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Theorem crth 12362

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 10213	. . . . . 6 ⊢ (𝑥 ∈ (0..^(𝑀 · 𝑁)) → 𝑥 ∈ ℤ)
2		crth.1	. . . . . 6 ⊢ 𝑆 = (0..^(𝑀 · 𝑁))
3	1, 2	eleq2s 2288	. . . . 5 ⊢ (𝑥 ∈ 𝑆 → 𝑥 ∈ ℤ)
4		simpr 110	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℤ) → 𝑥 ∈ ℤ)
5		crth.4	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1))
6	5	simp1d 1011	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℕ)
7	6	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℤ) → 𝑀 ∈ ℕ)
8		zmodfzo 10418	. . . . . . . 8 ⊢ ((𝑥 ∈ ℤ ∧ 𝑀 ∈ ℕ) → (𝑥 mod 𝑀) ∈ (0..^𝑀))
9	4, 7, 8	syl2anc 411	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℤ) → (𝑥 mod 𝑀) ∈ (0..^𝑀))
10	5	simp2d 1012	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℕ)
11	10	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℤ) → 𝑁 ∈ ℕ)
12		zmodfzo 10418	. . . . . . . 8 ⊢ ((𝑥 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝑥 mod 𝑁) ∈ (0..^𝑁))
13	4, 11, 12	syl2anc 411	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℤ) → (𝑥 mod 𝑁) ∈ (0..^𝑁))
14		opelxpi 4691	. . . . . . 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 411	. . . . . 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	eleqtrrdi 2287	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℤ) → ⟨(𝑥 mod 𝑀), (𝑥 mod 𝑁)⟩ ∈ 𝑇)
18	3, 17	sylan2 286	. . . 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 5712	. . 3 ⊢ (𝜑 → 𝐹:𝑆⟶𝑇)
21		simprl 529	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → 𝑦 ∈ 𝑆)
22		elfzoelz 10213	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (0..^(𝑀 · 𝑁)) → 𝑦 ∈ ℤ)
23	22, 2	eleq2s 2288	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ 𝑆 → 𝑦 ∈ ℤ)
24	23	ad2antrl 490	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → 𝑦 ∈ ℤ)
25		zq 9691	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℤ → 𝑦 ∈ ℚ)
26	24, 25	syl 14	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → 𝑦 ∈ ℚ)
27	6	adantr 276	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → 𝑀 ∈ ℕ)
28		nnq 9698	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℚ)
29	27, 28	syl 14	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → 𝑀 ∈ ℚ)
30	27	nngt0d 9026	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → 0 < 𝑀)
31	26, 29, 30	modqcld 10399	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑦 mod 𝑀) ∈ ℚ)
32	10	adantr 276	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → 𝑁 ∈ ℕ)
33		nnq 9698	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℚ)
34	32, 33	syl 14	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → 𝑁 ∈ ℚ)
35	32	nngt0d 9026	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → 0 < 𝑁)
36	26, 34, 35	modqcld 10399	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑦 mod 𝑁) ∈ ℚ)
37		opexg 4257	. . . . . . . . 9 ⊢ (((𝑦 mod 𝑀) ∈ ℚ ∧ (𝑦 mod 𝑁) ∈ ℚ) → ⟨(𝑦 mod 𝑀), (𝑦 mod 𝑁)⟩ ∈ V)
38	31, 36, 37	syl2anc 411	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ⟨(𝑦 mod 𝑀), (𝑦 mod 𝑁)⟩ ∈ V)
