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Theorem crth 11541

Description: The Chinese Remainder Theorem: the function that maps 𝑥 to its remainder classes mod 𝑀 and mod 𝑁 is 1-1 and onto when 𝑀 and 𝑁 are coprime. (Contributed by Mario Carneiro, 24-Feb-2014.) (Proof shortened by Mario Carneiro, 2-May-2016.)

**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))

**Assertion**
Ref	Expression
crth	⊢ (𝜑 → 𝐹:𝑆–1-1-onto→𝑇)

Distinct variable groups: 𝑥,𝑀 𝑥,𝑁 𝑥,𝑆 𝑥,𝑇 𝜑,𝑥 Allowed substitution hint: 𝐹(𝑥)

Proof of Theorem crth Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elfzoelz 9621	. . . . . 6 ⊢ (𝑥 ∈ (0..^(𝑀 · 𝑁)) → 𝑥 ∈ ℤ)
2		crth.1	. . . . . 6 ⊢ 𝑆 = (0..^(𝑀 · 𝑁))
3	1, 2	eleq2s 2183	. . . . 5 ⊢ (𝑥 ∈ 𝑆 → 𝑥 ∈ ℤ)
4		simpr 109	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℤ) → 𝑥 ∈ ℤ)
5		crth.4	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1))
6	5	simp1d 956	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℕ)
7	6	adantr 271	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℤ) → 𝑀 ∈ ℕ)
8		zmodfzo 9817	. . . . . . . 8 ⊢ ((𝑥 ∈ ℤ ∧ 𝑀 ∈ ℕ) → (𝑥 mod 𝑀) ∈ (0..^𝑀))
9	4, 7, 8	syl2anc 404	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℤ) → (𝑥 mod 𝑀) ∈ (0..^𝑀))
10	5	simp2d 957	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℕ)
11	10	adantr 271	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℤ) → 𝑁 ∈ ℕ)
12		zmodfzo 9817	. . . . . . . 8 ⊢ ((𝑥 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝑥 mod 𝑁) ∈ (0..^𝑁))
13	4, 11, 12	syl2anc 404	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℤ) → (𝑥 mod 𝑁) ∈ (0..^𝑁))
14		opelxpi 4485	. . . . . . 7 ⊢ (((𝑥 mod 𝑀) ∈ (0..^𝑀) ∧ (𝑥 mod 𝑁) ∈ (0..^𝑁)) → ⟨(𝑥 mod 𝑀), (𝑥 mod 𝑁)⟩ ∈ ((0..^𝑀) × (0..^𝑁)))
15	9, 13, 14	syl2anc 404	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℤ) → ⟨(𝑥 mod 𝑀), (𝑥 mod 𝑁)⟩ ∈ ((0..^𝑀) × (0..^𝑁)))
16		crth.2	. . . . . 6 ⊢ 𝑇 = ((0..^𝑀) × (0..^𝑁))
17	15, 16	syl6eleqr 2182	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℤ) → ⟨(𝑥 mod 𝑀), (𝑥 mod 𝑁)⟩ ∈ 𝑇)
18	3, 17	sylan2 281	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ⟨(𝑥 mod 𝑀), (𝑥 mod 𝑁)⟩ ∈ 𝑇)
19		crth.3	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝑆 ↦ ⟨(𝑥 mod 𝑀), (𝑥 mod 𝑁)⟩)
20	18, 19	fmptd 5468	. . 3 ⊢ (𝜑 → 𝐹:𝑆⟶𝑇)
21		simprl 499	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → 𝑦 ∈ 𝑆)
22		elfzoelz 9621	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (0..^(𝑀 · 𝑁)) → 𝑦 ∈ ℤ)
23	22, 2	eleq2s 2183	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ 𝑆 → 𝑦 ∈ ℤ)
24	23	ad2antrl 475	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → 𝑦 ∈ ℤ)
25		zq 9174	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℤ → 𝑦 ∈ ℚ)
26	24, 25	syl 14	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → 𝑦 ∈ ℚ)
