Theorem qredeq 16587

Description: Two equal reduced fractions have the same numerator and denominator. (Contributed by Jeff Hankins, 29-Sep-2013.)

Assertion

Ref	Expression
qredeq	⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1) ∧ (𝑀 / 𝑁) = (𝑃 / 𝑄)) → (𝑀 = 𝑃 ∧ 𝑁 = 𝑄))

Proof of Theorem qredeq

Step	Hyp	Ref	Expression
1		zcn 12495	. . . . . . . . . 10 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℂ)
2	1	adantr 480	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ) → 𝑀 ∈ ℂ)
3		nncn 12155	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ)
4	3	adantl 481	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℂ)
5		nnne0 12181	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → 𝑁 ≠ 0)
6	5	adantl 481	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ) → 𝑁 ≠ 0)
7	2, 4, 6	divcld 11919	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝑀 / 𝑁) ∈ ℂ)
8	7	3adant3 1132	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) → (𝑀 / 𝑁) ∈ ℂ)
9	8	adantr 480	. . . . . 6 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → (𝑀 / 𝑁) ∈ ℂ)
10		zcn 12495	. . . . . . . . . 10 ⊢ (𝑃 ∈ ℤ → 𝑃 ∈ ℂ)
11	10	adantr 480	. . . . . . . . 9 ⊢ ((𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ) → 𝑃 ∈ ℂ)
12		nncn 12155	. . . . . . . . . 10 ⊢ (𝑄 ∈ ℕ → 𝑄 ∈ ℂ)
13	12	adantl 481	. . . . . . . . 9 ⊢ ((𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ) → 𝑄 ∈ ℂ)
14		nnne0 12181	. . . . . . . . . 10 ⊢ (𝑄 ∈ ℕ → 𝑄 ≠ 0)
15	14	adantl 481	. . . . . . . . 9 ⊢ ((𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ) → 𝑄 ≠ 0)
16	11, 13, 15	divcld 11919	. . . . . . . 8 ⊢ ((𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ) → (𝑃 / 𝑄) ∈ ℂ)
17	16	3adant3 1132	. . . . . . 7 ⊢ ((𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1) → (𝑃 / 𝑄) ∈ ℂ)
18	17	adantl 481	. . . . . 6 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → (𝑃 / 𝑄) ∈ ℂ)
19	3	3ad2ant2 1134	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) → 𝑁 ∈ ℂ)
20	19	adantr 480	. . . . . 6 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → 𝑁 ∈ ℂ)
21	5	3ad2ant2 1134	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) → 𝑁 ≠ 0)
22	21	adantr 480	. . . . . 6 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → 𝑁 ≠ 0)
23	9, 18, 20, 22	mulcand 11772	. . . . 5 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → ((𝑁 · (𝑀 / 𝑁)) = (𝑁 · (𝑃 / 𝑄)) ↔ (𝑀 / 𝑁) = (𝑃 / 𝑄)))
24	2, 4, 6	divcan2d 11921	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝑁 · (𝑀 / 𝑁)) = 𝑀)
25	24	3adant3 1132	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) → (𝑁 · (𝑀 / 𝑁)) = 𝑀)
26	25	adantr 480	. . . . . 6 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → (𝑁 · (𝑀 / 𝑁)) = 𝑀)
27	26	eqeq1d 2731	. . . . 5 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → ((𝑁 · (𝑀 / 𝑁)) = (𝑁 · (𝑃 / 𝑄)) ↔ 𝑀 = (𝑁 · (𝑃 / 𝑄))))
28	23, 27	bitr3d 281	. . . 4 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → ((𝑀 / 𝑁) = (𝑃 / 𝑄) ↔ 𝑀 = (𝑁 · (𝑃 / 𝑄))))
29	1	3ad2ant1 1133	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) → 𝑀 ∈ ℂ)
30	29	adantr 480	. . . . . 6 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → 𝑀 ∈ ℂ)
31		mulcl 11112	. . . . . . 7 ⊢ ((𝑁 ∈ ℂ ∧ (𝑃 / 𝑄) ∈ ℂ) → (𝑁 · (𝑃 / 𝑄)) ∈ ℂ)
