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Theorem mpodvdsmulf1o 27175

Description: If 𝑀 and 𝑁 are two coprime integers, multiplication forms a bijection from the set of pairs ⟨𝑗, 𝑘⟩ where 𝑗 ∥ 𝑀 and 𝑘 ∥ 𝑁, to the set of divisors of 𝑀 · 𝑁. Version of dvdsmulf1o 27177 using maps-to notation, which does not require ax-mulf 11109. (Contributed by GG, 18-Apr-2025.)

**Hypotheses**
Ref	Expression
mpodvdsmulf1o.1	⊢ (𝜑 → 𝑀 ∈ ℕ)
mpodvdsmulf1o.2	⊢ (𝜑 → 𝑁 ∈ ℕ)
mpodvdsmulf1o.3	⊢ (𝜑 → (𝑀 gcd 𝑁) = 1)
mpodvdsmulf1o.x	⊢ 𝑋 = {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑀}
mpodvdsmulf1o.y	⊢ 𝑌 = {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}
mpodvdsmulf1o.z	⊢ 𝑍 = {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑀 · 𝑁)}

**Assertion**
Ref	Expression
mpodvdsmulf1o	⊢ (𝜑 → ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)–1-1-onto→𝑍)

Distinct variable groups: 𝑥,𝑦,𝑀 𝑥,𝑁,𝑦 Allowed substitution hints: 𝜑(𝑥,𝑦) 𝑋(𝑥,𝑦) 𝑌(𝑥,𝑦) 𝑍(𝑥,𝑦)

Proof of Theorem mpodvdsmulf1o Dummy variables 𝑤 𝑢 𝑣 𝑖 𝑗 𝑛 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mpomulf 11124	. . . . . . 7 ⊢ (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)):(ℂ × ℂ)⟶ℂ
2		ffn 6655	. . . . . . 7 ⊢ ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)):(ℂ × ℂ)⟶ℂ → (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) Fn (ℂ × ℂ))
3	1, 2	ax-mp 5	. . . . . 6 ⊢ (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) Fn (ℂ × ℂ)
4		mpodvdsmulf1o.x	. . . . . . . . 9 ⊢ 𝑋 = {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑀}
5	4	ssrab3 4013	. . . . . . . 8 ⊢ 𝑋 ⊆ ℕ
6		nnsscn 12170	. . . . . . . 8 ⊢ ℕ ⊆ ℂ
7	5, 6	sstri 3924	. . . . . . 7 ⊢ 𝑋 ⊆ ℂ
8		mpodvdsmulf1o.y	. . . . . . . . 9 ⊢ 𝑌 = {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}
9	8	ssrab3 4013	. . . . . . . 8 ⊢ 𝑌 ⊆ ℕ
10	9, 6	sstri 3924	. . . . . . 7 ⊢ 𝑌 ⊆ ℂ
11		xpss12 5633	. . . . . . 7 ⊢ ((𝑋 ⊆ ℂ ∧ 𝑌 ⊆ ℂ) → (𝑋 × 𝑌) ⊆ (ℂ × ℂ))
12	7, 10, 11	mp2an 698	. . . . . 6 ⊢ (𝑋 × 𝑌) ⊆ (ℂ × ℂ)
13		fnssres 6608	. . . . . 6 ⊢ (((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) Fn (ℂ × ℂ) ∧ (𝑋 × 𝑌) ⊆ (ℂ × ℂ)) → ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌))
14	3, 12, 13	mp2an 698	. . . . 5 ⊢ ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌)
15	14	a1i 11	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌))
16		ovres 7522	. . . . . . 7 ⊢ ((𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌) → (𝑖((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ↾ (𝑋 × 𝑌))𝑗) = (𝑖(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))𝑗))
17	16	adantl 482	. . . . . 6 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) → (𝑖((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ↾ (𝑋 × 𝑌))𝑗) = (𝑖(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))𝑗))
18	7	sseli 3911	. . . . . . . . . 10 ⊢ (𝑖 ∈ 𝑋 → 𝑖 ∈ ℂ)
19	18	adantr 481	. . . . . . . . 9 ⊢ ((𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌) → 𝑖 ∈ ℂ)
20	10	sseli 3911	. . . . . . . . . 10 ⊢ (𝑗 ∈ 𝑌 → 𝑗 ∈ ℂ)
21	20	adantl 482	. . . . . . . . 9 ⊢ ((𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌) → 𝑗 ∈ ℂ)
22		ovmpot 7517	. . . . . . . . . 10 ⊢ ((𝑖 ∈ ℂ ∧ 𝑗 ∈ ℂ) → (𝑖(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))𝑗) = (𝑖 · 𝑗))
23	22	eqcomd 2745	. . . . . . . . 9 ⊢ ((𝑖 ∈ ℂ ∧ 𝑗 ∈ ℂ) → (𝑖 · 𝑗) = (𝑖(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))𝑗))
24	19, 21, 23	syl2anc 590	. . . . . . . 8 ⊢ ((𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌) → (𝑖 · 𝑗) = (𝑖(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))𝑗))
25	24	adantl 482	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) → (𝑖 · 𝑗) = (𝑖(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))𝑗))
26	5	sseli 3911	. . . . . . . . . 10 ⊢ (𝑖 ∈ 𝑋 → 𝑖 ∈ ℕ)
27	26	ad2antrl 734	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) → 𝑖 ∈ ℕ)
28	9	sseli 3911	. . . . . . . . . 10 ⊢ (𝑗 ∈ 𝑌 → 𝑗 ∈ ℕ)
29	28	ad2antll 735	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) → 𝑗 ∈ ℕ)
30	27, 29	nnmulcld 12221	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) → (𝑖 · 𝑗) ∈ ℕ)
31		breq1 5075	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑗 → (𝑥 ∥ 𝑁 ↔ 𝑗 ∥ 𝑁))
32	31, 8	elrab2 3632	. . . . . . . . . . 11 ⊢ (𝑗 ∈ 𝑌 ↔ (𝑗 ∈ ℕ ∧ 𝑗 ∥ 𝑁))