27	6	adantr 271	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → 𝑀 ∈ ℕ)
28		nnq 9181	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℚ)
29	27, 28	syl 14	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → 𝑀 ∈ ℚ)
30	27	nngt0d 8529	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → 0 < 𝑀)
31	26, 29, 30	modqcld 9798	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑦 mod 𝑀) ∈ ℚ)
32	10	adantr 271	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → 𝑁 ∈ ℕ)
33		nnq 9181	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℚ)
34	32, 33	syl 14	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → 𝑁 ∈ ℚ)
35	32	nngt0d 8529	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → 0 < 𝑁)
36	26, 34, 35	modqcld 9798	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑦 mod 𝑁) ∈ ℚ)
37		opexg 4066	. . . . . . . . 9 ⊢ (((𝑦 mod 𝑀) ∈ ℚ ∧ (𝑦 mod 𝑁) ∈ ℚ) → ⟨(𝑦 mod 𝑀), (𝑦 mod 𝑁)⟩ ∈ V)
38	31, 36, 37	syl2anc 404	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ⟨(𝑦 mod 𝑀), (𝑦 mod 𝑁)⟩ ∈ V)
39		oveq1 5675	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑥 mod 𝑀) = (𝑦 mod 𝑀))
40		oveq1 5675	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑥 mod 𝑁) = (𝑦 mod 𝑁))
41	39, 40	opeq12d 3638	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ⟨(𝑥 mod 𝑀), (𝑥 mod 𝑁)⟩ = ⟨(𝑦 mod 𝑀), (𝑦 mod 𝑁)⟩)
42	41, 19	fvmptg 5395	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝑆 ∧ ⟨(𝑦 mod 𝑀), (𝑦 mod 𝑁)⟩ ∈ V) → (𝐹‘𝑦) = ⟨(𝑦 mod 𝑀), (𝑦 mod 𝑁)⟩)
43	21, 38, 42	syl2anc 404	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝐹‘𝑦) = ⟨(𝑦 mod 𝑀), (𝑦 mod 𝑁)⟩)
44		simprr 500	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → 𝑧 ∈ 𝑆)
45		elfzoelz 9621	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ (0..^(𝑀 · 𝑁)) → 𝑧 ∈ ℤ)
46	45, 2	eleq2s 2183	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ 𝑆 → 𝑧 ∈ ℤ)
47	44, 46	syl 14	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → 𝑧 ∈ ℤ)
48		zq 9174	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ℤ → 𝑧 ∈ ℚ)
49	47, 48	syl 14	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → 𝑧 ∈ ℚ)
50	49, 29, 30	modqcld 9798	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑧 mod 𝑀) ∈ ℚ)
51	49, 34, 35	modqcld 9798	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑧 mod 𝑁) ∈ ℚ)
52		opexg 4066	. . . . . . . . 9 ⊢ (((𝑧 mod 𝑀) ∈ ℚ ∧ (𝑧 mod 𝑁) ∈ ℚ) → ⟨(𝑧 mod 𝑀), (𝑧 mod 𝑁)⟩ ∈ V)
53	50, 51, 52	syl2anc 404	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ⟨(𝑧 mod 𝑀), (𝑧 mod 𝑁)⟩ ∈ V)
54		oveq1 5675	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (𝑥 mod 𝑀) = (𝑧 mod 𝑀))
55		oveq1 5675	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (𝑥 mod 𝑁) = (𝑧 mod 𝑁))
56	54, 55	opeq12d 3638	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → ⟨(𝑥 mod 𝑀), (𝑥 mod 𝑁)⟩ = ⟨(𝑧 mod 𝑀), (𝑧 mod 𝑁)⟩)
