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Theorem crth 16712

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 𝜑,𝑥 𝑥,𝑆 𝑥,𝑇 𝑥,𝑁 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 13630	. . . . . 6 ⊢ (𝑥 ∈ (0..^(𝑀 · 𝑁)) → 𝑥 ∈ ℤ)
2		crth.1	. . . . . 6 ⊢ 𝑆 = (0..^(𝑀 · 𝑁))
3	1, 2	eleq2s 2843	. . . . 5 ⊢ (𝑥 ∈ 𝑆 → 𝑥 ∈ ℤ)
4		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℤ) → 𝑥 ∈ ℤ)
5		crth.4	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1))
6	5	simp1d 1139	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℕ)
7	6	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℤ) → 𝑀 ∈ ℕ)
8		zmodfzo 13857	. . . . . . . 8 ⊢ ((𝑥 ∈ ℤ ∧ 𝑀 ∈ ℕ) → (𝑥 mod 𝑀) ∈ (0..^𝑀))
9	4, 7, 8	syl2anc 583	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℤ) → (𝑥 mod 𝑀) ∈ (0..^𝑀))
10	5	simp2d 1140	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℕ)
11	10	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℤ) → 𝑁 ∈ ℕ)
12		zmodfzo 13857	. . . . . . . 8 ⊢ ((𝑥 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝑥 mod 𝑁) ∈ (0..^𝑁))
13	4, 11, 12	syl2anc 583	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℤ) → (𝑥 mod 𝑁) ∈ (0..^𝑁))
14	9, 13	opelxpd 5706	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℤ) → ⟨(𝑥 mod 𝑀), (𝑥 mod 𝑁)⟩ ∈ ((0..^𝑀) × (0..^𝑁)))
15		crth.2	. . . . . 6 ⊢ 𝑇 = ((0..^𝑀) × (0..^𝑁))
16	14, 15	eleqtrrdi 2836	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℤ) → ⟨(𝑥 mod 𝑀), (𝑥 mod 𝑁)⟩ ∈ 𝑇)
17	3, 16	sylan2 592	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ⟨(𝑥 mod 𝑀), (𝑥 mod 𝑁)⟩ ∈ 𝑇)
18		crth.3	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝑆 ↦ ⟨(𝑥 mod 𝑀), (𝑥 mod 𝑁)⟩)
19	17, 18	fmptd 7106	. . 3 ⊢ (𝜑 → 𝐹:𝑆⟶𝑇)
20		oveq1 7409	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑥 mod 𝑀) = (𝑦 mod 𝑀))
21		oveq1 7409	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑥 mod 𝑁) = (𝑦 mod 𝑁))
22	20, 21	opeq12d 4874	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ⟨(𝑥 mod 𝑀), (𝑥 mod 𝑁)⟩ = ⟨(𝑦 mod 𝑀), (𝑦 mod 𝑁)⟩)
23		opex 5455	. . . . . . . . 9 ⊢ ⟨(𝑦 mod 𝑀), (𝑦 mod 𝑁)⟩ ∈ V
24	22, 18, 23	fvmpt 6989	. . . . . . . 8 ⊢ (𝑦 ∈ 𝑆 → (𝐹‘𝑦) = ⟨(𝑦 mod 𝑀), (𝑦 mod 𝑁)⟩)
25	24	ad2antrl 725	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝐹‘𝑦) = ⟨(𝑦 mod 𝑀), (𝑦 mod 𝑁)⟩)
26		oveq1 7409	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (𝑥 mod 𝑀) = (𝑧 mod 𝑀))