33	32	simprbi 498	. . . . . . . . . 10 ⊢ (𝑗 ∈ 𝑌 → 𝑗 ∥ 𝑁)
34	33	ad2antll 735	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) → 𝑗 ∥ 𝑁)
35		breq1 5075	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑖 → (𝑥 ∥ 𝑀 ↔ 𝑖 ∥ 𝑀))
36	35, 4	elrab2 3632	. . . . . . . . . . 11 ⊢ (𝑖 ∈ 𝑋 ↔ (𝑖 ∈ ℕ ∧ 𝑖 ∥ 𝑀))
37	36	simprbi 498	. . . . . . . . . 10 ⊢ (𝑖 ∈ 𝑋 → 𝑖 ∥ 𝑀)
38	37	ad2antrl 734	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) → 𝑖 ∥ 𝑀)
39	29	nnzd 12541	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) → 𝑗 ∈ ℤ)
40		mpodvdsmulf1o.2	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑁 ∈ ℕ)
41	40	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) → 𝑁 ∈ ℕ)
42	41	nnzd 12541	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) → 𝑁 ∈ ℤ)
43	27	nnzd 12541	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) → 𝑖 ∈ ℤ)
44		dvdscmul 16242	. . . . . . . . . . 11 ⊢ ((𝑗 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑖 ∈ ℤ) → (𝑗 ∥ 𝑁 → (𝑖 · 𝑗) ∥ (𝑖 · 𝑁)))
45	39, 42, 43, 44	syl3anc 1379	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) → (𝑗 ∥ 𝑁 → (𝑖 · 𝑗) ∥ (𝑖 · 𝑁)))
46		mpodvdsmulf1o.1	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑀 ∈ ℕ)
47	46	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) → 𝑀 ∈ ℕ)
48	47	nnzd 12541	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) → 𝑀 ∈ ℤ)
49		dvdsmulc 16243	. . . . . . . . . . 11 ⊢ ((𝑖 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑖 ∥ 𝑀 → (𝑖 · 𝑁) ∥ (𝑀 · 𝑁)))
50	43, 48, 42, 49	syl3anc 1379	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) → (𝑖 ∥ 𝑀 → (𝑖 · 𝑁) ∥ (𝑀 · 𝑁)))
51	30	nnzd 12541	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) → (𝑖 · 𝑗) ∈ ℤ)
52	43, 42	zmulcld 12630	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) → (𝑖 · 𝑁) ∈ ℤ)
53	48, 42	zmulcld 12630	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) → (𝑀 · 𝑁) ∈ ℤ)
54		dvdstr 16254	. . . . . . . . . . 11 ⊢ (((𝑖 · 𝑗) ∈ ℤ ∧ (𝑖 · 𝑁) ∈ ℤ ∧ (𝑀 · 𝑁) ∈ ℤ) → (((𝑖 · 𝑗) ∥ (𝑖 · 𝑁) ∧ (𝑖 · 𝑁) ∥ (𝑀 · 𝑁)) → (𝑖 · 𝑗) ∥ (𝑀 · 𝑁)))
55	51, 52, 53, 54	syl3anc 1379	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) → (((𝑖 · 𝑗) ∥ (𝑖 · 𝑁) ∧ (𝑖 · 𝑁) ∥ (𝑀 · 𝑁)) → (𝑖 · 𝑗) ∥ (𝑀 · 𝑁)))
56	45, 50, 55	syl2and 614	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) → ((𝑗 ∥ 𝑁 ∧ 𝑖 ∥ 𝑀) → (𝑖 · 𝑗) ∥ (𝑀 · 𝑁)))
57	34, 38, 56	mp2and 705	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) → (𝑖 · 𝑗) ∥ (𝑀 · 𝑁))
58		breq1 5075	. . . . . . . . 9 ⊢ (𝑥 = (𝑖 · 𝑗) → (𝑥 ∥ (𝑀 · 𝑁) ↔ (𝑖 · 𝑗) ∥ (𝑀 · 𝑁)))
59		mpodvdsmulf1o.z	. . . . . . . . 9 ⊢ 𝑍 = {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑀 · 𝑁)}
60	58, 59	elrab2 3632	. . . . . . . 8 ⊢ ((𝑖 · 𝑗) ∈ 𝑍 ↔ ((𝑖 · 𝑗) ∈ ℕ ∧ (𝑖 · 𝑗) ∥ (𝑀 · 𝑁)))
61	30, 57, 60	sylanbrc 589	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) → (𝑖 · 𝑗) ∈ 𝑍)
62	25, 61	eqeltrrd 2840	. . . . . 6 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) → (𝑖(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))𝑗) ∈ 𝑍)
63	17, 62	eqeltrd 2839	. . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) → (𝑖((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ↾ (𝑋 × 𝑌))𝑗) ∈ 𝑍)
64	63	ralrimivva 3182	. . . 4 ⊢ (𝜑 → ∀𝑖 ∈ 𝑋 ∀𝑗 ∈ 𝑌 (𝑖((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ↾ (𝑋 × 𝑌))𝑗) ∈ 𝑍)
65		ffnov 7482	. . . 4 ⊢ (((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)⟶𝑍 ↔ (((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌) ∧ ∀𝑖 ∈ 𝑋 ∀𝑗 ∈ 𝑌 (𝑖((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ↾ (𝑋 × 𝑌))𝑗) ∈ 𝑍))
66	15, 64, 65	sylanbrc 589	. . 3 ⊢ (𝜑 → ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)⟶𝑍)
67	19	ad2antlr 733	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ (𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌)) → 𝑖 ∈ ℂ)
68	21	ad2antlr 733	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ (𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌)) → 𝑗 ∈ ℂ)
69	67, 68, 22	syl2anc 590	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ (𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌)) → (𝑖(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))𝑗) = (𝑖 · 𝑗))
70	7	sseli 3911	. . . . . . . . . 10 ⊢ (𝑚 ∈ 𝑋 → 𝑚 ∈ ℂ)
71	70	ad2antrl 734	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ (𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌)) → 𝑚 ∈ ℂ)
