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Theorem phimullem 16118

Description: Lemma for phimul 16119. (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 13345	. . . . . 6 ⊢ (0..^𝑀) ∈ Fin
3		ssrab2 4058	. . . . . 6 ⊢ {𝑦 ∈ (0..^𝑀) ∣ (𝑦 gcd 𝑀) = 1} ⊆ (0..^𝑀)
4		ssfi 8740	. . . . . 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 690	. . . . 5 ⊢ {𝑦 ∈ (0..^𝑀) ∣ (𝑦 gcd 𝑀) = 1} ∈ Fin
6	1, 5	eqeltri 2911	. . . 4 ⊢ 𝑈 ∈ Fin
7		phimul.5	. . . . 5 ⊢ 𝑉 = {𝑦 ∈ (0..^𝑁) ∣ (𝑦 gcd 𝑁) = 1}
8		fzofi 13345	. . . . . 6 ⊢ (0..^𝑁) ∈ Fin
9		ssrab2 4058	. . . . . 6 ⊢ {𝑦 ∈ (0..^𝑁) ∣ (𝑦 gcd 𝑁) = 1} ⊆ (0..^𝑁)
10		ssfi 8740	. . . . . 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 690	. . . . 5 ⊢ {𝑦 ∈ (0..^𝑁) ∣ (𝑦 gcd 𝑁) = 1} ∈ Fin
12	7, 11	eqeltri 2911	. . . 4 ⊢ 𝑉 ∈ Fin
13		hashxp 13798	. . . 4 ⊢ ((𝑈 ∈ Fin ∧ 𝑉 ∈ Fin) → (♯‘(𝑈 × 𝑉)) = ((♯‘𝑈) · (♯‘𝑉)))
14	6, 12, 13	mp2an 690	. . 3 ⊢ (♯‘(𝑈 × 𝑉)) = ((♯‘𝑈) · (♯‘𝑉))
15		oveq1 7165	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑤 → (𝑦 gcd (𝑀 · 𝑁)) = (𝑤 gcd (𝑀 · 𝑁)))
16	15	eqeq1d 2825	. . . . . . . . . . . . 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 3685	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ 𝑊 ↔ (𝑤 ∈ 𝑆 ∧ (𝑤 gcd (𝑀 · 𝑁)) = 1))
19	18	simplbi 500	. . . . . . . . . . 11 ⊢ (𝑤 ∈ 𝑊 → 𝑤 ∈ 𝑆)
20		oveq1 7165	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑤 → (𝑥 mod 𝑀) = (𝑤 mod 𝑀))
21		oveq1 7165	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑤 → (𝑥 mod 𝑁) = (𝑤 mod 𝑁))
22	20, 21	opeq12d 4813	. . . . . . . . . . . 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 5358	. . . . . . . . . . . 12 ⊢ ⟨(𝑤 mod 𝑀), (𝑤 mod 𝑁)⟩ ∈ V
25	22, 23, 24	fvmpt 6770	. . . . . . . . . . 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 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → (𝐹‘𝑤) = ⟨(𝑤 mod 𝑀), (𝑤 mod 𝑁)⟩)
28		crth.1	. . . . . . . . . . . . . . 15 ⊢ 𝑆 = (0..^(𝑀 · 𝑁))
29	19, 28	eleqtrdi 2925	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ 𝑊 → 𝑤 ∈ (0..^(𝑀 · 𝑁)))
30	29	adantl 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → 𝑤 ∈ (0..^(𝑀 · 𝑁)))
31		elfzoelz 13041	. . . . . . . . . . . . 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 1138	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑀 ∈ ℕ)
35	34	adantr 483	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → 𝑀 ∈ ℕ)
36		zmodfzo 13265	. . . . . . . . . . . 12 ⊢ ((𝑤 ∈ ℤ ∧ 𝑀 ∈ ℕ) → (𝑤 mod 𝑀) ∈ (0..^𝑀))
37	32, 35, 36	syl2anc 586	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → (𝑤 mod 𝑀) ∈ (0..^𝑀))
38		modgcd 15882	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ ℤ ∧ 𝑀 ∈ ℕ) → ((𝑤 mod 𝑀) gcd 𝑀) = (𝑤 gcd 𝑀))