27		oveq1 7409	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (𝑥 mod 𝑁) = (𝑧 mod 𝑁))
28	26, 27	opeq12d 4874	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → ⟨(𝑥 mod 𝑀), (𝑥 mod 𝑁)⟩ = ⟨(𝑧 mod 𝑀), (𝑧 mod 𝑁)⟩)
29		opex 5455	. . . . . . . . 9 ⊢ ⟨(𝑧 mod 𝑀), (𝑧 mod 𝑁)⟩ ∈ V
30	28, 18, 29	fvmpt 6989	. . . . . . . 8 ⊢ (𝑧 ∈ 𝑆 → (𝐹‘𝑧) = ⟨(𝑧 mod 𝑀), (𝑧 mod 𝑁)⟩)
31	30	ad2antll 726	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝐹‘𝑧) = ⟨(𝑧 mod 𝑀), (𝑧 mod 𝑁)⟩)
32	25, 31	eqeq12d 2740	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝐹‘𝑦) = (𝐹‘𝑧) ↔ ⟨(𝑦 mod 𝑀), (𝑦 mod 𝑁)⟩ = ⟨(𝑧 mod 𝑀), (𝑧 mod 𝑁)⟩))
33		ovex 7435	. . . . . . 7 ⊢ (𝑦 mod 𝑀) ∈ V
34		ovex 7435	. . . . . . 7 ⊢ (𝑦 mod 𝑁) ∈ V
35	33, 34	opth 5467	. . . . . 6 ⊢ (⟨(𝑦 mod 𝑀), (𝑦 mod 𝑁)⟩ = ⟨(𝑧 mod 𝑀), (𝑧 mod 𝑁)⟩ ↔ ((𝑦 mod 𝑀) = (𝑧 mod 𝑀) ∧ (𝑦 mod 𝑁) = (𝑧 mod 𝑁)))
36	32, 35	bitrdi 287	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝐹‘𝑦) = (𝐹‘𝑧) ↔ ((𝑦 mod 𝑀) = (𝑧 mod 𝑀) ∧ (𝑦 mod 𝑁) = (𝑧 mod 𝑁))))
37	6	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → 𝑀 ∈ ℕ)
38	37	nnzd 12583	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → 𝑀 ∈ ℤ)
39	10	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → 𝑁 ∈ ℕ)
40	39	nnzd 12583	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → 𝑁 ∈ ℤ)
41		simprl 768	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → 𝑦 ∈ 𝑆)
42	41, 2	eleqtrdi 2835	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → 𝑦 ∈ (0..^(𝑀 · 𝑁)))
43		elfzoelz 13630	. . . . . . . . 9 ⊢ (𝑦 ∈ (0..^(𝑀 · 𝑁)) → 𝑦 ∈ ℤ)
44	42, 43	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → 𝑦 ∈ ℤ)
45		simprr 770	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → 𝑧 ∈ 𝑆)
46	45, 2	eleqtrdi 2835	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → 𝑧 ∈ (0..^(𝑀 · 𝑁)))
47		elfzoelz 13630	. . . . . . . . 9 ⊢ (𝑧 ∈ (0..^(𝑀 · 𝑁)) → 𝑧 ∈ ℤ)
48	46, 47	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → 𝑧 ∈ ℤ)
49	44, 48	zsubcld 12669	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑦 − 𝑧) ∈ ℤ)
50	5	simp3d 1141	. . . . . . . 8 ⊢ (𝜑 → (𝑀 gcd 𝑁) = 1)
51	50	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑀 gcd 𝑁) = 1)
52		coprmdvds2 16590	. . . . . . 7 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝑦 − 𝑧) ∈ ℤ) ∧ (𝑀 gcd 𝑁) = 1) → ((𝑀 ∥ (𝑦 − 𝑧) ∧ 𝑁 ∥ (𝑦 − 𝑧)) → (𝑀 · 𝑁) ∥ (𝑦 − 𝑧)))