72	10	sseli 3911	. . . . . . . . . 10 ⊢ (𝑛 ∈ 𝑌 → 𝑛 ∈ ℂ)
73	72	ad2antll 735	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ (𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌)) → 𝑛 ∈ ℂ)
74		ovmpot 7517	. . . . . . . . 9 ⊢ ((𝑚 ∈ ℂ ∧ 𝑛 ∈ ℂ) → (𝑚(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))𝑛) = (𝑚 · 𝑛))
75	71, 73, 74	syl2anc 590	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ (𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌)) → (𝑚(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))𝑛) = (𝑚 · 𝑛))
76	69, 75	eqeq12d 2755	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ (𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌)) → ((𝑖(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))𝑗) = (𝑚(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))𝑛) ↔ (𝑖 · 𝑗) = (𝑚 · 𝑛)))
77	27	adantr 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌) ∧ (𝑖 · 𝑗) = (𝑚 · 𝑛))) → 𝑖 ∈ ℕ)
78	77	nnnn0d 12489	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌) ∧ (𝑖 · 𝑗) = (𝑚 · 𝑛))) → 𝑖 ∈ ℕ₀)
79		simprll 784	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌) ∧ (𝑖 · 𝑗) = (𝑚 · 𝑛))) → 𝑚 ∈ 𝑋)
80	5, 79	sselid 3913	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌) ∧ (𝑖 · 𝑗) = (𝑚 · 𝑛))) → 𝑚 ∈ ℕ)
81	80	nnnn0d 12489	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌) ∧ (𝑖 · 𝑗) = (𝑚 · 𝑛))) → 𝑚 ∈ ℕ₀)
82	77	nnzd 12541	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌) ∧ (𝑖 · 𝑗) = (𝑚 · 𝑛))) → 𝑖 ∈ ℤ)
83	29	adantr 481	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌) ∧ (𝑖 · 𝑗) = (𝑚 · 𝑛))) → 𝑗 ∈ ℕ)
84	83	nnzd 12541	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌) ∧ (𝑖 · 𝑗) = (𝑚 · 𝑛))) → 𝑗 ∈ ℤ)
85		dvdsmul1 16237	. . . . . . . . . . . . 13 ⊢ ((𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ) → 𝑖 ∥ (𝑖 · 𝑗))
86	82, 84, 85	syl2anc 590	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌) ∧ (𝑖 · 𝑗) = (𝑚 · 𝑛))) → 𝑖 ∥ (𝑖 · 𝑗))
87		simprr 778	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌) ∧ (𝑖 · 𝑗) = (𝑚 · 𝑛))) → (𝑖 · 𝑗) = (𝑚 · 𝑛))
88	7, 79	sselid 3913	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌) ∧ (𝑖 · 𝑗) = (𝑚 · 𝑛))) → 𝑚 ∈ ℂ)
89		simprlr 785	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌) ∧ (𝑖 · 𝑗) = (𝑚 · 𝑛))) → 𝑛 ∈ 𝑌)
90	10, 89	sselid 3913	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌) ∧ (𝑖 · 𝑗) = (𝑚 · 𝑛))) → 𝑛 ∈ ℂ)
91	88, 90	mulcomd 11157	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌) ∧ (𝑖 · 𝑗) = (𝑚 · 𝑛))) → (𝑚 · 𝑛) = (𝑛 · 𝑚))
92	87, 91	eqtrd 2774	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌) ∧ (𝑖 · 𝑗) = (𝑚 · 𝑛))) → (𝑖 · 𝑗) = (𝑛 · 𝑚))
93	86, 92	breqtrd 5098	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌) ∧ (𝑖 · 𝑗) = (𝑚 · 𝑛))) → 𝑖 ∥ (𝑛 · 𝑚))
94	9, 89	sselid 3913	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌) ∧ (𝑖 · 𝑗) = (𝑚 · 𝑛))) → 𝑛 ∈ ℕ)
95	94	nnzd 12541	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌) ∧ (𝑖 · 𝑗) = (𝑚 · 𝑛))) → 𝑛 ∈ ℤ)
96	42	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌) ∧ (𝑖 · 𝑗) = (𝑚 · 𝑛))) → 𝑁 ∈ ℤ)
97	82, 96	gcdcomd 16474	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌) ∧ (𝑖 · 𝑗) = (𝑚 · 𝑛))) → (𝑖 gcd 𝑁) = (𝑁 gcd 𝑖))
98	48	adantr 481	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌) ∧ (𝑖 · 𝑗) = (𝑚 · 𝑛))) → 𝑀 ∈ ℤ)
99	40	nnzd 12541	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑁 ∈ ℤ)
100	46	nnzd 12541	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑀 ∈ ℤ)
101	99, 100	gcdcomd 16474	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑁 gcd 𝑀) = (𝑀 gcd 𝑁))
102		mpodvdsmulf1o.3	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑀 gcd 𝑁) = 1)
103	101, 102	eqtrd 2774	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑁 gcd 𝑀) = 1)
104	103	ad2antrr 732	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌) ∧ (𝑖 · 𝑗) = (𝑚 · 𝑛))) → (𝑁 gcd 𝑀) = 1)
105	38	adantr 481	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌) ∧ (𝑖 · 𝑗) = (𝑚 · 𝑛))) → 𝑖 ∥ 𝑀)
106		rpdvds 16620	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℤ ∧ 𝑖 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ ((𝑁 gcd 𝑀) = 1 ∧ 𝑖 ∥ 𝑀)) → (𝑁 gcd 𝑖) = 1)
107	96, 82, 98, 104, 105, 106	syl32anc 1386	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌) ∧ (𝑖 · 𝑗) = (𝑚 · 𝑛))) → (𝑁 gcd 𝑖) = 1)
