1446 . . . . . . 7 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((𝐶𝐵) gcd (𝐶 + 𝐵)) = ((𝐶𝐵) gcd ((𝐶 + 𝐵) + (1 · (𝐶𝐵)))))
83 ppncan 10728 . . . . . . . . . . 11 ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐶 + 𝐵) + (𝐶𝐵)) = (𝐶 + 𝐶))
84833anidm13 1401 . . . . . . . . . 10 ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐶 + 𝐵) + (𝐶𝐵)) = (𝐶 + 𝐶))
8561mulid2d 10457 . . . . . . . . . . 11 ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (1 · (𝐶𝐵)) = (𝐶𝐵))
8685oveq2d 6991 . . . . . . . . . 10 ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐶 + 𝐵) + (1 · (𝐶𝐵))) = ((𝐶 + 𝐵) + (𝐶𝐵)))
87 2times 11582 . . . . . . . . . . 11 (𝐶 ∈ ℂ → (2 · 𝐶) = (𝐶 + 𝐶))
8887adantr 473 . . . . . . . . . 10 ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (2 · 𝐶) = (𝐶 + 𝐶))
8984, 86, 883eqtr4d 2819 . . . . . . . . 9 ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐶 + 𝐵) + (1 · (𝐶𝐵))) = (2 · 𝐶))
9057, 58, 89syl2anc 576 . . . . . . . 8 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((𝐶 + 𝐵) + (1 · (𝐶𝐵))) = (2 · 𝐶))
9190oveq2d 6991 . . . . . . 7 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((𝐶𝐵) gcd ((𝐶 + 𝐵) + (1 · (𝐶𝐵)))) = ((𝐶𝐵) gcd (2 · 𝐶)))
9282, 91eqtrd 2809 . . . . . 6 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((𝐶𝐵) gcd (𝐶 + 𝐵)) = ((𝐶𝐵) gcd (2 · 𝐶)))
938nnzd 11898 . . . . . . . . 9 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 𝐶 ∈ ℤ)
94 zmulcl 11843 . . . . . . . . 9 ((2 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (2 · 𝐶) ∈ ℤ)
9518, 93, 94sylancr 579 . . . . . . . 8 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (2 · 𝐶) ∈ ℤ)
96 gcddvds 15711 . . . . . . . 8 (((𝐶𝐵) ∈ ℤ ∧ (2 · 𝐶) ∈ ℤ) → (((𝐶𝐵) gcd (2 · 𝐶)) ∥ (𝐶𝐵) ∧ ((𝐶𝐵) gcd (2 · 𝐶)) ∥ (2 · 𝐶)))
977, 95, 96syl2anc 576 . . . . . . 7 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (((𝐶𝐵) gcd (2 · 𝐶)) ∥ (𝐶𝐵) ∧ ((𝐶𝐵) gcd (2 · 𝐶)) ∥ (2 · 𝐶)))
9897simprd 488 . . . . . 6 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((𝐶𝐵) gcd (2 · 𝐶)) ∥ (2 · 𝐶))
9992, 98eqbrtrd 4948 . . . . 5 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((𝐶𝐵) gcd (𝐶 + 𝐵)) ∥ (2 · 𝐶))
100 nnaddcl 11462 . . . . . . . . . . . . . 14 ((𝐶 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐶 + 𝐵) ∈ ℕ)
101100nnne0d 11489 . . . . . . . . . . . . 13 ((𝐶 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐶 + 𝐵) ≠ 0)
102101ancoms 451 . . . . . . . . . . . 12 ((𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐶 + 𝐵) ≠ 0)
1031023adant1 1111 . . . . . . . . . . 11 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐶 + 𝐵) ≠ 0)
