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Theorem phimullem 12179

Description: Lemma for phimul 12180. (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,𝑀 𝑥,𝑁 𝑥,𝑆,𝑦 𝑥,𝑇 𝜑,𝑥,𝑦 𝑦,𝑀 𝑦,𝑁 Allowed substitution hints: 𝑇(𝑦) 𝑈(𝑥,𝑦) 𝐹(𝑥) 𝑉(𝑥,𝑦) 𝑊(𝑥,𝑦)

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

Step	Hyp	Ref	Expression
1		oveq1 5860	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑤 → (𝑦 gcd (𝑀 · 𝑁)) = (𝑤 gcd (𝑀 · 𝑁)))
2	1	eqeq1d 2179	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑤 → ((𝑦 gcd (𝑀 · 𝑁)) = 1 ↔ (𝑤 gcd (𝑀 · 𝑁)) = 1))
3		phimul.6	. . . . . . . . . . . . 13 ⊢ 𝑊 = {𝑦 ∈ 𝑆 ∣ (𝑦 gcd (𝑀 · 𝑁)) = 1}
4	2, 3	elrab2 2889	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ 𝑊 ↔ (𝑤 ∈ 𝑆 ∧ (𝑤 gcd (𝑀 · 𝑁)) = 1))
5	4	simplbi 272	. . . . . . . . . . 11 ⊢ (𝑤 ∈ 𝑊 → 𝑤 ∈ 𝑆)
6	5	adantl 275	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → 𝑤 ∈ 𝑆)
7		elfzoelz 10103	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ (0..^(𝑀 · 𝑁)) → 𝑤 ∈ ℤ)
8		crth.1	. . . . . . . . . . . . . . 15 ⊢ 𝑆 = (0..^(𝑀 · 𝑁))
9	7, 8	eleq2s 2265	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ 𝑆 → 𝑤 ∈ ℤ)
10	6, 9	syl 14	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → 𝑤 ∈ ℤ)
11		zq 9585	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ ℤ → 𝑤 ∈ ℚ)
12	10, 11	syl 14	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → 𝑤 ∈ ℚ)
13		crth.4	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1))
14	13	simp1d 1004	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑀 ∈ ℕ)
15	14	adantr 274	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → 𝑀 ∈ ℕ)
16		nnq 9592	. . . . . . . . . . . . 13 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℚ)
17	15, 16	syl 14	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → 𝑀 ∈ ℚ)
18	15	nngt0d 8922	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → 0 < 𝑀)
19	12, 17, 18	modqcld 10284	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → (𝑤 mod 𝑀) ∈ ℚ)
20	13	simp2d 1005	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑁 ∈ ℕ)
21	20	adantr 274	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → 𝑁 ∈ ℕ)
22		nnq 9592	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℚ)
23	21, 22	syl 14	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → 𝑁 ∈ ℚ)
24	21	nngt0d 8922	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → 0 < 𝑁)
25	12, 23, 24	modqcld 10284	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → (𝑤 mod 𝑁) ∈ ℚ)
26		opexg 4213	. . . . . . . . . . 11 ⊢ (((𝑤 mod 𝑀) ∈ ℚ ∧ (𝑤 mod 𝑁) ∈ ℚ) → ⟨(𝑤 mod 𝑀), (𝑤 mod 𝑁)⟩ ∈ V)
27	19, 25, 26	syl2anc 409	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → ⟨(𝑤 mod 𝑀), (𝑤 mod 𝑁)⟩ ∈ V)
28		oveq1 5860	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑤 → (𝑥 mod 𝑀) = (𝑤 mod 𝑀))
29		oveq1 5860	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑤 → (𝑥 mod 𝑁) = (𝑤 mod 𝑁))
30	28, 29	opeq12d 3773	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑤 → ⟨(𝑥 mod 𝑀), (𝑥 mod 𝑁)⟩ = ⟨(𝑤 mod 𝑀), (𝑤 mod 𝑁)⟩)
31		crth.3	. . . . . . . . . . 11 ⊢ 𝐹 = (𝑥 ∈ 𝑆 ↦ ⟨(𝑥 mod 𝑀), (𝑥 mod 𝑁)⟩)
32	30, 31	fvmptg 5572	. . . . . . . . . 10 ⊢ ((𝑤 ∈ 𝑆 ∧ ⟨(𝑤 mod 𝑀), (𝑤 mod 𝑁)⟩ ∈ V) → (𝐹‘𝑤) = ⟨(𝑤 mod 𝑀), (𝑤 mod 𝑁)⟩)
33	6, 27, 32	syl2anc 409	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → (𝐹‘𝑤) = ⟨(𝑤 mod 𝑀), (𝑤 mod 𝑁)⟩)
34	5, 8	eleqtrdi 2263	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ 𝑊 → 𝑤 ∈ (0..^(𝑀 · 𝑁)))
35	34	adantl 275	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → 𝑤 ∈ (0..^(𝑀 · 𝑁)))
36	35, 7	syl 14	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → 𝑤 ∈ ℤ)
37		zmodfzo 10303	. . . . . . . . . . . 12 ⊢ ((𝑤 ∈ ℤ ∧ 𝑀 ∈ ℕ) → (𝑤 mod 𝑀) ∈ (0..^𝑀))
