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Theorem phimullem 15701

Description: Lemma for phimul 15702. (Contributed by Mario Carneiro, 24-Feb-2014.)

**Hypotheses**
Ref	Expression
crth.1	⊢ 𝑆 = (0..^(𝑀 · 𝑁))
crth.2	⊢ 𝑇 = ((0..^𝑀) × (0..^𝑁))
crth.3	⊢ 𝐹 = (𝑥 ∈ 𝑆 ↦ ⟨(𝑥 mod 𝑀), (𝑥 mod 𝑁)⟩)
crth.4	⊢ (𝜑 → (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1))
phimul.4	⊢ 𝑈 = {𝑦 ∈ (0..^𝑀) ∣ (𝑦 gcd 𝑀) = 1}
phimul.5	⊢ 𝑉 = {𝑦 ∈ (0..^𝑁) ∣ (𝑦 gcd 𝑁) = 1}
phimul.6	⊢ 𝑊 = {𝑦 ∈ 𝑆 ∣ (𝑦 gcd (𝑀 · 𝑁)) = 1}

**Assertion**
Ref	Expression
phimullem	⊢ (𝜑 → (ϕ‘(𝑀 · 𝑁)) = ((ϕ‘𝑀) · (ϕ‘𝑁)))

Distinct variable groups: 𝑦,𝐹 𝑥,𝑦,𝑀 𝜑,𝑥,𝑦 𝑥,𝑆,𝑦 𝑥,𝑇 𝑥,𝑁,𝑦 Allowed substitution hints: 𝑇(𝑦) 𝑈(𝑥,𝑦) 𝐹(𝑥) 𝑉(𝑥,𝑦) 𝑊(𝑥,𝑦)

Proof of Theorem phimullem Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		phimul.4	. . . . 5 ⊢ 𝑈 = {𝑦 ∈ (0..^𝑀) ∣ (𝑦 gcd 𝑀) = 1}
2		fzofi 12997	. . . . . 6 ⊢ (0..^𝑀) ∈ Fin
3		ssrab2 3884	. . . . . 6 ⊢ {𝑦 ∈ (0..^𝑀) ∣ (𝑦 gcd 𝑀) = 1} ⊆ (0..^𝑀)
4		ssfi 8419	. . . . . 6 ⊢ (((0..^𝑀) ∈ Fin ∧ {𝑦 ∈ (0..^𝑀) ∣ (𝑦 gcd 𝑀) = 1} ⊆ (0..^𝑀)) → {𝑦 ∈ (0..^𝑀) ∣ (𝑦 gcd 𝑀) = 1} ∈ Fin)
5	2, 3, 4	mp2an 675	. . . . 5 ⊢ {𝑦 ∈ (0..^𝑀) ∣ (𝑦 gcd 𝑀) = 1} ∈ Fin
6	1, 5	eqeltri 2881	. . . 4 ⊢ 𝑈 ∈ Fin
7		phimul.5	. . . . 5 ⊢ 𝑉 = {𝑦 ∈ (0..^𝑁) ∣ (𝑦 gcd 𝑁) = 1}
8		fzofi 12997	. . . . . 6 ⊢ (0..^𝑁) ∈ Fin
9		ssrab2 3884	. . . . . 6 ⊢ {𝑦 ∈ (0..^𝑁) ∣ (𝑦 gcd 𝑁) = 1} ⊆ (0..^𝑁)
10		ssfi 8419	. . . . . 6 ⊢ (((0..^𝑁) ∈ Fin ∧ {𝑦 ∈ (0..^𝑁) ∣ (𝑦 gcd 𝑁) = 1} ⊆ (0..^𝑁)) → {𝑦 ∈ (0..^𝑁) ∣ (𝑦 gcd 𝑁) = 1} ∈ Fin)
11	8, 9, 10	mp2an 675	. . . . 5 ⊢ {𝑦 ∈ (0..^𝑁) ∣ (𝑦 gcd 𝑁) = 1} ∈ Fin
12	7, 11	eqeltri 2881	. . . 4 ⊢ 𝑉 ∈ Fin
13		hashxp 13438	. . . 4 ⊢ ((𝑈 ∈ Fin ∧ 𝑉 ∈ Fin) → (♯‘(𝑈 × 𝑉)) = ((♯‘𝑈) · (♯‘𝑉)))
14	6, 12, 13	mp2an 675	. . 3 ⊢ (♯‘(𝑈 × 𝑉)) = ((♯‘𝑈) · (♯‘𝑉))
15		oveq1 6881	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑤 → (𝑦 gcd (𝑀 · 𝑁)) = (𝑤 gcd (𝑀 · 𝑁)))
16	15	eqeq1d 2808	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑤 → ((𝑦 gcd (𝑀 · 𝑁)) = 1 ↔ (𝑤 gcd (𝑀 · 𝑁)) = 1))
17		phimul.6	. . . . . . . . . . . . 13 ⊢ 𝑊 = {𝑦 ∈ 𝑆 ∣ (𝑦 gcd (𝑀 · 𝑁)) = 1}
18	16, 17	elrab2 3562	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ 𝑊 ↔ (𝑤 ∈ 𝑆 ∧ (𝑤 gcd (𝑀 · 𝑁)) = 1))
19	18	simplbi 487	. . . . . . . . . . 11 ⊢ (𝑤 ∈ 𝑊 → 𝑤 ∈ 𝑆)
20		oveq1 6881	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑤 → (𝑥 mod 𝑀) = (𝑤 mod 𝑀))
21		oveq1 6881	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑤 → (𝑥 mod 𝑁) = (𝑤 mod 𝑁))
22	20, 21	opeq12d 4603	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑤 → ⟨(𝑥 mod 𝑀), (𝑥 mod 𝑁)⟩ = ⟨(𝑤 mod 𝑀), (𝑤 mod 𝑁)⟩)
23		crth.3	. . . . . . . . . . . 12 ⊢ 𝐹 = (𝑥 ∈ 𝑆 ↦ ⟨(𝑥 mod 𝑀), (𝑥 mod 𝑁)⟩)
24		opex 5122	. . . . . . . . . . . 12 ⊢ ⟨(𝑤 mod 𝑀), (𝑤 mod 𝑁)⟩ ∈ V
25	22, 23, 24	fvmpt 6503	. . . . . . . . . . 11 ⊢ (𝑤 ∈ 𝑆 → (𝐹‘𝑤) = ⟨(𝑤 mod 𝑀), (𝑤 mod 𝑁)⟩)
26	19, 25	syl 17	. . . . . . . . . 10 ⊢ (𝑤 ∈ 𝑊 → (𝐹‘𝑤) = ⟨(𝑤 mod 𝑀), (𝑤 mod 𝑁)⟩)
27	26	adantl 469	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → (𝐹‘𝑤) = ⟨(𝑤 mod 𝑀), (𝑤 mod 𝑁)⟩)
28		crth.1	. . . . . . . . . . . . . . 15 ⊢ 𝑆 = (0..^(𝑀 · 𝑁))
29	19, 28	syl6eleq 2895	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ 𝑊 → 𝑤 ∈ (0..^(𝑀 · 𝑁)))
30	29	adantl 469	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → 𝑤 ∈ (0..^(𝑀 · 𝑁)))