108	97, 107	eqtrd 2774	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌) ∧ (𝑖 · 𝑗) = (𝑚 · 𝑛))) → (𝑖 gcd 𝑁) = 1)
109		breq1 5075	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑛 → (𝑥 ∥ 𝑁 ↔ 𝑛 ∥ 𝑁))
110	109, 8	elrab2 3632	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ 𝑌 ↔ (𝑛 ∈ ℕ ∧ 𝑛 ∥ 𝑁))
111	110	simprbi 498	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ 𝑌 → 𝑛 ∥ 𝑁)
112	89, 111	syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌) ∧ (𝑖 · 𝑗) = (𝑚 · 𝑛))) → 𝑛 ∥ 𝑁)
113		rpdvds 16620	. . . . . . . . . . . 12 ⊢ (((𝑖 ∈ ℤ ∧ 𝑛 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ((𝑖 gcd 𝑁) = 1 ∧ 𝑛 ∥ 𝑁)) → (𝑖 gcd 𝑛) = 1)
114	82, 95, 96, 108, 112, 113	syl32anc 1386	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌) ∧ (𝑖 · 𝑗) = (𝑚 · 𝑛))) → (𝑖 gcd 𝑛) = 1)
115	80	nnzd 12541	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌) ∧ (𝑖 · 𝑗) = (𝑚 · 𝑛))) → 𝑚 ∈ ℤ)
116		coprmdvds 16613	. . . . . . . . . . . 12 ⊢ ((𝑖 ∈ ℤ ∧ 𝑛 ∈ ℤ ∧ 𝑚 ∈ ℤ) → ((𝑖 ∥ (𝑛 · 𝑚) ∧ (𝑖 gcd 𝑛) = 1) → 𝑖 ∥ 𝑚))
117	82, 95, 115, 116	syl3anc 1379	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌) ∧ (𝑖 · 𝑗) = (𝑚 · 𝑛))) → ((𝑖 ∥ (𝑛 · 𝑚) ∧ (𝑖 gcd 𝑛) = 1) → 𝑖 ∥ 𝑚))
118	93, 114, 117	mp2and 705	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌) ∧ (𝑖 · 𝑗) = (𝑚 · 𝑛))) → 𝑖 ∥ 𝑚)
119		dvdsmul1 16237	. . . . . . . . . . . . 13 ⊢ ((𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ) → 𝑚 ∥ (𝑚 · 𝑛))
120	115, 95, 119	syl2anc 590	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌) ∧ (𝑖 · 𝑗) = (𝑚 · 𝑛))) → 𝑚 ∥ (𝑚 · 𝑛))
121	77	nncnd 12181	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌) ∧ (𝑖 · 𝑗) = (𝑚 · 𝑛))) → 𝑖 ∈ ℂ)
122	83	nncnd 12181	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌) ∧ (𝑖 · 𝑗) = (𝑚 · 𝑛))) → 𝑗 ∈ ℂ)
123	121, 122	mulcomd 11157	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌) ∧ (𝑖 · 𝑗) = (𝑚 · 𝑛))) → (𝑖 · 𝑗) = (𝑗 · 𝑖))
124	87, 123	eqtr3d 2776	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌) ∧ (𝑖 · 𝑗) = (𝑚 · 𝑛))) → (𝑚 · 𝑛) = (𝑗 · 𝑖))
125	120, 124	breqtrd 5098	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌) ∧ (𝑖 · 𝑗) = (𝑚 · 𝑛))) → 𝑚 ∥ (𝑗 · 𝑖))
126	115, 96	gcdcomd 16474	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌) ∧ (𝑖 · 𝑗) = (𝑚 · 𝑛))) → (𝑚 gcd 𝑁) = (𝑁 gcd 𝑚))
127		breq1 5075	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑚 → (𝑥 ∥ 𝑀 ↔ 𝑚 ∥ 𝑀))
128	127, 4	elrab2 3632	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ 𝑋 ↔ (𝑚 ∈ ℕ ∧ 𝑚 ∥ 𝑀))
129	128	simprbi 498	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ 𝑋 → 𝑚 ∥ 𝑀)
130	79, 129	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌) ∧ (𝑖 · 𝑗) = (𝑚 · 𝑛))) → 𝑚 ∥ 𝑀)
131		rpdvds 16620	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℤ ∧ 𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ ((𝑁 gcd 𝑀) = 1 ∧ 𝑚 ∥ 𝑀)) → (𝑁 gcd 𝑚) = 1)
132	96, 115, 98, 104, 130, 131	syl32anc 1386	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌) ∧ (𝑖 · 𝑗) = (𝑚 · 𝑛))) → (𝑁 gcd 𝑚) = 1)
133	126, 132	eqtrd 2774	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌) ∧ (𝑖 · 𝑗) = (𝑚 · 𝑛))) → (𝑚 gcd 𝑁) = 1)
134	34	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌) ∧ (𝑖 · 𝑗) = (𝑚 · 𝑛))) → 𝑗 ∥ 𝑁)
135		rpdvds 16620	. . . . . . . . . . . 12 ⊢ (((𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ((𝑚 gcd 𝑁) = 1 ∧ 𝑗 ∥ 𝑁)) → (𝑚 gcd 𝑗) = 1)
136	115, 84, 96, 133, 134, 135	syl32anc 1386	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌) ∧ (𝑖 · 𝑗) = (𝑚 · 𝑛))) → (𝑚 gcd 𝑗) = 1)
137		coprmdvds 16613	. . . . . . . . . . . 12 ⊢ ((𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ ∧ 𝑖 ∈ ℤ) → ((𝑚 ∥ (𝑗 · 𝑖) ∧ (𝑚 gcd 𝑗) = 1) → 𝑚 ∥ 𝑖))
138	115, 84, 82, 137	syl3anc 1379	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌) ∧ (𝑖 · 𝑗) = (𝑚 · 𝑛))) → ((𝑚 ∥ (𝑗 · 𝑖) ∧ (𝑚 gcd 𝑗) = 1) → 𝑚 ∥ 𝑖))
139	125, 136, 138	mp2and 705	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌) ∧ (𝑖 · 𝑗) = (𝑚 · 𝑛))) → 𝑚 ∥ 𝑖)
140		dvdseq 16274	. . . . . . . . . 10 ⊢ (((𝑖 ∈ ℕ₀ ∧ 𝑚 ∈ ℕ₀) ∧ (𝑖 ∥ 𝑚 ∧ 𝑚 ∥ 𝑖)) → 𝑖 = 𝑚)
141	78, 81, 118, 139, 140	syl22anc 844	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌) ∧ (𝑖 · 𝑗) = (𝑚 · 𝑛))) → 𝑖 = 𝑚)
142	77	nnne0d 12218	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌) ∧ (𝑖 · 𝑗) = (𝑚 · 𝑛))) → 𝑖 ≠ 0)
