Theorem qredeq 11813

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 9083	. . . . . . . . . 10 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℂ)
2	1	adantr 274	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ) → 𝑀 ∈ ℂ)
3		nncn 8752	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ)
4	3	adantl 275	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℂ)
5		nnap0 8773	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → 𝑁 # 0)
6	5	adantl 275	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ) → 𝑁 # 0)
7	2, 4, 6	divclapd 8574	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝑀 / 𝑁) ∈ ℂ)
8	7	3adant3 1002	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) → (𝑀 / 𝑁) ∈ ℂ)
9	8	adantr 274	. . . . . 6 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → (𝑀 / 𝑁) ∈ ℂ)
10		zcn 9083	. . . . . . . . . 10 ⊢ (𝑃 ∈ ℤ → 𝑃 ∈ ℂ)
11	10	adantr 274	. . . . . . . . 9 ⊢ ((𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ) → 𝑃 ∈ ℂ)
12		nncn 8752	. . . . . . . . . 10 ⊢ (𝑄 ∈ ℕ → 𝑄 ∈ ℂ)
13	12	adantl 275	. . . . . . . . 9 ⊢ ((𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ) → 𝑄 ∈ ℂ)
14		nnap0 8773	. . . . . . . . . 10 ⊢ (𝑄 ∈ ℕ → 𝑄 # 0)
15	14	adantl 275	. . . . . . . . 9 ⊢ ((𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ) → 𝑄 # 0)
16	11, 13, 15	divclapd 8574	. . . . . . . 8 ⊢ ((𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ) → (𝑃 / 𝑄) ∈ ℂ)
17	16	3adant3 1002	. . . . . . 7 ⊢ ((𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1) → (𝑃 / 𝑄) ∈ ℂ)
18	17	adantl 275	. . . . . 6 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → (𝑃 / 𝑄) ∈ ℂ)
19	3	3ad2ant2 1004	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) → 𝑁 ∈ ℂ)
20	19	adantr 274	. . . . . 6 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → 𝑁 ∈ ℂ)
21	5	3ad2ant2 1004	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) → 𝑁 # 0)
22	21	adantr 274	. . . . . 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 8446	. . . . 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 8576	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝑁 · (𝑀 / 𝑁)) = 𝑀)
25	24	3adant3 1002	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) → (𝑁 · (𝑀 / 𝑁)) = 𝑀)
26	25	adantr 274	. . . . . 6 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → (𝑁 · (𝑀 / 𝑁)) = 𝑀)
27	26	eqeq1d 2149	. . . . 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 189	. . . 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 1003	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) → 𝑀 ∈ ℂ)
30	29	adantr 274	. . . . . 6 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → 𝑀 ∈ ℂ)
31		mulcl 7771	. . . . . . 7 ⊢ ((𝑁 ∈ ℂ ∧ (𝑃 / 𝑄) ∈ ℂ) → (𝑁 · (𝑃 / 𝑄)) ∈ ℂ)
32	19, 17, 31	syl2an 287	. . . . . 6 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → (𝑁 · (𝑃 / 𝑄)) ∈ ℂ)
33	12	3ad2ant2 1004	. . . . . . 7 ⊢ ((𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1) → 𝑄 ∈ ℂ)
34	33	adantl 275	. . . . . 6 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → 𝑄 ∈ ℂ)
35	14	3ad2ant2 1004	. . . . . . 7 ⊢ ((𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1) → 𝑄 # 0)
36	35	adantl 275	. . . . . 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 8447	. . . . 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 7813	. . . . . . 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 8575	. . . . . . . . . 10 ⊢ ((𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ) → ((𝑃 / 𝑄) · 𝑄) = 𝑃)
40	39	3adant3 1002	. . . . . . . . 9 ⊢ ((𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1) → ((𝑃 / 𝑄) · 𝑄) = 𝑃)
41	40	adantl 275	. . . . . . . 8 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → ((𝑃 / 𝑄) · 𝑄) = 𝑃)
42	41	oveq2d 5798	. . . . . . 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 2173	. . . . . 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 2152	. . . . 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 189	. . . 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 187	. . 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 9097	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ)
48	47	3ad2ant2 1004	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) → 𝑁 ∈ ℤ)