39	32, 35, 38	syl2anc 586	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → ((𝑤 mod 𝑀) gcd 𝑀) = (𝑤 gcd 𝑀))
40	35	nnzd 12089	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → 𝑀 ∈ ℤ)
41		gcddvds 15854	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑤 ∈ ℤ ∧ 𝑀 ∈ ℤ) → ((𝑤 gcd 𝑀) ∥ 𝑤 ∧ (𝑤 gcd 𝑀) ∥ 𝑀))
42	32, 40, 41	syl2anc 586	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → ((𝑤 gcd 𝑀) ∥ 𝑤 ∧ (𝑤 gcd 𝑀) ∥ 𝑀))
43	42	simpld 497	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → (𝑤 gcd 𝑀) ∥ 𝑤)
44		nnne0 11674	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑀 ∈ ℕ → 𝑀 ≠ 0)
45		simpr 487	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑤 = 0 ∧ 𝑀 = 0) → 𝑀 = 0)
46	45	necon3ai 3043	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑀 ≠ 0 → ¬ (𝑤 = 0 ∧ 𝑀 = 0))
47	35, 44, 46	3syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → ¬ (𝑤 = 0 ∧ 𝑀 = 0))
48		gcdn0cl 15853	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑤 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ ¬ (𝑤 = 0 ∧ 𝑀 = 0)) → (𝑤 gcd 𝑀) ∈ ℕ)
49	32, 40, 47, 48	syl21anc 835	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → (𝑤 gcd 𝑀) ∈ ℕ)
50	49	nnzd 12089	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → (𝑤 gcd 𝑀) ∈ ℤ)
51	33	simp2d 1139	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑁 ∈ ℕ)
52	51	adantr 483	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → 𝑁 ∈ ℕ)
53	52	nnzd 12089	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → 𝑁 ∈ ℤ)
54	42	simprd 498	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → (𝑤 gcd 𝑀) ∥ 𝑀)
55	50, 40, 53, 54	dvdsmultr1d 15650	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → (𝑤 gcd 𝑀) ∥ (𝑀 · 𝑁))
56	35, 52	nnmulcld 11693	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → (𝑀 · 𝑁) ∈ ℕ)
57	56	nnzd 12089	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → (𝑀 · 𝑁) ∈ ℤ)
58		nnne0 11674	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑀 · 𝑁) ∈ ℕ → (𝑀 · 𝑁) ≠ 0)
59		simpr 487	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑤 = 0 ∧ (𝑀 · 𝑁) = 0) → (𝑀 · 𝑁) = 0)
60	59	necon3ai 3043	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑀 · 𝑁) ≠ 0 → ¬ (𝑤 = 0 ∧ (𝑀 · 𝑁) = 0))
61	56, 58, 60	3syl 18	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → ¬ (𝑤 = 0 ∧ (𝑀 · 𝑁) = 0))
62		dvdslegcd 15855	. . . . . . . . . . . . . . . 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))))
63	50, 32, 57, 61, 62	syl31anc 1369	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → (((𝑤 gcd 𝑀) ∥ 𝑤 ∧ (𝑤 gcd 𝑀) ∥ (𝑀 · 𝑁)) → (𝑤 gcd 𝑀) ≤ (𝑤 gcd (𝑀 · 𝑁))))
64	43, 55, 63	mp2and 697	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → (𝑤 gcd 𝑀) ≤ (𝑤 gcd (𝑀 · 𝑁)))
65	18	simprbi 499	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ 𝑊 → (𝑤 gcd (𝑀 · 𝑁)) = 1)