143	141	oveq1d 7371	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌) ∧ (𝑖 · 𝑗) = (𝑚 · 𝑛))) → (𝑖 · 𝑛) = (𝑚 · 𝑛))
144	87, 143	eqtr4d 2777	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌) ∧ (𝑖 · 𝑗) = (𝑚 · 𝑛))) → (𝑖 · 𝑗) = (𝑖 · 𝑛))
145	122, 90, 121, 142, 144	mulcanad 11776	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌) ∧ (𝑖 · 𝑗) = (𝑚 · 𝑛))) → 𝑗 = 𝑛)
146	141, 145	opeq12d 4812	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌) ∧ (𝑖 · 𝑗) = (𝑚 · 𝑛))) → ⟨𝑖, 𝑗⟩ = ⟨𝑚, 𝑛⟩)
147	146	expr 457	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ (𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌)) → ((𝑖 · 𝑗) = (𝑚 · 𝑛) → ⟨𝑖, 𝑗⟩ = ⟨𝑚, 𝑛⟩))
148	76, 147	sylbid 241	. . . . . 6 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ (𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌)) → ((𝑖(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))𝑗) = (𝑚(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))𝑛) → ⟨𝑖, 𝑗⟩ = ⟨𝑚, 𝑛⟩))
149	148	ralrimivva 3182	. . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) → ∀𝑚 ∈ 𝑋 ∀𝑛 ∈ 𝑌 ((𝑖(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))𝑗) = (𝑚(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))𝑛) → ⟨𝑖, 𝑗⟩ = ⟨𝑚, 𝑛⟩))
150	149	ralrimivva 3182	. . . 4 ⊢ (𝜑 → ∀𝑖 ∈ 𝑋 ∀𝑗 ∈ 𝑌 ∀𝑚 ∈ 𝑋 ∀𝑛 ∈ 𝑌 ((𝑖(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))𝑗) = (𝑚(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))𝑛) → ⟨𝑖, 𝑗⟩ = ⟨𝑚, 𝑛⟩))
151		fvres 6846	. . . . . . . . 9 ⊢ (𝑢 ∈ (𝑋 × 𝑌) → (((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ↾ (𝑋 × 𝑌))‘𝑢) = ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑢))
152		fvres 6846	. . . . . . . . 9 ⊢ (𝑣 ∈ (𝑋 × 𝑌) → (((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ↾ (𝑋 × 𝑌))‘𝑣) = ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑣))
153	151, 152	eqeqan12d 2753	. . . . . . . 8 ⊢ ((𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) → ((((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ↾ (𝑋 × 𝑌))‘𝑢) = (((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ↾ (𝑋 × 𝑌))‘𝑣) ↔ ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑢) = ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑣)))
154	153	imbi1d 342	. . . . . . 7 ⊢ ((𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) → (((((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ↾ (𝑋 × 𝑌))‘𝑢) = (((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ↾ (𝑋 × 𝑌))‘𝑣) → 𝑢 = 𝑣) ↔ (((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑢) = ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑣) → 𝑢 = 𝑣)))
155	154	ralbidva 3160	. . . . . 6 ⊢ (𝑢 ∈ (𝑋 × 𝑌) → (∀𝑣 ∈ (𝑋 × 𝑌)((((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ↾ (𝑋 × 𝑌))‘𝑢) = (((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ↾ (𝑋 × 𝑌))‘𝑣) → 𝑢 = 𝑣) ↔ ∀𝑣 ∈ (𝑋 × 𝑌)(((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑢) = ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑣) → 𝑢 = 𝑣)))
156	155	ralbiia 3083	. . . . 5 ⊢ (∀𝑢 ∈ (𝑋 × 𝑌)∀𝑣 ∈ (𝑋 × 𝑌)((((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ↾ (𝑋 × 𝑌))‘𝑢) = (((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ↾ (𝑋 × 𝑌))‘𝑣) → 𝑢 = 𝑣) ↔ ∀𝑢 ∈ (𝑋 × 𝑌)∀𝑣 ∈ (𝑋 × 𝑌)(((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑢) = ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑣) → 𝑢 = 𝑣))
157		fveq2 6827	. . . . . . . . . . 11 ⊢ (𝑣 = ⟨𝑚, 𝑛⟩ → ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑣) = ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘⟨𝑚, 𝑛⟩))
158		df-ov 7359	. . . . . . . . . . 11 ⊢ (𝑚(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))𝑛) = ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘⟨𝑚, 𝑛⟩)
159	157, 158	eqtr4di 2792	. . . . . . . . . 10 ⊢ (𝑣 = ⟨𝑚, 𝑛⟩ → ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑣) = (𝑚(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))𝑛))
160	159	eqeq2d 2750	. . . . . . . . 9 ⊢ (𝑣 = ⟨𝑚, 𝑛⟩ → (((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑢) = ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑣) ↔ ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑢) = (𝑚(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))𝑛)))
161		eqeq2 2751	. . . . . . . . 9 ⊢ (𝑣 = ⟨𝑚, 𝑛⟩ → (𝑢 = 𝑣 ↔ 𝑢 = ⟨𝑚, 𝑛⟩))