49		simp2 983	. . . . . . . . 9 ⊢ ((𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1) → 𝑄 ∈ ℕ)
50	48, 49	anim12i 336	. . . . . . . 8 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → (𝑁 ∈ ℤ ∧ 𝑄 ∈ ℕ))
51	50	adantr 274	. . . . . . 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 274	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → 𝑁 ∈ ℤ)
53		simpl1 985	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → 𝑀 ∈ ℤ)
54		nnz 9097	. . . . . . . . . . . 12 ⊢ (𝑄 ∈ ℕ → 𝑄 ∈ ℤ)
55	54	3ad2ant2 1004	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1) → 𝑄 ∈ ℤ)
56	55	adantl 275	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → 𝑄 ∈ ℤ)
57	52, 53, 56	3jca 1162	. . . . . . . . 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 274	. . . . . . . 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 982	. . . . . . . . . . . 12 ⊢ ((𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1) → 𝑃 ∈ ℤ)
60		dvdsmul1 11551	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℤ ∧ 𝑃 ∈ ℤ) → 𝑁 ∥ (𝑁 · 𝑃))
61	48, 59, 60	syl2an 287	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → 𝑁 ∥ (𝑁 · 𝑃))
62	61	adantr 274	. . . . . . . . . 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 109	. . . . . . . . . 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 3964	. . . . . . . . 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 11698	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑁 gcd 𝑀) = (𝑀 gcd 𝑁))
66	47, 65	sylan 281	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℤ) → (𝑁 gcd 𝑀) = (𝑀 gcd 𝑁))
67	66	ancoms 266	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝑁 gcd 𝑀) = (𝑀 gcd 𝑁))
68	67	3adant3 1002	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) → (𝑁 gcd 𝑀) = (𝑀 gcd 𝑁))
69		simp3 984	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) → (𝑀 gcd 𝑁) = 1)
70	68, 69	eqtrd 2173	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) → (𝑁 gcd 𝑀) = 1)
71	70	ad2antrr 480	. . . . . . . . 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 304	. . . . . . . 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 11809	. . . . . . . 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 11578	. . . . . . 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 983	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) → 𝑁 ∈ ℕ)
78	55, 77	anim12i 336	. . . . . . . . 9 ⊢ (((𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1)) → (𝑄 ∈ ℤ ∧ 𝑁 ∈ ℕ))
79	78	ancoms 266	. . . . . . . 8 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → (𝑄 ∈ ℤ ∧ 𝑁 ∈ ℕ))
80	79	adantr 274	. . . . . . 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 988	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → 𝑃 ∈ ℤ)
82	56, 81, 52	3jca 1162	. . . . . . . . 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 274	. . . . . . . 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 982	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) → 𝑀 ∈ ℤ)
85		dvdsmul2 11552	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℤ ∧ 𝑄 ∈ ℤ) → 𝑄 ∥ (𝑀 · 𝑄))
86	84, 55, 85	syl2an 287	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → 𝑄 ∥ (𝑀 · 𝑄))
87	86	adantr 274	. . . . . . . . . 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 1003	. . . . . . . . . . . . 13 ⊢ ((𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1) → 𝑃 ∈ ℂ)
89		mulcom 7773	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℂ ∧ 𝑃 ∈ ℂ) → (𝑁 · 𝑃) = (𝑃 · 𝑁))
90	19, 88, 89	syl2an 287	. . . . . . . . . . . 12 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → (𝑁 · 𝑃) = (𝑃 · 𝑁))
91	90	adantr 274	. . . . . . . . . . 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 2173	. . . . . . . . . 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 3962	. . . . . . . . 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 11698	. . . . . . . . . . . . . 14 ⊢ ((𝑄 ∈ ℤ ∧ 𝑃 ∈ ℤ) → (𝑄 gcd 𝑃) = (𝑃 gcd 𝑄))
95	54, 94	sylan 281	. . . . . . . . . . . . 13 ⊢ ((𝑄 ∈ ℕ ∧ 𝑃 ∈ ℤ) → (𝑄 gcd 𝑃) = (𝑃 gcd 𝑄))
96	95	ancoms 266	. . . . . . . . . . . 12 ⊢ ((𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ) → (𝑄 gcd 𝑃) = (𝑃 gcd 𝑄))
97	96	3adant3 1002	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1) → (𝑄 gcd 𝑃) = (𝑃 gcd 𝑄))
