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Theorem hashgcdlem 16721

Description: A correspondence between elements of specific GCD and relative primes in a smaller ring. (Contributed by Stefan O'Rear, 12-Sep-2015.)

**Hypotheses**
Ref	Expression
hashgcdlem.a	⊢ 𝐴 = {𝑦 ∈ (0..^(𝑀 / 𝑁)) ∣ (𝑦 gcd (𝑀 / 𝑁)) = 1}
hashgcdlem.b	⊢ 𝐵 = {𝑧 ∈ (0..^𝑀) ∣ (𝑧 gcd 𝑀) = 𝑁}
hashgcdlem.f	⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝑥 · 𝑁))

**Assertion**
Ref	Expression
hashgcdlem	⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) → 𝐹:𝐴–1-1-onto→𝐵)

Distinct variable groups: 𝑥,𝑦,𝑀 𝑥,𝑧,𝑀 𝑥,𝐴 𝑥,𝐵 𝑥,𝑁,𝑦 𝑧,𝑁 Allowed substitution hints: 𝐴(𝑦,𝑧) 𝐵(𝑦,𝑧) 𝐹(𝑥,𝑦,𝑧)

Proof of Theorem hashgcdlem Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		hashgcdlem.f	. 2 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝑥 · 𝑁))
2		oveq1 7416	. . . . 5 ⊢ (𝑦 = 𝑥 → (𝑦 gcd (𝑀 / 𝑁)) = (𝑥 gcd (𝑀 / 𝑁)))
3	2	eqeq1d 2735	. . . 4 ⊢ (𝑦 = 𝑥 → ((𝑦 gcd (𝑀 / 𝑁)) = 1 ↔ (𝑥 gcd (𝑀 / 𝑁)) = 1))
4		hashgcdlem.a	. . . 4 ⊢ 𝐴 = {𝑦 ∈ (0..^(𝑀 / 𝑁)) ∣ (𝑦 gcd (𝑀 / 𝑁)) = 1}
5	3, 4	elrab2 3687	. . 3 ⊢ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ (𝑥 gcd (𝑀 / 𝑁)) = 1))
6		elfzonn0 13677	. . . . . . 7 ⊢ (𝑥 ∈ (0..^(𝑀 / 𝑁)) → 𝑥 ∈ ℕ₀)
7	6	ad2antrl 727	. . . . . 6 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ (𝑥 gcd (𝑀 / 𝑁)) = 1)) → 𝑥 ∈ ℕ₀)
8		nnnn0 12479	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ₀)
9	8	3ad2ant2 1135	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) → 𝑁 ∈ ℕ₀)
10	9	adantr 482	. . . . . 6 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ (𝑥 gcd (𝑀 / 𝑁)) = 1)) → 𝑁 ∈ ℕ₀)
11	7, 10	nn0mulcld 12537	. . . . 5 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ (𝑥 gcd (𝑀 / 𝑁)) = 1)) → (𝑥 · 𝑁) ∈ ℕ₀)
12		simpl1 1192	. . . . 5 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ (𝑥 gcd (𝑀 / 𝑁)) = 1)) → 𝑀 ∈ ℕ)
13		elfzolt2 13641	. . . . . . 7 ⊢ (𝑥 ∈ (0..^(𝑀 / 𝑁)) → 𝑥 < (𝑀 / 𝑁))
14	13	ad2antrl 727	. . . . . 6 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ (𝑥 gcd (𝑀 / 𝑁)) = 1)) → 𝑥 < (𝑀 / 𝑁))
15		elfzoelz 13632	. . . . . . . . 9 ⊢ (𝑥 ∈ (0..^(𝑀 / 𝑁)) → 𝑥 ∈ ℤ)
16	15	ad2antrl 727	. . . . . . . 8 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ (𝑥 gcd (𝑀 / 𝑁)) = 1)) → 𝑥 ∈ ℤ)
17	16	zred 12666	. . . . . . 7 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ (𝑥 gcd (𝑀 / 𝑁)) = 1)) → 𝑥 ∈ ℝ)
18		nnre 12219	. . . . . . . . 9 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℝ)
19	18	3ad2ant1 1134	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) → 𝑀 ∈ ℝ)