162	160, 161	imbi12d 345	. . . . . . . 8 ⊢ (𝑣 = ⟨𝑚, 𝑛⟩ → ((((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑢) = ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑣) → 𝑢 = 𝑣) ↔ (((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑢) = (𝑚(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))𝑛) → 𝑢 = ⟨𝑚, 𝑛⟩)))
163	162	ralxp 5783	. . . . . . 7 ⊢ (∀𝑣 ∈ (𝑋 × 𝑌)(((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑢) = ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑣) → 𝑢 = 𝑣) ↔ ∀𝑚 ∈ 𝑋 ∀𝑛 ∈ 𝑌 (((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑢) = (𝑚(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))𝑛) → 𝑢 = ⟨𝑚, 𝑛⟩))
164		fveq2 6827	. . . . . . . . . . 11 ⊢ (𝑢 = ⟨𝑖, 𝑗⟩ → ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑢) = ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘⟨𝑖, 𝑗⟩))
165		df-ov 7359	. . . . . . . . . . 11 ⊢ (𝑖(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))𝑗) = ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘⟨𝑖, 𝑗⟩)
166	164, 165	eqtr4di 2792	. . . . . . . . . 10 ⊢ (𝑢 = ⟨𝑖, 𝑗⟩ → ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑢) = (𝑖(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))𝑗))
167	166	eqeq1d 2741	. . . . . . . . 9 ⊢ (𝑢 = ⟨𝑖, 𝑗⟩ → (((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑢) = (𝑚(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))𝑛) ↔ (𝑖(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))𝑗) = (𝑚(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))𝑛)))
168		eqeq1 2743	. . . . . . . . 9 ⊢ (𝑢 = ⟨𝑖, 𝑗⟩ → (𝑢 = ⟨𝑚, 𝑛⟩ ↔ ⟨𝑖, 𝑗⟩ = ⟨𝑚, 𝑛⟩))
169	167, 168	imbi12d 345	. . . . . . . 8 ⊢ (𝑢 = ⟨𝑖, 𝑗⟩ → ((((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑢) = (𝑚(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))𝑛) → 𝑢 = ⟨𝑚, 𝑛⟩) ↔ ((𝑖(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))𝑗) = (𝑚(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))𝑛) → ⟨𝑖, 𝑗⟩ = ⟨𝑚, 𝑛⟩)))
170	169	2ralbidv 3203	. . . . . . 7 ⊢ (𝑢 = ⟨𝑖, 𝑗⟩ → (∀𝑚 ∈ 𝑋 ∀𝑛 ∈ 𝑌 (((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑢) = (𝑚(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))𝑛) → 𝑢 = ⟨𝑚, 𝑛⟩) ↔ ∀𝑚 ∈ 𝑋 ∀𝑛 ∈ 𝑌 ((𝑖(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))𝑗) = (𝑚(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))𝑛) → ⟨𝑖, 𝑗⟩ = ⟨𝑚, 𝑛⟩)))
171	163, 170	bitrid 284	. . . . . 6 ⊢ (𝑢 = ⟨𝑖, 𝑗⟩ → (∀𝑣 ∈ (𝑋 × 𝑌)(((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑢) = ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑣) → 𝑢 = 𝑣) ↔ ∀𝑚 ∈ 𝑋 ∀𝑛 ∈ 𝑌 ((𝑖(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))𝑗) = (𝑚(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))𝑛) → ⟨𝑖, 𝑗⟩ = ⟨𝑚, 𝑛⟩)))
172	171	ralxp 5783	. . . . 5 ⊢ (∀𝑢 ∈ (𝑋 × 𝑌)∀𝑣 ∈ (𝑋 × 𝑌)(((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑢) = ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑣) → 𝑢 = 𝑣) ↔ ∀𝑖 ∈ 𝑋 ∀𝑗 ∈ 𝑌 ∀𝑚 ∈ 𝑋 ∀𝑛 ∈ 𝑌 ((𝑖(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))𝑗) = (𝑚(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))𝑛) → ⟨𝑖, 𝑗⟩ = ⟨𝑚, 𝑛⟩))
173	156, 172	bitri 276	. . . 4 ⊢ (∀𝑢 ∈ (𝑋 × 𝑌)∀𝑣 ∈ (𝑋 × 𝑌)((((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ↾ (𝑋 × 𝑌))‘𝑢) = (((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ↾ (𝑋 × 𝑌))‘𝑣) → 𝑢 = 𝑣) ↔ ∀𝑖 ∈ 𝑋 ∀𝑗 ∈ 𝑌 ∀𝑚 ∈ 𝑋 ∀𝑛 ∈ 𝑌 ((𝑖(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))𝑗) = (𝑚(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))𝑛) → ⟨𝑖, 𝑗⟩ = ⟨𝑚, 𝑛⟩))
174	150, 173	sylibr 235	. . 3 ⊢ (𝜑 → ∀𝑢 ∈ (𝑋 × 𝑌)∀𝑣 ∈ (𝑋 × 𝑌)((((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ↾ (𝑋 × 𝑌))‘𝑢) = (((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ↾ (𝑋 × 𝑌))‘𝑣) → 𝑢 = 𝑣))
175		dff13 7198	. . 3 ⊢ (((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)–1-1→𝑍 ↔ (((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)⟶𝑍 ∧ ∀𝑢 ∈ (𝑋 × 𝑌)∀𝑣 ∈ (𝑋 × 𝑌)((((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ↾ (𝑋 × 𝑌))‘𝑢) = (((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ↾ (𝑋 × 𝑌))‘𝑣) → 𝑢 = 𝑣)))
176	66, 174, 175	sylanbrc 589	. 2 ⊢ (𝜑 → ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)–1-1→𝑍)
177		breq1 5075	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑤 → (𝑥 ∥ (𝑀 · 𝑁) ↔ 𝑤 ∥ (𝑀 · 𝑁)))
178	177, 59	elrab2 3632	. . . . . . . . . . 11 ⊢ (𝑤 ∈ 𝑍 ↔ (𝑤 ∈ ℕ ∧ 𝑤 ∥ (𝑀 · 𝑁)))