31		elfzoelz 12694	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ (0..^(𝑀 · 𝑁)) → 𝑤 ∈ ℤ)
32	30, 31	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → 𝑤 ∈ ℤ)
33		crth.4	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1))
34	33	simp1d 1165	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑀 ∈ ℕ)
35	34	adantr 468	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → 𝑀 ∈ ℕ)
36		zmodfzo 12917	. . . . . . . . . . . 12 ⊢ ((𝑤 ∈ ℤ ∧ 𝑀 ∈ ℕ) → (𝑤 mod 𝑀) ∈ (0..^𝑀))
37	32, 35, 36	syl2anc 575	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → (𝑤 mod 𝑀) ∈ (0..^𝑀))
38		modgcd 15472	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ ℤ ∧ 𝑀 ∈ ℕ) → ((𝑤 mod 𝑀) gcd 𝑀) = (𝑤 gcd 𝑀))
39	32, 35, 38	syl2anc 575	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → ((𝑤 mod 𝑀) gcd 𝑀) = (𝑤 gcd 𝑀))
40	35	nnzd 11747	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → 𝑀 ∈ ℤ)
41		gcddvds 15444	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑤 ∈ ℤ ∧ 𝑀 ∈ ℤ) → ((𝑤 gcd 𝑀) ∥ 𝑤 ∧ (𝑤 gcd 𝑀) ∥ 𝑀))
42	32, 40, 41	syl2anc 575	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → ((𝑤 gcd 𝑀) ∥ 𝑤 ∧ (𝑤 gcd 𝑀) ∥ 𝑀))
43	42	simpld 484	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → (𝑤 gcd 𝑀) ∥ 𝑤)
44	42	simprd 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → (𝑤 gcd 𝑀) ∥ 𝑀)
45	33	simp2d 1166	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑁 ∈ ℕ)
46	45	adantr 468	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → 𝑁 ∈ ℕ)
47	46	nnzd 11747	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → 𝑁 ∈ ℤ)
48		dvdsmul1 15226	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑀 ∥ (𝑀 · 𝑁))
49	40, 47, 48	syl2anc 575	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → 𝑀 ∥ (𝑀 · 𝑁))
50		nnne0 11339	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑀 ∈ ℕ → 𝑀 ≠ 0)
51		simpr 473	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑤 = 0 ∧ 𝑀 = 0) → 𝑀 = 0)
52	51	necon3ai 3003	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑀 ≠ 0 → ¬ (𝑤 = 0 ∧ 𝑀 = 0))
53	35, 50, 52	3syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → ¬ (𝑤 = 0 ∧ 𝑀 = 0))
54		gcdn0cl 15443	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑤 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ ¬ (𝑤 = 0 ∧ 𝑀 = 0)) → (𝑤 gcd 𝑀) ∈ ℕ)
55	32, 40, 53, 54	syl21anc 857	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → (𝑤 gcd 𝑀) ∈ ℕ)
56	55	nnzd 11747	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → (𝑤 gcd 𝑀) ∈ ℤ)
57	35, 46	nnmulcld 11354	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → (𝑀 · 𝑁) ∈ ℕ)
58	57	nnzd 11747	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → (𝑀 · 𝑁) ∈ ℤ)
59		dvdstr 15241	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑤 gcd 𝑀) ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ (𝑀 · 𝑁) ∈ ℤ) → (((𝑤 gcd 𝑀) ∥ 𝑀 ∧ 𝑀 ∥ (𝑀 · 𝑁)) → (𝑤 gcd 𝑀) ∥ (𝑀 · 𝑁)))
60	56, 40, 58, 59	syl3anc 1483	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → (((𝑤 gcd 𝑀) ∥ 𝑀 ∧ 𝑀 ∥ (𝑀 · 𝑁)) → (𝑤 gcd 𝑀) ∥ (𝑀 · 𝑁)))
61	44, 49, 60	mp2and 682	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → (𝑤 gcd 𝑀) ∥ (𝑀 · 𝑁))
62		nnne0 11339	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑀 · 𝑁) ∈ ℕ → (𝑀 · 𝑁) ≠ 0)