53	38, 40, 49, 51, 52	syl31anc 1370	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑀 ∥ (𝑦 − 𝑧) ∧ 𝑁 ∥ (𝑦 − 𝑧)) → (𝑀 · 𝑁) ∥ (𝑦 − 𝑧)))
54		moddvds 16207	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ ∧ 𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ) → ((𝑦 mod 𝑀) = (𝑧 mod 𝑀) ↔ 𝑀 ∥ (𝑦 − 𝑧)))
55	37, 44, 48, 54	syl3anc 1368	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑦 mod 𝑀) = (𝑧 mod 𝑀) ↔ 𝑀 ∥ (𝑦 − 𝑧)))
56		moddvds 16207	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ) → ((𝑦 mod 𝑁) = (𝑧 mod 𝑁) ↔ 𝑁 ∥ (𝑦 − 𝑧)))
57	39, 44, 48, 56	syl3anc 1368	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑦 mod 𝑁) = (𝑧 mod 𝑁) ↔ 𝑁 ∥ (𝑦 − 𝑧)))
58	55, 57	anbi12d 630	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (((𝑦 mod 𝑀) = (𝑧 mod 𝑀) ∧ (𝑦 mod 𝑁) = (𝑧 mod 𝑁)) ↔ (𝑀 ∥ (𝑦 − 𝑧) ∧ 𝑁 ∥ (𝑦 − 𝑧))))
59	44	zred 12664	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → 𝑦 ∈ ℝ)
60	37, 39	nnmulcld 12263	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑀 · 𝑁) ∈ ℕ)
61	60	nnrpd 13012	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑀 · 𝑁) ∈ ℝ⁺)
62		elfzole1 13638	. . . . . . . . . 10 ⊢ (𝑦 ∈ (0..^(𝑀 · 𝑁)) → 0 ≤ 𝑦)
63	42, 62	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → 0 ≤ 𝑦)
64		elfzolt2 13639	. . . . . . . . . 10 ⊢ (𝑦 ∈ (0..^(𝑀 · 𝑁)) → 𝑦 < (𝑀 · 𝑁))
65	42, 64	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → 𝑦 < (𝑀 · 𝑁))
66		modid 13859	. . . . . . . . 9 ⊢ (((𝑦 ∈ ℝ ∧ (𝑀 · 𝑁) ∈ ℝ⁺) ∧ (0 ≤ 𝑦 ∧ 𝑦 < (𝑀 · 𝑁))) → (𝑦 mod (𝑀 · 𝑁)) = 𝑦)
67	59, 61, 63, 65, 66	syl22anc 836	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑦 mod (𝑀 · 𝑁)) = 𝑦)
68	48	zred 12664	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → 𝑧 ∈ ℝ)
69		elfzole1 13638	. . . . . . . . . 10 ⊢ (𝑧 ∈ (0..^(𝑀 · 𝑁)) → 0 ≤ 𝑧)
70	46, 69	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → 0 ≤ 𝑧)
71		elfzolt2 13639	. . . . . . . . . 10 ⊢ (𝑧 ∈ (0..^(𝑀 · 𝑁)) → 𝑧 < (𝑀 · 𝑁))
72	46, 71	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → 𝑧 < (𝑀 · 𝑁))
73		modid 13859	. . . . . . . . 9 ⊢ (((𝑧 ∈ ℝ ∧ (𝑀 · 𝑁) ∈ ℝ⁺) ∧ (0 ≤ 𝑧 ∧ 𝑧 < (𝑀 · 𝑁))) → (𝑧 mod (𝑀 · 𝑁)) = 𝑧)
74	68, 61, 70, 72, 73	syl22anc 836	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑧 mod (𝑀 · 𝑁)) = 𝑧)
75	67, 74	eqeq12d 2740	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑦 mod (𝑀 · 𝑁)) = (𝑧 mod (𝑀 · 𝑁)) ↔ 𝑦 = 𝑧))