20	19	adantr 482	. . . . . . 7 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ (𝑥 gcd (𝑀 / 𝑁)) = 1)) → 𝑀 ∈ ℝ)
21		nnre 12219	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ)
22		nngt0 12243	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → 0 < 𝑁)
23	21, 22	jca 513	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → (𝑁 ∈ ℝ ∧ 0 < 𝑁))
24	23	3ad2ant2 1135	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) → (𝑁 ∈ ℝ ∧ 0 < 𝑁))
25	24	adantr 482	. . . . . . 7 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ (𝑥 gcd (𝑀 / 𝑁)) = 1)) → (𝑁 ∈ ℝ ∧ 0 < 𝑁))
26		ltmuldiv 12087	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ ∧ 𝑀 ∈ ℝ ∧ (𝑁 ∈ ℝ ∧ 0 < 𝑁)) → ((𝑥 · 𝑁) < 𝑀 ↔ 𝑥 < (𝑀 / 𝑁)))
27	17, 20, 25, 26	syl3anc 1372	. . . . . 6 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ (𝑥 gcd (𝑀 / 𝑁)) = 1)) → ((𝑥 · 𝑁) < 𝑀 ↔ 𝑥 < (𝑀 / 𝑁)))
28	14, 27	mpbird 257	. . . . 5 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ (𝑥 gcd (𝑀 / 𝑁)) = 1)) → (𝑥 · 𝑁) < 𝑀)
29		elfzo0 13673	. . . . 5 ⊢ ((𝑥 · 𝑁) ∈ (0..^𝑀) ↔ ((𝑥 · 𝑁) ∈ ℕ₀ ∧ 𝑀 ∈ ℕ ∧ (𝑥 · 𝑁) < 𝑀))
30	11, 12, 28, 29	syl3anbrc 1344	. . . 4 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ (𝑥 gcd (𝑀 / 𝑁)) = 1)) → (𝑥 · 𝑁) ∈ (0..^𝑀))
31		nncn 12220	. . . . . . . . . 10 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℂ)
32	31	3ad2ant1 1134	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) → 𝑀 ∈ ℂ)
33		nncn 12220	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ)
34	33	3ad2ant2 1135	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) → 𝑁 ∈ ℂ)
35		nnne0 12246	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → 𝑁 ≠ 0)
36	35	3ad2ant2 1135	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) → 𝑁 ≠ 0)
37	32, 34, 36	divcan1d 11991	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) → ((𝑀 / 𝑁) · 𝑁) = 𝑀)
38	37	adantr 482	. . . . . . 7 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ (𝑥 gcd (𝑀 / 𝑁)) = 1)) → ((𝑀 / 𝑁) · 𝑁) = 𝑀)
39	38	eqcomd 2739	. . . . . 6 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ (𝑥 gcd (𝑀 / 𝑁)) = 1)) → 𝑀 = ((𝑀 / 𝑁) · 𝑁))
40	39	oveq2d 7425	. . . . 5 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ (𝑥 gcd (𝑀 / 𝑁)) = 1)) → ((𝑥 · 𝑁) gcd 𝑀) = ((𝑥 · 𝑁) gcd ((𝑀 / 𝑁) · 𝑁)))
41		nndivdvds 16206	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑁 ∥ 𝑀 ↔ (𝑀 / 𝑁) ∈ ℕ))
42	41	biimp3a 1470	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) → (𝑀 / 𝑁) ∈ ℕ)
43	42	nnzd 12585	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) → (𝑀 / 𝑁) ∈ ℤ)