63		simpr 473	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑤 = 0 ∧ (𝑀 · 𝑁) = 0) → (𝑀 · 𝑁) = 0)
64	63	necon3ai 3003	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑀 · 𝑁) ≠ 0 → ¬ (𝑤 = 0 ∧ (𝑀 · 𝑁) = 0))
65	57, 62, 64	3syl 18	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → ¬ (𝑤 = 0 ∧ (𝑀 · 𝑁) = 0))
66		dvdslegcd 15445	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑤 gcd 𝑀) ∈ ℤ ∧ 𝑤 ∈ ℤ ∧ (𝑀 · 𝑁) ∈ ℤ) ∧ ¬ (𝑤 = 0 ∧ (𝑀 · 𝑁) = 0)) → (((𝑤 gcd 𝑀) ∥ 𝑤 ∧ (𝑤 gcd 𝑀) ∥ (𝑀 · 𝑁)) → (𝑤 gcd 𝑀) ≤ (𝑤 gcd (𝑀 · 𝑁))))
67	56, 32, 58, 65, 66	syl31anc 1485	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → (((𝑤 gcd 𝑀) ∥ 𝑤 ∧ (𝑤 gcd 𝑀) ∥ (𝑀 · 𝑁)) → (𝑤 gcd 𝑀) ≤ (𝑤 gcd (𝑀 · 𝑁))))
68	43, 61, 67	mp2and 682	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → (𝑤 gcd 𝑀) ≤ (𝑤 gcd (𝑀 · 𝑁)))
69	18	simprbi 486	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ 𝑊 → (𝑤 gcd (𝑀 · 𝑁)) = 1)
70	69	adantl 469	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → (𝑤 gcd (𝑀 · 𝑁)) = 1)
71	68, 70	breqtrd 4870	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → (𝑤 gcd 𝑀) ≤ 1)
72		nnle1eq1 11334	. . . . . . . . . . . . . 14 ⊢ ((𝑤 gcd 𝑀) ∈ ℕ → ((𝑤 gcd 𝑀) ≤ 1 ↔ (𝑤 gcd 𝑀) = 1))
73	55, 72	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → ((𝑤 gcd 𝑀) ≤ 1 ↔ (𝑤 gcd 𝑀) = 1))
74	71, 73	mpbid 223	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → (𝑤 gcd 𝑀) = 1)
75	39, 74	eqtrd 2840	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → ((𝑤 mod 𝑀) gcd 𝑀) = 1)
76		oveq1 6881	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝑤 mod 𝑀) → (𝑦 gcd 𝑀) = ((𝑤 mod 𝑀) gcd 𝑀))
77	76	eqeq1d 2808	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝑤 mod 𝑀) → ((𝑦 gcd 𝑀) = 1 ↔ ((𝑤 mod 𝑀) gcd 𝑀) = 1))
78	77, 1	elrab2 3562	. . . . . . . . . . 11 ⊢ ((𝑤 mod 𝑀) ∈ 𝑈 ↔ ((𝑤 mod 𝑀) ∈ (0..^𝑀) ∧ ((𝑤 mod 𝑀) gcd 𝑀) = 1))
79	37, 75, 78	sylanbrc 574	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → (𝑤 mod 𝑀) ∈ 𝑈)
80		zmodfzo 12917	. . . . . . . . . . . 12 ⊢ ((𝑤 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝑤 mod 𝑁) ∈ (0..^𝑁))
81	32, 46, 80	syl2anc 575	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → (𝑤 mod 𝑁) ∈ (0..^𝑁))
82		modgcd 15472	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ ℤ ∧ 𝑁 ∈ ℕ) → ((𝑤 mod 𝑁) gcd 𝑁) = (𝑤 gcd 𝑁))
83	32, 46, 82	syl2anc 575	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → ((𝑤 mod 𝑁) gcd 𝑁) = (𝑤 gcd 𝑁))
84		gcddvds 15444	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑤 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑤 gcd 𝑁) ∥ 𝑤 ∧ (𝑤 gcd 𝑁) ∥ 𝑁))
85	32, 47, 84	syl2anc 575	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → ((𝑤 gcd 𝑁) ∥ 𝑤 ∧ (𝑤 gcd 𝑁) ∥ 𝑁))
86	85	simpld 484	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → (𝑤 gcd 𝑁) ∥ 𝑤)
87	85	simprd 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → (𝑤 gcd 𝑁) ∥ 𝑁)
88		dvdsmul2 15227	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑁 ∥ (𝑀 · 𝑁))
89	40, 47, 88	syl2anc 575	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → 𝑁 ∥ (𝑀 · 𝑁))
90		nnne0 11339	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ → 𝑁 ≠ 0)
91		simpr 473	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑤 = 0 ∧ 𝑁 = 0) → 𝑁 = 0)