98		simp3 984	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1) → (𝑃 gcd 𝑄) = 1)
99	97, 98	eqtrd 2173	. . . . . . . . . 10 ⊢ ((𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1) → (𝑄 gcd 𝑃) = 1)
100	99	ad2antlr 481	. . . . . . . . 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 304	. . . . . . . 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 11809	. . . . . . . 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 11578	. . . . . . 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 8751	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ)
107	106	3ad2ant2 1004	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) → 𝑁 ∈ ℝ)
108	107	ad2antrr 480	. . . . . . 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 8751	. . . . . . . . 9 ⊢ (𝑄 ∈ ℕ → 𝑄 ∈ ℝ)
110	109	3ad2ant2 1004	. . . . . . . 8 ⊢ ((𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1) → 𝑄 ∈ ℝ)
111	110	ad2antlr 481	. . . . . . 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 7903	. . . . . 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 929	. . . . 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 5790	. . . . . . . . . 10 ⊢ (𝑁 = 𝑄 → (𝑀 · 𝑁) = (𝑀 · 𝑄))
115	114	eqeq1d 2149	. . . . . . . . 9 ⊢ (𝑁 = 𝑄 → ((𝑀 · 𝑁) = (𝑁 · 𝑃) ↔ (𝑀 · 𝑄) = (𝑁 · 𝑃)))
116	115	anbi2d 460	. . . . . . . 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 7773	. . . . . . . . . . . . . 14 ⊢ ((𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ) → (𝑀 · 𝑁) = (𝑁 · 𝑀))
118	1, 3, 117	syl2an 287	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝑀 · 𝑁) = (𝑁 · 𝑀))
119	118	3adant3 1002	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) → (𝑀 · 𝑁) = (𝑁 · 𝑀))
120	119	adantr 274	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → (𝑀 · 𝑁) = (𝑁 · 𝑀))
121	120	eqeq1d 2149	. . . . . . . . . 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 275	. . . . . . . . . . 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 8446	. . . . . . . . . 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 187	. . . . . . . . 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 294	. . . . . . . 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	syl6bir 163	. . . . . . 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 324	. . . . 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 114	. . 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 149	. 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 1179	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 103   ∧ w3a 963   = wceq 1332   ∈ wcel 1481   class class class wbr 3937  (class class class)co 5782  ℂcc 7642  ℝcr 7643  0cc0 7644  1c1 7645   · cmul 7649   ≤ cle 7825   # cap 8367   / cdiv 8456  ℕcn 8744  ℤcz 9078   ∥ cdvds 11529   gcd cgcd 11671

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-iinf 4510  ax-cnex 7735  ax-resscn 7736  ax-1cn 7737  ax-1re 7738  ax-icn 7739  ax-addcl 7740  ax-addrcl 7741  ax-mulcl 7742  ax-mulrcl 7743  ax-addcom 7744  ax-mulcom 7745  ax-addass 7746  ax-mulass 7747  ax-distr 7748  ax-i2m1 7749  ax-0lt1 7750  ax-1rid 7751  ax-0id 7752  ax-rnegex 7753  ax-precex 7754  ax-cnre 7755  ax-pre-ltirr 7756  ax-pre-ltwlin 7757  ax-pre-lttrn 7758  ax-pre-apti 7759  ax-pre-ltadd 7760  ax-pre-mulgt0 7761  ax-pre-mulext 7762  ax-arch 7763  ax-caucvg 7764

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rmo 2425  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-if 3480  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-id 4223  df-po 4226  df-iso 4227  df-iord 4296  df-on 4298  df-ilim 4299  df-suc 4301  df-iom 4513  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-riota 5738  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-recs 6210  df-frec 6296  df-sup 6879  df-pnf 7826  df-mnf 7827  df-xr 7828  df-ltxr 7829  df-le 7830  df-sub 7959  df-neg 7960  df-reap 8361  df-ap 8368  df-div 8457  df-inn 8745  df-2 8803  df-3 8804  df-4 8805  df-n0 9002  df-z 9079  df-uz 9351  df-q 9439  df-rp 9471  df-fz 9822  df-fzo 9951  df-fl 10074  df-mod 10127  df-seqfrec 10250  df-exp 10324  df-cj 10646  df-re 10647  df-im 10648  df-rsqrt 10802  df-abs 10803  df-dvds 11530  df-gcd 11672

This theorem is referenced by:  qredeu  11814