57	56, 19	fvmptg 5395	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝑆 ∧ ⟨(𝑧 mod 𝑀), (𝑧 mod 𝑁)⟩ ∈ V) → (𝐹‘𝑧) = ⟨(𝑧 mod 𝑀), (𝑧 mod 𝑁)⟩)
58	44, 53, 57	syl2anc 404	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝐹‘𝑧) = ⟨(𝑧 mod 𝑀), (𝑧 mod 𝑁)⟩)
59	43, 58	eqeq12d 2103	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝐹‘𝑦) = (𝐹‘𝑧) ↔ ⟨(𝑦 mod 𝑀), (𝑦 mod 𝑁)⟩ = ⟨(𝑧 mod 𝑀), (𝑧 mod 𝑁)⟩))
60		opthg 4076	. . . . . . 7 ⊢ (((𝑦 mod 𝑀) ∈ ℚ ∧ (𝑦 mod 𝑁) ∈ ℚ) → (⟨(𝑦 mod 𝑀), (𝑦 mod 𝑁)⟩ = ⟨(𝑧 mod 𝑀), (𝑧 mod 𝑁)⟩ ↔ ((𝑦 mod 𝑀) = (𝑧 mod 𝑀) ∧ (𝑦 mod 𝑁) = (𝑧 mod 𝑁))))
61	31, 36, 60	syl2anc 404	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (⟨(𝑦 mod 𝑀), (𝑦 mod 𝑁)⟩ = ⟨(𝑧 mod 𝑀), (𝑧 mod 𝑁)⟩ ↔ ((𝑦 mod 𝑀) = (𝑧 mod 𝑀) ∧ (𝑦 mod 𝑁) = (𝑧 mod 𝑁))))
62	59, 61	bitrd 187	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝐹‘𝑦) = (𝐹‘𝑧) ↔ ((𝑦 mod 𝑀) = (𝑧 mod 𝑀) ∧ (𝑦 mod 𝑁) = (𝑧 mod 𝑁))))
63	27	nnzd 8930	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → 𝑀 ∈ ℤ)
64	32	nnzd 8930	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → 𝑁 ∈ ℤ)
65	21, 2	syl6eleq 2181	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → 𝑦 ∈ (0..^(𝑀 · 𝑁)))
66	65, 22	syl 14	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → 𝑦 ∈ ℤ)
67	44, 2	syl6eleq 2181	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → 𝑧 ∈ (0..^(𝑀 · 𝑁)))
68	67, 45	syl 14	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → 𝑧 ∈ ℤ)
69	66, 68	zsubcld 8936	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑦 − 𝑧) ∈ ℤ)
70	5	simp3d 958	. . . . . . . 8 ⊢ (𝜑 → (𝑀 gcd 𝑁) = 1)
71	70	adantr 271	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑀 gcd 𝑁) = 1)
72		coprmdvds2 11416	. . . . . . 7 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝑦 − 𝑧) ∈ ℤ) ∧ (𝑀 gcd 𝑁) = 1) → ((𝑀 ∥ (𝑦 − 𝑧) ∧ 𝑁 ∥ (𝑦 − 𝑧)) → (𝑀 · 𝑁) ∥ (𝑦 − 𝑧)))
73	63, 64, 69, 71, 72	syl31anc 1178	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑀 ∥ (𝑦 − 𝑧) ∧ 𝑁 ∥ (𝑦 − 𝑧)) → (𝑀 · 𝑁) ∥ (𝑦 − 𝑧)))
74		moddvds 11146	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ ∧ 𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ) → ((𝑦 mod 𝑀) = (𝑧 mod 𝑀) ↔ 𝑀 ∥ (𝑦 − 𝑧)))
75	27, 66, 68, 74	syl3anc 1175	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑦 mod 𝑀) = (𝑧 mod 𝑀) ↔ 𝑀 ∥ (𝑦 − 𝑧)))
76		moddvds 11146	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ) → ((𝑦 mod 𝑁) = (𝑧 mod 𝑁) ↔ 𝑁 ∥ (𝑦 − 𝑧)))
77	32, 66, 68, 76	syl3anc 1175	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑦 mod 𝑁) = (𝑧 mod 𝑁) ↔ 𝑁 ∥ (𝑦 − 𝑧)))
78	75, 77	anbi12d 458	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (((𝑦 mod 𝑀) = (𝑧 mod 𝑀) ∧ (𝑦 mod 𝑁) = (𝑧 mod 𝑁)) ↔ (𝑀 ∥ (𝑦 − 𝑧) ∧ 𝑁 ∥ (𝑦 − 𝑧))))
79		qmulcl 9185	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℚ ∧ 𝑁 ∈ ℚ) → (𝑀 · 𝑁) ∈ ℚ)
80	29, 34, 79	syl2anc 404	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑀 · 𝑁) ∈ ℚ)