92	91	necon3ai 3003	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ≠ 0 → ¬ (𝑤 = 0 ∧ 𝑁 = 0))
93	46, 90, 92	3syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → ¬ (𝑤 = 0 ∧ 𝑁 = 0))
94		gcdn0cl 15443	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑤 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑤 = 0 ∧ 𝑁 = 0)) → (𝑤 gcd 𝑁) ∈ ℕ)
95	32, 47, 93, 94	syl21anc 857	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → (𝑤 gcd 𝑁) ∈ ℕ)
96	95	nnzd 11747	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → (𝑤 gcd 𝑁) ∈ ℤ)
97		dvdstr 15241	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑤 gcd 𝑁) ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝑀 · 𝑁) ∈ ℤ) → (((𝑤 gcd 𝑁) ∥ 𝑁 ∧ 𝑁 ∥ (𝑀 · 𝑁)) → (𝑤 gcd 𝑁) ∥ (𝑀 · 𝑁)))
98	96, 47, 58, 97	syl3anc 1483	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → (((𝑤 gcd 𝑁) ∥ 𝑁 ∧ 𝑁 ∥ (𝑀 · 𝑁)) → (𝑤 gcd 𝑁) ∥ (𝑀 · 𝑁)))
99	87, 89, 98	mp2and 682	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → (𝑤 gcd 𝑁) ∥ (𝑀 · 𝑁))
100		dvdslegcd 15445	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑤 gcd 𝑁) ∈ ℤ ∧ 𝑤 ∈ ℤ ∧ (𝑀 · 𝑁) ∈ ℤ) ∧ ¬ (𝑤 = 0 ∧ (𝑀 · 𝑁) = 0)) → (((𝑤 gcd 𝑁) ∥ 𝑤 ∧ (𝑤 gcd 𝑁) ∥ (𝑀 · 𝑁)) → (𝑤 gcd 𝑁) ≤ (𝑤 gcd (𝑀 · 𝑁))))
101	96, 32, 58, 65, 100	syl31anc 1485	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → (((𝑤 gcd 𝑁) ∥ 𝑤 ∧ (𝑤 gcd 𝑁) ∥ (𝑀 · 𝑁)) → (𝑤 gcd 𝑁) ≤ (𝑤 gcd (𝑀 · 𝑁))))
102	86, 99, 101	mp2and 682	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → (𝑤 gcd 𝑁) ≤ (𝑤 gcd (𝑀 · 𝑁)))
103	102, 70	breqtrd 4870	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → (𝑤 gcd 𝑁) ≤ 1)
104		nnle1eq1 11334	. . . . . . . . . . . . . 14 ⊢ ((𝑤 gcd 𝑁) ∈ ℕ → ((𝑤 gcd 𝑁) ≤ 1 ↔ (𝑤 gcd 𝑁) = 1))
105	95, 104	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → ((𝑤 gcd 𝑁) ≤ 1 ↔ (𝑤 gcd 𝑁) = 1))
106	103, 105	mpbid 223	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → (𝑤 gcd 𝑁) = 1)
107	83, 106	eqtrd 2840	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → ((𝑤 mod 𝑁) gcd 𝑁) = 1)
108		oveq1 6881	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝑤 mod 𝑁) → (𝑦 gcd 𝑁) = ((𝑤 mod 𝑁) gcd 𝑁))
109	108	eqeq1d 2808	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝑤 mod 𝑁) → ((𝑦 gcd 𝑁) = 1 ↔ ((𝑤 mod 𝑁) gcd 𝑁) = 1))
110	109, 7	elrab2 3562	. . . . . . . . . . 11 ⊢ ((𝑤 mod 𝑁) ∈ 𝑉 ↔ ((𝑤 mod 𝑁) ∈ (0..^𝑁) ∧ ((𝑤 mod 𝑁) gcd 𝑁) = 1))
111	81, 107, 110	sylanbrc 574	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → (𝑤 mod 𝑁) ∈ 𝑉)
112		opelxpi 5348	. . . . . . . . . 10 ⊢ (((𝑤 mod 𝑀) ∈ 𝑈 ∧ (𝑤 mod 𝑁) ∈ 𝑉) → ⟨(𝑤 mod 𝑀), (𝑤 mod 𝑁)⟩ ∈ (𝑈 × 𝑉))
113	79, 111, 112	syl2anc 575	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → ⟨(𝑤 mod 𝑀), (𝑤 mod 𝑁)⟩ ∈ (𝑈 × 𝑉))
114	27, 113	eqeltrd 2885	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → (𝐹‘𝑤) ∈ (𝑈 × 𝑉))
115	114	ralrimiva 3154	. . . . . . 7 ⊢ (𝜑 → ∀𝑤 ∈ 𝑊 (𝐹‘𝑤) ∈ (𝑈 × 𝑉))
116		crth.2	. . . . . . . . . 10 ⊢ 𝑇 = ((0..^𝑀) × (0..^𝑁))
117	28, 116, 23, 33	crth 15700	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:𝑆–1-1-onto→𝑇)
118		f1ofn 6354	. . . . . . . . 9 ⊢ (𝐹:𝑆–1-1-onto→𝑇 → 𝐹 Fn 𝑆)
119		fnfun 6199	. . . . . . . . 9 ⊢ (𝐹 Fn 𝑆 → Fun 𝐹)
120	117, 118, 119	3syl 18	. . . . . . . 8 ⊢ (𝜑 → Fun 𝐹)
121		ssrab2 3884	. . . . . . . . . 10 ⊢ {𝑦 ∈ 𝑆 ∣ (𝑦 gcd (𝑀 · 𝑁)) = 1} ⊆ 𝑆