39		oveq1 5925	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑥 mod 𝑀) = (𝑦 mod 𝑀))
40		oveq1 5925	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑥 mod 𝑁) = (𝑦 mod 𝑁))
41	39, 40	opeq12d 3812	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ⟨(𝑥 mod 𝑀), (𝑥 mod 𝑁)⟩ = ⟨(𝑦 mod 𝑀), (𝑦 mod 𝑁)⟩)
42	41, 19	fvmptg 5633	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝑆 ∧ ⟨(𝑦 mod 𝑀), (𝑦 mod 𝑁)⟩ ∈ V) → (𝐹‘𝑦) = ⟨(𝑦 mod 𝑀), (𝑦 mod 𝑁)⟩)
43	21, 38, 42	syl2anc 411	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝐹‘𝑦) = ⟨(𝑦 mod 𝑀), (𝑦 mod 𝑁)⟩)
44		simprr 531	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → 𝑧 ∈ 𝑆)
45		elfzoelz 10213	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ (0..^(𝑀 · 𝑁)) → 𝑧 ∈ ℤ)
46	45, 2	eleq2s 2288	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ 𝑆 → 𝑧 ∈ ℤ)
47	44, 46	syl 14	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → 𝑧 ∈ ℤ)
48		zq 9691	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ℤ → 𝑧 ∈ ℚ)
49	47, 48	syl 14	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → 𝑧 ∈ ℚ)
50	49, 29, 30	modqcld 10399	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑧 mod 𝑀) ∈ ℚ)
51	49, 34, 35	modqcld 10399	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑧 mod 𝑁) ∈ ℚ)
52		opexg 4257	. . . . . . . . 9 ⊢ (((𝑧 mod 𝑀) ∈ ℚ ∧ (𝑧 mod 𝑁) ∈ ℚ) → ⟨(𝑧 mod 𝑀), (𝑧 mod 𝑁)⟩ ∈ V)
53	50, 51, 52	syl2anc 411	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ⟨(𝑧 mod 𝑀), (𝑧 mod 𝑁)⟩ ∈ V)
54		oveq1 5925	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (𝑥 mod 𝑀) = (𝑧 mod 𝑀))
55		oveq1 5925	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (𝑥 mod 𝑁) = (𝑧 mod 𝑁))
56	54, 55	opeq12d 3812	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → ⟨(𝑥 mod 𝑀), (𝑥 mod 𝑁)⟩ = ⟨(𝑧 mod 𝑀), (𝑧 mod 𝑁)⟩)
57	56, 19	fvmptg 5633	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝑆 ∧ ⟨(𝑧 mod 𝑀), (𝑧 mod 𝑁)⟩ ∈ V) → (𝐹‘𝑧) = ⟨(𝑧 mod 𝑀), (𝑧 mod 𝑁)⟩)
58	44, 53, 57	syl2anc 411	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝐹‘𝑧) = ⟨(𝑧 mod 𝑀), (𝑧 mod 𝑁)⟩)
59	43, 58	eqeq12d 2208	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝐹‘𝑦) = (𝐹‘𝑧) ↔ ⟨(𝑦 mod 𝑀), (𝑦 mod 𝑁)⟩ = ⟨(𝑧 mod 𝑀), (𝑧 mod 𝑁)⟩))
60		opthg 4267	. . . . . . 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 411	. . . . . 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 188	. . . . 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 9438	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → 𝑀 ∈ ℤ)
64	32	nnzd 9438	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → 𝑁 ∈ ℤ)
65	21, 2	eleqtrdi 2286	. . . . . . . . 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	eleqtrdi 2286	. . . . . . . . 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 9444	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑦 − 𝑧) ∈ ℤ)
70	5	simp3d 1013	. . . . . . . 8 ⊢ (𝜑 → (𝑀 gcd 𝑁) = 1)
71	70	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑀 gcd 𝑁) = 1)