38	36, 15, 37	syl2anc 409	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → (𝑤 mod 𝑀) ∈ (0..^𝑀))
39		modgcd 11946	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ ℤ ∧ 𝑀 ∈ ℕ) → ((𝑤 mod 𝑀) gcd 𝑀) = (𝑤 gcd 𝑀))
40	36, 15, 39	syl2anc 409	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → ((𝑤 mod 𝑀) gcd 𝑀) = (𝑤 gcd 𝑀))
41	15	nnzd 9333	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → 𝑀 ∈ ℤ)
42		gcddvds 11918	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑤 ∈ ℤ ∧ 𝑀 ∈ ℤ) → ((𝑤 gcd 𝑀) ∥ 𝑤 ∧ (𝑤 gcd 𝑀) ∥ 𝑀))
43	36, 41, 42	syl2anc 409	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → ((𝑤 gcd 𝑀) ∥ 𝑤 ∧ (𝑤 gcd 𝑀) ∥ 𝑀))
44	43	simpld 111	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → (𝑤 gcd 𝑀) ∥ 𝑤)
45		nnne0 8906	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑀 ∈ ℕ → 𝑀 ≠ 0)
46		simpr 109	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑤 = 0 ∧ 𝑀 = 0) → 𝑀 = 0)
47	46	necon3ai 2389	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑀 ≠ 0 → ¬ (𝑤 = 0 ∧ 𝑀 = 0))
48	15, 45, 47	3syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → ¬ (𝑤 = 0 ∧ 𝑀 = 0))
49		gcdn0cl 11917	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑤 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ ¬ (𝑤 = 0 ∧ 𝑀 = 0)) → (𝑤 gcd 𝑀) ∈ ℕ)
50	36, 41, 48, 49	syl21anc 1232	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → (𝑤 gcd 𝑀) ∈ ℕ)
51	50	nnzd 9333	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → (𝑤 gcd 𝑀) ∈ ℤ)
52	21	nnzd 9333	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → 𝑁 ∈ ℤ)
53	43	simprd 113	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → (𝑤 gcd 𝑀) ∥ 𝑀)
54	51, 41, 52, 53	dvdsmultr1d 11794	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → (𝑤 gcd 𝑀) ∥ (𝑀 · 𝑁))
55	15, 21	nnmulcld 8927	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → (𝑀 · 𝑁) ∈ ℕ)
56	55	nnzd 9333	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → (𝑀 · 𝑁) ∈ ℤ)
57		nnne0 8906	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑀 · 𝑁) ∈ ℕ → (𝑀 · 𝑁) ≠ 0)
58		simpr 109	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑤 = 0 ∧ (𝑀 · 𝑁) = 0) → (𝑀 · 𝑁) = 0)
59	58	necon3ai 2389	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑀 · 𝑁) ≠ 0 → ¬ (𝑤 = 0 ∧ (𝑀 · 𝑁) = 0))
60	55, 57, 59	3syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → ¬ (𝑤 = 0 ∧ (𝑀 · 𝑁) = 0))
61		dvdslegcd 11919	. . . . . . . . . . . . . . . 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))))
62	51, 36, 56, 60, 61	syl31anc 1236	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → (((𝑤 gcd 𝑀) ∥ 𝑤 ∧ (𝑤 gcd 𝑀) ∥ (𝑀 · 𝑁)) → (𝑤 gcd 𝑀) ≤ (𝑤 gcd (𝑀 · 𝑁))))
63	44, 54, 62	mp2and 431	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → (𝑤 gcd 𝑀) ≤ (𝑤 gcd (𝑀 · 𝑁)))
64	4	simprbi 273	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ 𝑊 → (𝑤 gcd (𝑀 · 𝑁)) = 1)
65	64	adantl 275	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → (𝑤 gcd (𝑀 · 𝑁)) = 1)
66	63, 65	breqtrd 4015	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → (𝑤 gcd 𝑀) ≤ 1)
67		nnle1eq1 8902	. . . . . . . . . . . . . 14 ⊢ ((𝑤 gcd 𝑀) ∈ ℕ → ((𝑤 gcd 𝑀) ≤ 1 ↔ (𝑤 gcd 𝑀) = 1))
68	50, 67	syl 14	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → ((𝑤 gcd 𝑀) ≤ 1 ↔ (𝑤 gcd 𝑀) = 1))
69	66, 68	mpbid 146	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → (𝑤 gcd 𝑀) = 1)
70	40, 69	eqtrd 2203	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → ((𝑤 mod 𝑀) gcd 𝑀) = 1)
71		oveq1 5860	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝑤 mod 𝑀) → (𝑦 gcd 𝑀) = ((𝑤 mod 𝑀) gcd 𝑀))
72	71	eqeq1d 2179	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝑤 mod 𝑀) → ((𝑦 gcd 𝑀) = 1 ↔ ((𝑤 mod 𝑀) gcd 𝑀) = 1))