44	43	adantr 482	. . . . . 6 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ (𝑥 gcd (𝑀 / 𝑁)) = 1)) → (𝑀 / 𝑁) ∈ ℤ)
45		mulgcdr 16492	. . . . . 6 ⊢ ((𝑥 ∈ ℤ ∧ (𝑀 / 𝑁) ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → ((𝑥 · 𝑁) gcd ((𝑀 / 𝑁) · 𝑁)) = ((𝑥 gcd (𝑀 / 𝑁)) · 𝑁))
46	16, 44, 10, 45	syl3anc 1372	. . . . 5 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ (𝑥 gcd (𝑀 / 𝑁)) = 1)) → ((𝑥 · 𝑁) gcd ((𝑀 / 𝑁) · 𝑁)) = ((𝑥 gcd (𝑀 / 𝑁)) · 𝑁))
47		simprr 772	. . . . . . 7 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ (𝑥 gcd (𝑀 / 𝑁)) = 1)) → (𝑥 gcd (𝑀 / 𝑁)) = 1)
48	47	oveq1d 7424	. . . . . 6 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ (𝑥 gcd (𝑀 / 𝑁)) = 1)) → ((𝑥 gcd (𝑀 / 𝑁)) · 𝑁) = (1 · 𝑁))
49	34	mullidd 11232	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) → (1 · 𝑁) = 𝑁)
50	49	adantr 482	. . . . . 6 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ (𝑥 gcd (𝑀 / 𝑁)) = 1)) → (1 · 𝑁) = 𝑁)
51	48, 50	eqtrd 2773	. . . . 5 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ (𝑥 gcd (𝑀 / 𝑁)) = 1)) → ((𝑥 gcd (𝑀 / 𝑁)) · 𝑁) = 𝑁)
52	40, 46, 51	3eqtrd 2777	. . . 4 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ (𝑥 gcd (𝑀 / 𝑁)) = 1)) → ((𝑥 · 𝑁) gcd 𝑀) = 𝑁)
53		oveq1 7416	. . . . . 6 ⊢ (𝑧 = (𝑥 · 𝑁) → (𝑧 gcd 𝑀) = ((𝑥 · 𝑁) gcd 𝑀))
54	53	eqeq1d 2735	. . . . 5 ⊢ (𝑧 = (𝑥 · 𝑁) → ((𝑧 gcd 𝑀) = 𝑁 ↔ ((𝑥 · 𝑁) gcd 𝑀) = 𝑁))
55		hashgcdlem.b	. . . . 5 ⊢ 𝐵 = {𝑧 ∈ (0..^𝑀) ∣ (𝑧 gcd 𝑀) = 𝑁}
56	54, 55	elrab2 3687	. . . 4 ⊢ ((𝑥 · 𝑁) ∈ 𝐵 ↔ ((𝑥 · 𝑁) ∈ (0..^𝑀) ∧ ((𝑥 · 𝑁) gcd 𝑀) = 𝑁))
57	30, 52, 56	sylanbrc 584	. . 3 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ (𝑥 gcd (𝑀 / 𝑁)) = 1)) → (𝑥 · 𝑁) ∈ 𝐵)
58	5, 57	sylan2b 595	. 2 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ 𝑥 ∈ 𝐴) → (𝑥 · 𝑁) ∈ 𝐵)
59		oveq1 7416	. . . . 5 ⊢ (𝑧 = 𝑤 → (𝑧 gcd 𝑀) = (𝑤 gcd 𝑀))
60	59	eqeq1d 2735	. . . 4 ⊢ (𝑧 = 𝑤 → ((𝑧 gcd 𝑀) = 𝑁 ↔ (𝑤 gcd 𝑀) = 𝑁))
61	60, 55	elrab2 3687	. . 3 ⊢ (𝑤 ∈ 𝐵 ↔ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁))
62		simprr 772	. . . . . . . 8 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → (𝑤 gcd 𝑀) = 𝑁)
63		elfzoelz 13632	. . . . . . . . . . 11 ⊢ (𝑤 ∈ (0..^𝑀) → 𝑤 ∈ ℤ)
64	63	ad2antrl 727	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → 𝑤 ∈ ℤ)
65		simpl1 1192	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → 𝑀 ∈ ℕ)
66	65	nnzd 12585	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → 𝑀 ∈ ℤ)
67		gcddvds 16444	. . . . . . . . . 10 ⊢ ((𝑤 ∈ ℤ ∧ 𝑀 ∈ ℤ) → ((𝑤 gcd 𝑀) ∥ 𝑤 ∧ (𝑤 gcd 𝑀) ∥ 𝑀))
