Theorem qredeq 12028

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 9196	. . . . . . . . . 10 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℂ)
2	1	adantr 274	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ) → 𝑀 ∈ ℂ)
3		nncn 8865	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ)
4	3	adantl 275	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℂ)
5		nnap0 8886	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → 𝑁 # 0)
6	5	adantl 275	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ) → 𝑁 # 0)
7	2, 4, 6	divclapd 8686	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝑀 / 𝑁) ∈ ℂ)
8	7	3adant3 1007	. . . . . . 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 9196	. . . . . . . . . 10 ⊢ (𝑃 ∈ ℤ → 𝑃 ∈ ℂ)
11	10	adantr 274	. . . . . . . . 9 ⊢ ((𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ) → 𝑃 ∈ ℂ)
12		nncn 8865	. . . . . . . . . 10 ⊢ (𝑄 ∈ ℕ → 𝑄 ∈ ℂ)
13	12	adantl 275	. . . . . . . . 9 ⊢ ((𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ) → 𝑄 ∈ ℂ)
14		nnap0 8886	. . . . . . . . . 10 ⊢ (𝑄 ∈ ℕ → 𝑄 # 0)
15	14	adantl 275	. . . . . . . . 9 ⊢ ((𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ) → 𝑄 # 0)
16	11, 13, 15	divclapd 8686	. . . . . . . 8 ⊢ ((𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ) → (𝑃 / 𝑄) ∈ ℂ)
17	16	3adant3 1007	. . . . . . 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 1009	. . . . . . 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 1009	. . . . . . 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 8558	. . . . 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 8688	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝑁 · (𝑀 / 𝑁)) = 𝑀)
25	24	3adant3 1007	. . . . . . 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 2174	. . . . 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 1008	. . . . . . 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 7880	. . . . . . 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 1009	. . . . . . 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 1009	. . . . . . 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 8559	. . . . 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 7922	. . . . . . 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 8687	. . . . . . . . . 10 ⊢ ((𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ) → ((𝑃 / 𝑄) · 𝑄) = 𝑃)
40	39	3adant3 1007	. . . . . . . . 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 5858	. . . . . . 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 2198	. . . . . 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 2177	. . . . 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 9210	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ)
48	47	3ad2ant2 1009	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) → 𝑁 ∈ ℤ)
49		simp2 988	. . . . . . . . 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 990	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → 𝑀 ∈ ℤ)
54		nnz 9210	. . . . . . . . . . . 12 ⊢ (𝑄 ∈ ℕ → 𝑄 ∈ ℤ)
55	54	3ad2ant2 1009	. . . . . . . . . . 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 1167	. . . . . . . . 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 987	. . . . . . . . . . . 12 ⊢ ((𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1) → 𝑃 ∈ ℤ)
60		dvdsmul1 11753	. . . . . . . . . . . 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 4010	. . . . . . . . 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 11906	. . . . . . . . . . . . . 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 1007	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) → (𝑁 gcd 𝑀) = (𝑀 gcd 𝑁))
69		simp3 989	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) → (𝑀 gcd 𝑁) = 1)
70	68, 69	eqtrd 2198	. . . . . . . . . 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 12024	. . . . . . . 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 11782	. . . . . . 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 988	. . . . . . . . . 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 993	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) ∧ (𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1)) → 𝑃 ∈ ℤ)
82	56, 81, 52	3jca 1167	. . . . . . . . 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 987	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1) → 𝑀 ∈ ℤ)
85		dvdsmul2 11754	. . . . . . . . . . . 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 1008	. . . . . . . . . . . . 13 ⊢ ((𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1) → 𝑃 ∈ ℂ)
89		mulcom 7882	. . . . . . . . . . . . 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 2198	. . . . . . . . . 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 4008	. . . . . . . . 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 11906	. . . . . . . . . . . . . 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 1007	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1) → (𝑄 gcd 𝑃) = (𝑃 gcd 𝑄))
98		simp3 989	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ ∧ (𝑃 gcd 𝑄) = 1) → (𝑃 gcd 𝑄) = 1)
99	97, 98	eqtrd 2198	. . . . . . . . . 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 12024	. . . . . . . 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 11782	. . . . . . 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 8864	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ)
107	106	3ad2ant2 1009	. . . . . . . 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 8864	. . . . . . . . 9 ⊢ (𝑄 ∈ ℕ → 𝑄 ∈ ℝ)
110	109	3ad2ant2 1009	. . . . . . . 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 8014	. . . . . 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 934	. . . . 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 5850	. . . . . . . . . 10 ⊢ (𝑁 = 𝑄 → (𝑀 · 𝑁) = (𝑀 · 𝑄))
115	114	eqeq1d 2174	. . . . . . . . 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 7882	. . . . . . . . . . . . . 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 1007	. . . . . . . . . . . 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 2174	. . . . . . . . . 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 8558	. . . . . . . . . 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 1190	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 968   = wceq 1343   ∈ wcel 2136   class class class wbr 3982  (class class class)co 5842  ℂcc 7751  ℝcr 7752  0cc0 7753  1c1 7754   · cmul 7758   ≤ cle 7934   # cap 8479   / cdiv 8568  ℕcn 8857  ℤcz 9191   ∥ cdvds 11727   gcd cgcd 11875

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871  ax-arch 7872  ax-caucvg 7873

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-ilim 4347  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-frec 6359  df-sup 6949  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-div 8569  df-inn 8858  df-2 8916  df-3 8917  df-4 8918  df-n0 9115  df-z 9192  df-uz 9467  df-q 9558  df-rp 9590  df-fz 9945  df-fzo 10078  df-fl 10205  df-mod 10258  df-seqfrec 10381  df-exp 10455  df-cj 10784  df-re 10785  df-im 10786  df-rsqrt 10940  df-abs 10941  df-dvds 11728  df-gcd 11876

This theorem is referenced by:  qredeu  12029