81		elfzole1 9629	. . . . . . . . . 10 ⊢ (𝑦 ∈ (0..^(𝑀 · 𝑁)) → 0 ≤ 𝑦)
82	65, 81	syl 14	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → 0 ≤ 𝑦)
83		elfzolt2 9630	. . . . . . . . . 10 ⊢ (𝑦 ∈ (0..^(𝑀 · 𝑁)) → 𝑦 < (𝑀 · 𝑁))
84	65, 83	syl 14	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → 𝑦 < (𝑀 · 𝑁))
85		modqid 9819	. . . . . . . . 9 ⊢ (((𝑦 ∈ ℚ ∧ (𝑀 · 𝑁) ∈ ℚ) ∧ (0 ≤ 𝑦 ∧ 𝑦 < (𝑀 · 𝑁))) → (𝑦 mod (𝑀 · 𝑁)) = 𝑦)
86	26, 80, 82, 84, 85	syl22anc 1176	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑦 mod (𝑀 · 𝑁)) = 𝑦)
87		elfzole1 9629	. . . . . . . . . 10 ⊢ (𝑧 ∈ (0..^(𝑀 · 𝑁)) → 0 ≤ 𝑧)
88	67, 87	syl 14	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → 0 ≤ 𝑧)
89		elfzolt2 9630	. . . . . . . . . 10 ⊢ (𝑧 ∈ (0..^(𝑀 · 𝑁)) → 𝑧 < (𝑀 · 𝑁))
90	67, 89	syl 14	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → 𝑧 < (𝑀 · 𝑁))
91		modqid 9819	. . . . . . . . 9 ⊢ (((𝑧 ∈ ℚ ∧ (𝑀 · 𝑁) ∈ ℚ) ∧ (0 ≤ 𝑧 ∧ 𝑧 < (𝑀 · 𝑁))) → (𝑧 mod (𝑀 · 𝑁)) = 𝑧)
92	49, 80, 88, 90, 91	syl22anc 1176	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑧 mod (𝑀 · 𝑁)) = 𝑧)
93	86, 92	eqeq12d 2103	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑦 mod (𝑀 · 𝑁)) = (𝑧 mod (𝑀 · 𝑁)) ↔ 𝑦 = 𝑧))
94	27, 32	nnmulcld 8534	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑀 · 𝑁) ∈ ℕ)
95		moddvds 11146	. . . . . . . 8 ⊢ (((𝑀 · 𝑁) ∈ ℕ ∧ 𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ) → ((𝑦 mod (𝑀 · 𝑁)) = (𝑧 mod (𝑀 · 𝑁)) ↔ (𝑀 · 𝑁) ∥ (𝑦 − 𝑧)))
96	94, 66, 68, 95	syl3anc 1175	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑦 mod (𝑀 · 𝑁)) = (𝑧 mod (𝑀 · 𝑁)) ↔ (𝑀 · 𝑁) ∥ (𝑦 − 𝑧)))
97	93, 96	bitr3d 189	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑦 = 𝑧 ↔ (𝑀 · 𝑁) ∥ (𝑦 − 𝑧)))
98	73, 78, 97	3imtr4d 202	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (((𝑦 mod 𝑀) = (𝑧 mod 𝑀) ∧ (𝑦 mod 𝑁) = (𝑧 mod 𝑁)) → 𝑦 = 𝑧))
99	62, 98	sylbid 149	. . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝐹‘𝑦) = (𝐹‘𝑧) → 𝑦 = 𝑧))
100	99	ralrimivva 2456	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ((𝐹‘𝑦) = (𝐹‘𝑧) → 𝑦 = 𝑧))
101		dff13 5563	. . 3 ⊢ (𝐹:𝑆–1-1→𝑇 ↔ (𝐹:𝑆⟶𝑇 ∧ ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ((𝐹‘𝑦) = (𝐹‘𝑧) → 𝑦 = 𝑧)))
102	20, 100, 101	sylanbrc 409	. 2 ⊢ (𝜑 → 𝐹:𝑆–1-1→𝑇)
103		nnnn0 8743	. . . . . 6 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℕ₀)
104		nnnn0 8743	. . . . . 6 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ₀)
105		hashfzo0 10294	. . . . . . . . 9 ⊢ (𝑀 ∈ ℕ₀ → (♯‘(0..^𝑀)) = 𝑀)
106		hashfzo0 10294	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ₀ → (♯‘(0..^𝑁)) = 𝑁)
107	105, 106	oveqan12d 5687	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → ((♯‘(0..^𝑀)) · (♯‘(0..^𝑁))) = (𝑀 · 𝑁))
108		0z 8824	. . . . . . . . . 10 ⊢ 0 ∈ ℤ
109		simpl 108	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → 𝑀 ∈ ℕ₀)
