Theorem qredeq 12874

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 9649	. . . . . . . . . 10 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℂ)
2	1	adantr 276	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ) → 𝑀 ∈ ℂ)
3		nncn 9312	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ)
4	3	adantl 277	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℂ)
5		nnap0 9333	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → 𝑁 # 0)
6	5	adantl 277	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ) → 𝑁 # 0)
7	2, 4, 6	divclapd 9120	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝑀 / 𝑁) ∈ ℂ)
8	7	3adant3 1048	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) → (𝑀 / 𝑁) ∈ ℂ)
9	8	adantr 276	. . . . . 6 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → (𝑀 / 𝑁) ∈ ℂ)
10		zcn 9649	. . . . . . . . . 10 ⊢ (𝑃 ∈ ℤ → 𝑃 ∈ ℂ)
11	10	adantr 276	. . . . . . . . 9 ⊢ ((𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ) → 𝑃 ∈ ℂ)
12		nncn 9312	. . . . . . . . . 10 ⊢ (𝑄 ∈ ℕ → 𝑄 ∈ ℂ)
13	12	adantl 277	. . . . . . . . 9 ⊢ ((𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ) → 𝑄 ∈ ℂ)
14		nnap0 9333	. . . . . . . . . 10 ⊢ (𝑄 ∈ ℕ → 𝑄 # 0)
15	14	adantl 277	. . . . . . . . 9 ⊢ ((𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ) → 𝑄 # 0)
16	11, 13, 15	divclapd 9120	. . . . . . . 8 ⊢ ((𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ) → (𝑃 / 𝑄) ∈ ℂ)
17	16	3adant3 1048	. . . . . . 7 ⊢ ((𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1) → (𝑃 / 𝑄) ∈ ℂ)
18	17	adantl 277	. . . . . 6 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → (𝑃 / 𝑄) ∈ ℂ)
19	3	3ad2ant2 1050	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) → 𝑁 ∈ ℂ)
20	19	adantr 276	. . . . . 6 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → 𝑁 ∈ ℂ)
21	5	3ad2ant2 1050	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) → 𝑁 # 0)
22	21	adantr 276	. . . . . 6 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → 𝑁 # 0)
23	9, 18, 20, 22	mulcanapd 8989	. . . . 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	divcanap2d 9122	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝑁 · (𝑀 / 𝑁)) = 𝑀)
25	24	3adant3 1048	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) → (𝑁 · (𝑀 / 𝑁)) = 𝑀)
26	25	adantr 276	. . . . . 6 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → (𝑁 · (𝑀 / 𝑁)) = 𝑀)
27	26	eqeq1d 2247	. . . . 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 190	. . . 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 1049	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) → 𝑀 ∈ ℂ)
30	29	adantr 276	. . . . . 6 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → 𝑀 ∈ ℂ)
31		mulcl 8306	. . . . . . 7 ⊢ ((𝑁 ∈ ℂ ∧ (𝑃 / 𝑄) ∈ ℂ) → (𝑁 · (𝑃 / 𝑄)) ∈ ℂ)
32	19, 17, 31	syl2an 289	. . . . . 6 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → (𝑁 · (𝑃 / 𝑄)) ∈ ℂ)
33	12	3ad2ant2 1050	. . . . . . 7 ⊢ ((𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1) → 𝑄 ∈ ℂ)
34	33	adantl 277	. . . . . 6 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → 𝑄 ∈ ℂ)
35	14	3ad2ant2 1050	. . . . . . 7 ⊢ ((𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1) → 𝑄 # 0)
36	35	adantl 277	. . . . . 6 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → 𝑄 # 0)
37	30, 32, 34, 36	mulcanap2d 8990	. . . . 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 8349	. . . . . . 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	divcanap1d 9121	. . . . . . . . . 10 ⊢ ((𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ) → ((𝑃 / 𝑄) · 𝑄) = 𝑃)
40	39	3adant3 1048	. . . . . . . . 9 ⊢ ((𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1) → ((𝑃 / 𝑄) · 𝑄) = 𝑃)
41	40	adantl 277	. . . . . . . 8 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → ((𝑃 / 𝑄) · 𝑄) = 𝑃)
42	41	oveq2d 6101	. . . . . . 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 2271	. . . . . 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 2250	. . . . 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 190	. . . 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 188	. . 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 9663	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ)