179	178	simplbi 497	. . . . . . . . . 10 ⊢ (𝑤 ∈ 𝑍 → 𝑤 ∈ ℕ)
180	179	adantl 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → 𝑤 ∈ ℕ)
181	180	nnzd 12541	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → 𝑤 ∈ ℤ)
182	46	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → 𝑀 ∈ ℕ)
183	182	nnzd 12541	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → 𝑀 ∈ ℤ)
184	182	nnne0d 12218	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → 𝑀 ≠ 0)
185		simpr 485	. . . . . . . . . 10 ⊢ ((𝑤 = 0 ∧ 𝑀 = 0) → 𝑀 = 0)
186	185	necon3ai 2959	. . . . . . . . 9 ⊢ (𝑀 ≠ 0 → ¬ (𝑤 = 0 ∧ 𝑀 = 0))
187	184, 186	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → ¬ (𝑤 = 0 ∧ 𝑀 = 0))
188		gcdn0cl 16462	. . . . . . . 8 ⊢ (((𝑤 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ ¬ (𝑤 = 0 ∧ 𝑀 = 0)) → (𝑤 gcd 𝑀) ∈ ℕ)
189	181, 183, 187, 188	syl21anc 843	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → (𝑤 gcd 𝑀) ∈ ℕ)
190		gcddvds 16463	. . . . . . . . 9 ⊢ ((𝑤 ∈ ℤ ∧ 𝑀 ∈ ℤ) → ((𝑤 gcd 𝑀) ∥ 𝑤 ∧ (𝑤 gcd 𝑀) ∥ 𝑀))
191	181, 183, 190	syl2anc 590	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → ((𝑤 gcd 𝑀) ∥ 𝑤 ∧ (𝑤 gcd 𝑀) ∥ 𝑀))
192	191	simprd 496	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → (𝑤 gcd 𝑀) ∥ 𝑀)
193		breq1 5075	. . . . . . . 8 ⊢ (𝑥 = (𝑤 gcd 𝑀) → (𝑥 ∥ 𝑀 ↔ (𝑤 gcd 𝑀) ∥ 𝑀))
194	193, 4	elrab2 3632	. . . . . . 7 ⊢ ((𝑤 gcd 𝑀) ∈ 𝑋 ↔ ((𝑤 gcd 𝑀) ∈ ℕ ∧ (𝑤 gcd 𝑀) ∥ 𝑀))
195	189, 192, 194	sylanbrc 589	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → (𝑤 gcd 𝑀) ∈ 𝑋)
196	40	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → 𝑁 ∈ ℕ)
197	196	nnzd 12541	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → 𝑁 ∈ ℤ)
198	196	nnne0d 12218	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → 𝑁 ≠ 0)
199		simpr 485	. . . . . . . . . 10 ⊢ ((𝑤 = 0 ∧ 𝑁 = 0) → 𝑁 = 0)
200	199	necon3ai 2959	. . . . . . . . 9 ⊢ (𝑁 ≠ 0 → ¬ (𝑤 = 0 ∧ 𝑁 = 0))
201	198, 200	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → ¬ (𝑤 = 0 ∧ 𝑁 = 0))
202		gcdn0cl 16462	. . . . . . . 8 ⊢ (((𝑤 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑤 = 0 ∧ 𝑁 = 0)) → (𝑤 gcd 𝑁) ∈ ℕ)
203	181, 197, 201, 202	syl21anc 843	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → (𝑤 gcd 𝑁) ∈ ℕ)
204		gcddvds 16463	. . . . . . . . 9 ⊢ ((𝑤 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑤 gcd 𝑁) ∥ 𝑤 ∧ (𝑤 gcd 𝑁) ∥ 𝑁))
205	181, 197, 204	syl2anc 590	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → ((𝑤 gcd 𝑁) ∥ 𝑤 ∧ (𝑤 gcd 𝑁) ∥ 𝑁))
206	205	simprd 496	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → (𝑤 gcd 𝑁) ∥ 𝑁)
207		breq1 5075	. . . . . . . 8 ⊢ (𝑥 = (𝑤 gcd 𝑁) → (𝑥 ∥ 𝑁 ↔ (𝑤 gcd 𝑁) ∥ 𝑁))
208	207, 8	elrab2 3632	. . . . . . 7 ⊢ ((𝑤 gcd 𝑁) ∈ 𝑌 ↔ ((𝑤 gcd 𝑁) ∈ ℕ ∧ (𝑤 gcd 𝑁) ∥ 𝑁))
209	203, 206, 208	sylanbrc 589	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → (𝑤 gcd 𝑁) ∈ 𝑌)
210	195, 209	opelxpd 5657	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → ⟨(𝑤 gcd 𝑀), (𝑤 gcd 𝑁)⟩ ∈ (𝑋 × 𝑌))
211	210	fvresd 6847	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → (((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ↾ (𝑋 × 𝑌))‘⟨(𝑤 gcd 𝑀), (𝑤 gcd 𝑁)⟩) = ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘⟨(𝑤 gcd 𝑀), (𝑤 gcd 𝑁)⟩))
212		df-ov 7359	. . . . . . . 8 ⊢ ((𝑤 gcd 𝑀)(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))(𝑤 gcd 𝑁)) = ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘⟨(𝑤 gcd 𝑀), (𝑤 gcd 𝑁)⟩)
213	189	nncnd 12181	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → (𝑤 gcd 𝑀) ∈ ℂ)
214	203	nncnd 12181	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → (𝑤 gcd 𝑁) ∈ ℂ)
215		ovmpot 7517	. . . . . . . . 9 ⊢ (((𝑤 gcd 𝑀) ∈ ℂ ∧ (𝑤 gcd 𝑁) ∈ ℂ) → ((𝑤 gcd 𝑀)(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))(𝑤 gcd 𝑁)) = ((𝑤 gcd 𝑀) · (𝑤 gcd 𝑁)))
216	213, 214, 215	syl2anc 590	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → ((𝑤 gcd 𝑀)(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))(𝑤 gcd 𝑁)) = ((𝑤 gcd 𝑀) · (𝑤 gcd 𝑁)))
217	212, 216	eqtr3id 2788	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘⟨(𝑤 gcd 𝑀), (𝑤 gcd 𝑁)⟩) = ((𝑤 gcd 𝑀) · (𝑤 gcd 𝑁)))
218		df-ov 7359	. . . . . . . 8 ⊢ ((𝑤 gcd 𝑀) · (𝑤 gcd 𝑁)) = ( · ‘⟨(𝑤 gcd 𝑀), (𝑤 gcd 𝑁)⟩)
219	218	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → ((𝑤 gcd 𝑀) · (𝑤 gcd 𝑁)) = ( · ‘⟨(𝑤 gcd 𝑀), (𝑤 gcd 𝑁)⟩))