1041033ad2ant1 1114 . . . . . . . . . 10 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (𝐶 + 𝐵) ≠ 0)
105104neneqd 2967 . . . . . . . . 9 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ¬ (𝐶 + 𝐵) = 0)
106105intnand 481 . . . . . . . 8 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ¬ ((𝐶𝐵) = 0 ∧ (𝐶 + 𝐵) = 0))
107 gcdn0cl 15710 . . . . . . . 8 ((((𝐶𝐵) ∈ ℤ ∧ (𝐶 + 𝐵) ∈ ℤ) ∧ ¬ ((𝐶𝐵) = 0 ∧ (𝐶 + 𝐵) = 0)) → ((𝐶𝐵) gcd (𝐶 + 𝐵)) ∈ ℕ)
1087, 11, 106, 107syl21anc 826 . . . . . . 7 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((𝐶𝐵) gcd (𝐶 + 𝐵)) ∈ ℕ)
109108nnzd 11898 . . . . . 6 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((𝐶𝐵) gcd (𝐶 + 𝐵)) ∈ ℤ)
110 dvdsgcd 15747 . . . . . 6 ((((𝐶𝐵) gcd (𝐶 + 𝐵)) ∈ ℤ ∧ (2 · 𝐵) ∈ ℤ ∧ (2 · 𝐶) ∈ ℤ) → ((((𝐶𝐵) gcd (𝐶 + 𝐵)) ∥ (2 · 𝐵) ∧ ((𝐶𝐵) gcd (𝐶 + 𝐵)) ∥ (2 · 𝐶)) → ((𝐶𝐵) gcd (𝐶 + 𝐵)) ∥ ((2 · 𝐵) gcd (2 · 𝐶))))
111109, 75, 95, 110syl3anc 1352 . . . . 5 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((((𝐶𝐵) gcd (𝐶 + 𝐵)) ∥ (2 · 𝐵) ∧ ((𝐶𝐵) gcd (𝐶 + 𝐵)) ∥ (2 · 𝐶)) → ((𝐶𝐵) gcd (𝐶 + 𝐵)) ∥ ((2 · 𝐵) gcd (2 · 𝐶))))
11279, 99, 111mp2and 687 . . . 4 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((𝐶𝐵) gcd (𝐶 + 𝐵)) ∥ ((2 · 𝐵) gcd (2 · 𝐶)))
113 2nn0 11725 . . . . . 6 2 ∈ ℕ0
114 mulgcd 15751 . . . . . 6 ((2 ∈ ℕ0𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → ((2 · 𝐵) gcd (2 · 𝐶)) = (2 · (𝐵 gcd 𝐶)))
115113, 73, 93, 114mp3an2i 1446 . . . . 5 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((2 · 𝐵) gcd (2 · 𝐶)) = (2 · (𝐵 gcd 𝐶)))
116 pythagtriplem3 16010 . . . . . . 7 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (𝐵 gcd 𝐶) = 1)
117116oveq2d 6991 . . . . . 6 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (2 · (𝐵 gcd 𝐶)) = (2 · 1))
118 2t1e2 11609 . . . . . 6 (2 · 1) = 2
119117, 118syl6eq 2825 . . . . 5 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (2 · (𝐵 gcd 𝐶)) = 2)
120115, 119eqtrd 2809 . . . 4 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((2 · 𝐵) gcd (2 · 𝐶)) = 2)
121112, 120breqtrd 4952 . . 3 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((𝐶𝐵) gcd (𝐶 + 𝐵)) ∥ 2)
122 dvdsprime 15886 . . . 4 ((2 ∈ ℙ ∧ ((𝐶𝐵) gcd (𝐶 + 𝐵)) ∈ ℕ) → (((𝐶𝐵) gcd (𝐶 + 𝐵)) ∥ 2 ↔ (((𝐶𝐵) gcd (𝐶 + 𝐵)) = 2 ∨ ((𝐶𝐵) gcd (𝐶 + 𝐵)) = 1)))
12347, 108, 122sylancr 579 . . 3 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (((𝐶𝐵) gcd (𝐶 + 𝐵)) ∥ 2 ↔ (((𝐶𝐵) gcd (𝐶 + 𝐵)) = 2 ∨ ((𝐶𝐵) gcd (𝐶 + 𝐵)) = 1)))
124121, 123mpbid 224 . 2 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (((𝐶𝐵) gcd (𝐶 + 𝐵)) = 2 ∨ ((𝐶𝐵) gcd (𝐶 + 𝐵)) = 1))
125 orel1 873 . 2 (¬ ((𝐶𝐵) gcd (𝐶 + 𝐵)) = 2 → ((((𝐶𝐵) gcd (𝐶 + 𝐵)) = 2 ∨ ((𝐶𝐵) gcd (𝐶 + 𝐵)) = 1) → ((𝐶𝐵) gcd (𝐶 + 𝐵)) = 1))