32	19, 17, 31	syl2an 596	. . . . . 6 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → (𝑁 · (𝑃 / 𝑄)) ∈ ℂ)
33	12	3ad2ant2 1134	. . . . . . 7 ⊢ ((𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1) → 𝑄 ∈ ℂ)
34	33	adantl 481	. . . . . 6 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → 𝑄 ∈ ℂ)
35	14	3ad2ant2 1134	. . . . . . 7 ⊢ ((𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1) → 𝑄 ≠ 0)
36	35	adantl 481	. . . . . 6 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → 𝑄 ≠ 0)
37	30, 32, 34, 36	mulcan2d 11773	. . . . 5 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → ((𝑀 · 𝑄) = ((𝑁 · (𝑃 / 𝑄)) · 𝑄) ↔ 𝑀 = (𝑁 · (𝑃 / 𝑄))))
38	20, 18, 34	mulassd 11157	. . . . . . 7 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → ((𝑁 · (𝑃 / 𝑄)) · 𝑄) = (𝑁 · ((𝑃 / 𝑄) · 𝑄)))
39	11, 13, 15	divcan1d 11920	. . . . . . . . . 10 ⊢ ((𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ) → ((𝑃 / 𝑄) · 𝑄) = 𝑃)
40	39	3adant3 1132	. . . . . . . . 9 ⊢ ((𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1) → ((𝑃 / 𝑄) · 𝑄) = 𝑃)
41	40	adantl 481	. . . . . . . 8 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → ((𝑃 / 𝑄) · 𝑄) = 𝑃)
42	41	oveq2d 7369	. . . . . . 7 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → (𝑁 · ((𝑃 / 𝑄) · 𝑄)) = (𝑁 · 𝑃))
43	38, 42	eqtrd 2764	. . . . . 6 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → ((𝑁 · (𝑃 / 𝑄)) · 𝑄) = (𝑁 · 𝑃))
44	43	eqeq2d 2740	. . . . 5 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → ((𝑀 · 𝑄) = ((𝑁 · (𝑃 / 𝑄)) · 𝑄) ↔ (𝑀 · 𝑄) = (𝑁 · 𝑃)))
45	37, 44	bitr3d 281	. . . 4 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → (𝑀 = (𝑁 · (𝑃 / 𝑄)) ↔ (𝑀 · 𝑄) = (𝑁 · 𝑃)))
46	28, 45	bitrd 279	. . 3 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → ((𝑀 / 𝑁) = (𝑃 / 𝑄) ↔ (𝑀 · 𝑄) = (𝑁 · 𝑃)))
47		nnz 12511	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ)
48	47	3ad2ant2 1134	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) → 𝑁 ∈ ℤ)
49		simp2 1137	. . . . . . . . 9 ⊢ ((𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1) → 𝑄 ∈ ℕ)
50	48, 49	anim12i 613	. . . . . . . 8 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → (𝑁 ∈ ℤ ∧ 𝑄 ∈ ℕ))
51	50	adantr 480	. . . . . . 7 ⊢ ((((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) ∧ (𝑀 · 𝑄) = (𝑁 · 𝑃)) → (𝑁 ∈ ℤ ∧ 𝑄 ∈ ℕ))
52	48	adantr 480	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → 𝑁 ∈ ℤ)
53		simpl1 1192	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → 𝑀 ∈ ℤ)
54		nnz 12511	. . . . . . . . . . . 12 ⊢ (𝑄 ∈ ℕ → 𝑄 ∈ ℤ)
55	54	3ad2ant2 1134	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1) → 𝑄 ∈ ℤ)
56	55	adantl 481	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → 𝑄 ∈ ℤ)
57	52, 53, 56	3jca 1128	. . . . . . . . 9 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑄 ∈ ℤ))
58	57	adantr 480	. . . . . . . 8 ⊢ ((((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) ∧ (𝑀 · 𝑄) = (𝑁 · 𝑃)) → (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑄 ∈ ℤ))
59		simp1 1136	. . . . . . . . . . . 12 ⊢ ((𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1) → 𝑃 ∈ ℤ)
60		dvdsmul1 16207	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℤ ∧ 𝑃 ∈ ℤ) → 𝑁 ∥ (𝑁 · 𝑃))
61	48, 59, 60	syl2an 596	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → 𝑁 ∥ (𝑁 · 𝑃))