122	17, 121	eqsstri 3832	. . . . . . . . 9 ⊢ 𝑊 ⊆ 𝑆
123		fndm 6201	. . . . . . . . . 10 ⊢ (𝐹 Fn 𝑆 → dom 𝐹 = 𝑆)
124	117, 118, 123	3syl 18	. . . . . . . . 9 ⊢ (𝜑 → dom 𝐹 = 𝑆)
125	122, 124	syl5sseqr 3851	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ⊆ dom 𝐹)
126		funimass4 6468	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ 𝑊 ⊆ dom 𝐹) → ((𝐹 “ 𝑊) ⊆ (𝑈 × 𝑉) ↔ ∀𝑤 ∈ 𝑊 (𝐹‘𝑤) ∈ (𝑈 × 𝑉)))
127	120, 125, 126	syl2anc 575	. . . . . . 7 ⊢ (𝜑 → ((𝐹 “ 𝑊) ⊆ (𝑈 × 𝑉) ↔ ∀𝑤 ∈ 𝑊 (𝐹‘𝑤) ∈ (𝑈 × 𝑉)))
128	115, 127	mpbird 248	. . . . . 6 ⊢ (𝜑 → (𝐹 “ 𝑊) ⊆ (𝑈 × 𝑉))
129	1, 3	eqsstri 3832	. . . . . . . . . . 11 ⊢ 𝑈 ⊆ (0..^𝑀)
130	7, 9	eqsstri 3832	. . . . . . . . . . 11 ⊢ 𝑉 ⊆ (0..^𝑁)
131		xpss12 5326	. . . . . . . . . . 11 ⊢ ((𝑈 ⊆ (0..^𝑀) ∧ 𝑉 ⊆ (0..^𝑁)) → (𝑈 × 𝑉) ⊆ ((0..^𝑀) × (0..^𝑁)))
132	129, 130, 131	mp2an 675	. . . . . . . . . 10 ⊢ (𝑈 × 𝑉) ⊆ ((0..^𝑀) × (0..^𝑁))
133	132, 116	sseqtr4i 3835	. . . . . . . . 9 ⊢ (𝑈 × 𝑉) ⊆ 𝑇
134	133	sseli 3794	. . . . . . . 8 ⊢ (𝑧 ∈ (𝑈 × 𝑉) → 𝑧 ∈ 𝑇)
135		f1ocnvfv2 6757	. . . . . . . 8 ⊢ ((𝐹:𝑆–1-1-onto→𝑇 ∧ 𝑧 ∈ 𝑇) → (𝐹‘(^◡𝐹‘𝑧)) = 𝑧)
136	117, 134, 135	syl2an 585	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑈 × 𝑉)) → (𝐹‘(^◡𝐹‘𝑧)) = 𝑧)
137		f1ocnv 6365	. . . . . . . . . . 11 ⊢ (𝐹:𝑆–1-1-onto→𝑇 → ^◡𝐹:𝑇–1-1-onto→𝑆)
138		f1of 6353	. . . . . . . . . . 11 ⊢ (^◡𝐹:𝑇–1-1-onto→𝑆 → ^◡𝐹:𝑇⟶𝑆)
139	117, 137, 138	3syl 18	. . . . . . . . . 10 ⊢ (𝜑 → ^◡𝐹:𝑇⟶𝑆)
140		ffvelrn 6579	. . . . . . . . . 10 ⊢ ((^◡𝐹:𝑇⟶𝑆 ∧ 𝑧 ∈ 𝑇) → (^◡𝐹‘𝑧) ∈ 𝑆)
141	139, 134, 140	syl2an 585	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑈 × 𝑉)) → (^◡𝐹‘𝑧) ∈ 𝑆)
142	141, 28	syl6eleq 2895	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑈 × 𝑉)) → (^◡𝐹‘𝑧) ∈ (0..^(𝑀 · 𝑁)))
143		elfzoelz 12694	. . . . . . . . . . . . 13 ⊢ ((^◡𝐹‘𝑧) ∈ (0..^(𝑀 · 𝑁)) → (^◡𝐹‘𝑧) ∈ ℤ)
144	142, 143	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑈 × 𝑉)) → (^◡𝐹‘𝑧) ∈ ℤ)
145	34	adantr 468	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑈 × 𝑉)) → 𝑀 ∈ ℕ)
146		modgcd 15472	. . . . . . . . . . . 12 ⊢ (((^◡𝐹‘𝑧) ∈ ℤ ∧ 𝑀 ∈ ℕ) → (((^◡𝐹‘𝑧) mod 𝑀) gcd 𝑀) = ((^◡𝐹‘𝑧) gcd 𝑀))
147	144, 145, 146	syl2anc 575	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑈 × 𝑉)) → (((^◡𝐹‘𝑧) mod 𝑀) gcd 𝑀) = ((^◡𝐹‘𝑧) gcd 𝑀))
148		oveq1 6881	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑤 = (^◡𝐹‘𝑧) → (𝑤 mod 𝑀) = ((^◡𝐹‘𝑧) mod 𝑀))
149		oveq1 6881	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑤 = (^◡𝐹‘𝑧) → (𝑤 mod 𝑁) = ((^◡𝐹‘𝑧) mod 𝑁))
150	148, 149	opeq12d 4603	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 = (^◡𝐹‘𝑧) → ⟨(𝑤 mod 𝑀), (𝑤 mod 𝑁)⟩ = ⟨((^◡𝐹‘𝑧) mod 𝑀), ((^◡𝐹‘𝑧) mod 𝑁)⟩)
151	22	cbvmptv 4944	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ 𝑆 ↦ ⟨(𝑥 mod 𝑀), (𝑥 mod 𝑁)⟩) = (𝑤 ∈ 𝑆 ↦ ⟨(𝑤 mod 𝑀), (𝑤 mod 𝑁)⟩)
152	23, 151	eqtri 2828	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝐹 = (𝑤 ∈ 𝑆 ↦ ⟨(𝑤 mod 𝑀), (𝑤 mod 𝑁)⟩)
153		opex 5122	. . . . . . . . . . . . . . . . . . 19 ⊢ ⟨((^◡𝐹‘𝑧) mod 𝑀), ((^◡𝐹‘𝑧) mod 𝑁)⟩ ∈ V
154	150, 152, 153	fvmpt 6503	. . . . . . . . . . . . . . . . . 18 ⊢ ((^◡𝐹‘𝑧) ∈ 𝑆 → (𝐹‘(^◡𝐹‘𝑧)) = ⟨((^◡𝐹‘𝑧) mod 𝑀), ((^◡𝐹‘𝑧) mod 𝑁)⟩)
155	141, 154	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑈 × 𝑉)) → (𝐹‘(^◡𝐹‘𝑧)) = ⟨((^◡𝐹‘𝑧) mod 𝑀), ((^◡𝐹‘𝑧) mod 𝑁)⟩)