68	64, 66, 67	syl2anc 585	. . . . . . . . 9 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → ((𝑤 gcd 𝑀) ∥ 𝑤 ∧ (𝑤 gcd 𝑀) ∥ 𝑀))
69	68	simpld 496	. . . . . . . 8 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → (𝑤 gcd 𝑀) ∥ 𝑤)
70	62, 69	eqbrtrrd 5173	. . . . . . 7 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → 𝑁 ∥ 𝑤)
71		nnz 12579	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ)
72	71	3ad2ant2 1135	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) → 𝑁 ∈ ℤ)
73	72	adantr 482	. . . . . . . 8 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → 𝑁 ∈ ℤ)
74	36	adantr 482	. . . . . . . 8 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → 𝑁 ≠ 0)
75		dvdsval2 16200	. . . . . . . 8 ⊢ ((𝑁 ∈ ℤ ∧ 𝑁 ≠ 0 ∧ 𝑤 ∈ ℤ) → (𝑁 ∥ 𝑤 ↔ (𝑤 / 𝑁) ∈ ℤ))
76	73, 74, 64, 75	syl3anc 1372	. . . . . . 7 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → (𝑁 ∥ 𝑤 ↔ (𝑤 / 𝑁) ∈ ℤ))
77	70, 76	mpbid 231	. . . . . 6 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → (𝑤 / 𝑁) ∈ ℤ)
78		elfzofz 13648	. . . . . . . . 9 ⊢ (𝑤 ∈ (0..^𝑀) → 𝑤 ∈ (0...𝑀))
79	78	ad2antrl 727	. . . . . . . 8 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → 𝑤 ∈ (0...𝑀))
80		elfznn0 13594	. . . . . . . 8 ⊢ (𝑤 ∈ (0...𝑀) → 𝑤 ∈ ℕ₀)
81		nn0re 12481	. . . . . . . . 9 ⊢ (𝑤 ∈ ℕ₀ → 𝑤 ∈ ℝ)
82		nn0ge0 12497	. . . . . . . . 9 ⊢ (𝑤 ∈ ℕ₀ → 0 ≤ 𝑤)
83	81, 82	jca 513	. . . . . . . 8 ⊢ (𝑤 ∈ ℕ₀ → (𝑤 ∈ ℝ ∧ 0 ≤ 𝑤))
84	79, 80, 83	3syl 18	. . . . . . 7 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → (𝑤 ∈ ℝ ∧ 0 ≤ 𝑤))
85	24	adantr 482	. . . . . . 7 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → (𝑁 ∈ ℝ ∧ 0 < 𝑁))
86		divge0 12083	. . . . . . 7 ⊢ (((𝑤 ∈ ℝ ∧ 0 ≤ 𝑤) ∧ (𝑁 ∈ ℝ ∧ 0 < 𝑁)) → 0 ≤ (𝑤 / 𝑁))
87	84, 85, 86	syl2anc 585	. . . . . 6 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → 0 ≤ (𝑤 / 𝑁))
88		elnn0z 12571	. . . . . 6 ⊢ ((𝑤 / 𝑁) ∈ ℕ₀ ↔ ((𝑤 / 𝑁) ∈ ℤ ∧ 0 ≤ (𝑤 / 𝑁)))
89	77, 87, 88	sylanbrc 584	. . . . 5 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → (𝑤 / 𝑁) ∈ ℕ₀)
90	42	adantr 482	. . . . 5 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → (𝑀 / 𝑁) ∈ ℕ)
91		elfzolt2 13641	. . . . . . 7 ⊢ (𝑤 ∈ (0..^𝑀) → 𝑤 < 𝑀)
92	91	ad2antrl 727	. . . . . 6 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → 𝑤 < 𝑀)
93	64	zred 12666	. . . . . . 7 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → 𝑤 ∈ ℝ)
94	19	adantr 482	. . . . . . 7 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → 𝑀 ∈ ℝ)
95		ltdiv1 12078	. . . . . . 7 ⊢ ((𝑤 ∈ ℝ ∧ 𝑀 ∈ ℝ ∧ (𝑁 ∈ ℝ ∧ 0 < 𝑁)) → (𝑤 < 𝑀 ↔ (𝑤 / 𝑁) < (𝑀 / 𝑁)))