110	109	nn0zd 8929	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → 𝑀 ∈ ℤ)
111		fzofig 9902	. . . . . . . . . 10 ⊢ ((0 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (0..^𝑀) ∈ Fin)
112	108, 110, 111	sylancr 406	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (0..^𝑀) ∈ Fin)
113		nn0z 8833	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℤ)
114	113	adantl 272	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → 𝑁 ∈ ℤ)
115		fzofig 9902	. . . . . . . . . 10 ⊢ ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (0..^𝑁) ∈ Fin)
116	108, 114, 115	sylancr 406	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (0..^𝑁) ∈ Fin)
117		hashxp 10297	. . . . . . . . 9 ⊢ (((0..^𝑀) ∈ Fin ∧ (0..^𝑁) ∈ Fin) → (♯‘((0..^𝑀) × (0..^𝑁))) = ((♯‘(0..^𝑀)) · (♯‘(0..^𝑁))))
118	112, 116, 117	syl2anc 404	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (♯‘((0..^𝑀) × (0..^𝑁))) = ((♯‘(0..^𝑀)) · (♯‘(0..^𝑁))))
119		nn0mulcl 8772	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝑀 · 𝑁) ∈ ℕ₀)
120		hashfzo0 10294	. . . . . . . . 9 ⊢ ((𝑀 · 𝑁) ∈ ℕ₀ → (♯‘(0..^(𝑀 · 𝑁))) = (𝑀 · 𝑁))
121	119, 120	syl 14	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (♯‘(0..^(𝑀 · 𝑁))) = (𝑀 · 𝑁))
122	107, 118, 121	3eqtr4rd 2132	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (♯‘(0..^(𝑀 · 𝑁))) = (♯‘((0..^𝑀) × (0..^𝑁))))
123	119	nn0zd 8929	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝑀 · 𝑁) ∈ ℤ)
124		fzofig 9902	. . . . . . . . 9 ⊢ ((0 ∈ ℤ ∧ (𝑀 · 𝑁) ∈ ℤ) → (0..^(𝑀 · 𝑁)) ∈ Fin)
125	108, 123, 124	sylancr 406	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (0..^(𝑀 · 𝑁)) ∈ Fin)
126		xpfi 6696	. . . . . . . . 9 ⊢ (((0..^𝑀) ∈ Fin ∧ (0..^𝑁) ∈ Fin) → ((0..^𝑀) × (0..^𝑁)) ∈ Fin)
127	112, 116, 126	syl2anc 404	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → ((0..^𝑀) × (0..^𝑁)) ∈ Fin)
128		hashen 10255	. . . . . . . 8 ⊢ (((0..^(𝑀 · 𝑁)) ∈ Fin ∧ ((0..^𝑀) × (0..^𝑁)) ∈ Fin) → ((♯‘(0..^(𝑀 · 𝑁))) = (♯‘((0..^𝑀) × (0..^𝑁))) ↔ (0..^(𝑀 · 𝑁)) ≈ ((0..^𝑀) × (0..^𝑁))))
129	125, 127, 128	syl2anc 404	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → ((♯‘(0..^(𝑀 · 𝑁))) = (♯‘((0..^𝑀) × (0..^𝑁))) ↔ (0..^(𝑀 · 𝑁)) ≈ ((0..^𝑀) × (0..^𝑁))))
130	122, 129	mpbid 146	. . . . . 6 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (0..^(𝑀 · 𝑁)) ≈ ((0..^𝑀) × (0..^𝑁)))
131	103, 104, 130	syl2an 284	. . . . 5 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (0..^(𝑀 · 𝑁)) ≈ ((0..^𝑀) × (0..^𝑁)))
132	6, 10, 131	syl2anc 404	. . . 4 ⊢ (𝜑 → (0..^(𝑀 · 𝑁)) ≈ ((0..^𝑀) × (0..^𝑁)))
133	132, 2, 16	3brtr4g 3885	. . 3 ⊢ (𝜑 → 𝑆 ≈ 𝑇)
134	6	nnnn0d 8789	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
135	10	nnnn0d 8789	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
136	134, 135, 127	syl2anc 404	. . . 4 ⊢ (𝜑 → ((0..^𝑀) × (0..^𝑁)) ∈ Fin)