76		moddvds 16207	. . . . . . . 8 ⊢ (((𝑀 · 𝑁) ∈ ℕ ∧ 𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ) → ((𝑦 mod (𝑀 · 𝑁)) = (𝑧 mod (𝑀 · 𝑁)) ↔ (𝑀 · 𝑁) ∥ (𝑦 − 𝑧)))
77	60, 44, 48, 76	syl3anc 1368	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑦 mod (𝑀 · 𝑁)) = (𝑧 mod (𝑀 · 𝑁)) ↔ (𝑀 · 𝑁) ∥ (𝑦 − 𝑧)))
78	75, 77	bitr3d 281	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑦 = 𝑧 ↔ (𝑀 · 𝑁) ∥ (𝑦 − 𝑧)))
79	53, 58, 78	3imtr4d 294	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (((𝑦 mod 𝑀) = (𝑧 mod 𝑀) ∧ (𝑦 mod 𝑁) = (𝑧 mod 𝑁)) → 𝑦 = 𝑧))
80	36, 79	sylbid 239	. . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝐹‘𝑦) = (𝐹‘𝑧) → 𝑦 = 𝑧))
81	80	ralrimivva 3192	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ((𝐹‘𝑦) = (𝐹‘𝑧) → 𝑦 = 𝑧))
82		dff13 7247	. . 3 ⊢ (𝐹:𝑆–1-1→𝑇 ↔ (𝐹:𝑆⟶𝑇 ∧ ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ((𝐹‘𝑦) = (𝐹‘𝑧) → 𝑦 = 𝑧)))
83	19, 81, 82	sylanbrc 582	. 2 ⊢ (𝜑 → 𝐹:𝑆–1-1→𝑇)
84		nnnn0 12477	. . . . . 6 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℕ₀)
85		nnnn0 12477	. . . . . 6 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ₀)
86		nn0mulcl 12506	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝑀 · 𝑁) ∈ ℕ₀)
87		hashfzo0 14388	. . . . . . . . 9 ⊢ ((𝑀 · 𝑁) ∈ ℕ₀ → (♯‘(0..^(𝑀 · 𝑁))) = (𝑀 · 𝑁))
88	86, 87	syl 17	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (♯‘(0..^(𝑀 · 𝑁))) = (𝑀 · 𝑁))
89		fzofi 13937	. . . . . . . . . 10 ⊢ (0..^𝑀) ∈ Fin
90		fzofi 13937	. . . . . . . . . 10 ⊢ (0..^𝑁) ∈ Fin
91		hashxp 14392	. . . . . . . . . 10 ⊢ (((0..^𝑀) ∈ Fin ∧ (0..^𝑁) ∈ Fin) → (♯‘((0..^𝑀) × (0..^𝑁))) = ((♯‘(0..^𝑀)) · (♯‘(0..^𝑁))))
92	89, 90, 91	mp2an 689	. . . . . . . . 9 ⊢ (♯‘((0..^𝑀) × (0..^𝑁))) = ((♯‘(0..^𝑀)) · (♯‘(0..^𝑁)))
93		hashfzo0 14388	. . . . . . . . . 10 ⊢ (𝑀 ∈ ℕ₀ → (♯‘(0..^𝑀)) = 𝑀)
94		hashfzo0 14388	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ₀ → (♯‘(0..^𝑁)) = 𝑁)
95	93, 94	oveqan12d 7421	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → ((♯‘(0..^𝑀)) · (♯‘(0..^𝑁))) = (𝑀 · 𝑁))
96	92, 95	eqtrid 2776	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (♯‘((0..^𝑀) × (0..^𝑁))) = (𝑀 · 𝑁))
97	88, 96	eqtr4d 2767	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (♯‘(0..^(𝑀 · 𝑁))) = (♯‘((0..^𝑀) × (0..^𝑁))))
98		fzofi 13937	. . . . . . . 8 ⊢ (0..^(𝑀 · 𝑁)) ∈ Fin