156	136, 155	eqtr3d 2842	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑈 × 𝑉)) → 𝑧 = ⟨((^◡𝐹‘𝑧) mod 𝑀), ((^◡𝐹‘𝑧) mod 𝑁)⟩)
157		simpr 473	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑈 × 𝑉)) → 𝑧 ∈ (𝑈 × 𝑉))
158	156, 157	eqeltrrd 2886	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑈 × 𝑉)) → ⟨((^◡𝐹‘𝑧) mod 𝑀), ((^◡𝐹‘𝑧) mod 𝑁)⟩ ∈ (𝑈 × 𝑉))
159		opelxp 5346	. . . . . . . . . . . . . . 15 ⊢ (⟨((^◡𝐹‘𝑧) mod 𝑀), ((^◡𝐹‘𝑧) mod 𝑁)⟩ ∈ (𝑈 × 𝑉) ↔ (((^◡𝐹‘𝑧) mod 𝑀) ∈ 𝑈 ∧ ((^◡𝐹‘𝑧) mod 𝑁) ∈ 𝑉))
160	158, 159	sylib 209	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑈 × 𝑉)) → (((^◡𝐹‘𝑧) mod 𝑀) ∈ 𝑈 ∧ ((^◡𝐹‘𝑧) mod 𝑁) ∈ 𝑉))
161	160	simpld 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑈 × 𝑉)) → ((^◡𝐹‘𝑧) mod 𝑀) ∈ 𝑈)
162		oveq1 6881	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = ((^◡𝐹‘𝑧) mod 𝑀) → (𝑦 gcd 𝑀) = (((^◡𝐹‘𝑧) mod 𝑀) gcd 𝑀))
163	162	eqeq1d 2808	. . . . . . . . . . . . . 14 ⊢ (𝑦 = ((^◡𝐹‘𝑧) mod 𝑀) → ((𝑦 gcd 𝑀) = 1 ↔ (((^◡𝐹‘𝑧) mod 𝑀) gcd 𝑀) = 1))
164	163, 1	elrab2 3562	. . . . . . . . . . . . 13 ⊢ (((^◡𝐹‘𝑧) mod 𝑀) ∈ 𝑈 ↔ (((^◡𝐹‘𝑧) mod 𝑀) ∈ (0..^𝑀) ∧ (((^◡𝐹‘𝑧) mod 𝑀) gcd 𝑀) = 1))
165	161, 164	sylib 209	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑈 × 𝑉)) → (((^◡𝐹‘𝑧) mod 𝑀) ∈ (0..^𝑀) ∧ (((^◡𝐹‘𝑧) mod 𝑀) gcd 𝑀) = 1))
166	165	simprd 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑈 × 𝑉)) → (((^◡𝐹‘𝑧) mod 𝑀) gcd 𝑀) = 1)
167	147, 166	eqtr3d 2842	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑈 × 𝑉)) → ((^◡𝐹‘𝑧) gcd 𝑀) = 1)
168	45	adantr 468	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑈 × 𝑉)) → 𝑁 ∈ ℕ)
169		modgcd 15472	. . . . . . . . . . . 12 ⊢ (((^◡𝐹‘𝑧) ∈ ℤ ∧ 𝑁 ∈ ℕ) → (((^◡𝐹‘𝑧) mod 𝑁) gcd 𝑁) = ((^◡𝐹‘𝑧) gcd 𝑁))
170	144, 168, 169	syl2anc 575	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑈 × 𝑉)) → (((^◡𝐹‘𝑧) mod 𝑁) gcd 𝑁) = ((^◡𝐹‘𝑧) gcd 𝑁))
171	160	simprd 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑈 × 𝑉)) → ((^◡𝐹‘𝑧) mod 𝑁) ∈ 𝑉)
172		oveq1 6881	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = ((^◡𝐹‘𝑧) mod 𝑁) → (𝑦 gcd 𝑁) = (((^◡𝐹‘𝑧) mod 𝑁) gcd 𝑁))
173	172	eqeq1d 2808	. . . . . . . . . . . . . 14 ⊢ (𝑦 = ((^◡𝐹‘𝑧) mod 𝑁) → ((𝑦 gcd 𝑁) = 1 ↔ (((^◡𝐹‘𝑧) mod 𝑁) gcd 𝑁) = 1))
174	173, 7	elrab2 3562	. . . . . . . . . . . . 13 ⊢ (((^◡𝐹‘𝑧) mod 𝑁) ∈ 𝑉 ↔ (((^◡𝐹‘𝑧) mod 𝑁) ∈ (0..^𝑁) ∧ (((^◡𝐹‘𝑧) mod 𝑁) gcd 𝑁) = 1))
175	171, 174	sylib 209	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑈 × 𝑉)) → (((^◡𝐹‘𝑧) mod 𝑁) ∈ (0..^𝑁) ∧ (((^◡𝐹‘𝑧) mod 𝑁) gcd 𝑁) = 1))
176	175	simprd 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑈 × 𝑉)) → (((^◡𝐹‘𝑧) mod 𝑁) gcd 𝑁) = 1)
177	170, 176	eqtr3d 2842	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑈 × 𝑉)) → ((^◡𝐹‘𝑧) gcd 𝑁) = 1)
178	34	nnzd 11747	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑀 ∈ ℤ)
179	178	adantr 468	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑈 × 𝑉)) → 𝑀 ∈ ℤ)
180	45	nnzd 11747	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℤ)
181	180	adantr 468	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑈 × 𝑉)) → 𝑁 ∈ ℤ)
182		rpmul 15591	. . . . . . . . . . 11 ⊢ (((^◡𝐹‘𝑧) ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((((^◡𝐹‘𝑧) gcd 𝑀) = 1 ∧ ((^◡𝐹‘𝑧) gcd 𝑁) = 1) → ((^◡𝐹‘𝑧) gcd (𝑀 · 𝑁)) = 1))