66	65	adantl 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → (𝑤 gcd (𝑀 · 𝑁)) = 1)
67	64, 66	breqtrd 5094	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → (𝑤 gcd 𝑀) ≤ 1)
68		nnle1eq1 11670	. . . . . . . . . . . . . 14 ⊢ ((𝑤 gcd 𝑀) ∈ ℕ → ((𝑤 gcd 𝑀) ≤ 1 ↔ (𝑤 gcd 𝑀) = 1))
69	49, 68	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → ((𝑤 gcd 𝑀) ≤ 1 ↔ (𝑤 gcd 𝑀) = 1))
70	67, 69	mpbid 234	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → (𝑤 gcd 𝑀) = 1)
71	39, 70	eqtrd 2858	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → ((𝑤 mod 𝑀) gcd 𝑀) = 1)
72		oveq1 7165	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝑤 mod 𝑀) → (𝑦 gcd 𝑀) = ((𝑤 mod 𝑀) gcd 𝑀))
73	72	eqeq1d 2825	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝑤 mod 𝑀) → ((𝑦 gcd 𝑀) = 1 ↔ ((𝑤 mod 𝑀) gcd 𝑀) = 1))
74	73, 1	elrab2 3685	. . . . . . . . . . 11 ⊢ ((𝑤 mod 𝑀) ∈ 𝑈 ↔ ((𝑤 mod 𝑀) ∈ (0..^𝑀) ∧ ((𝑤 mod 𝑀) gcd 𝑀) = 1))
75	37, 71, 74	sylanbrc 585	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → (𝑤 mod 𝑀) ∈ 𝑈)
76		zmodfzo 13265	. . . . . . . . . . . 12 ⊢ ((𝑤 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝑤 mod 𝑁) ∈ (0..^𝑁))
77	32, 52, 76	syl2anc 586	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → (𝑤 mod 𝑁) ∈ (0..^𝑁))
78		modgcd 15882	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ ℤ ∧ 𝑁 ∈ ℕ) → ((𝑤 mod 𝑁) gcd 𝑁) = (𝑤 gcd 𝑁))
79	32, 52, 78	syl2anc 586	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → ((𝑤 mod 𝑁) gcd 𝑁) = (𝑤 gcd 𝑁))
80		gcddvds 15854	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑤 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑤 gcd 𝑁) ∥ 𝑤 ∧ (𝑤 gcd 𝑁) ∥ 𝑁))
81	32, 53, 80	syl2anc 586	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → ((𝑤 gcd 𝑁) ∥ 𝑤 ∧ (𝑤 gcd 𝑁) ∥ 𝑁))
82	81	simpld 497	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → (𝑤 gcd 𝑁) ∥ 𝑤)
83	81	simprd 498	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → (𝑤 gcd 𝑁) ∥ 𝑁)
84		dvdsmul2 15634	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑁 ∥ (𝑀 · 𝑁))
85	40, 53, 84	syl2anc 586	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → 𝑁 ∥ (𝑀 · 𝑁))
86		nnne0 11674	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ → 𝑁 ≠ 0)
87		simpr 487	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑤 = 0 ∧ 𝑁 = 0) → 𝑁 = 0)
88	87	necon3ai 3043	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ≠ 0 → ¬ (𝑤 = 0 ∧ 𝑁 = 0))
89	52, 86, 88	3syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → ¬ (𝑤 = 0 ∧ 𝑁 = 0))
90		gcdn0cl 15853	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑤 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑤 = 0 ∧ 𝑁 = 0)) → (𝑤 gcd 𝑁) ∈ ℕ)