62	61	adantr 480	. . . . . . . . . 10 ⊢ ((((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) ∧ (𝑀 · 𝑄) = (𝑁 · 𝑃)) → 𝑁 ∥ (𝑁 · 𝑃))
63		simpr 484	. . . . . . . . . 10 ⊢ ((((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) ∧ (𝑀 · 𝑄) = (𝑁 · 𝑃)) → (𝑀 · 𝑄) = (𝑁 · 𝑃))
64	62, 63	breqtrrd 5123	. . . . . . . . 9 ⊢ ((((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) ∧ (𝑀 · 𝑄) = (𝑁 · 𝑃)) → 𝑁 ∥ (𝑀 · 𝑄))
65		gcdcom 16443	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑁 gcd 𝑀) = (𝑀 gcd 𝑁))
66	47, 65	sylan 580	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℤ) → (𝑁 gcd 𝑀) = (𝑀 gcd 𝑁))
67	66	ancoms 458	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝑁 gcd 𝑀) = (𝑀 gcd 𝑁))
68	67	3adant3 1132	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) → (𝑁 gcd 𝑀) = (𝑀 gcd 𝑁))
69		simp3 1138	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) → (𝑀 gcd 𝑁) = 1)
70	68, 69	eqtrd 2764	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) → (𝑁 gcd 𝑀) = 1)
71	70	ad2antrr 726	. . . . . . . . 9 ⊢ ((((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) ∧ (𝑀 · 𝑄) = (𝑁 · 𝑃)) → (𝑁 gcd 𝑀) = 1)
72	64, 71	jca 511	. . . . . . . 8 ⊢ ((((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) ∧ (𝑀 · 𝑄) = (𝑁 · 𝑃)) → (𝑁 ∥ (𝑀 · 𝑄) ∧ (𝑁 gcd 𝑀) = 1))
73		coprmdvds 16583	. . . . . . . 8 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑄 ∈ ℤ) → ((𝑁 ∥ (𝑀 · 𝑄) ∧ (𝑁 gcd 𝑀) = 1) → 𝑁 ∥ 𝑄))
74	58, 72, 73	sylc 65	. . . . . . 7 ⊢ ((((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) ∧ (𝑀 · 𝑄) = (𝑁 · 𝑃)) → 𝑁 ∥ 𝑄)
75		dvdsle 16240	. . . . . . 7 ⊢ ((𝑁 ∈ ℤ ∧ 𝑄 ∈ ℕ) → (𝑁 ∥ 𝑄 → 𝑁 ≤ 𝑄))
76	51, 74, 75	sylc 65	. . . . . 6 ⊢ ((((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) ∧ (𝑀 · 𝑄) = (𝑁 · 𝑃)) → 𝑁 ≤ 𝑄)
77		simp2 1137	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) → 𝑁 ∈ ℕ)
78	55, 77	anim12i 613	. . . . . . . . 9 ⊢ (((𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1)) → (𝑄 ∈ ℤ ∧ 𝑁 ∈ ℕ))
79	78	ancoms 458	. . . . . . . 8 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → (𝑄 ∈ ℤ ∧ 𝑁 ∈ ℕ))
80	79	adantr 480	. . . . . . 7 ⊢ ((((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) ∧ (𝑀 · 𝑄) = (𝑁 · 𝑃)) → (𝑄 ∈ ℤ ∧ 𝑁 ∈ ℕ))
81		simpr1 1195	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → 𝑃 ∈ ℤ)
82	56, 81, 52	3jca 1128	. . . . . . . . 9 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → (𝑄 ∈ ℤ ∧ 𝑃 ∈ ℤ ∧ 𝑁 ∈ ℤ))
83	82	adantr 480	. . . . . . . 8 ⊢ ((((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) ∧ (𝑀 · 𝑄) = (𝑁 · 𝑃)) → (𝑄 ∈ ℤ ∧ 𝑃 ∈ ℤ ∧ 𝑁 ∈ ℤ))
84		simp1 1136	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) → 𝑀 ∈ ℤ)
85		dvdsmul2 16208	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℤ ∧ 𝑄 ∈ ℤ) → 𝑄 ∥ (𝑀 · 𝑄))
86	84, 55, 85	syl2an 596	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → 𝑄 ∥ (𝑀 · 𝑄))
87	86	adantr 480	. . . . . . . . . 10 ⊢ ((((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) ∧ (𝑀 · 𝑄) = (𝑁 · 𝑃)) → 𝑄 ∥ (𝑀 · 𝑄))
88	10	3ad2ant1 1133	. . . . . . . . . . . . 13 ⊢ ((𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1) → 𝑃 ∈ ℂ)
89		mulcom 11114	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℂ ∧ 𝑃 ∈ ℂ) → (𝑁 · 𝑃) = (𝑃 · 𝑁))