183	144, 179, 181, 182	syl3anc 1483	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑈 × 𝑉)) → ((((^◡𝐹‘𝑧) gcd 𝑀) = 1 ∧ ((^◡𝐹‘𝑧) gcd 𝑁) = 1) → ((^◡𝐹‘𝑧) gcd (𝑀 · 𝑁)) = 1))
184	167, 177, 183	mp2and 682	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑈 × 𝑉)) → ((^◡𝐹‘𝑧) gcd (𝑀 · 𝑁)) = 1)
185		oveq1 6881	. . . . . . . . . . 11 ⊢ (𝑦 = (^◡𝐹‘𝑧) → (𝑦 gcd (𝑀 · 𝑁)) = ((^◡𝐹‘𝑧) gcd (𝑀 · 𝑁)))
186	185	eqeq1d 2808	. . . . . . . . . 10 ⊢ (𝑦 = (^◡𝐹‘𝑧) → ((𝑦 gcd (𝑀 · 𝑁)) = 1 ↔ ((^◡𝐹‘𝑧) gcd (𝑀 · 𝑁)) = 1))
187	186, 17	elrab2 3562	. . . . . . . . 9 ⊢ ((^◡𝐹‘𝑧) ∈ 𝑊 ↔ ((^◡𝐹‘𝑧) ∈ 𝑆 ∧ ((^◡𝐹‘𝑧) gcd (𝑀 · 𝑁)) = 1))
188	141, 184, 187	sylanbrc 574	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑈 × 𝑉)) → (^◡𝐹‘𝑧) ∈ 𝑊)
189		funfvima2 6718	. . . . . . . . . 10 ⊢ ((Fun 𝐹 ∧ 𝑊 ⊆ dom 𝐹) → ((^◡𝐹‘𝑧) ∈ 𝑊 → (𝐹‘(^◡𝐹‘𝑧)) ∈ (𝐹 “ 𝑊)))
190	120, 125, 189	syl2anc 575	. . . . . . . . 9 ⊢ (𝜑 → ((^◡𝐹‘𝑧) ∈ 𝑊 → (𝐹‘(^◡𝐹‘𝑧)) ∈ (𝐹 “ 𝑊)))
191	190	imp 395	. . . . . . . 8 ⊢ ((𝜑 ∧ (^◡𝐹‘𝑧) ∈ 𝑊) → (𝐹‘(^◡𝐹‘𝑧)) ∈ (𝐹 “ 𝑊))
192	188, 191	syldan 581	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑈 × 𝑉)) → (𝐹‘(^◡𝐹‘𝑧)) ∈ (𝐹 “ 𝑊))
193	136, 192	eqeltrrd 2886	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑈 × 𝑉)) → 𝑧 ∈ (𝐹 “ 𝑊))
194	128, 193	eqelssd 3819	. . . . 5 ⊢ (𝜑 → (𝐹 “ 𝑊) = (𝑈 × 𝑉))
195		f1of1 6352	. . . . . . 7 ⊢ (𝐹:𝑆–1-1-onto→𝑇 → 𝐹:𝑆–1-1→𝑇)
196	117, 195	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐹:𝑆–1-1→𝑇)
197		fzofi 12997	. . . . . . . . . 10 ⊢ (0..^(𝑀 · 𝑁)) ∈ Fin
198	28, 197	eqeltri 2881	. . . . . . . . 9 ⊢ 𝑆 ∈ Fin
199		ssfi 8419	. . . . . . . . 9 ⊢ ((𝑆 ∈ Fin ∧ 𝑊 ⊆ 𝑆) → 𝑊 ∈ Fin)
200	198, 122, 199	mp2an 675	. . . . . . . 8 ⊢ 𝑊 ∈ Fin
201	200	elexi 3407	. . . . . . 7 ⊢ 𝑊 ∈ V
202	201	f1imaen 8254	. . . . . 6 ⊢ ((𝐹:𝑆–1-1→𝑇 ∧ 𝑊 ⊆ 𝑆) → (𝐹 “ 𝑊) ≈ 𝑊)
203	196, 122, 202	sylancl 576	. . . . 5 ⊢ (𝜑 → (𝐹 “ 𝑊) ≈ 𝑊)
204	194, 203	eqbrtrrd 4868	. . . 4 ⊢ (𝜑 → (𝑈 × 𝑉) ≈ 𝑊)
205		xpfi 8470	. . . . . 6 ⊢ ((𝑈 ∈ Fin ∧ 𝑉 ∈ Fin) → (𝑈 × 𝑉) ∈ Fin)
206	6, 12, 205	mp2an 675	. . . . 5 ⊢ (𝑈 × 𝑉) ∈ Fin
207		hashen 13355	. . . . 5 ⊢ (((𝑈 × 𝑉) ∈ Fin ∧ 𝑊 ∈ Fin) → ((♯‘(𝑈 × 𝑉)) = (♯‘𝑊) ↔ (𝑈 × 𝑉) ≈ 𝑊))
208	206, 200, 207	mp2an 675	. . . 4 ⊢ ((♯‘(𝑈 × 𝑉)) = (♯‘𝑊) ↔ (𝑈 × 𝑉) ≈ 𝑊)
209	204, 208	sylibr 225	. . 3 ⊢ (𝜑 → (♯‘(𝑈 × 𝑉)) = (♯‘𝑊))
210	14, 209	syl5reqr 2855	. 2 ⊢ (𝜑 → (♯‘𝑊) = ((♯‘𝑈) · (♯‘𝑉)))
211	34, 45	nnmulcld 11354	. . 3 ⊢ (𝜑 → (𝑀 · 𝑁) ∈ ℕ)
212		dfphi2 15696	. . . 4 ⊢ ((𝑀 · 𝑁) ∈ ℕ → (ϕ‘(𝑀 · 𝑁)) = (♯‘{𝑦 ∈ (0..^(𝑀 · 𝑁)) ∣ (𝑦 gcd (𝑀 · 𝑁)) = 1}))
213		rabeq 3382	. . . . . . 7 ⊢ (𝑆 = (0..^(𝑀 · 𝑁)) → {𝑦 ∈ 𝑆 ∣ (𝑦 gcd (𝑀 · 𝑁)) = 1} = {𝑦 ∈ (0..^(𝑀 · 𝑁)) ∣ (𝑦 gcd (𝑀 · 𝑁)) = 1})
214	28, 213	ax-mp 5	. . . . . 6 ⊢ {𝑦 ∈ 𝑆 ∣ (𝑦 gcd (𝑀 · 𝑁)) = 1} = {𝑦 ∈ (0..^(𝑀 · 𝑁)) ∣ (𝑦 gcd (𝑀 · 𝑁)) = 1}
215	17, 214	eqtri 2828	. . . . 5 ⊢ 𝑊 = {𝑦 ∈ (0..^(𝑀 · 𝑁)) ∣ (𝑦 gcd (𝑀 · 𝑁)) = 1}
216	215	fveq2i 6411	. . . 4 ⊢ (♯‘𝑊) = (♯‘{𝑦 ∈ (0..^(𝑀 · 𝑁)) ∣ (𝑦 gcd (𝑀 · 𝑁)) = 1})
217	212, 216	syl6eqr 2858	. . 3 ⊢ ((𝑀 · 𝑁) ∈ ℕ → (ϕ‘(𝑀 · 𝑁)) = (♯‘𝑊))
218	211, 217	syl 17	. 2 ⊢ (𝜑 → (ϕ‘(𝑀 · 𝑁)) = (♯‘𝑊))
219		dfphi2 15696	. . . . 5 ⊢ (𝑀 ∈ ℕ → (ϕ‘𝑀) = (♯‘{𝑦 ∈ (0..^𝑀) ∣ (𝑦 gcd 𝑀) = 1}))
220	1	fveq2i 6411	. . . . 5 ⊢ (♯‘𝑈) = (♯‘{𝑦 ∈ (0..^𝑀) ∣ (𝑦 gcd 𝑀) = 1})
