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Theorem dvdsmulgcd 15907

Description: A divisibility equivalent for odmulg 18685. (Contributed by Stefan O'Rear, 6-Sep-2015.)

**Assertion**
Ref	Expression
dvdsmulgcd	⊢ ((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (𝐴 ∥ (𝐵 · 𝐶) ↔ 𝐴 ∥ (𝐵 · (𝐶 gcd 𝐴))))

Proof of Theorem dvdsmulgcd Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simplr 767	. . . 4 ⊢ (((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · 𝐶)) → 𝐶 ∈ ℤ)
2		dvdszrcl 15614	. . . . . 6 ⊢ (𝐴 ∥ (𝐵 · 𝐶) → (𝐴 ∈ ℤ ∧ (𝐵 · 𝐶) ∈ ℤ))
3	2	adantl 484	. . . . 5 ⊢ (((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · 𝐶)) → (𝐴 ∈ ℤ ∧ (𝐵 · 𝐶) ∈ ℤ))
4	3	simpld 497	. . . 4 ⊢ (((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · 𝐶)) → 𝐴 ∈ ℤ)
5		bezout 15893	. . . 4 ⊢ ((𝐶 ∈ ℤ ∧ 𝐴 ∈ ℤ) → ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ (𝐶 gcd 𝐴) = ((𝐶 · 𝑥) + (𝐴 · 𝑦)))
6	1, 4, 5	syl2anc 586	. . 3 ⊢ (((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · 𝐶)) → ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ (𝐶 gcd 𝐴) = ((𝐶 · 𝑥) + (𝐴 · 𝑦)))
7	4	adantr 483	. . . . . . 7 ⊢ ((((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · 𝐶)) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → 𝐴 ∈ ℤ)
8		simplll 773	. . . . . . . 8 ⊢ ((((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · 𝐶)) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → 𝐵 ∈ ℤ)
9		simpllr 774	. . . . . . . . 9 ⊢ ((((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · 𝐶)) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → 𝐶 ∈ ℤ)
10		simprl 769	. . . . . . . . 9 ⊢ ((((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · 𝐶)) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → 𝑥 ∈ ℤ)
11	9, 10	zmulcld 12096	. . . . . . . 8 ⊢ ((((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · 𝐶)) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → (𝐶 · 𝑥) ∈ ℤ)
12	8, 11	zmulcld 12096	. . . . . . 7 ⊢ ((((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · 𝐶)) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → (𝐵 · (𝐶 · 𝑥)) ∈ ℤ)
13		simprr 771	. . . . . . . . 9 ⊢ ((((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · 𝐶)) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → 𝑦 ∈ ℤ)
14	7, 13	zmulcld 12096	. . . . . . . 8 ⊢ ((((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · 𝐶)) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → (𝐴 · 𝑦) ∈ ℤ)
15	8, 14	zmulcld 12096	. . . . . . 7 ⊢ ((((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · 𝐶)) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → (𝐵 · (𝐴 · 𝑦)) ∈ ℤ)
16	8, 9	zmulcld 12096	. . . . . . . . 9 ⊢ ((((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · 𝐶)) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → (𝐵 · 𝐶) ∈ ℤ)
17		simplr 767	. . . . . . . . 9 ⊢ ((((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · 𝐶)) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → 𝐴 ∥ (𝐵 · 𝐶))
18	7, 16, 10, 17	dvdsmultr1d 15650	. . . . . . . 8 ⊢ ((((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · 𝐶)) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → 𝐴 ∥ ((𝐵 · 𝐶) · 𝑥))
19	8	zcnd 12091	. . . . . . . . 9 ⊢ ((((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · 𝐶)) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → 𝐵 ∈ ℂ)
20	9	zcnd 12091	. . . . . . . . 9 ⊢ ((((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · 𝐶)) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → 𝐶 ∈ ℂ)