96	93, 94, 85, 95	syl3anc 1372	. . . . . 6 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → (𝑤 < 𝑀 ↔ (𝑤 / 𝑁) < (𝑀 / 𝑁)))
97	92, 96	mpbid 231	. . . . 5 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → (𝑤 / 𝑁) < (𝑀 / 𝑁))
98		elfzo0 13673	. . . . 5 ⊢ ((𝑤 / 𝑁) ∈ (0..^(𝑀 / 𝑁)) ↔ ((𝑤 / 𝑁) ∈ ℕ₀ ∧ (𝑀 / 𝑁) ∈ ℕ ∧ (𝑤 / 𝑁) < (𝑀 / 𝑁)))
99	89, 90, 97, 98	syl3anbrc 1344	. . . 4 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → (𝑤 / 𝑁) ∈ (0..^(𝑀 / 𝑁)))
100	62	oveq1d 7424	. . . . 5 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → ((𝑤 gcd 𝑀) / 𝑁) = (𝑁 / 𝑁))
101		simpl2 1193	. . . . . 6 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → 𝑁 ∈ ℕ)
102		simpl3 1194	. . . . . 6 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → 𝑁 ∥ 𝑀)
103		gcddiv 16493	. . . . . 6 ⊢ (((𝑤 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ (𝑁 ∥ 𝑤 ∧ 𝑁 ∥ 𝑀)) → ((𝑤 gcd 𝑀) / 𝑁) = ((𝑤 / 𝑁) gcd (𝑀 / 𝑁)))
104	64, 66, 101, 70, 102, 103	syl32anc 1379	. . . . 5 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → ((𝑤 gcd 𝑀) / 𝑁) = ((𝑤 / 𝑁) gcd (𝑀 / 𝑁)))
105	34, 36	dividd 11988	. . . . . 6 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) → (𝑁 / 𝑁) = 1)
106	105	adantr 482	. . . . 5 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → (𝑁 / 𝑁) = 1)
107	100, 104, 106	3eqtr3d 2781	. . . 4 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → ((𝑤 / 𝑁) gcd (𝑀 / 𝑁)) = 1)
108		oveq1 7416	. . . . . 6 ⊢ (𝑦 = (𝑤 / 𝑁) → (𝑦 gcd (𝑀 / 𝑁)) = ((𝑤 / 𝑁) gcd (𝑀 / 𝑁)))
109	108	eqeq1d 2735	. . . . 5 ⊢ (𝑦 = (𝑤 / 𝑁) → ((𝑦 gcd (𝑀 / 𝑁)) = 1 ↔ ((𝑤 / 𝑁) gcd (𝑀 / 𝑁)) = 1))
110	109, 4	elrab2 3687	. . . 4 ⊢ ((𝑤 / 𝑁) ∈ 𝐴 ↔ ((𝑤 / 𝑁) ∈ (0..^(𝑀 / 𝑁)) ∧ ((𝑤 / 𝑁) gcd (𝑀 / 𝑁)) = 1))
111	99, 107, 110	sylanbrc 584	. . 3 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → (𝑤 / 𝑁) ∈ 𝐴)
112	61, 111	sylan2b 595	. 2 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ 𝑤 ∈ 𝐵) → (𝑤 / 𝑁) ∈ 𝐴)
113	5	simplbi 499	. . . 4 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ (0..^(𝑀 / 𝑁)))
114	61	simplbi 499	. . . 4 ⊢ (𝑤 ∈ 𝐵 → 𝑤 ∈ (0..^𝑀))
115	113, 114	anim12i 614	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) → (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ 𝑤 ∈ (0..^𝑀)))
116	63	ad2antll 728	. . . . . . . 8 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ 𝑤 ∈ (0..^𝑀))) → 𝑤 ∈ ℤ)
117	116	zcnd 12667	. . . . . . 7 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ 𝑤 ∈ (0..^𝑀))) → 𝑤 ∈ ℂ)
118	34	adantr 482	. . . . . . 7 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ 𝑤 ∈ (0..^𝑀))) → 𝑁 ∈ ℂ)