91	32, 53, 89, 90	syl21anc 835	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → (𝑤 gcd 𝑁) ∈ ℕ)
92	91	nnzd 12089	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → (𝑤 gcd 𝑁) ∈ ℤ)
93		dvdstr 15648	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑤 gcd 𝑁) ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝑀 · 𝑁) ∈ ℤ) → (((𝑤 gcd 𝑁) ∥ 𝑁 ∧ 𝑁 ∥ (𝑀 · 𝑁)) → (𝑤 gcd 𝑁) ∥ (𝑀 · 𝑁)))
94	92, 53, 57, 93	syl3anc 1367	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → (((𝑤 gcd 𝑁) ∥ 𝑁 ∧ 𝑁 ∥ (𝑀 · 𝑁)) → (𝑤 gcd 𝑁) ∥ (𝑀 · 𝑁)))
95	83, 85, 94	mp2and 697	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → (𝑤 gcd 𝑁) ∥ (𝑀 · 𝑁))
96		dvdslegcd 15855	. . . . . . . . . . . . . . . 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))))
97	92, 32, 57, 61, 96	syl31anc 1369	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → (((𝑤 gcd 𝑁) ∥ 𝑤 ∧ (𝑤 gcd 𝑁) ∥ (𝑀 · 𝑁)) → (𝑤 gcd 𝑁) ≤ (𝑤 gcd (𝑀 · 𝑁))))
98	82, 95, 97	mp2and 697	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → (𝑤 gcd 𝑁) ≤ (𝑤 gcd (𝑀 · 𝑁)))
99	98, 66	breqtrd 5094	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → (𝑤 gcd 𝑁) ≤ 1)
100		nnle1eq1 11670	. . . . . . . . . . . . . 14 ⊢ ((𝑤 gcd 𝑁) ∈ ℕ → ((𝑤 gcd 𝑁) ≤ 1 ↔ (𝑤 gcd 𝑁) = 1))
101	91, 100	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → ((𝑤 gcd 𝑁) ≤ 1 ↔ (𝑤 gcd 𝑁) = 1))
102	99, 101	mpbid 234	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → (𝑤 gcd 𝑁) = 1)
103	79, 102	eqtrd 2858	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → ((𝑤 mod 𝑁) gcd 𝑁) = 1)
104		oveq1 7165	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝑤 mod 𝑁) → (𝑦 gcd 𝑁) = ((𝑤 mod 𝑁) gcd 𝑁))
105	104	eqeq1d 2825	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝑤 mod 𝑁) → ((𝑦 gcd 𝑁) = 1 ↔ ((𝑤 mod 𝑁) gcd 𝑁) = 1))
106	105, 7	elrab2 3685	. . . . . . . . . . 11 ⊢ ((𝑤 mod 𝑁) ∈ 𝑉 ↔ ((𝑤 mod 𝑁) ∈ (0..^𝑁) ∧ ((𝑤 mod 𝑁) gcd 𝑁) = 1))
107	77, 103, 106	sylanbrc 585	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → (𝑤 mod 𝑁) ∈ 𝑉)
108	75, 107	opelxpd 5595	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → ⟨(𝑤 mod 𝑀), (𝑤 mod 𝑁)⟩ ∈ (𝑈 × 𝑉))
109	27, 108	eqeltrd 2915	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → (𝐹‘𝑤) ∈ (𝑈 × 𝑉))
110	109	ralrimiva 3184	. . . . . . 7 ⊢ (𝜑 → ∀𝑤 ∈ 𝑊 (𝐹‘𝑤) ∈ (𝑈 × 𝑉))
111		crth.2	. . . . . . . . . 10 ⊢ 𝑇 = ((0..^𝑀) × (0..^𝑁))
112	28, 111, 23, 33	crth 16117	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:𝑆–1-1-onto→𝑇)
113		f1ofn 6618	. . . . . . . . 9 ⊢ (𝐹:𝑆–1-1-onto→𝑇 → 𝐹 Fn 𝑆)
114		fnfun 6455	. . . . . . . . 9 ⊢ (𝐹 Fn 𝑆 → Fun 𝐹)
115	112, 113, 114	3syl 18	. . . . . . . 8 ⊢ (𝜑 → Fun 𝐹)
116	17	ssrab3 4059	. . . . . . . . 9 ⊢ 𝑊 ⊆ 𝑆
117		fndm 6457	. . . . . . . . . 10 ⊢ (𝐹 Fn 𝑆 → dom 𝐹 = 𝑆)