21	10	zcnd 12091	. . . . . . . . 9 ⊢ ((((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · 𝐶)) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → 𝑥 ∈ ℂ)
22	19, 20, 21	mulassd 10666	. . . . . . . 8 ⊢ ((((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · 𝐶)) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → ((𝐵 · 𝐶) · 𝑥) = (𝐵 · (𝐶 · 𝑥)))
23	18, 22	breqtrd 5094	. . . . . . 7 ⊢ ((((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · 𝐶)) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → 𝐴 ∥ (𝐵 · (𝐶 · 𝑥)))
24	8, 13	zmulcld 12096	. . . . . . . . 9 ⊢ ((((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · 𝐶)) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → (𝐵 · 𝑦) ∈ ℤ)
25		dvdsmul1 15633	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 · 𝑦) ∈ ℤ) → 𝐴 ∥ (𝐴 · (𝐵 · 𝑦)))
26	7, 24, 25	syl2anc 586	. . . . . . . 8 ⊢ ((((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · 𝐶)) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → 𝐴 ∥ (𝐴 · (𝐵 · 𝑦)))
27	7	zcnd 12091	. . . . . . . . 9 ⊢ ((((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · 𝐶)) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → 𝐴 ∈ ℂ)
28	13	zcnd 12091	. . . . . . . . 9 ⊢ ((((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · 𝐶)) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → 𝑦 ∈ ℂ)
29	19, 27, 28	mul12d 10851	. . . . . . . 8 ⊢ ((((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · 𝐶)) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → (𝐵 · (𝐴 · 𝑦)) = (𝐴 · (𝐵 · 𝑦)))
30	26, 29	breqtrrd 5096	. . . . . . 7 ⊢ ((((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · 𝐶)) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → 𝐴 ∥ (𝐵 · (𝐴 · 𝑦)))
31		dvds2add 15645	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 · (𝐶 · 𝑥)) ∈ ℤ ∧ (𝐵 · (𝐴 · 𝑦)) ∈ ℤ) → ((𝐴 ∥ (𝐵 · (𝐶 · 𝑥)) ∧ 𝐴 ∥ (𝐵 · (𝐴 · 𝑦))) → 𝐴 ∥ ((𝐵 · (𝐶 · 𝑥)) + (𝐵 · (𝐴 · 𝑦)))))
32	31	imp 409	. . . . . . 7 ⊢ (((𝐴 ∈ ℤ ∧ (𝐵 · (𝐶 · 𝑥)) ∈ ℤ ∧ (𝐵 · (𝐴 · 𝑦)) ∈ ℤ) ∧ (𝐴 ∥ (𝐵 · (𝐶 · 𝑥)) ∧ 𝐴 ∥ (𝐵 · (𝐴 · 𝑦)))) → 𝐴 ∥ ((𝐵 · (𝐶 · 𝑥)) + (𝐵 · (𝐴 · 𝑦))))
33	7, 12, 15, 23, 30, 32	syl32anc 1374	. . . . . 6 ⊢ ((((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · 𝐶)) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → 𝐴 ∥ ((𝐵 · (𝐶 · 𝑥)) + (𝐵 · (𝐴 · 𝑦))))
34	11	zcnd 12091	. . . . . . 7 ⊢ ((((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · 𝐶)) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → (𝐶 · 𝑥) ∈ ℂ)
35	14	zcnd 12091	. . . . . . 7 ⊢ ((((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · 𝐶)) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → (𝐴 · 𝑦) ∈ ℂ)
36	19, 34, 35	adddid 10667	. . . . . 6 ⊢ ((((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · 𝐶)) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → (𝐵 · ((𝐶 · 𝑥) + (𝐴 · 𝑦))) = ((𝐵 · (𝐶 · 𝑥)) + (𝐵 · (𝐴 · 𝑦))))
37	33, 36	breqtrrd 5096	. . . . 5 ⊢ ((((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · 𝐶)) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → 𝐴 ∥ (𝐵 · ((𝐶 · 𝑥) + (𝐴 · 𝑦))))
38		oveq2 7166	. . . . . 6 ⊢ ((𝐶 gcd 𝐴) = ((𝐶 · 𝑥) + (𝐴 · 𝑦)) → (𝐵 · (𝐶 gcd 𝐴)) = (𝐵 · ((𝐶 · 𝑥) + (𝐴 · 𝑦))))
39	38	breq2d 5080	. . . . 5 ⊢ ((𝐶 gcd 𝐴) = ((𝐶 · 𝑥) + (𝐴 · 𝑦)) → (𝐴 ∥ (𝐵 · (𝐶 gcd 𝐴)) ↔ 𝐴 ∥ (𝐵 · ((𝐶 · 𝑥) + (𝐴 · 𝑦)))))
40	37, 39	syl5ibrcom 249	. . . 4 ⊢ ((((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · 𝐶)) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → ((𝐶 gcd 𝐴) = ((𝐶 · 𝑥) + (𝐴 · 𝑦)) → 𝐴 ∥ (𝐵 · (𝐶 gcd 𝐴))))