73		phimul.4	. . . . . . . . . . . 12 ⊢ 𝑈 = {𝑦 ∈ (0..^𝑀) ∣ (𝑦 gcd 𝑀) = 1}
74	72, 73	elrab2 2889	. . . . . . . . . . 11 ⊢ ((𝑤 mod 𝑀) ∈ 𝑈 ↔ ((𝑤 mod 𝑀) ∈ (0..^𝑀) ∧ ((𝑤 mod 𝑀) gcd 𝑀) = 1))
75	38, 70, 74	sylanbrc 415	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → (𝑤 mod 𝑀) ∈ 𝑈)
76		zmodfzo 10303	. . . . . . . . . . . 12 ⊢ ((𝑤 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝑤 mod 𝑁) ∈ (0..^𝑁))
77	36, 21, 76	syl2anc 409	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → (𝑤 mod 𝑁) ∈ (0..^𝑁))
78		modgcd 11946	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ ℤ ∧ 𝑁 ∈ ℕ) → ((𝑤 mod 𝑁) gcd 𝑁) = (𝑤 gcd 𝑁))
79	36, 21, 78	syl2anc 409	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → ((𝑤 mod 𝑁) gcd 𝑁) = (𝑤 gcd 𝑁))
80		gcddvds 11918	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑤 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑤 gcd 𝑁) ∥ 𝑤 ∧ (𝑤 gcd 𝑁) ∥ 𝑁))
81	36, 52, 80	syl2anc 409	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → ((𝑤 gcd 𝑁) ∥ 𝑤 ∧ (𝑤 gcd 𝑁) ∥ 𝑁))
82	81	simpld 111	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → (𝑤 gcd 𝑁) ∥ 𝑤)
83	81	simprd 113	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → (𝑤 gcd 𝑁) ∥ 𝑁)
84		dvdsmul2 11776	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑁 ∥ (𝑀 · 𝑁))
85	41, 52, 84	syl2anc 409	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → 𝑁 ∥ (𝑀 · 𝑁))
86		nnne0 8906	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ → 𝑁 ≠ 0)
87		simpr 109	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑤 = 0 ∧ 𝑁 = 0) → 𝑁 = 0)
88	87	necon3ai 2389	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ≠ 0 → ¬ (𝑤 = 0 ∧ 𝑁 = 0))
89	21, 86, 88	3syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → ¬ (𝑤 = 0 ∧ 𝑁 = 0))
90		gcdn0cl 11917	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑤 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑤 = 0 ∧ 𝑁 = 0)) → (𝑤 gcd 𝑁) ∈ ℕ)
91	36, 52, 89, 90	syl21anc 1232	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → (𝑤 gcd 𝑁) ∈ ℕ)
92	91	nnzd 9333	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → (𝑤 gcd 𝑁) ∈ ℤ)
93		dvdstr 11790	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑤 gcd 𝑁) ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝑀 · 𝑁) ∈ ℤ) → (((𝑤 gcd 𝑁) ∥ 𝑁 ∧ 𝑁 ∥ (𝑀 · 𝑁)) → (𝑤 gcd 𝑁) ∥ (𝑀 · 𝑁)))
94	92, 52, 56, 93	syl3anc 1233	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → (((𝑤 gcd 𝑁) ∥ 𝑁 ∧ 𝑁 ∥ (𝑀 · 𝑁)) → (𝑤 gcd 𝑁) ∥ (𝑀 · 𝑁)))
95	83, 85, 94	mp2and 431	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → (𝑤 gcd 𝑁) ∥ (𝑀 · 𝑁))
96		dvdslegcd 11919	. . . . . . . . . . . . . . . 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, 36, 56, 60, 96	syl31anc 1236	. . . . . . . . . . . . . . 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 431	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → (𝑤 gcd 𝑁) ≤ (𝑤 gcd (𝑀 · 𝑁)))
99	98, 65	breqtrd 4015	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → (𝑤 gcd 𝑁) ≤ 1)
100		nnle1eq1 8902	. . . . . . . . . . . . . 14 ⊢ ((𝑤 gcd 𝑁) ∈ ℕ → ((𝑤 gcd 𝑁) ≤ 1 ↔ (𝑤 gcd 𝑁) = 1))
101	91, 100	syl 14	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → ((𝑤 gcd 𝑁) ≤ 1 ↔ (𝑤 gcd 𝑁) = 1))
102	99, 101	mpbid 146	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → (𝑤 gcd 𝑁) = 1)
103	79, 102	eqtrd 2203	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → ((𝑤 mod 𝑁) gcd 𝑁) = 1)
104		oveq1 5860	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝑤 mod 𝑁) → (𝑦 gcd 𝑁) = ((𝑤 mod 𝑁) gcd 𝑁))
105	104	eqeq1d 2179	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝑤 mod 𝑁) → ((𝑦 gcd 𝑁) = 1 ↔ ((𝑤 mod 𝑁) gcd 𝑁) = 1))