220	211, 217, 219	3eqtrd 2778	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → (((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ↾ (𝑋 × 𝑌))‘⟨(𝑤 gcd 𝑀), (𝑤 gcd 𝑁)⟩) = ( · ‘⟨(𝑤 gcd 𝑀), (𝑤 gcd 𝑁)⟩))
221	102	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → (𝑀 gcd 𝑁) = 1)
222		rpmulgcd2 16616	. . . . . . . 8 ⊢ (((𝑤 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 gcd 𝑁) = 1) → (𝑤 gcd (𝑀 · 𝑁)) = ((𝑤 gcd 𝑀) · (𝑤 gcd 𝑁)))
223	181, 183, 197, 221, 222	syl31anc 1381	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → (𝑤 gcd (𝑀 · 𝑁)) = ((𝑤 gcd 𝑀) · (𝑤 gcd 𝑁)))
224	223, 218	eqtrdi 2790	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → (𝑤 gcd (𝑀 · 𝑁)) = ( · ‘⟨(𝑤 gcd 𝑀), (𝑤 gcd 𝑁)⟩))
225	178	simprbi 498	. . . . . . . 8 ⊢ (𝑤 ∈ 𝑍 → 𝑤 ∥ (𝑀 · 𝑁))
226	225	adantl 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → 𝑤 ∥ (𝑀 · 𝑁))
227	46, 40	nnmulcld 12221	. . . . . . . 8 ⊢ (𝜑 → (𝑀 · 𝑁) ∈ ℕ)
228		gcdeq 16513	. . . . . . . 8 ⊢ ((𝑤 ∈ ℕ ∧ (𝑀 · 𝑁) ∈ ℕ) → ((𝑤 gcd (𝑀 · 𝑁)) = 𝑤 ↔ 𝑤 ∥ (𝑀 · 𝑁)))
229	179, 227, 228	syl2anr 603	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → ((𝑤 gcd (𝑀 · 𝑁)) = 𝑤 ↔ 𝑤 ∥ (𝑀 · 𝑁)))
230	226, 229	mpbird 258	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → (𝑤 gcd (𝑀 · 𝑁)) = 𝑤)
231	220, 224, 230	3eqtr2rd 2781	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → 𝑤 = (((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ↾ (𝑋 × 𝑌))‘⟨(𝑤 gcd 𝑀), (𝑤 gcd 𝑁)⟩))
232		fveq2 6827	. . . . . 6 ⊢ (𝑢 = ⟨(𝑤 gcd 𝑀), (𝑤 gcd 𝑁)⟩ → (((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ↾ (𝑋 × 𝑌))‘𝑢) = (((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ↾ (𝑋 × 𝑌))‘⟨(𝑤 gcd 𝑀), (𝑤 gcd 𝑁)⟩))
233	232	rspceeqv 3583	. . . . 5 ⊢ ((⟨(𝑤 gcd 𝑀), (𝑤 gcd 𝑁)⟩ ∈ (𝑋 × 𝑌) ∧ 𝑤 = (((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ↾ (𝑋 × 𝑌))‘⟨(𝑤 gcd 𝑀), (𝑤 gcd 𝑁)⟩)) → ∃𝑢 ∈ (𝑋 × 𝑌)𝑤 = (((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ↾ (𝑋 × 𝑌))‘𝑢))
234	210, 231, 233	syl2anc 590	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → ∃𝑢 ∈ (𝑋 × 𝑌)𝑤 = (((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ↾ (𝑋 × 𝑌))‘𝑢))
235	234	ralrimiva 3131	. . 3 ⊢ (𝜑 → ∀𝑤 ∈ 𝑍 ∃𝑢 ∈ (𝑋 × 𝑌)𝑤 = (((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ↾ (𝑋 × 𝑌))‘𝑢))
236		dffo3 7043	. . 3 ⊢ (((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)–onto→𝑍 ↔ (((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)⟶𝑍 ∧ ∀𝑤 ∈ 𝑍 ∃𝑢 ∈ (𝑋 × 𝑌)𝑤 = (((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ↾ (𝑋 × 𝑌))‘𝑢)))
237	66, 235, 236	sylanbrc 589	. 2 ⊢ (𝜑 → ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)–onto→𝑍)
238		df-f1o 6492	. 2 ⊢ (((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)–1-1-onto→𝑍 ↔ (((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)–1-1→𝑍 ∧ ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)–onto→𝑍))
239	176, 237, 238	sylanbrc 589	1 ⊢ (𝜑 → ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)–1-1-onto→𝑍)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1547 ∈ wcel 2119 ≠ wne 2934 ∀wral 3053 ∃wrex 3063 {crab 3391 ⊆ wss 3883 ⟨cop 4561 class class class wbr 5072 × cxp 5616 ↾ cres 5620 Fn wfn 6480 ⟶wf 6481 –1-1→wf1 6482 –onto→wfo 6483 –1-1-onto→wf1o 6484 ‘cfv 6485 (class class class)co 7356 ∈ cmpo 7358 ℂcc 11027 0cc0 11029 1c1 11030 · cmul 11034 ℕcn 12165 ℕ₀cn0 12428 ℤcz 12515 ∥ cdvds 16212 gcd cgcd 16454

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-pre-sup 11107

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-op 4562 df-uni 4839 df-iun 4923 df-br 5073 df-opab 5135 df-mpt 5154 df-tr 5180 df-id 5513 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5571 df-we 5573 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-pred 6252 df-ord 6313 df-on 6314 df-lim 6315 df-suc 6316 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-er 8633 df-en 8884 df-dom 8885 df-sdom 8886 df-sup 9345 df-inf 9346 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12166 df-2 12235 df-3 12236 df-n0 12429 df-z 12516 df-uz 12780 df-rp 12934 df-fl 13742 df-mod 13820 df-seq 13955 df-exp 14015 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-dvds 16213 df-gcd 16455

This theorem is referenced by: fsumdvdsmul 27176

W3C validator