41	40	rexlimdvva 3296	. . 3 ⊢ (((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · 𝐶)) → (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ (𝐶 gcd 𝐴) = ((𝐶 · 𝑥) + (𝐴 · 𝑦)) → 𝐴 ∥ (𝐵 · (𝐶 gcd 𝐴))))
42	6, 41	mpd 15	. 2 ⊢ (((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · 𝐶)) → 𝐴 ∥ (𝐵 · (𝐶 gcd 𝐴)))
43		dvdszrcl 15614	. . . . 5 ⊢ (𝐴 ∥ (𝐵 · (𝐶 gcd 𝐴)) → (𝐴 ∈ ℤ ∧ (𝐵 · (𝐶 gcd 𝐴)) ∈ ℤ))
44	43	adantl 484	. . . 4 ⊢ (((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · (𝐶 gcd 𝐴))) → (𝐴 ∈ ℤ ∧ (𝐵 · (𝐶 gcd 𝐴)) ∈ ℤ))
45	44	simpld 497	. . 3 ⊢ (((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · (𝐶 gcd 𝐴))) → 𝐴 ∈ ℤ)
46	44	simprd 498	. . 3 ⊢ (((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · (𝐶 gcd 𝐴))) → (𝐵 · (𝐶 gcd 𝐴)) ∈ ℤ)
47		zmulcl 12034	. . . 4 ⊢ ((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (𝐵 · 𝐶) ∈ ℤ)
48	47	adantr 483	. . 3 ⊢ (((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · (𝐶 gcd 𝐴))) → (𝐵 · 𝐶) ∈ ℤ)
49		simpr 487	. . 3 ⊢ (((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · (𝐶 gcd 𝐴))) → 𝐴 ∥ (𝐵 · (𝐶 gcd 𝐴)))
50		simplr 767	. . . . . 6 ⊢ (((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · (𝐶 gcd 𝐴))) → 𝐶 ∈ ℤ)
51		gcddvds 15854	. . . . . 6 ⊢ ((𝐶 ∈ ℤ ∧ 𝐴 ∈ ℤ) → ((𝐶 gcd 𝐴) ∥ 𝐶 ∧ (𝐶 gcd 𝐴) ∥ 𝐴))
52	50, 45, 51	syl2anc 586	. . . . 5 ⊢ (((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · (𝐶 gcd 𝐴))) → ((𝐶 gcd 𝐴) ∥ 𝐶 ∧ (𝐶 gcd 𝐴) ∥ 𝐴))
53	52	simpld 497	. . . 4 ⊢ (((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · (𝐶 gcd 𝐴))) → (𝐶 gcd 𝐴) ∥ 𝐶)
54	50, 45	gcdcld 15859	. . . . . 6 ⊢ (((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · (𝐶 gcd 𝐴))) → (𝐶 gcd 𝐴) ∈ ℕ₀)
55	54	nn0zd 12088	. . . . 5 ⊢ (((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · (𝐶 gcd 𝐴))) → (𝐶 gcd 𝐴) ∈ ℤ)
56		simpll 765	. . . . 5 ⊢ (((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · (𝐶 gcd 𝐴))) → 𝐵 ∈ ℤ)
57		dvdscmul 15638	. . . . 5 ⊢ (((𝐶 gcd 𝐴) ∈ ℤ ∧ 𝐶 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐶 gcd 𝐴) ∥ 𝐶 → (𝐵 · (𝐶 gcd 𝐴)) ∥ (𝐵 · 𝐶)))
58	55, 50, 56, 57	syl3anc 1367	. . . 4 ⊢ (((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · (𝐶 gcd 𝐴))) → ((𝐶 gcd 𝐴) ∥ 𝐶 → (𝐵 · (𝐶 gcd 𝐴)) ∥ (𝐵 · 𝐶)))
59	53, 58	mpd 15	. . 3 ⊢ (((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · (𝐶 gcd 𝐴))) → (𝐵 · (𝐶 gcd 𝐴)) ∥ (𝐵 · 𝐶))
60		dvdstr 15648	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 · (𝐶 gcd 𝐴)) ∈ ℤ ∧ (𝐵 · 𝐶) ∈ ℤ) → ((𝐴 ∥ (𝐵 · (𝐶 gcd 𝐴)) ∧ (𝐵 · (𝐶 gcd 𝐴)) ∥ (𝐵 · 𝐶)) → 𝐴 ∥ (𝐵 · 𝐶)))
61	60	imp 409	. . 3 ⊢ (((𝐴 ∈ ℤ ∧ (𝐵 · (𝐶 gcd 𝐴)) ∈ ℤ ∧ (𝐵 · 𝐶) ∈ ℤ) ∧ (𝐴 ∥ (𝐵 · (𝐶 gcd 𝐴)) ∧ (𝐵 · (𝐶 gcd 𝐴)) ∥ (𝐵 · 𝐶))) → 𝐴 ∥ (𝐵 · 𝐶))
62	45, 46, 48, 49, 59, 61	syl32anc 1374	. 2 ⊢ (((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 · (𝐶 gcd 𝐴))) → 𝐴 ∥ (𝐵 · 𝐶))
63	42, 62	impbida 799	1 ⊢ ((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (𝐴 ∥ (𝐵 · 𝐶) ↔ 𝐴 ∥ (𝐵 · (𝐶 gcd 𝐴))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ∃wrex 3141 class class class wbr 5068 (class class class)co 7158 + caddc 10542 · cmul 10544 ℤcz 11984 ∥ cdvds 15609 gcd cgcd 15845

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-sup 8908 df-inf 8909 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-n0 11901 df-z 11985 df-uz 12247 df-rp 12393 df-fl 13165 df-mod 13241 df-seq 13373 df-exp 13433 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-dvds 15610 df-gcd 15846

This theorem is referenced by: coprmdvds 15999 odmulg 18685 fpprwpprb 43912

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