118	112, 113, 117	3syl 18	. . . . . . . . 9 ⊢ (𝜑 → dom 𝐹 = 𝑆)
119	116, 118	sseqtrrid 4022	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ⊆ dom 𝐹)
120		funimass4 6732	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ 𝑊 ⊆ dom 𝐹) → ((𝐹 “ 𝑊) ⊆ (𝑈 × 𝑉) ↔ ∀𝑤 ∈ 𝑊 (𝐹‘𝑤) ∈ (𝑈 × 𝑉)))
121	115, 119, 120	syl2anc 586	. . . . . . 7 ⊢ (𝜑 → ((𝐹 “ 𝑊) ⊆ (𝑈 × 𝑉) ↔ ∀𝑤 ∈ 𝑊 (𝐹‘𝑤) ∈ (𝑈 × 𝑉)))
122	110, 121	mpbird 259	. . . . . 6 ⊢ (𝜑 → (𝐹 “ 𝑊) ⊆ (𝑈 × 𝑉))
123	1	ssrab3 4059	. . . . . . . . . . 11 ⊢ 𝑈 ⊆ (0..^𝑀)
124	7	ssrab3 4059	. . . . . . . . . . 11 ⊢ 𝑉 ⊆ (0..^𝑁)
125		xpss12 5572	. . . . . . . . . . 11 ⊢ ((𝑈 ⊆ (0..^𝑀) ∧ 𝑉 ⊆ (0..^𝑁)) → (𝑈 × 𝑉) ⊆ ((0..^𝑀) × (0..^𝑁)))
126	123, 124, 125	mp2an 690	. . . . . . . . . 10 ⊢ (𝑈 × 𝑉) ⊆ ((0..^𝑀) × (0..^𝑁))
127	126, 111	sseqtrri 4006	. . . . . . . . 9 ⊢ (𝑈 × 𝑉) ⊆ 𝑇
128	127	sseli 3965	. . . . . . . 8 ⊢ (𝑧 ∈ (𝑈 × 𝑉) → 𝑧 ∈ 𝑇)
129		f1ocnvfv2 7036	. . . . . . . 8 ⊢ ((𝐹:𝑆–1-1-onto→𝑇 ∧ 𝑧 ∈ 𝑇) → (𝐹‘(^◡𝐹‘𝑧)) = 𝑧)
130	112, 128, 129	syl2an 597	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑈 × 𝑉)) → (𝐹‘(^◡𝐹‘𝑧)) = 𝑧)
131		f1ocnv 6629	. . . . . . . . . . 11 ⊢ (𝐹:𝑆–1-1-onto→𝑇 → ^◡𝐹:𝑇–1-1-onto→𝑆)
132		f1of 6617	. . . . . . . . . . 11 ⊢ (^◡𝐹:𝑇–1-1-onto→𝑆 → ^◡𝐹:𝑇⟶𝑆)
133	112, 131, 132	3syl 18	. . . . . . . . . 10 ⊢ (𝜑 → ^◡𝐹:𝑇⟶𝑆)
134		ffvelrn 6851	. . . . . . . . . 10 ⊢ ((^◡𝐹:𝑇⟶𝑆 ∧ 𝑧 ∈ 𝑇) → (^◡𝐹‘𝑧) ∈ 𝑆)
135	133, 128, 134	syl2an 597	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑈 × 𝑉)) → (^◡𝐹‘𝑧) ∈ 𝑆)
136	135, 28	eleqtrdi 2925	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑈 × 𝑉)) → (^◡𝐹‘𝑧) ∈ (0..^(𝑀 · 𝑁)))
137		elfzoelz 13041	. . . . . . . . . . . . 13 ⊢ ((^◡𝐹‘𝑧) ∈ (0..^(𝑀 · 𝑁)) → (^◡𝐹‘𝑧) ∈ ℤ)
138	136, 137	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑈 × 𝑉)) → (^◡𝐹‘𝑧) ∈ ℤ)
139	34	adantr 483	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑈 × 𝑉)) → 𝑀 ∈ ℕ)
140		modgcd 15882	. . . . . . . . . . . 12 ⊢ (((^◡𝐹‘𝑧) ∈ ℤ ∧ 𝑀 ∈ ℕ) → (((^◡𝐹‘𝑧) mod 𝑀) gcd 𝑀) = ((^◡𝐹‘𝑧) gcd 𝑀))
141	138, 139, 140	syl2anc 586	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑈 × 𝑉)) → (((^◡𝐹‘𝑧) mod 𝑀) gcd 𝑀) = ((^◡𝐹‘𝑧) gcd 𝑀))
142		oveq1 7165	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑤 = (^◡𝐹‘𝑧) → (𝑤 mod 𝑀) = ((^◡𝐹‘𝑧) mod 𝑀))
143		oveq1 7165	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑤 = (^◡𝐹‘𝑧) → (𝑤 mod 𝑁) = ((^◡𝐹‘𝑧) mod 𝑁))
144	142, 143	opeq12d 4813	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 = (^◡𝐹‘𝑧) → ⟨(𝑤 mod 𝑀), (𝑤 mod 𝑁)⟩ = ⟨((^◡𝐹‘𝑧) mod 𝑀), ((^◡𝐹‘𝑧) mod 𝑁)⟩)
145	22	cbvmptv 5171	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ 𝑆 ↦ ⟨(𝑥 mod 𝑀), (𝑥 mod 𝑁)⟩) = (𝑤 ∈ 𝑆 ↦ ⟨(𝑤 mod 𝑀), (𝑤 mod 𝑁)⟩)
146	23, 145	eqtri 2846	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝐹 = (𝑤 ∈ 𝑆 ↦ ⟨(𝑤 mod 𝑀), (𝑤 mod 𝑁)⟩)