72		coprmdvds2 12231	. . . . . . 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 1252	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑀 ∥ (𝑦 − 𝑧) ∧ 𝑁 ∥ (𝑦 − 𝑧)) → (𝑀 · 𝑁) ∥ (𝑦 − 𝑧)))
74		moddvds 11942	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ ∧ 𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ) → ((𝑦 mod 𝑀) = (𝑧 mod 𝑀) ↔ 𝑀 ∥ (𝑦 − 𝑧)))
75	27, 66, 68, 74	syl3anc 1249	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑦 mod 𝑀) = (𝑧 mod 𝑀) ↔ 𝑀 ∥ (𝑦 − 𝑧)))
76		moddvds 11942	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ) → ((𝑦 mod 𝑁) = (𝑧 mod 𝑁) ↔ 𝑁 ∥ (𝑦 − 𝑧)))
77	32, 66, 68, 76	syl3anc 1249	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑦 mod 𝑁) = (𝑧 mod 𝑁) ↔ 𝑁 ∥ (𝑦 − 𝑧)))
78	75, 77	anbi12d 473	. . . . . 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 9702	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℚ ∧ 𝑁 ∈ ℚ) → (𝑀 · 𝑁) ∈ ℚ)
80	29, 34, 79	syl2anc 411	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑀 · 𝑁) ∈ ℚ)
81		elfzole1 10222	. . . . . . . . . 10 ⊢ (𝑦 ∈ (0..^(𝑀 · 𝑁)) → 0 ≤ 𝑦)
82	65, 81	syl 14	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → 0 ≤ 𝑦)
83		elfzolt2 10223	. . . . . . . . . 10 ⊢ (𝑦 ∈ (0..^(𝑀 · 𝑁)) → 𝑦 < (𝑀 · 𝑁))
84	65, 83	syl 14	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → 𝑦 < (𝑀 · 𝑁))
85		modqid 10420	. . . . . . . . 9 ⊢ (((𝑦 ∈ ℚ ∧ (𝑀 · 𝑁) ∈ ℚ) ∧ (0 ≤ 𝑦 ∧ 𝑦 < (𝑀 · 𝑁))) → (𝑦 mod (𝑀 · 𝑁)) = 𝑦)
86	26, 80, 82, 84, 85	syl22anc 1250	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑦 mod (𝑀 · 𝑁)) = 𝑦)
87		elfzole1 10222	. . . . . . . . . 10 ⊢ (𝑧 ∈ (0..^(𝑀 · 𝑁)) → 0 ≤ 𝑧)
88	67, 87	syl 14	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → 0 ≤ 𝑧)
89		elfzolt2 10223	. . . . . . . . . 10 ⊢ (𝑧 ∈ (0..^(𝑀 · 𝑁)) → 𝑧 < (𝑀 · 𝑁))
90	67, 89	syl 14	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → 𝑧 < (𝑀 · 𝑁))
91		modqid 10420	. . . . . . . . 9 ⊢ (((𝑧 ∈ ℚ ∧ (𝑀 · 𝑁) ∈ ℚ) ∧ (0 ≤ 𝑧 ∧ 𝑧 < (𝑀 · 𝑁))) → (𝑧 mod (𝑀 · 𝑁)) = 𝑧)
92	49, 80, 88, 90, 91	syl22anc 1250	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑧 mod (𝑀 · 𝑁)) = 𝑧)
93	86, 92	eqeq12d 2208	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑦 mod (𝑀 · 𝑁)) = (𝑧 mod (𝑀 · 𝑁)) ↔ 𝑦 = 𝑧))
94	27, 32	nnmulcld 9031	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑀 · 𝑁) ∈ ℕ)
95		moddvds 11942	. . . . . . . 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 1249	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑦 mod (𝑀 · 𝑁)) = (𝑧 mod (𝑀 · 𝑁)) ↔ (𝑀 · 𝑁) ∥ (𝑦 − 𝑧)))
97	93, 96	bitr3d 190	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑦 = 𝑧 ↔ (𝑀 · 𝑁) ∥ (𝑦 − 𝑧)))
98	73, 78, 97	3imtr4d 203	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (((𝑦 mod 𝑀) = (𝑧 mod 𝑀) ∧ (𝑦 mod 𝑁) = (𝑧 mod 𝑁)) → 𝑦 = 𝑧))
99	62, 98	sylbid 150	. . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝐹‘𝑦) = (𝐹‘𝑧) → 𝑦 = 𝑧))
100	99	ralrimivva 2576	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ((𝐹‘𝑦) = (𝐹‘𝑧) → 𝑦 = 𝑧))