106		phimul.5	. . . . . . . . . . . 12 ⊢ 𝑉 = {𝑦 ∈ (0..^𝑁) ∣ (𝑦 gcd 𝑁) = 1}
107	105, 106	elrab2 2889	. . . . . . . . . . 11 ⊢ ((𝑤 mod 𝑁) ∈ 𝑉 ↔ ((𝑤 mod 𝑁) ∈ (0..^𝑁) ∧ ((𝑤 mod 𝑁) gcd 𝑁) = 1))
108	77, 103, 107	sylanbrc 415	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → (𝑤 mod 𝑁) ∈ 𝑉)
109		opelxpi 4643	. . . . . . . . . 10 ⊢ (((𝑤 mod 𝑀) ∈ 𝑈 ∧ (𝑤 mod 𝑁) ∈ 𝑉) → ⟨(𝑤 mod 𝑀), (𝑤 mod 𝑁)⟩ ∈ (𝑈 × 𝑉))
110	75, 108, 109	syl2anc 409	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → ⟨(𝑤 mod 𝑀), (𝑤 mod 𝑁)⟩ ∈ (𝑈 × 𝑉))
111	33, 110	eqeltrd 2247	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑊) → (𝐹‘𝑤) ∈ (𝑈 × 𝑉))
112	111	ralrimiva 2543	. . . . . . 7 ⊢ (𝜑 → ∀𝑤 ∈ 𝑊 (𝐹‘𝑤) ∈ (𝑈 × 𝑉))
113		crth.2	. . . . . . . . . 10 ⊢ 𝑇 = ((0..^𝑀) × (0..^𝑁))
114	8, 113, 31, 13	crth 12178	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:𝑆–1-1-onto→𝑇)
115		f1ofn 5443	. . . . . . . . 9 ⊢ (𝐹:𝑆–1-1-onto→𝑇 → 𝐹 Fn 𝑆)
116		fnfun 5295	. . . . . . . . 9 ⊢ (𝐹 Fn 𝑆 → Fun 𝐹)
117	114, 115, 116	3syl 17	. . . . . . . 8 ⊢ (𝜑 → Fun 𝐹)
118		ssrab2 3232	. . . . . . . . . 10 ⊢ {𝑦 ∈ 𝑆 ∣ (𝑦 gcd (𝑀 · 𝑁)) = 1} ⊆ 𝑆
119	3, 118	eqsstri 3179	. . . . . . . . 9 ⊢ 𝑊 ⊆ 𝑆
120		fndm 5297	. . . . . . . . . 10 ⊢ (𝐹 Fn 𝑆 → dom 𝐹 = 𝑆)
121	114, 115, 120	3syl 17	. . . . . . . . 9 ⊢ (𝜑 → dom 𝐹 = 𝑆)
122	119, 121	sseqtrrid 3198	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ⊆ dom 𝐹)
123		funimass4 5547	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ 𝑊 ⊆ dom 𝐹) → ((𝐹 “ 𝑊) ⊆ (𝑈 × 𝑉) ↔ ∀𝑤 ∈ 𝑊 (𝐹‘𝑤) ∈ (𝑈 × 𝑉)))
124	117, 122, 123	syl2anc 409	. . . . . . 7 ⊢ (𝜑 → ((𝐹 “ 𝑊) ⊆ (𝑈 × 𝑉) ↔ ∀𝑤 ∈ 𝑊 (𝐹‘𝑤) ∈ (𝑈 × 𝑉)))
125	112, 124	mpbird 166	. . . . . 6 ⊢ (𝜑 → (𝐹 “ 𝑊) ⊆ (𝑈 × 𝑉))
126		ssrab2 3232	. . . . . . . . . . . . . 14 ⊢ {𝑦 ∈ (0..^𝑀) ∣ (𝑦 gcd 𝑀) = 1} ⊆ (0..^𝑀)
127	73, 126	eqsstri 3179	. . . . . . . . . . . . 13 ⊢ 𝑈 ⊆ (0..^𝑀)
128		ssrab2 3232	. . . . . . . . . . . . . 14 ⊢ {𝑦 ∈ (0..^𝑁) ∣ (𝑦 gcd 𝑁) = 1} ⊆ (0..^𝑁)
129	106, 128	eqsstri 3179	. . . . . . . . . . . . 13 ⊢ 𝑉 ⊆ (0..^𝑁)
130		xpss12 4718	. . . . . . . . . . . . 13 ⊢ ((𝑈 ⊆ (0..^𝑀) ∧ 𝑉 ⊆ (0..^𝑁)) → (𝑈 × 𝑉) ⊆ ((0..^𝑀) × (0..^𝑁)))
131	127, 129, 130	mp2an 424	. . . . . . . . . . . 12 ⊢ (𝑈 × 𝑉) ⊆ ((0..^𝑀) × (0..^𝑁))
132	131, 113	sseqtrri 3182	. . . . . . . . . . 11 ⊢ (𝑈 × 𝑉) ⊆ 𝑇
133	132	sseli 3143	. . . . . . . . . 10 ⊢ (𝑧 ∈ (𝑈 × 𝑉) → 𝑧 ∈ 𝑇)
134		f1ocnvfv2 5757	. . . . . . . . . 10 ⊢ ((𝐹:𝑆–1-1-onto→𝑇 ∧ 𝑧 ∈ 𝑇) → (𝐹‘(^◡𝐹‘𝑧)) = 𝑧)
135	114, 133, 134	syl2an 287	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑈 × 𝑉)) → (𝐹‘(^◡𝐹‘𝑧)) = 𝑧)
136		f1ocnv 5455	. . . . . . . . . . . . 13 ⊢ (𝐹:𝑆–1-1-onto→𝑇 → ^◡𝐹:𝑇–1-1-onto→𝑆)
137		f1of 5442	. . . . . . . . . . . . 13 ⊢ (^◡𝐹:𝑇–1-1-onto→𝑆 → ^◡𝐹:𝑇⟶𝑆)
138	114, 136, 137	3syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → ^◡𝐹:𝑇⟶𝑆)
139		ffvelrn 5629	. . . . . . . . . . . 12 ⊢ ((^◡𝐹:𝑇⟶𝑆 ∧ 𝑧 ∈ 𝑇) → (^◡𝐹‘𝑧) ∈ 𝑆)
140	138, 133, 139	syl2an 287	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑈 × 𝑉)) → (^◡𝐹‘𝑧) ∈ 𝑆)
141	140, 8	eleqtrdi 2263	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑈 × 𝑉)) → (^◡𝐹‘𝑧) ∈ (0..^(𝑀 · 𝑁)))
142		elfzoelz 10103	. . . . . . . . . . . . . . 15 ⊢ ((^◡𝐹‘𝑧) ∈ (0..^(𝑀 · 𝑁)) → (^◡𝐹‘𝑧) ∈ ℤ)
143	141, 142	syl 14	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑈 × 𝑉)) → (^◡𝐹‘𝑧) ∈ ℤ)
144	14	adantr 274	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑈 × 𝑉)) → 𝑀 ∈ ℕ)
145		modgcd 11946	. . . . . . . . . . . . . 14 ⊢ (((^◡𝐹‘𝑧) ∈ ℤ ∧ 𝑀 ∈ ℕ) → (((^◡𝐹‘𝑧) mod 𝑀) gcd 𝑀) = ((^◡𝐹‘𝑧) gcd 𝑀))
146	143, 144, 145	syl2anc 409	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑈 × 𝑉)) → (((^◡𝐹‘𝑧) mod 𝑀) gcd 𝑀) = ((^◡𝐹‘𝑧) gcd 𝑀))
147		zq 9585	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((^◡𝐹‘𝑧) ∈ ℤ → (^◡𝐹‘𝑧) ∈ ℚ)