90	19, 88, 89	syl2an 596	. . . . . . . . . . . 12 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → (𝑁 · 𝑃) = (𝑃 · 𝑁))
91	90	adantr 480	. . . . . . . . . . 11 ⊢ ((((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) ∧ (𝑀 · 𝑄) = (𝑁 · 𝑃)) → (𝑁 · 𝑃) = (𝑃 · 𝑁))
92	63, 91	eqtrd 2764	. . . . . . . . . 10 ⊢ ((((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) ∧ (𝑀 · 𝑄) = (𝑁 · 𝑃)) → (𝑀 · 𝑄) = (𝑃 · 𝑁))
93	87, 92	breqtrd 5121	. . . . . . . . 9 ⊢ ((((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) ∧ (𝑀 · 𝑄) = (𝑁 · 𝑃)) → 𝑄 ∥ (𝑃 · 𝑁))
94		gcdcom 16443	. . . . . . . . . . . . . 14 ⊢ ((𝑄 ∈ ℤ ∧ 𝑃 ∈ ℤ) → (𝑄 gcd 𝑃) = (𝑃 gcd 𝑄))
95	54, 94	sylan 580	. . . . . . . . . . . . 13 ⊢ ((𝑄 ∈ ℕ ∧ 𝑃 ∈ ℤ) → (𝑄 gcd 𝑃) = (𝑃 gcd 𝑄))
96	95	ancoms 458	. . . . . . . . . . . 12 ⊢ ((𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ) → (𝑄 gcd 𝑃) = (𝑃 gcd 𝑄))
97	96	3adant3 1132	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1) → (𝑄 gcd 𝑃) = (𝑃 gcd 𝑄))
98		simp3 1138	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1) → (𝑃 gcd 𝑄) = 1)
99	97, 98	eqtrd 2764	. . . . . . . . . 10 ⊢ ((𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1) → (𝑄 gcd 𝑃) = 1)
100	99	ad2antlr 727	. . . . . . . . 9 ⊢ ((((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) ∧ (𝑀 · 𝑄) = (𝑁 · 𝑃)) → (𝑄 gcd 𝑃) = 1)
101	93, 100	jca 511	. . . . . . . 8 ⊢ ((((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) ∧ (𝑀 · 𝑄) = (𝑁 · 𝑃)) → (𝑄 ∥ (𝑃 · 𝑁) ∧ (𝑄 gcd 𝑃) = 1))
102		coprmdvds 16583	. . . . . . . 8 ⊢ ((𝑄 ∈ ℤ ∧ 𝑃 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑄 ∥ (𝑃 · 𝑁) ∧ (𝑄 gcd 𝑃) = 1) → 𝑄 ∥ 𝑁))
103	83, 101, 102	sylc 65	. . . . . . 7 ⊢ ((((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) ∧ (𝑀 · 𝑄) = (𝑁 · 𝑃)) → 𝑄 ∥ 𝑁)
104		dvdsle 16240	. . . . . . 7 ⊢ ((𝑄 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝑄 ∥ 𝑁 → 𝑄 ≤ 𝑁))
105	80, 103, 104	sylc 65	. . . . . 6 ⊢ ((((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) ∧ (𝑀 · 𝑄) = (𝑁 · 𝑃)) → 𝑄 ≤ 𝑁)
106		nnre 12154	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ)
107	106	3ad2ant2 1134	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) → 𝑁 ∈ ℝ)
108	107	ad2antrr 726	. . . . . . 7 ⊢ ((((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) ∧ (𝑀 · 𝑄) = (𝑁 · 𝑃)) → 𝑁 ∈ ℝ)
109		nnre 12154	. . . . . . . . 9 ⊢ (𝑄 ∈ ℕ → 𝑄 ∈ ℝ)
110	109	3ad2ant2 1134	. . . . . . . 8 ⊢ ((𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1) → 𝑄 ∈ ℝ)
111	110	ad2antlr 727	. . . . . . 7 ⊢ ((((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) ∧ (𝑀 · 𝑄) = (𝑁 · 𝑃)) → 𝑄 ∈ ℝ)
112	108, 111	letri3d 11277	. . . . . 6 ⊢ ((((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) ∧ (𝑀 · 𝑄) = (𝑁 · 𝑃)) → (𝑁 = 𝑄 ↔ (𝑁 ≤ 𝑄 ∧ 𝑄 ≤ 𝑁)))
113	76, 105, 112	mpbir2and 713	. . . . 5 ⊢ ((((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) ∧ (𝑀 · 𝑄) = (𝑁 · 𝑃)) → 𝑁 = 𝑄)
114		oveq2 7361	. . . . . . . . . 10 ⊢ (𝑁 = 𝑄 → (𝑀 · 𝑁) = (𝑀 · 𝑄))
115	114	eqeq1d 2731	. . . . . . . . 9 ⊢ (𝑁 = 𝑄 → ((𝑀 · 𝑁) = (𝑁 · 𝑃) ↔ (𝑀 · 𝑄) = (𝑁 · 𝑃)))
116	115	anbi2d 630	. . . . . . . 8 ⊢ (𝑁 = 𝑄 → ((((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) ∧ (𝑀 · 𝑁) = (𝑁 · 𝑃)) ↔ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) ∧ (𝑀 · 𝑄) = (𝑁 · 𝑃))))