101		dff13 5811	. . 3 ⊢ (𝐹:𝑆–1-1→𝑇 ↔ (𝐹:𝑆⟶𝑇 ∧ ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ((𝐹‘𝑦) = (𝐹‘𝑧) → 𝑦 = 𝑧)))
102	20, 100, 101	sylanbrc 417	. 2 ⊢ (𝜑 → 𝐹:𝑆–1-1→𝑇)
103		nnnn0 9247	. . . . . 6 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℕ₀)
104		nnnn0 9247	. . . . . 6 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ₀)
105		hashfzo0 10894	. . . . . . . . 9 ⊢ (𝑀 ∈ ℕ₀ → (♯‘(0..^𝑀)) = 𝑀)
106		hashfzo0 10894	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ₀ → (♯‘(0..^𝑁)) = 𝑁)
107	105, 106	oveqan12d 5937	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → ((♯‘(0..^𝑀)) · (♯‘(0..^𝑁))) = (𝑀 · 𝑁))
108		0z 9328	. . . . . . . . . 10 ⊢ 0 ∈ ℤ
109		simpl 109	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → 𝑀 ∈ ℕ₀)
110	109	nn0zd 9437	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → 𝑀 ∈ ℤ)
111		fzofig 10503	. . . . . . . . . 10 ⊢ ((0 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (0..^𝑀) ∈ Fin)
112	108, 110, 111	sylancr 414	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (0..^𝑀) ∈ Fin)
113		nn0z 9337	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℤ)
114	113	adantl 277	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → 𝑁 ∈ ℤ)
115		fzofig 10503	. . . . . . . . . 10 ⊢ ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (0..^𝑁) ∈ Fin)
116	108, 114, 115	sylancr 414	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (0..^𝑁) ∈ Fin)
117		hashxp 10897	. . . . . . . . 9 ⊢ (((0..^𝑀) ∈ Fin ∧ (0..^𝑁) ∈ Fin) → (♯‘((0..^𝑀) × (0..^𝑁))) = ((♯‘(0..^𝑀)) · (♯‘(0..^𝑁))))
118	112, 116, 117	syl2anc 411	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (♯‘((0..^𝑀) × (0..^𝑁))) = ((♯‘(0..^𝑀)) · (♯‘(0..^𝑁))))
119		nn0mulcl 9276	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝑀 · 𝑁) ∈ ℕ₀)
120		hashfzo0 10894	. . . . . . . . 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 2237	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (♯‘(0..^(𝑀 · 𝑁))) = (♯‘((0..^𝑀) × (0..^𝑁))))
123	119	nn0zd 9437	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝑀 · 𝑁) ∈ ℤ)
124		fzofig 10503	. . . . . . . . 9 ⊢ ((0 ∈ ℤ ∧ (𝑀 · 𝑁) ∈ ℤ) → (0..^(𝑀 · 𝑁)) ∈ Fin)
125	108, 123, 124	sylancr 414	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (0..^(𝑀 · 𝑁)) ∈ Fin)
126		xpfi 6986	. . . . . . . . 9 ⊢ (((0..^𝑀) ∈ Fin ∧ (0..^𝑁) ∈ Fin) → ((0..^𝑀) × (0..^𝑁)) ∈ Fin)
127	112, 116, 126	syl2anc 411	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → ((0..^𝑀) × (0..^𝑁)) ∈ Fin)
128		hashen 10855	. . . . . . . 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 411	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → ((♯‘(0..^(𝑀 · 𝑁))) = (♯‘((0..^𝑀) × (0..^𝑁))) ↔ (0..^(𝑀 · 𝑁)) ≈ ((0..^𝑀) × (0..^𝑁))))
130	122, 129	mpbid 147	. . . . . 6 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (0..^(𝑀 · 𝑁)) ≈ ((0..^𝑀) × (0..^𝑁)))
131	103, 104, 130	syl2an 289	. . . . 5 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (0..^(𝑀 · 𝑁)) ≈ ((0..^𝑀) × (0..^𝑁)))