48	47	3ad2ant2 1050	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) → 𝑁 ∈ ℤ)
49		simp2 1029	. . . . . . . . 9 ⊢ ((𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1) → 𝑄 ∈ ℕ)
50	48, 49	anim12i 338	. . . . . . . 8 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → (𝑁 ∈ ℤ ∧ 𝑄 ∈ ℕ))
51	50	adantr 276	. . . . . . 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 276	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → 𝑁 ∈ ℤ)
53		simpl1 1031	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → 𝑀 ∈ ℤ)
54		nnz 9663	. . . . . . . . . . . 12 ⊢ (𝑄 ∈ ℕ → 𝑄 ∈ ℤ)
55	54	3ad2ant2 1050	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1) → 𝑄 ∈ ℤ)
56	55	adantl 277	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → 𝑄 ∈ ℤ)
57	52, 53, 56	3jca 1208	. . . . . . . . 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 276	. . . . . . . 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 1028	. . . . . . . . . . . 12 ⊢ ((𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1) → 𝑃 ∈ ℤ)
60		dvdsmul1 12580	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℤ ∧ 𝑃 ∈ ℤ) → 𝑁 ∥ (𝑁 · 𝑃))
61	48, 59, 60	syl2an 289	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → 𝑁 ∥ (𝑁 · 𝑃))
62	61	adantr 276	. . . . . . . . . 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 110	. . . . . . . . . 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 4158	. . . . . . . . 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 12750	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑁 gcd 𝑀) = (𝑀 gcd 𝑁))
66	47, 65	sylan 283	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℤ) → (𝑁 gcd 𝑀) = (𝑀 gcd 𝑁))
67	66	ancoms 268	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝑁 gcd 𝑀) = (𝑀 gcd 𝑁))
68	67	3adant3 1048	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) → (𝑁 gcd 𝑀) = (𝑀 gcd 𝑁))
69		simp3 1030	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) → (𝑀 gcd 𝑁) = 1)
70	68, 69	eqtrd 2271	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) → (𝑁 gcd 𝑀) = 1)
71	70	ad2antrr 492	. . . . . . . . 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 306	. . . . . . . 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 12870	. . . . . . . 8 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑄 ∈ ℤ) → ((𝑁 ∥ (𝑀 · 𝑄) ∧ (𝑁 gcd 𝑀) = 1) → 𝑁 ∥ 𝑄))
74	58, 72, 73	sylc 62	. . . . . . 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 12611	. . . . . . 7 ⊢ ((𝑁 ∈ ℤ ∧ 𝑄 ∈ ℕ) → (𝑁 ∥ 𝑄 → 𝑁 ≤ 𝑄))
76	51, 74, 75	sylc 62	. . . . . 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 1029	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) → 𝑁 ∈ ℕ)
78	55, 77	anim12i 338	. . . . . . . . 9 ⊢ (((𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1)) → (𝑄 ∈ ℤ ∧ 𝑁 ∈ ℕ))
79	78	ancoms 268	. . . . . . . 8 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → (𝑄 ∈ ℤ ∧ 𝑁 ∈ ℕ))
80	79	adantr 276	. . . . . . 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 1034	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → 𝑃 ∈ ℤ)
82	56, 81, 52	3jca 1208	. . . . . . . . 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 276	. . . . . . . 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 1028	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) → 𝑀 ∈ ℤ)
85		dvdsmul2 12581	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℤ ∧ 𝑄 ∈ ℤ) → 𝑄 ∥ (𝑀 · 𝑄))
86	84, 55, 85	syl2an 289	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → 𝑄 ∥ (𝑀 · 𝑄))
87	86	adantr 276	. . . . . . . . . 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 1049	. . . . . . . . . . . . 13 ⊢ ((𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1) → 𝑃 ∈ ℂ)
89		mulcom 8308	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℂ ∧ 𝑃 ∈ ℂ) → (𝑁 · 𝑃) = (𝑃 · 𝑁))
90	19, 88, 89	syl2an 289	. . . . . . . . . . . 12 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → (𝑁 · 𝑃) = (𝑃 · 𝑁))
91	90	adantr 276	. . . . . . . . . . 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 2271	. . . . . . . . . 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 4156	. . . . . . . . 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 12750	. . . . . . . . . . . . . 14 ⊢ ((𝑄 ∈ ℤ ∧ 𝑃 ∈ ℤ) → (𝑄 gcd 𝑃) = (𝑃 gcd 𝑄))