119	36	adantr 482	. . . . . . 7 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ 𝑤 ∈ (0..^𝑀))) → 𝑁 ≠ 0)
120	117, 118, 119	divcan1d 11991	. . . . . 6 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ 𝑤 ∈ (0..^𝑀))) → ((𝑤 / 𝑁) · 𝑁) = 𝑤)
121	120	eqcomd 2739	. . . . 5 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ 𝑤 ∈ (0..^𝑀))) → 𝑤 = ((𝑤 / 𝑁) · 𝑁))
122		oveq1 7416	. . . . . 6 ⊢ (𝑥 = (𝑤 / 𝑁) → (𝑥 · 𝑁) = ((𝑤 / 𝑁) · 𝑁))
123	122	eqeq2d 2744	. . . . 5 ⊢ (𝑥 = (𝑤 / 𝑁) → (𝑤 = (𝑥 · 𝑁) ↔ 𝑤 = ((𝑤 / 𝑁) · 𝑁)))
124	121, 123	syl5ibrcom 246	. . . 4 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ 𝑤 ∈ (0..^𝑀))) → (𝑥 = (𝑤 / 𝑁) → 𝑤 = (𝑥 · 𝑁)))
125	15	ad2antrl 727	. . . . . . . 8 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ 𝑤 ∈ (0..^𝑀))) → 𝑥 ∈ ℤ)
126	125	zcnd 12667	. . . . . . 7 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ 𝑤 ∈ (0..^𝑀))) → 𝑥 ∈ ℂ)
127	126, 118, 119	divcan4d 11996	. . . . . 6 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ 𝑤 ∈ (0..^𝑀))) → ((𝑥 · 𝑁) / 𝑁) = 𝑥)
128	127	eqcomd 2739	. . . . 5 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ 𝑤 ∈ (0..^𝑀))) → 𝑥 = ((𝑥 · 𝑁) / 𝑁))
129		oveq1 7416	. . . . . 6 ⊢ (𝑤 = (𝑥 · 𝑁) → (𝑤 / 𝑁) = ((𝑥 · 𝑁) / 𝑁))
130	129	eqeq2d 2744	. . . . 5 ⊢ (𝑤 = (𝑥 · 𝑁) → (𝑥 = (𝑤 / 𝑁) ↔ 𝑥 = ((𝑥 · 𝑁) / 𝑁)))
131	128, 130	syl5ibrcom 246	. . . 4 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ 𝑤 ∈ (0..^𝑀))) → (𝑤 = (𝑥 · 𝑁) → 𝑥 = (𝑤 / 𝑁)))
132	124, 131	impbid 211	. . 3 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ 𝑤 ∈ (0..^𝑀))) → (𝑥 = (𝑤 / 𝑁) ↔ 𝑤 = (𝑥 · 𝑁)))
133	115, 132	sylan2 594	. 2 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵)) → (𝑥 = (𝑤 / 𝑁) ↔ 𝑤 = (𝑥 · 𝑁)))
134	1, 58, 112, 133	f1o2d 7660	1 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) → 𝐹:𝐴–1-1-onto→𝐵)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 {crab 3433 class class class wbr 5149 ↦ cmpt 5232 –1-1-onto→wf1o 6543 (class class class)co 7409 ℂcc 11108 ℝcr 11109 0cc0 11110 1c1 11111 · cmul 11115 < clt 11248 ≤ cle 11249 / cdiv 11871 ℕcn 12212 ℕ₀cn0 12472 ℤcz 12558 ...cfz 13484 ..^cfzo 13627 ∥ cdvds 16197 gcd cgcd 16435

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-sup 9437 df-inf 9438 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-n0 12473 df-z 12559 df-uz 12823 df-rp 12975 df-fz 13485 df-fzo 13628 df-fl 13757 df-mod 13835 df-seq 13967 df-exp 14028 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-dvds 16198 df-gcd 16436

This theorem is referenced by: hashgcdeq 16722

W3C validator