117		mulcom 11114	. . . . . . . . . . . . . 14 ⊢ ((𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ) → (𝑀 · 𝑁) = (𝑁 · 𝑀))
118	1, 3, 117	syl2an 596	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝑀 · 𝑁) = (𝑁 · 𝑀))
119	118	3adant3 1132	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) → (𝑀 · 𝑁) = (𝑁 · 𝑀))
120	119	adantr 480	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → (𝑀 · 𝑁) = (𝑁 · 𝑀))
121	120	eqeq1d 2731	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → ((𝑀 · 𝑁) = (𝑁 · 𝑃) ↔ (𝑁 · 𝑀) = (𝑁 · 𝑃)))
122	88	adantl 481	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → 𝑃 ∈ ℂ)
123	30, 122, 20, 22	mulcand 11772	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → ((𝑁 · 𝑀) = (𝑁 · 𝑃) ↔ 𝑀 = 𝑃))
124	121, 123	bitrd 279	. . . . . . . . 9 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → ((𝑀 · 𝑁) = (𝑁 · 𝑃) ↔ 𝑀 = 𝑃))
125	124	biimpa 476	. . . . . . . 8 ⊢ ((((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) ∧ (𝑀 · 𝑁) = (𝑁 · 𝑃)) → 𝑀 = 𝑃)
126	116, 125	biimtrrdi 254	. . . . . . 7 ⊢ (𝑁 = 𝑄 → ((((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) ∧ (𝑀 · 𝑄) = (𝑁 · 𝑃)) → 𝑀 = 𝑃))
127	126	com12 32	. . . . . 6 ⊢ ((((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) ∧ (𝑀 · 𝑄) = (𝑁 · 𝑃)) → (𝑁 = 𝑄 → 𝑀 = 𝑃))
128	127	ancrd 551	. . . . 5 ⊢ ((((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) ∧ (𝑀 · 𝑄) = (𝑁 · 𝑃)) → (𝑁 = 𝑄 → (𝑀 = 𝑃 ∧ 𝑁 = 𝑄)))
129	113, 128	mpd 15	. . . 4 ⊢ ((((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) ∧ (𝑀 · 𝑄) = (𝑁 · 𝑃)) → (𝑀 = 𝑃 ∧ 𝑁 = 𝑄))
130	129	ex 412	. . 3 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → ((𝑀 · 𝑄) = (𝑁 · 𝑃) → (𝑀 = 𝑃 ∧ 𝑁 = 𝑄)))
131	46, 130	sylbid 240	. 2 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → ((𝑀 / 𝑁) = (𝑃 / 𝑄) → (𝑀 = 𝑃 ∧ 𝑁 = 𝑄)))
132	131	3impia 1117	1 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1) ∧ (𝑀 / 𝑁) = (𝑃 / 𝑄)) → (𝑀 = 𝑃 ∧ 𝑁 = 𝑄))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2925   class class class wbr 5095  (class class class)co 7353  ℂcc 11026  ℝcr 11027  0cc0 11028  1c1 11029   · cmul 11033   ≤ cle 11169   / cdiv 11796  ℕcn 12147  ℤcz 12490   ∥ cdvds 16182   gcd cgcd 16424

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675  ax-cnex 11084  ax-resscn 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-addrcl 11089  ax-mulcl 11090  ax-mulrcl 11091  ax-mulcom 11092  ax-addass 11093  ax-mulass 11094  ax-distr 11095  ax-i2m1 11096  ax-1ne0 11097  ax-1rid 11098  ax-rnegex 11099  ax-rrecex 11100  ax-cnre 11101  ax-pre-lttri 11102  ax-pre-lttrn 11103  ax-pre-ltadd 11104  ax-pre-mulgt0 11105  ax-pre-sup 11106

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3345  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7310  df-ov 7356  df-oprab 7357  df-mpo 7358  df-om 7807  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-er 8632  df-en 8880  df-dom 8881  df-sdom 8882  df-sup 9351  df-inf 9352  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11368  df-neg 11369  df-div 11797  df-nn 12148  df-2 12210  df-3 12211  df-n0 12404  df-z 12491  df-uz 12755  df-rp 12913  df-fl 13715  df-mod 13793  df-seq 13928  df-exp 13988  df-cj 15025  df-re 15026  df-im 15027  df-sqrt 15161  df-abs 15162  df-dvds 16183  df-gcd 16425

This theorem is referenced by:  qredeu  16588