137	16, 136	syl5eqel 2175	. . 3 ⊢ (𝜑 → 𝑇 ∈ Fin)
138		f1finf1o 6712	. . 3 ⊢ ((𝑆 ≈ 𝑇 ∧ 𝑇 ∈ Fin) → (𝐹:𝑆–1-1→𝑇 ↔ 𝐹:𝑆–1-1-onto→𝑇))
139	133, 137, 138	syl2anc 404	. 2 ⊢ (𝜑 → (𝐹:𝑆–1-1→𝑇 ↔ 𝐹:𝑆–1-1-onto→𝑇))
140	102, 139	mpbid 146	1 ⊢ (𝜑 → 𝐹:𝑆–1-1-onto→𝑇)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∧ w3a 925 = wceq 1290 ∈ wcel 1439 ∀wral 2360 Vcvv 2622 ⟨cop 3455 class class class wbr 3853 ↦ cmpt 3907 × cxp 4452 ⟶wf 5026 –1-1→wf1 5027 –1-1-onto→wf1o 5029 ‘cfv 5030 (class class class)co 5668 ≈ cen 6511 Fincfn 6513 0cc0 7413 1c1 7414 · cmul 7418 < clt 7585 ≤ cle 7586 − cmin 7716 ℕcn 8485 ℕ₀cn0 8736 ℤcz 8813 ℚcq 9167 ..^cfzo 9616 mod cmo 9792 ♯chash 10246 ∥ cdvds 11137 gcd cgcd 11279

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-13 1450 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-coll 3962 ax-sep 3965 ax-nul 3973 ax-pow 4017 ax-pr 4047 ax-un 4271 ax-setind 4368 ax-iinf 4418 ax-cnex 7499 ax-resscn 7500 ax-1cn 7501 ax-1re 7502 ax-icn 7503 ax-addcl 7504 ax-addrcl 7505 ax-mulcl 7506 ax-mulrcl 7507 ax-addcom 7508 ax-mulcom 7509 ax-addass 7510 ax-mulass 7511 ax-distr 7512 ax-i2m1 7513 ax-0lt1 7514 ax-1rid 7515 ax-0id 7516 ax-rnegex 7517 ax-precex 7518 ax-cnre 7519 ax-pre-ltirr 7520 ax-pre-ltwlin 7521 ax-pre-lttrn 7522 ax-pre-apti 7523 ax-pre-ltadd 7524 ax-pre-mulgt0 7525 ax-pre-mulext 7526 ax-arch 7527 ax-caucvg 7528

This theorem depends on definitions: df-bi 116 df-dc 782 df-3or 926 df-3an 927 df-tru 1293 df-fal 1296 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ne 2257 df-nel 2352 df-ral 2365 df-rex 2366 df-reu 2367 df-rmo 2368 df-rab 2369 df-v 2624 df-sbc 2844 df-csb 2937 df-dif 3004 df-un 3006 df-in 3008 df-ss 3015 df-nul 3290 df-if 3400 df-pw 3437 df-sn 3458 df-pr 3459 df-op 3461 df-uni 3662 df-int 3697 df-iun 3740 df-br 3854 df-opab 3908 df-mpt 3909 df-tr 3945 df-id 4131 df-po 4134 df-iso 4135 df-iord 4204 df-on 4206 df-ilim 4207 df-suc 4209 df-iom 4421 df-xp 4460 df-rel 4461 df-cnv 4462 df-co 4463 df-dm 4464 df-rn 4465 df-res 4466 df-ima 4467 df-iota 4995 df-fun 5032 df-fn 5033 df-f 5034 df-f1 5035 df-fo 5036 df-f1o 5037 df-fv 5038 df-riota 5624 df-ov 5671 df-oprab 5672 df-mpt2 5673 df-1st 5927 df-2nd 5928 df-recs 6086 df-irdg 6151 df-frec 6172 df-1o 6197 df-oadd 6201 df-er 6308 df-en 6514 df-dom 6515 df-fin 6516 df-sup 6735 df-pnf 7587 df-mnf 7588 df-xr 7589 df-ltxr 7590 df-le 7591 df-sub 7718 df-neg 7719 df-reap 8115 df-ap 8122 df-div 8203 df-inn 8486 df-2 8544 df-3 8545 df-4 8546 df-n0 8737 df-z 8814 df-uz 9083 df-q 9168 df-rp 9198 df-fz 9488 df-fzo 9617 df-fl 9740 df-mod 9793 df-iseq 9916 df-seq3 9917 df-exp 10018 df-ihash 10247 df-cj 10339 df-re 10340 df-im 10341 df-rsqrt 10494 df-abs 10495 df-dvds 11138 df-gcd 11280

This theorem is referenced by: phimullem 11542

W3C validator