147		opex 5358	. . . . . . . . . . . . . . . . . . 19 ⊢ ⟨((^◡𝐹‘𝑧) mod 𝑀), ((^◡𝐹‘𝑧) mod 𝑁)⟩ ∈ V
148	144, 146, 147	fvmpt 6770	. . . . . . . . . . . . . . . . . 18 ⊢ ((^◡𝐹‘𝑧) ∈ 𝑆 → (𝐹‘(^◡𝐹‘𝑧)) = ⟨((^◡𝐹‘𝑧) mod 𝑀), ((^◡𝐹‘𝑧) mod 𝑁)⟩)
149	135, 148	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑈 × 𝑉)) → (𝐹‘(^◡𝐹‘𝑧)) = ⟨((^◡𝐹‘𝑧) mod 𝑀), ((^◡𝐹‘𝑧) mod 𝑁)⟩)
150	130, 149	eqtr3d 2860	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑈 × 𝑉)) → 𝑧 = ⟨((^◡𝐹‘𝑧) mod 𝑀), ((^◡𝐹‘𝑧) mod 𝑁)⟩)
151		simpr 487	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑈 × 𝑉)) → 𝑧 ∈ (𝑈 × 𝑉))
152	150, 151	eqeltrrd 2916	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑈 × 𝑉)) → ⟨((^◡𝐹‘𝑧) mod 𝑀), ((^◡𝐹‘𝑧) mod 𝑁)⟩ ∈ (𝑈 × 𝑉))
153		opelxp 5593	. . . . . . . . . . . . . . 15 ⊢ (⟨((^◡𝐹‘𝑧) mod 𝑀), ((^◡𝐹‘𝑧) mod 𝑁)⟩ ∈ (𝑈 × 𝑉) ↔ (((^◡𝐹‘𝑧) mod 𝑀) ∈ 𝑈 ∧ ((^◡𝐹‘𝑧) mod 𝑁) ∈ 𝑉))
154	152, 153	sylib 220	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑈 × 𝑉)) → (((^◡𝐹‘𝑧) mod 𝑀) ∈ 𝑈 ∧ ((^◡𝐹‘𝑧) mod 𝑁) ∈ 𝑉))
155	154	simpld 497	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑈 × 𝑉)) → ((^◡𝐹‘𝑧) mod 𝑀) ∈ 𝑈)
156		oveq1 7165	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = ((^◡𝐹‘𝑧) mod 𝑀) → (𝑦 gcd 𝑀) = (((^◡𝐹‘𝑧) mod 𝑀) gcd 𝑀))
157	156	eqeq1d 2825	. . . . . . . . . . . . . 14 ⊢ (𝑦 = ((^◡𝐹‘𝑧) mod 𝑀) → ((𝑦 gcd 𝑀) = 1 ↔ (((^◡𝐹‘𝑧) mod 𝑀) gcd 𝑀) = 1))
158	157, 1	elrab2 3685	. . . . . . . . . . . . 13 ⊢ (((^◡𝐹‘𝑧) mod 𝑀) ∈ 𝑈 ↔ (((^◡𝐹‘𝑧) mod 𝑀) ∈ (0..^𝑀) ∧ (((^◡𝐹‘𝑧) mod 𝑀) gcd 𝑀) = 1))
159	155, 158	sylib 220	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑈 × 𝑉)) → (((^◡𝐹‘𝑧) mod 𝑀) ∈ (0..^𝑀) ∧ (((^◡𝐹‘𝑧) mod 𝑀) gcd 𝑀) = 1))
160	159	simprd 498	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑈 × 𝑉)) → (((^◡𝐹‘𝑧) mod 𝑀) gcd 𝑀) = 1)
161	141, 160	eqtr3d 2860	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑈 × 𝑉)) → ((^◡𝐹‘𝑧) gcd 𝑀) = 1)
162	51	adantr 483	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑈 × 𝑉)) → 𝑁 ∈ ℕ)
163		modgcd 15882	. . . . . . . . . . . 12 ⊢ (((^◡𝐹‘𝑧) ∈ ℤ ∧ 𝑁 ∈ ℕ) → (((^◡𝐹‘𝑧) mod 𝑁) gcd 𝑁) = ((^◡𝐹‘𝑧) gcd 𝑁))
164	138, 162, 163	syl2anc 586	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑈 × 𝑉)) → (((^◡𝐹‘𝑧) mod 𝑁) gcd 𝑁) = ((^◡𝐹‘𝑧) gcd 𝑁))
165	154	simprd 498	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑈 × 𝑉)) → ((^◡𝐹‘𝑧) mod 𝑁) ∈ 𝑉)
166		oveq1 7165	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = ((^◡𝐹‘𝑧) mod 𝑁) → (𝑦 gcd 𝑁) = (((^◡𝐹‘𝑧) mod 𝑁) gcd 𝑁))
167	166	eqeq1d 2825	. . . . . . . . . . . . . 14 ⊢ (𝑦 = ((^◡𝐹‘𝑧) mod 𝑁) → ((𝑦 gcd 𝑁) = 1 ↔ (((^◡𝐹‘𝑧) mod 𝑁) gcd 𝑁) = 1))
168	167, 7	elrab2 3685	. . . . . . . . . . . . 13 ⊢ (((^◡𝐹‘𝑧) mod 𝑁) ∈ 𝑉 ↔ (((^◡𝐹‘𝑧) mod 𝑁) ∈ (0..^𝑁) ∧ (((^◡𝐹‘𝑧) mod 𝑁) gcd 𝑁) = 1))