221	219, 220	syl6eqr 2858	. . . 4 ⊢ (𝑀 ∈ ℕ → (ϕ‘𝑀) = (♯‘𝑈))
222	34, 221	syl 17	. . 3 ⊢ (𝜑 → (ϕ‘𝑀) = (♯‘𝑈))
223		dfphi2 15696	. . . . 5 ⊢ (𝑁 ∈ ℕ → (ϕ‘𝑁) = (♯‘{𝑦 ∈ (0..^𝑁) ∣ (𝑦 gcd 𝑁) = 1}))
224	7	fveq2i 6411	. . . . 5 ⊢ (♯‘𝑉) = (♯‘{𝑦 ∈ (0..^𝑁) ∣ (𝑦 gcd 𝑁) = 1})
225	223, 224	syl6eqr 2858	. . . 4 ⊢ (𝑁 ∈ ℕ → (ϕ‘𝑁) = (♯‘𝑉))
226	45, 225	syl 17	. . 3 ⊢ (𝜑 → (ϕ‘𝑁) = (♯‘𝑉))
227	222, 226	oveq12d 6892	. 2 ⊢ (𝜑 → ((ϕ‘𝑀) · (ϕ‘𝑁)) = ((♯‘𝑈) · (♯‘𝑉)))
228	210, 218, 227	3eqtr4d 2850	1 ⊢ (𝜑 → (ϕ‘(𝑀 · 𝑁)) = ((ϕ‘𝑀) · (ϕ‘𝑁)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 197 ∧ wa 384 ∧ w3a 1100 = wceq 1637 ∈ wcel 2156 ≠ wne 2978 ∀wral 3096 {crab 3100 ⊆ wss 3769 ⟨cop 4376 class class class wbr 4844 ↦ cmpt 4923 × cxp 5309 ^◡ccnv 5310 dom cdm 5311 “ cima 5314 Fun wfun 6095 Fn wfn 6096 ⟶wf 6097 –1-1→wf1 6098 –1-1-onto→wf1o 6100 ‘cfv 6101 (class class class)co 6874 ≈ cen 8189 Fincfn 8192 0cc0 10221 1c1 10222 · cmul 10226 ≤ cle 10360 ℕcn 11305 ℤcz 11643 ..^cfzo 12689 mod cmo 12892 ♯chash 13337 ∥ cdvds 15203 gcd cgcd 15435 ϕcphi 15686

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1877 ax-4 1894 ax-5 2001 ax-6 2068 ax-7 2104 ax-8 2158 ax-9 2165 ax-10 2185 ax-11 2201 ax-12 2214 ax-13 2420 ax-ext 2784 ax-rep 4964 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5096 ax-un 7179 ax-cnex 10277 ax-resscn 10278 ax-1cn 10279 ax-icn 10280 ax-addcl 10281 ax-addrcl 10282 ax-mulcl 10283 ax-mulrcl 10284 ax-mulcom 10285 ax-addass 10286 ax-mulass 10287 ax-distr 10288 ax-i2m1 10289 ax-1ne0 10290 ax-1rid 10291 ax-rnegex 10292 ax-rrecex 10293 ax-cnre 10294 ax-pre-lttri 10295 ax-pre-lttrn 10296 ax-pre-ltadd 10297 ax-pre-mulgt0 10298 ax-pre-sup 10299

This theorem depends on definitions: df-bi 198 df-an 385 df-or 866 df-3or 1101 df-3an 1102 df-tru 1641 df-ex 1860 df-nf 1864 df-sb 2061 df-eu 2634 df-mo 2635 df-clab 2793 df-cleq 2799 df-clel 2802 df-nfc 2937 df-ne 2979 df-nel 3082 df-ral 3101 df-rex 3102 df-reu 3103 df-rmo 3104 df-rab 3105 df-v 3393 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4117 df-if 4280 df-pw 4353 df-sn 4371 df-pr 4373 df-tp 4375 df-op 4377 df-uni 4631 df-int 4670 df-iun 4714 df-br 4845 df-opab 4907 df-mpt 4924 df-tr 4947 df-id 5219 df-eprel 5224 df-po 5232 df-so 5233 df-fr 5270 df-we 5272 df-xp 5317 df-rel 5318 df-cnv 5319 df-co 5320 df-dm 5321 df-rn 5322 df-res 5323 df-ima 5324 df-pred 5893 df-ord 5939 df-on 5940 df-lim 5941 df-suc 5942 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-riota 6835 df-ov 6877 df-oprab 6878 df-mpt2 6879 df-om 7296 df-1st 7398 df-2nd 7399 df-wrecs 7642 df-recs 7704 df-rdg 7742 df-1o 7796 df-oadd 7800 df-er 7979 df-map 8094 df-en 8193 df-dom 8194 df-sdom 8195 df-fin 8196 df-sup 8587 df-inf 8588 df-card 9048 df-cda 9275 df-pnf 10361 df-mnf 10362 df-xr 10363 df-ltxr 10364 df-le 10365 df-sub 10553 df-neg 10554 df-div 10970 df-nn 11306 df-2 11364 df-3 11365 df-n0 11560 df-xnn0 11630 df-z 11644 df-uz 11905 df-rp 12047 df-fz 12550 df-fzo 12690 df-fl 12817 df-mod 12893 df-seq 13025 df-exp 13084 df-hash 13338 df-cj 14062 df-re 14063 df-im 14064 df-sqrt 14198 df-abs 14199 df-dvds 15204 df-gcd 15436 df-phi 15688

This theorem is referenced by: phimul 15702

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