132	6, 10, 131	syl2anc 411	. . . 4 ⊢ (𝜑 → (0..^(𝑀 · 𝑁)) ≈ ((0..^𝑀) × (0..^𝑁)))
133	132, 2, 16	3brtr4g 4063	. . 3 ⊢ (𝜑 → 𝑆 ≈ 𝑇)
134	6	nnnn0d 9293	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
135	10	nnnn0d 9293	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
136	134, 135, 127	syl2anc 411	. . . 4 ⊢ (𝜑 → ((0..^𝑀) × (0..^𝑁)) ∈ Fin)
137	16, 136	eqeltrid 2280	. . 3 ⊢ (𝜑 → 𝑇 ∈ Fin)
138		f1finf1o 7006	. . 3 ⊢ ((𝑆 ≈ 𝑇 ∧ 𝑇 ∈ Fin) → (𝐹:𝑆–1-1→𝑇 ↔ 𝐹:𝑆–1-1-onto→𝑇))
139	133, 137, 138	syl2anc 411	. 2 ⊢ (𝜑 → (𝐹:𝑆–1-1→𝑇 ↔ 𝐹:𝑆–1-1-onto→𝑇))
140	102, 139	mpbid 147	1 ⊢ (𝜑 → 𝐹:𝑆–1-1-onto→𝑇)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∧ w3a 980 = wceq 1364 ∈ wcel 2164 ∀wral 2472 Vcvv 2760 ⟨cop 3621 class class class wbr 4029 ↦ cmpt 4090 × cxp 4657 ⟶wf 5250 –1-1→wf1 5251 –1-1-onto→wf1o 5253 ‘cfv 5254 (class class class)co 5918 ≈ cen 6792 Fincfn 6794 0cc0 7872 1c1 7873 · cmul 7877 < clt 8054 ≤ cle 8055 − cmin 8190 ℕcn 8982 ℕ₀cn0 9240 ℤcz 9317 ℚcq 9684 ..^cfzo 10208 mod cmo 10393 ♯chash 10846 ∥ cdvds 11930 gcd cgcd 12079

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-coll 4144 ax-sep 4147 ax-nul 4155 ax-pow 4203 ax-pr 4238 ax-un 4464 ax-setind 4569 ax-iinf 4620 ax-cnex 7963 ax-resscn 7964 ax-1cn 7965 ax-1re 7966 ax-icn 7967 ax-addcl 7968 ax-addrcl 7969 ax-mulcl 7970 ax-mulrcl 7971 ax-addcom 7972 ax-mulcom 7973 ax-addass 7974 ax-mulass 7975 ax-distr 7976 ax-i2m1 7977 ax-0lt1 7978 ax-1rid 7979 ax-0id 7980 ax-rnegex 7981 ax-precex 7982 ax-cnre 7983 ax-pre-ltirr 7984 ax-pre-ltwlin 7985 ax-pre-lttrn 7986 ax-pre-apti 7987 ax-pre-ltadd 7988 ax-pre-mulgt0 7989 ax-pre-mulext 7990 ax-arch 7991 ax-caucvg 7992

This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-reu 2479 df-rmo 2480 df-rab 2481 df-v 2762 df-sbc 2986 df-csb 3081 df-dif 3155 df-un 3157 df-in 3159 df-ss 3166 df-nul 3447 df-if 3558 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-int 3871 df-iun 3914 df-br 4030 df-opab 4091 df-mpt 4092 df-tr 4128 df-id 4324 df-po 4327 df-iso 4328 df-iord 4397 df-on 4399 df-ilim 4400 df-suc 4402 df-iom 4623 df-xp 4665 df-rel 4666 df-cnv 4667 df-co 4668 df-dm 4669 df-rn 4670 df-res 4671 df-ima 4672 df-iota 5215 df-fun 5256 df-fn 5257 df-f 5258 df-f1 5259 df-fo 5260 df-f1o 5261 df-fv 5262 df-riota 5873 df-ov 5921 df-oprab 5922 df-mpo 5923 df-1st 6193 df-2nd 6194 df-recs 6358 df-irdg 6423 df-frec 6444 df-1o 6469 df-oadd 6473 df-er 6587 df-en 6795 df-dom 6796 df-fin 6797 df-sup 7043 df-pnf 8056 df-mnf 8057 df-xr 8058 df-ltxr 8059 df-le 8060 df-sub 8192 df-neg 8193 df-reap 8594 df-ap 8601 df-div 8692 df-inn 8983 df-2 9041 df-3 9042 df-4 9043 df-n0 9241 df-z 9318 df-uz 9593 df-q 9685 df-rp 9720 df-fz 10075 df-fzo 10209 df-fl 10339 df-mod 10394 df-seqfrec 10519 df-exp 10610 df-ihash 10847 df-cj 10986 df-re 10987 df-im 10988 df-rsqrt 11142 df-abs 11143 df-dvds 11931 df-gcd 12080

This theorem is referenced by: phimullem 12363

W3C validator