169	165, 168	sylib 220	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑈 × 𝑉)) → (((^◡𝐹‘𝑧) mod 𝑁) ∈ (0..^𝑁) ∧ (((^◡𝐹‘𝑧) mod 𝑁) gcd 𝑁) = 1))
170	169	simprd 498	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑈 × 𝑉)) → (((^◡𝐹‘𝑧) mod 𝑁) gcd 𝑁) = 1)
171	164, 170	eqtr3d 2860	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑈 × 𝑉)) → ((^◡𝐹‘𝑧) gcd 𝑁) = 1)
172	34	nnzd 12089	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑀 ∈ ℤ)
173	172	adantr 483	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑈 × 𝑉)) → 𝑀 ∈ ℤ)
174	51	nnzd 12089	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℤ)
175	174	adantr 483	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑈 × 𝑉)) → 𝑁 ∈ ℤ)
176		rpmul 16005	. . . . . . . . . . 11 ⊢ (((^◡𝐹‘𝑧) ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((((^◡𝐹‘𝑧) gcd 𝑀) = 1 ∧ ((^◡𝐹‘𝑧) gcd 𝑁) = 1) → ((^◡𝐹‘𝑧) gcd (𝑀 · 𝑁)) = 1))
177	138, 173, 175, 176	syl3anc 1367	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑈 × 𝑉)) → ((((^◡𝐹‘𝑧) gcd 𝑀) = 1 ∧ ((^◡𝐹‘𝑧) gcd 𝑁) = 1) → ((^◡𝐹‘𝑧) gcd (𝑀 · 𝑁)) = 1))
178	161, 171, 177	mp2and 697	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑈 × 𝑉)) → ((^◡𝐹‘𝑧) gcd (𝑀 · 𝑁)) = 1)
179		oveq1 7165	. . . . . . . . . . 11 ⊢ (𝑦 = (^◡𝐹‘𝑧) → (𝑦 gcd (𝑀 · 𝑁)) = ((^◡𝐹‘𝑧) gcd (𝑀 · 𝑁)))
180	179	eqeq1d 2825	. . . . . . . . . 10 ⊢ (𝑦 = (^◡𝐹‘𝑧) → ((𝑦 gcd (𝑀 · 𝑁)) = 1 ↔ ((^◡𝐹‘𝑧) gcd (𝑀 · 𝑁)) = 1))
181	180, 17	elrab2 3685	. . . . . . . . 9 ⊢ ((^◡𝐹‘𝑧) ∈ 𝑊 ↔ ((^◡𝐹‘𝑧) ∈ 𝑆 ∧ ((^◡𝐹‘𝑧) gcd (𝑀 · 𝑁)) = 1))
182	135, 178, 181	sylanbrc 585	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑈 × 𝑉)) → (^◡𝐹‘𝑧) ∈ 𝑊)
183		funfvima2 6995	. . . . . . . . . 10 ⊢ ((Fun 𝐹 ∧ 𝑊 ⊆ dom 𝐹) → ((^◡𝐹‘𝑧) ∈ 𝑊 → (𝐹‘(^◡𝐹‘𝑧)) ∈ (𝐹 “ 𝑊)))
184	115, 119, 183	syl2anc 586	. . . . . . . . 9 ⊢ (𝜑 → ((^◡𝐹‘𝑧) ∈ 𝑊 → (𝐹‘(^◡𝐹‘𝑧)) ∈ (𝐹 “ 𝑊)))
185	184	imp 409	. . . . . . . 8 ⊢ ((𝜑 ∧ (^◡𝐹‘𝑧) ∈ 𝑊) → (𝐹‘(^◡𝐹‘𝑧)) ∈ (𝐹 “ 𝑊))
186	182, 185	syldan 593	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑈 × 𝑉)) → (𝐹‘(^◡𝐹‘𝑧)) ∈ (𝐹 “ 𝑊))
187	130, 186	eqeltrrd 2916	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑈 × 𝑉)) → 𝑧 ∈ (𝐹 “ 𝑊))
188	122, 187	eqelssd 3990	. . . . 5 ⊢ (𝜑 → (𝐹 “ 𝑊) = (𝑈 × 𝑉))
189		f1of1 6616	. . . . . . 7 ⊢ (𝐹:𝑆–1-1-onto→𝑇 → 𝐹:𝑆–1-1→𝑇)
190	112, 189	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐹:𝑆–1-1→𝑇)
191		fzofi 13345	. . . . . . . . . 10 ⊢ (0..^(𝑀 · 𝑁)) ∈ Fin
192	28, 191	eqeltri 2911	. . . . . . . . 9 ⊢ 𝑆 ∈ Fin
193		ssfi 8740	. . . . . . . . 9 ⊢ ((𝑆 ∈ Fin ∧ 𝑊 ⊆ 𝑆) → 𝑊 ∈ Fin)
194	192, 116, 193	mp2an 690	. . . . . . . 8 ⊢ 𝑊 ∈ Fin
195	194	elexi 3515	. . . . . . 7 ⊢ 𝑊 ∈ V
196	195	f1imaen 8573	. . . . . 6 ⊢ ((𝐹:𝑆–1-1→𝑇 ∧ 𝑊 ⊆ 𝑆) → (𝐹 “ 𝑊) ≈ 𝑊)