95	54, 94	sylan 283	. . . . . . . . . . . . 13 ⊢ ((𝑄 ∈ ℕ ∧ 𝑃 ∈ ℤ) → (𝑄 gcd 𝑃) = (𝑃 gcd 𝑄))
96	95	ancoms 268	. . . . . . . . . . . 12 ⊢ ((𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ) → (𝑄 gcd 𝑃) = (𝑃 gcd 𝑄))
97	96	3adant3 1048	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1) → (𝑄 gcd 𝑃) = (𝑃 gcd 𝑄))
98		simp3 1030	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1) → (𝑃 gcd 𝑄) = 1)
99	97, 98	eqtrd 2271	. . . . . . . . . 10 ⊢ ((𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1) → (𝑄 gcd 𝑃) = 1)
100	99	ad2antlr 493	. . . . . . . . 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 306	. . . . . . . 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 12870	. . . . . . . 8 ⊢ ((𝑄 ∈ ℤ ∧ 𝑃 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑄 ∥ (𝑃 · 𝑁) ∧ (𝑄 gcd 𝑃) = 1) → 𝑄 ∥ 𝑁))
103	83, 101, 102	sylc 62	. . . . . . 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 12611	. . . . . . 7 ⊢ ((𝑄 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝑄 ∥ 𝑁 → 𝑄 ≤ 𝑁))
105	80, 103, 104	sylc 62	. . . . . 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 9311	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ)
107	106	3ad2ant2 1050	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) → 𝑁 ∈ ℝ)
108	107	ad2antrr 492	. . . . . . 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 9311	. . . . . . . . 9 ⊢ (𝑄 ∈ ℕ → 𝑄 ∈ ℝ)
110	109	3ad2ant2 1050	. . . . . . . 8 ⊢ ((𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1) → 𝑄 ∈ ℝ)
111	110	ad2antlr 493	. . . . . . 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 8441	. . . . . 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 957	. . . . 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 6093	. . . . . . . . . 10 ⊢ (𝑁 = 𝑄 → (𝑀 · 𝑁) = (𝑀 · 𝑄))
115	114	eqeq1d 2247	. . . . . . . . 9 ⊢ (𝑁 = 𝑄 → ((𝑀 · 𝑁) = (𝑁 · 𝑃) ↔ (𝑀 · 𝑄) = (𝑁 · 𝑃)))
116	115	anbi2d 468	. . . . . . . 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 8308	. . . . . . . . . . . . . 14 ⊢ ((𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ) → (𝑀 · 𝑁) = (𝑁 · 𝑀))
118	1, 3, 117	syl2an 289	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝑀 · 𝑁) = (𝑁 · 𝑀))
119	118	3adant3 1048	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) → (𝑀 · 𝑁) = (𝑁 · 𝑀))
120	119	adantr 276	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → (𝑀 · 𝑁) = (𝑁 · 𝑀))
121	120	eqeq1d 2247	. . . . . . . . . 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 277	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → 𝑃 ∈ ℂ)
123	30, 122, 20, 22	mulcanapd 8989	. . . . . . . . . 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 188	. . . . . . . . 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 296	. . . . . . . 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 164	. . . . . . 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 30	. . . . . 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 326	. . . . 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 13	. . . 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 115	. . 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 150	. 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 1231	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))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ∧ w3a 1009   = wceq 1402   ∈ wcel 2209   class class class wbr 4130  (class class class)co 6085  ℂcc 8177  ℝcr 8178  0cc0 8179  1c1 8180   · cmul 8184   ≤ cle 8361   # cap 8909   / cdiv 9002  ℕcn 9304  ℤcz 9644   ∥ cdvds 12554   gcd cgcd 12730

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297  ax-arch 8298  ax-caucvg 8299

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-sup 7324  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8903  df-ap 8910  df-div 9003  df-inn 9305  df-2 9363  df-3 9364  df-4 9365  df-n0 9564  df-z 9645  df-uz 9922  df-q 10020  df-rp 10055  df-fz 10412  df-fzo 10550  df-fl 10705  df-mod 10760  df-seqfrec 10885  df-exp 10976  df-cj 11607  df-re 11608  df-im 11609  df-rsqrt 11764  df-abs 11765  df-dvds 12555  df-gcd 12731

This theorem is used by:  qredeu  12875