12653, 124, 125sylc 65 1 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((𝐶𝐵) gcd (𝐶 + 𝐵)) = 1)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 387  wo 834  w3a 1069   = wceq 1508  wcel 2051  wne 2962   class class class wbr 4926  (class class class)co 6975  cc 10332  0cc0 10334  1c1 10335   + caddc 10337   · cmul 10339  cmin 10669  -cneg 10670  cn 11438  2c2 11494  0cn0 11706  cz 11792  cexp 13243  cdvds 15466   gcd cgcd 15702  cprime 15870
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2745  ax-sep 5057  ax-nul 5064  ax-pow 5116  ax-pr 5183  ax-un 7278  ax-cnex 10390  ax-resscn 10391  ax-1cn 10392  ax-icn 10393  ax-addcl 10394  ax-addrcl 10395  ax-mulcl 10396  ax-mulrcl 10397  ax-mulcom 10398  ax-addass 10399  ax-mulass 10400  ax-distr 10401  ax-i2m1 10402  ax-1ne0 10403  ax-1rid 10404  ax-rnegex 10405  ax-rrecex 10406  ax-cnre 10407  ax-pre-lttri 10408  ax-pre-lttrn 10409  ax-pre-ltadd 10410  ax-pre-mulgt0 10411  ax-pre-sup 10412
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3or 1070  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2754  df-cleq 2766  df-clel 2841  df-nfc 2913  df-ne 2963  df-nel 3069  df-ral 3088  df-rex 3089  df-reu 3090  df-rmo 3091  df-rab 3092  df-v 3412  df-sbc 3677  df-csb 3782  df-dif 3827  df-un 3829  df-in 3831  df-ss 3838  df-pss 3840  df-nul 4174  df-if 4346  df-pw 4419  df-sn 4437  df-pr 4439  df-tp 4441  df-op 4443  df-uni 4710  df-iun 4791  df-br 4927  df-opab 4989  df-mpt 5006  df-tr 5028  df-id 5309  df-eprel 5314  df-po 5323  df-so 5324  df-fr 5363  df-we 5365  df-xp 5410  df-rel 5411  df-cnv 5412  df-co 5413  df-dm 5414  df-rn 5415  df-res 5416  df-ima 5417  df-pred 5984  df-ord 6030  df-on 6031  df-lim 6032  df-suc 6033  df-iota 6150  df-fun 6188  df-fn 6189  df-f 6190  df-f1 6191  df-fo 6192  df-f1o 6193  df-fv 6194  df-riota 6936  df-ov 6978  df-oprab 6979  df-mpo 6980  df-om 7396  df-1st 7500  df-2nd 7501  df-wrecs 7749  df-recs 7811  df-rdg 7849  df-1o 7904  df-2o 7905  df-er 8088  df-en 8306  df-dom 8307  df-sdom 8308  df-fin 8309  df-sup 8700  df-inf 8701  df-pnf 10475  df-mnf 10476  df-xr 10477  df-ltxr 10478  df-le 10479  df-sub 10671  df-neg 10672  df-div 11098  df-nn 11439  df-2 11502  df-3 11503  df-n0 11707  df-z 11793  df-uz 12058  df-rp 12204  df-fz 12708  df-fl 12976  df-mod 13052  df-seq 13184  df-exp 13244  df-cj 14318  df-re 14319  df-im 14320  df-sqrt 14454  df-abs 14455  df-dvds 15467  df-gcd 15703  df-prm 15871
This theorem is referenced by:  pythagtriplem6  16013  pythagtriplem7  16014
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