148	143, 147	syl 14	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑈 × 𝑉)) → (^◡𝐹‘𝑧) ∈ ℚ)
149	144, 16	syl 14	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑈 × 𝑉)) → 𝑀 ∈ ℚ)
150	144	nngt0d 8922	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑈 × 𝑉)) → 0 < 𝑀)
151	148, 149, 150	modqcld 10284	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑈 × 𝑉)) → ((^◡𝐹‘𝑧) mod 𝑀) ∈ ℚ)
152	20	adantr 274	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑈 × 𝑉)) → 𝑁 ∈ ℕ)
153	152, 22	syl 14	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑈 × 𝑉)) → 𝑁 ∈ ℚ)
154	152	nngt0d 8922	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑈 × 𝑉)) → 0 < 𝑁)
155	148, 153, 154	modqcld 10284	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑈 × 𝑉)) → ((^◡𝐹‘𝑧) mod 𝑁) ∈ ℚ)
156		opexg 4213	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((^◡𝐹‘𝑧) mod 𝑀) ∈ ℚ ∧ ((^◡𝐹‘𝑧) mod 𝑁) ∈ ℚ) → ⟨((^◡𝐹‘𝑧) mod 𝑀), ((^◡𝐹‘𝑧) mod 𝑁)⟩ ∈ V)
157	151, 155, 156	syl2anc 409	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑈 × 𝑉)) → ⟨((^◡𝐹‘𝑧) mod 𝑀), ((^◡𝐹‘𝑧) mod 𝑁)⟩ ∈ V)
158		oveq1 5860	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑤 = (^◡𝐹‘𝑧) → (𝑤 mod 𝑀) = ((^◡𝐹‘𝑧) mod 𝑀))
159		oveq1 5860	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑤 = (^◡𝐹‘𝑧) → (𝑤 mod 𝑁) = ((^◡𝐹‘𝑧) mod 𝑁))
160	158, 159	opeq12d 3773	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑤 = (^◡𝐹‘𝑧) → ⟨(𝑤 mod 𝑀), (𝑤 mod 𝑁)⟩ = ⟨((^◡𝐹‘𝑧) mod 𝑀), ((^◡𝐹‘𝑧) mod 𝑁)⟩)
161	30	cbvmptv 4085	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ∈ 𝑆 ↦ ⟨(𝑥 mod 𝑀), (𝑥 mod 𝑁)⟩) = (𝑤 ∈ 𝑆 ↦ ⟨(𝑤 mod 𝑀), (𝑤 mod 𝑁)⟩)
162	31, 161	eqtri 2191	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝐹 = (𝑤 ∈ 𝑆 ↦ ⟨(𝑤 mod 𝑀), (𝑤 mod 𝑁)⟩)
163	160, 162	fvmptg 5572	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((^◡𝐹‘𝑧) ∈ 𝑆 ∧ ⟨((^◡𝐹‘𝑧) mod 𝑀), ((^◡𝐹‘𝑧) mod 𝑁)⟩ ∈ V) → (𝐹‘(^◡𝐹‘𝑧)) = ⟨((^◡𝐹‘𝑧) mod 𝑀), ((^◡𝐹‘𝑧) mod 𝑁)⟩)
164	140, 157, 163	syl2anc 409	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑈 × 𝑉)) → (𝐹‘(^◡𝐹‘𝑧)) = ⟨((^◡𝐹‘𝑧) mod 𝑀), ((^◡𝐹‘𝑧) mod 𝑁)⟩)
165	135, 164	eqtr3d 2205	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑈 × 𝑉)) → 𝑧 = ⟨((^◡𝐹‘𝑧) mod 𝑀), ((^◡𝐹‘𝑧) mod 𝑁)⟩)
166		simpr 109	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑈 × 𝑉)) → 𝑧 ∈ (𝑈 × 𝑉))
167	165, 166	eqeltrrd 2248	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑈 × 𝑉)) → ⟨((^◡𝐹‘𝑧) mod 𝑀), ((^◡𝐹‘𝑧) mod 𝑁)⟩ ∈ (𝑈 × 𝑉))
168		opelxp 4641	. . . . . . . . . . . . . . . . 17 ⊢ (⟨((^◡𝐹‘𝑧) mod 𝑀), ((^◡𝐹‘𝑧) mod 𝑁)⟩ ∈ (𝑈 × 𝑉) ↔ (((^◡𝐹‘𝑧) mod 𝑀) ∈ 𝑈 ∧ ((^◡𝐹‘𝑧) mod 𝑁) ∈ 𝑉))
169	167, 168	sylib 121	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑈 × 𝑉)) → (((^◡𝐹‘𝑧) mod 𝑀) ∈ 𝑈 ∧ ((^◡𝐹‘𝑧) mod 𝑁) ∈ 𝑉))
170	169	simpld 111	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑈 × 𝑉)) → ((^◡𝐹‘𝑧) mod 𝑀) ∈ 𝑈)
171		oveq1 5860	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = ((^◡𝐹‘𝑧) mod 𝑀) → (𝑦 gcd 𝑀) = (((^◡𝐹‘𝑧) mod 𝑀) gcd 𝑀))
172	171	eqeq1d 2179	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = ((^◡𝐹‘𝑧) mod 𝑀) → ((𝑦 gcd 𝑀) = 1 ↔ (((^◡𝐹‘𝑧) mod 𝑀) gcd 𝑀) = 1))
173	172, 73	elrab2 2889	. . . . . . . . . . . . . . 15 ⊢ (((^◡𝐹‘𝑧) mod 𝑀) ∈ 𝑈 ↔ (((^◡𝐹‘𝑧) mod 𝑀) ∈ (0..^𝑀) ∧ (((^◡𝐹‘𝑧) mod 𝑀) gcd 𝑀) = 1))
174	170, 173	sylib 121	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑈 × 𝑉)) → (((^◡𝐹‘𝑧) mod 𝑀) ∈ (0..^𝑀) ∧ (((^◡𝐹‘𝑧) mod 𝑀) gcd 𝑀) = 1))
175	174	simprd 113	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑈 × 𝑉)) → (((^◡𝐹‘𝑧) mod 𝑀) gcd 𝑀) = 1)
176	146, 175	eqtr3d 2205	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑈 × 𝑉)) → ((^◡𝐹‘𝑧) gcd 𝑀) = 1)