99		xpfi 9314	. . . . . . . . 9 ⊢ (((0..^𝑀) ∈ Fin ∧ (0..^𝑁) ∈ Fin) → ((0..^𝑀) × (0..^𝑁)) ∈ Fin)
100	89, 90, 99	mp2an 689	. . . . . . . 8 ⊢ ((0..^𝑀) × (0..^𝑁)) ∈ Fin
101		hashen 14305	. . . . . . . 8 ⊢ (((0..^(𝑀 · 𝑁)) ∈ Fin ∧ ((0..^𝑀) × (0..^𝑁)) ∈ Fin) → ((♯‘(0..^(𝑀 · 𝑁))) = (♯‘((0..^𝑀) × (0..^𝑁))) ↔ (0..^(𝑀 · 𝑁)) ≈ ((0..^𝑀) × (0..^𝑁))))
102	98, 100, 101	mp2an 689	. . . . . . 7 ⊢ ((♯‘(0..^(𝑀 · 𝑁))) = (♯‘((0..^𝑀) × (0..^𝑁))) ↔ (0..^(𝑀 · 𝑁)) ≈ ((0..^𝑀) × (0..^𝑁)))
103	97, 102	sylib 217	. . . . . 6 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (0..^(𝑀 · 𝑁)) ≈ ((0..^𝑀) × (0..^𝑁)))
104	84, 85, 103	syl2an 595	. . . . 5 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (0..^(𝑀 · 𝑁)) ≈ ((0..^𝑀) × (0..^𝑁)))
105	6, 10, 104	syl2anc 583	. . . 4 ⊢ (𝜑 → (0..^(𝑀 · 𝑁)) ≈ ((0..^𝑀) × (0..^𝑁)))
106	105, 2, 15	3brtr4g 5173	. . 3 ⊢ (𝜑 → 𝑆 ≈ 𝑇)
107	15, 100	eqeltri 2821	. . 3 ⊢ 𝑇 ∈ Fin
108		f1finf1o 9268	. . 3 ⊢ ((𝑆 ≈ 𝑇 ∧ 𝑇 ∈ Fin) → (𝐹:𝑆–1-1→𝑇 ↔ 𝐹:𝑆–1-1-onto→𝑇))
109	106, 107, 108	sylancl 585	. 2 ⊢ (𝜑 → (𝐹:𝑆–1-1→𝑇 ↔ 𝐹:𝑆–1-1-onto→𝑇))
110	83, 109	mpbid 231	1 ⊢ (𝜑 → 𝐹:𝑆–1-1-onto→𝑇)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∀wral 3053 ⟨cop 4627 class class class wbr 5139 ↦ cmpt 5222 × cxp 5665 ⟶wf 6530 –1-1→wf1 6531 –1-1-onto→wf1o 6533 ‘cfv 6534 (class class class)co 7402 ≈ cen 8933 Fincfn 8936 ℝcr 11106 0cc0 11107 1c1 11108 · cmul 11112 < clt 11246 ≤ cle 11247 − cmin 11442 ℕcn 12210 ℕ₀cn0 12470 ℤcz 12556 ℝ⁺crp 12972 ..^cfzo 13625 mod cmo 13832 ♯chash 14288 ∥ cdvds 16196 gcd cgcd 16434

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-pre-sup 11185

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-int 4942 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-1st 7969 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-oadd 8466 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-sup 9434 df-inf 9435 df-dju 9893 df-card 9931 df-pnf 11248 df-mnf 11249 df-xr 11250 df-ltxr 11251 df-le 11252 df-sub 11444 df-neg 11445 df-div 11870 df-nn 12211 df-2 12273 df-3 12274 df-n0 12471 df-z 12557 df-uz 12821 df-rp 12973 df-fz 13483 df-fzo 13626 df-fl 13755 df-mod 13833 df-seq 13965 df-exp 14026 df-hash 14289 df-cj 15044 df-re 15045 df-im 15046 df-sqrt 15180 df-abs 15181 df-dvds 16197 df-gcd 16435

This theorem is referenced by: phimullem 16713

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