197	190, 116, 196	sylancl 588	. . . . 5 ⊢ (𝜑 → (𝐹 “ 𝑊) ≈ 𝑊)
198	188, 197	eqbrtrrd 5092	. . . 4 ⊢ (𝜑 → (𝑈 × 𝑉) ≈ 𝑊)
199		xpfi 8791	. . . . . 6 ⊢ ((𝑈 ∈ Fin ∧ 𝑉 ∈ Fin) → (𝑈 × 𝑉) ∈ Fin)
200	6, 12, 199	mp2an 690	. . . . 5 ⊢ (𝑈 × 𝑉) ∈ Fin
201		hashen 13710	. . . . 5 ⊢ (((𝑈 × 𝑉) ∈ Fin ∧ 𝑊 ∈ Fin) → ((♯‘(𝑈 × 𝑉)) = (♯‘𝑊) ↔ (𝑈 × 𝑉) ≈ 𝑊))
202	200, 194, 201	mp2an 690	. . . 4 ⊢ ((♯‘(𝑈 × 𝑉)) = (♯‘𝑊) ↔ (𝑈 × 𝑉) ≈ 𝑊)
203	198, 202	sylibr 236	. . 3 ⊢ (𝜑 → (♯‘(𝑈 × 𝑉)) = (♯‘𝑊))
204	14, 203	syl5reqr 2873	. 2 ⊢ (𝜑 → (♯‘𝑊) = ((♯‘𝑈) · (♯‘𝑉)))
205	34, 51	nnmulcld 11693	. . 3 ⊢ (𝜑 → (𝑀 · 𝑁) ∈ ℕ)
206		dfphi2 16113	. . . 4 ⊢ ((𝑀 · 𝑁) ∈ ℕ → (ϕ‘(𝑀 · 𝑁)) = (♯‘{𝑦 ∈ (0..^(𝑀 · 𝑁)) ∣ (𝑦 gcd (𝑀 · 𝑁)) = 1}))
207	28	rabeqi 3484	. . . . . 6 ⊢ {𝑦 ∈ 𝑆 ∣ (𝑦 gcd (𝑀 · 𝑁)) = 1} = {𝑦 ∈ (0..^(𝑀 · 𝑁)) ∣ (𝑦 gcd (𝑀 · 𝑁)) = 1}
208	17, 207	eqtri 2846	. . . . 5 ⊢ 𝑊 = {𝑦 ∈ (0..^(𝑀 · 𝑁)) ∣ (𝑦 gcd (𝑀 · 𝑁)) = 1}
209	208	fveq2i 6675	. . . 4 ⊢ (♯‘𝑊) = (♯‘{𝑦 ∈ (0..^(𝑀 · 𝑁)) ∣ (𝑦 gcd (𝑀 · 𝑁)) = 1})
210	206, 209	syl6eqr 2876	. . 3 ⊢ ((𝑀 · 𝑁) ∈ ℕ → (ϕ‘(𝑀 · 𝑁)) = (♯‘𝑊))
211	205, 210	syl 17	. 2 ⊢ (𝜑 → (ϕ‘(𝑀 · 𝑁)) = (♯‘𝑊))
212		dfphi2 16113	. . . . 5 ⊢ (𝑀 ∈ ℕ → (ϕ‘𝑀) = (♯‘{𝑦 ∈ (0..^𝑀) ∣ (𝑦 gcd 𝑀) = 1}))
213	1	fveq2i 6675	. . . . 5 ⊢ (♯‘𝑈) = (♯‘{𝑦 ∈ (0..^𝑀) ∣ (𝑦 gcd 𝑀) = 1})
214	212, 213	syl6eqr 2876	. . . 4 ⊢ (𝑀 ∈ ℕ → (ϕ‘𝑀) = (♯‘𝑈))
215	34, 214	syl 17	. . 3 ⊢ (𝜑 → (ϕ‘𝑀) = (♯‘𝑈))
216		dfphi2 16113	. . . . 5 ⊢ (𝑁 ∈ ℕ → (ϕ‘𝑁) = (♯‘{𝑦 ∈ (0..^𝑁) ∣ (𝑦 gcd 𝑁) = 1}))
217	7	fveq2i 6675	. . . . 5 ⊢ (♯‘𝑉) = (♯‘{𝑦 ∈ (0..^𝑁) ∣ (𝑦 gcd 𝑁) = 1})
218	216, 217	syl6eqr 2876	. . . 4 ⊢ (𝑁 ∈ ℕ → (ϕ‘𝑁) = (♯‘𝑉))
219	51, 218	syl 17	. . 3 ⊢ (𝜑 → (ϕ‘𝑁) = (♯‘𝑉))
220	215, 219	oveq12d 7176	. 2 ⊢ (𝜑 → ((ϕ‘𝑀) · (ϕ‘𝑁)) = ((♯‘𝑈) · (♯‘𝑉)))
221	204, 211, 220	3eqtr4d 2868	1 ⊢ (𝜑 → (ϕ‘(𝑀 · 𝑁)) = ((ϕ‘𝑀) · (ϕ‘𝑁)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 ∀wral 3140 {crab 3144 ⊆ wss 3938 ⟨cop 4575 class class class wbr 5068 ↦ cmpt 5148 × cxp 5555 ^◡ccnv 5556 dom cdm 5557 “ cima 5560 Fun wfun 6351 Fn wfn 6352 ⟶wf 6353 –1-1→wf1 6354 –1-1-onto→wf1o 6356 ‘cfv 6357 (class class class)co 7158 ≈ cen 8508 Fincfn 8511 0cc0 10539 1c1 10540 · cmul 10544 ≤ cle 10678 ℕcn 11640 ℤcz 11984 ..^cfzo 13036 mod cmo 13240 ♯chash 13693 ∥ cdvds 15609 gcd cgcd 15845 ϕcphi 16103

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-sup 8908 df-inf 8909 df-dju 9332 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-n0 11901 df-xnn0 11971 df-z 11985 df-uz 12247 df-rp 12393 df-fz 12896 df-fzo 13037 df-fl 13165 df-mod 13241 df-seq 13373 df-exp 13433 df-hash 13694 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-dvds 15610 df-gcd 15846 df-phi 16105

This theorem is referenced by: phimul 16119

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