177		modgcd 11946	. . . . . . . . . . . . . 14 ⊢ (((^◡𝐹‘𝑧) ∈ ℤ ∧ 𝑁 ∈ ℕ) → (((^◡𝐹‘𝑧) mod 𝑁) gcd 𝑁) = ((^◡𝐹‘𝑧) gcd 𝑁))
178	143, 152, 177	syl2anc 409	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑈 × 𝑉)) → (((^◡𝐹‘𝑧) mod 𝑁) gcd 𝑁) = ((^◡𝐹‘𝑧) gcd 𝑁))
179	169	simprd 113	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑈 × 𝑉)) → ((^◡𝐹‘𝑧) mod 𝑁) ∈ 𝑉)
180		oveq1 5860	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = ((^◡𝐹‘𝑧) mod 𝑁) → (𝑦 gcd 𝑁) = (((^◡𝐹‘𝑧) mod 𝑁) gcd 𝑁))
181	180	eqeq1d 2179	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = ((^◡𝐹‘𝑧) mod 𝑁) → ((𝑦 gcd 𝑁) = 1 ↔ (((^◡𝐹‘𝑧) mod 𝑁) gcd 𝑁) = 1))
182	181, 106	elrab2 2889	. . . . . . . . . . . . . . 15 ⊢ (((^◡𝐹‘𝑧) mod 𝑁) ∈ 𝑉 ↔ (((^◡𝐹‘𝑧) mod 𝑁) ∈ (0..^𝑁) ∧ (((^◡𝐹‘𝑧) mod 𝑁) gcd 𝑁) = 1))
183	179, 182	sylib 121	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑈 × 𝑉)) → (((^◡𝐹‘𝑧) mod 𝑁) ∈ (0..^𝑁) ∧ (((^◡𝐹‘𝑧) mod 𝑁) gcd 𝑁) = 1))
184	183	simprd 113	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑈 × 𝑉)) → (((^◡𝐹‘𝑧) mod 𝑁) gcd 𝑁) = 1)
185	178, 184	eqtr3d 2205	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑈 × 𝑉)) → ((^◡𝐹‘𝑧) gcd 𝑁) = 1)
186	14	nnzd 9333	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑀 ∈ ℤ)
187	186	adantr 274	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑈 × 𝑉)) → 𝑀 ∈ ℤ)
188	20	nnzd 9333	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑁 ∈ ℤ)
189	188	adantr 274	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑈 × 𝑉)) → 𝑁 ∈ ℤ)
190		rpmul 12052	. . . . . . . . . . . . 13 ⊢ (((^◡𝐹‘𝑧) ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((((^◡𝐹‘𝑧) gcd 𝑀) = 1 ∧ ((^◡𝐹‘𝑧) gcd 𝑁) = 1) → ((^◡𝐹‘𝑧) gcd (𝑀 · 𝑁)) = 1))
191	143, 187, 189, 190	syl3anc 1233	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑈 × 𝑉)) → ((((^◡𝐹‘𝑧) gcd 𝑀) = 1 ∧ ((^◡𝐹‘𝑧) gcd 𝑁) = 1) → ((^◡𝐹‘𝑧) gcd (𝑀 · 𝑁)) = 1))
192	176, 185, 191	mp2and 431	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑈 × 𝑉)) → ((^◡𝐹‘𝑧) gcd (𝑀 · 𝑁)) = 1)
193		oveq1 5860	. . . . . . . . . . . . 13 ⊢ (𝑦 = (^◡𝐹‘𝑧) → (𝑦 gcd (𝑀 · 𝑁)) = ((^◡𝐹‘𝑧) gcd (𝑀 · 𝑁)))
194	193	eqeq1d 2179	. . . . . . . . . . . 12 ⊢ (𝑦 = (^◡𝐹‘𝑧) → ((𝑦 gcd (𝑀 · 𝑁)) = 1 ↔ ((^◡𝐹‘𝑧) gcd (𝑀 · 𝑁)) = 1))
195	194, 3	elrab2 2889	. . . . . . . . . . 11 ⊢ ((^◡𝐹‘𝑧) ∈ 𝑊 ↔ ((^◡𝐹‘𝑧) ∈ 𝑆 ∧ ((^◡𝐹‘𝑧) gcd (𝑀 · 𝑁)) = 1))
196	140, 192, 195	sylanbrc 415	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑈 × 𝑉)) → (^◡𝐹‘𝑧) ∈ 𝑊)
197		funfvima2 5728	. . . . . . . . . . . 12 ⊢ ((Fun 𝐹 ∧ 𝑊 ⊆ dom 𝐹) → ((^◡𝐹‘𝑧) ∈ 𝑊 → (𝐹‘(^◡𝐹‘𝑧)) ∈ (𝐹 “ 𝑊)))
198	117, 122, 197	syl2anc 409	. . . . . . . . . . 11 ⊢ (𝜑 → ((^◡𝐹‘𝑧) ∈ 𝑊 → (𝐹‘(^◡𝐹‘𝑧)) ∈ (𝐹 “ 𝑊)))
199	198	imp 123	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (^◡𝐹‘𝑧) ∈ 𝑊) → (𝐹‘(^◡𝐹‘𝑧)) ∈ (𝐹 “ 𝑊))
200	196, 199	syldan 280	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑈 × 𝑉)) → (𝐹‘(^◡𝐹‘𝑧)) ∈ (𝐹 “ 𝑊))
201	135, 200	eqeltrrd 2248	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑈 × 𝑉)) → 𝑧 ∈ (𝐹 “ 𝑊))
202	201	ex 114	. . . . . . 7 ⊢ (𝜑 → (𝑧 ∈ (𝑈 × 𝑉) → 𝑧 ∈ (𝐹 “ 𝑊)))
203	202	ssrdv 3153	. . . . . 6 ⊢ (𝜑 → (𝑈 × 𝑉) ⊆ (𝐹 “ 𝑊))
204	125, 203	eqssd 3164	. . . . 5 ⊢ (𝜑 → (𝐹 “ 𝑊) = (𝑈 × 𝑉))
205		f1of1 5441	. . . . . . 7 ⊢ (𝐹:𝑆–1-1-onto→𝑇 → 𝐹:𝑆–1-1→𝑇)
206	114, 205	syl 14	. . . . . 6 ⊢ (𝜑 → 𝐹:𝑆–1-1→𝑇)
207	119	a1i 9	. . . . . 6 ⊢ (𝜑 → 𝑊 ⊆ 𝑆)
208		0z 9223	. . . . . . . . . 10 ⊢ 0 ∈ ℤ
209	186, 188	zmulcld 9340	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀 · 𝑁) ∈ ℤ)
210		fzofig 10388	. . . . . . . . . 10 ⊢ ((0 ∈ ℤ ∧ (𝑀 · 𝑁) ∈ ℤ) → (0..^(𝑀 · 𝑁)) ∈ Fin)
211	208, 209, 210	sylancr 412	. . . . . . . . 9 ⊢ (𝜑 → (0..^(𝑀 · 𝑁)) ∈ Fin)
212	8, 211	eqeltrid 2257	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ Fin)
213		elfzoelz 10103	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (0..^(𝑀 · 𝑁)) → 𝑦 ∈ ℤ)
214	213, 8	eleq2s 2265	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝑆 → 𝑦 ∈ ℤ)
215	214	adantl 275	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → 𝑦 ∈ ℤ)
216	209	adantr 274	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (𝑀 · 𝑁) ∈ ℤ)
217	215, 216	gcdcld 11923	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (𝑦 gcd (𝑀 · 𝑁)) ∈ ℕ₀)
218	217	nn0zd 9332	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (𝑦 gcd (𝑀 · 𝑁)) ∈ ℤ)
219		1zzd 9239	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → 1 ∈ ℤ)
220		zdceq 9287	. . . . . . . . . 10 ⊢ (((𝑦 gcd (𝑀 · 𝑁)) ∈ ℤ ∧ 1 ∈ ℤ) → DECID (𝑦 gcd (𝑀 · 𝑁)) = 1)
221	218, 219, 220	syl2anc 409	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → DECID (𝑦 gcd (𝑀 · 𝑁)) = 1)
222	221	ralrimiva 2543	. . . . . . . 8 ⊢ (𝜑 → ∀𝑦 ∈ 𝑆 DECID (𝑦 gcd (𝑀 · 𝑁)) = 1)
223	212, 222	ssfirab 6911	. . . . . . 7 ⊢ (𝜑 → {𝑦 ∈ 𝑆 ∣ (𝑦 gcd (𝑀 · 𝑁)) = 1} ∈ Fin)
224	3, 223	eqeltrid 2257	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ Fin)
225		f1imaeng 6770	. . . . . 6 ⊢ ((𝐹:𝑆–1-1→𝑇 ∧ 𝑊 ⊆ 𝑆 ∧ 𝑊 ∈ Fin) → (𝐹 “ 𝑊) ≈ 𝑊)
226	206, 207, 224, 225	syl3anc 1233	. . . . 5 ⊢ (𝜑 → (𝐹 “ 𝑊) ≈ 𝑊)
227	204, 226	eqbrtrrd 4013	. . . 4 ⊢ (𝜑 → (𝑈 × 𝑉) ≈ 𝑊)
228		fzofig 10388	. . . . . . . . 9 ⊢ ((0 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (0..^𝑀) ∈ Fin)
229	208, 186, 228	sylancr 412	. . . . . . . 8 ⊢ (𝜑 → (0..^𝑀) ∈ Fin)
230		elfzoelz 10103	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (0..^𝑀) → 𝑦 ∈ ℤ)
231	230	adantl 275	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (0..^𝑀)) → 𝑦 ∈ ℤ)
232	186	adantr 274	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (0..^𝑀)) → 𝑀 ∈ ℤ)
233	231, 232	gcdcld 11923	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (0..^𝑀)) → (𝑦 gcd 𝑀) ∈ ℕ₀)
234	233	nn0zd 9332	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (0..^𝑀)) → (𝑦 gcd 𝑀) ∈ ℤ)
235		1z 9238	. . . . . . . . . 10 ⊢ 1 ∈ ℤ
236		zdceq 9287	. . . . . . . . . 10 ⊢ (((𝑦 gcd 𝑀) ∈ ℤ ∧ 1 ∈ ℤ) → DECID (𝑦 gcd 𝑀) = 1)
237	234, 235, 236	sylancl 411	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (0..^𝑀)) → DECID (𝑦 gcd 𝑀) = 1)
238	237	ralrimiva 2543	. . . . . . . 8 ⊢ (𝜑 → ∀𝑦 ∈ (0..^𝑀)DECID (𝑦 gcd 𝑀) = 1)
239	229, 238	ssfirab 6911	. . . . . . 7 ⊢ (𝜑 → {𝑦 ∈ (0..^𝑀) ∣ (𝑦 gcd 𝑀) = 1} ∈ Fin)
240	73, 239	eqeltrid 2257	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ Fin)
241		fzofig 10388	. . . . . . . . 9 ⊢ ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (0..^𝑁) ∈ Fin)
242	208, 188, 241	sylancr 412	. . . . . . . 8 ⊢ (𝜑 → (0..^𝑁) ∈ Fin)
243		elfzoelz 10103	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (0..^𝑁) → 𝑦 ∈ ℤ)
244	243	adantl 275	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (0..^𝑁)) → 𝑦 ∈ ℤ)
245	188	adantr 274	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (0..^𝑁)) → 𝑁 ∈ ℤ)
246	244, 245	gcdcld 11923	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (0..^𝑁)) → (𝑦 gcd 𝑁) ∈ ℕ₀)
247	246	nn0zd 9332	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (0..^𝑁)) → (𝑦 gcd 𝑁) ∈ ℤ)
248		1zzd 9239	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (0..^𝑁)) → 1 ∈ ℤ)
249		zdceq 9287	. . . . . . . . . 10 ⊢ (((𝑦 gcd 𝑁) ∈ ℤ ∧ 1 ∈ ℤ) → DECID (𝑦 gcd 𝑁) = 1)
250	247, 248, 249	syl2anc 409	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (0..^𝑁)) → DECID (𝑦 gcd 𝑁) = 1)
251	250	ralrimiva 2543	. . . . . . . 8 ⊢ (𝜑 → ∀𝑦 ∈ (0..^𝑁)DECID (𝑦 gcd 𝑁) = 1)
252	242, 251	ssfirab 6911	. . . . . . 7 ⊢ (𝜑 → {𝑦 ∈ (0..^𝑁) ∣ (𝑦 gcd 𝑁) = 1} ∈ Fin)
253	106, 252	eqeltrid 2257	. . . . . 6 ⊢ (𝜑 → 𝑉 ∈ Fin)
254		xpfi 6907	. . . . . 6 ⊢ ((𝑈 ∈ Fin ∧ 𝑉 ∈ Fin) → (𝑈 × 𝑉) ∈ Fin)
255	240, 253, 254	syl2anc 409	. . . . 5 ⊢ (𝜑 → (𝑈 × 𝑉) ∈ Fin)
256		hashen 10718	. . . . 5 ⊢ (((𝑈 × 𝑉) ∈ Fin ∧ 𝑊 ∈ Fin) → ((♯‘(𝑈 × 𝑉)) = (♯‘𝑊) ↔ (𝑈 × 𝑉) ≈ 𝑊))
257	255, 224, 256	syl2anc 409	. . . 4 ⊢ (𝜑 → ((♯‘(𝑈 × 𝑉)) = (♯‘𝑊) ↔ (𝑈 × 𝑉) ≈ 𝑊))
258	227, 257	mpbird 166	. . 3 ⊢ (𝜑 → (♯‘(𝑈 × 𝑉)) = (♯‘𝑊))
259		hashxp 10761	. . . 4 ⊢ ((𝑈 ∈ Fin ∧ 𝑉 ∈ Fin) → (♯‘(𝑈 × 𝑉)) = ((♯‘𝑈) · (♯‘𝑉)))
260	240, 253, 259	syl2anc 409	. . 3 ⊢ (𝜑 → (♯‘(𝑈 × 𝑉)) = ((♯‘𝑈) · (♯‘𝑉)))
261	258, 260	eqtr3d 2205	. 2 ⊢ (𝜑 → (♯‘𝑊) = ((♯‘𝑈) · (♯‘𝑉)))
262	14, 20	nnmulcld 8927	. . 3 ⊢ (𝜑 → (𝑀 · 𝑁) ∈ ℕ)
263		dfphi2 12174	. . . 4 ⊢ ((𝑀 · 𝑁) ∈ ℕ → (ϕ‘(𝑀 · 𝑁)) = (♯‘{𝑦 ∈ (0..^(𝑀 · 𝑁)) ∣ (𝑦 gcd (𝑀 · 𝑁)) = 1}))
264	8	rabeqi 2723	. . . . . 6 ⊢ {𝑦 ∈ 𝑆 ∣ (𝑦 gcd (𝑀 · 𝑁)) = 1} = {𝑦 ∈ (0..^(𝑀 · 𝑁)) ∣ (𝑦 gcd (𝑀 · 𝑁)) = 1}
265	3, 264	eqtri 2191	. . . . 5 ⊢ 𝑊 = {𝑦 ∈ (0..^(𝑀 · 𝑁)) ∣ (𝑦 gcd (𝑀 · 𝑁)) = 1}
266	265	fveq2i 5499	. . . 4 ⊢ (♯‘𝑊) = (♯‘{𝑦 ∈ (0..^(𝑀 · 𝑁)) ∣ (𝑦 gcd (𝑀 · 𝑁)) = 1})
267	263, 266	eqtr4di 2221	. . 3 ⊢ ((𝑀 · 𝑁) ∈ ℕ → (ϕ‘(𝑀 · 𝑁)) = (♯‘𝑊))
268	262, 267	syl 14	. 2 ⊢ (𝜑 → (ϕ‘(𝑀 · 𝑁)) = (♯‘𝑊))
269		dfphi2 12174	. . . . 5 ⊢ (𝑀 ∈ ℕ → (ϕ‘𝑀) = (♯‘{𝑦 ∈ (0..^𝑀) ∣ (𝑦 gcd 𝑀) = 1}))
270	73	fveq2i 5499	. . . . 5 ⊢ (♯‘𝑈) = (♯‘{𝑦 ∈ (0..^𝑀) ∣ (𝑦 gcd 𝑀) = 1})
271	269, 270	eqtr4di 2221	. . . 4 ⊢ (𝑀 ∈ ℕ → (ϕ‘𝑀) = (♯‘𝑈))
272	14, 271	syl 14	. . 3 ⊢ (𝜑 → (ϕ‘𝑀) = (♯‘𝑈))
273		dfphi2 12174	. . . . 5 ⊢ (𝑁 ∈ ℕ → (ϕ‘𝑁) = (♯‘{𝑦 ∈ (0..^𝑁) ∣ (𝑦 gcd 𝑁) = 1}))
274	106	fveq2i 5499	. . . . 5 ⊢ (♯‘𝑉) = (♯‘{𝑦 ∈ (0..^𝑁) ∣ (𝑦 gcd 𝑁) = 1})
275	273, 274	eqtr4di 2221	. . . 4 ⊢ (𝑁 ∈ ℕ → (ϕ‘𝑁) = (♯‘𝑉))
276	20, 275	syl 14	. . 3 ⊢ (𝜑 → (ϕ‘𝑁) = (♯‘𝑉))
277	272, 276	oveq12d 5871	. 2 ⊢ (𝜑 → ((ϕ‘𝑀) · (ϕ‘𝑁)) = ((♯‘𝑈) · (♯‘𝑉)))
278	261, 268, 277	3eqtr4d 2213	1 ⊢ (𝜑 → (ϕ‘(𝑀 · 𝑁)) = ((ϕ‘𝑀) · (ϕ‘𝑁)))

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ↔ wb 104 DECID wdc 829 ∧ w3a 973 = wceq 1348 ∈ wcel 2141 ≠ wne 2340 ∀wral 2448 {crab 2452 Vcvv 2730 ⊆ wss 3121 ⟨cop 3586 class class class wbr 3989 ↦ cmpt 4050 × cxp 4609 ^◡ccnv 4610 dom cdm 4611 “ cima 4614 Fun wfun 5192 Fn wfn 5193 ⟶wf 5194 –1-1→wf1 5195 –1-1-onto→wf1o 5197 ‘cfv 5198 (class class class)co 5853 ≈ cen 6716 Fincfn 6718 0cc0 7774 1c1 7775 · cmul 7779 ≤ cle 7955 ℕcn 8878 ℤcz 9212 ℚcq 9578 ..^cfzo 10098 mod cmo 10278 ♯chash 10709 ∥ cdvds 11749 gcd cgcd 11897 ϕcphi 12163

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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-mulrcl 7873 ax-addcom 7874 ax-mulcom 7875 ax-addass 7876 ax-mulass 7877 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-1rid 7881 ax-0id 7882 ax-rnegex 7883 ax-precex 7884 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-apti 7889 ax-pre-ltadd 7890 ax-pre-mulgt0 7891 ax-pre-mulext 7892 ax-arch 7893 ax-caucvg 7894

This theorem depends on definitions: df-bi 116 df-stab 826 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-if 3527 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-id 4278 df-po 4281 df-iso 4282 df-iord 4351 df-on 4353 df-ilim 4354 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-recs 6284 df-irdg 6349 df-frec 6370 df-1o 6395 df-oadd 6399 df-er 6513 df-en 6719 df-dom 6720 df-fin 6721 df-sup 6961 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-reap 8494 df-ap 8501 df-div 8590 df-inn 8879 df-2 8937 df-3 8938 df-4 8939 df-n0 9136 df-z 9213 df-uz 9488 df-q 9579 df-rp 9611 df-fz 9966 df-fzo 10099 df-fl 10226 df-mod 10279 df-seqfrec 10402 df-exp 10476 df-ihash 10710 df-cj 10806 df-re 10807 df-im 10808 df-rsqrt 10962 df-abs 10963 df-dvds 11750 df-gcd 11898 df-phi 12165

This theorem is referenced by: phimul 12180

W3C validator