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Theorem hashgcdlem 15493

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 6657	. . . . 5 ⊢ (𝑦 = 𝑥 → (𝑦 gcd (𝑀 / 𝑁)) = (𝑥 gcd (𝑀 / 𝑁)))
3	2	eqeq1d 2624	. . . 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 3366	. . 3 ⊢ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ (𝑥 gcd (𝑀 / 𝑁)) = 1))
6		elfzonn0 12512	. . . . . . 7 ⊢ (𝑥 ∈ (0..^(𝑀 / 𝑁)) → 𝑥 ∈ ℕ₀)
7	6	ad2antrl 764	. . . . . 6 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ (𝑥 gcd (𝑀 / 𝑁)) = 1)) → 𝑥 ∈ ℕ₀)
8		nnnn0 11299	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ₀)
9	8	3ad2ant2 1083	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) → 𝑁 ∈ ℕ₀)
10	9	adantr 481	. . . . . 6 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ (𝑥 gcd (𝑀 / 𝑁)) = 1)) → 𝑁 ∈ ℕ₀)
11	7, 10	nn0mulcld 11356	. . . . 5 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ (𝑥 gcd (𝑀 / 𝑁)) = 1)) → (𝑥 · 𝑁) ∈ ℕ₀)
12		simpl1 1064	. . . . 5 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ (𝑥 gcd (𝑀 / 𝑁)) = 1)) → 𝑀 ∈ ℕ)
13		elfzolt2 12479	. . . . . . 7 ⊢ (𝑥 ∈ (0..^(𝑀 / 𝑁)) → 𝑥 < (𝑀 / 𝑁))
14	13	ad2antrl 764	. . . . . 6 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ (𝑥 gcd (𝑀 / 𝑁)) = 1)) → 𝑥 < (𝑀 / 𝑁))
15		elfzoelz 12470	. . . . . . . . 9 ⊢ (𝑥 ∈ (0..^(𝑀 / 𝑁)) → 𝑥 ∈ ℤ)
16	15	ad2antrl 764	. . . . . . . 8 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ (𝑥 gcd (𝑀 / 𝑁)) = 1)) → 𝑥 ∈ ℤ)
17	16	zred 11482	. . . . . . 7 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ (𝑥 gcd (𝑀 / 𝑁)) = 1)) → 𝑥 ∈ ℝ)
18		nnre 11027	. . . . . . . . 9 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℝ)
19	18	3ad2ant1 1082	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) → 𝑀 ∈ ℝ)
20	19	adantr 481	. . . . . . 7 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ (𝑥 gcd (𝑀 / 𝑁)) = 1)) → 𝑀 ∈ ℝ)
21		nnre 11027	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ)
22		nngt0 11049	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → 0 < 𝑁)
23	21, 22	jca 554	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → (𝑁 ∈ ℝ ∧ 0 < 𝑁))
24	23	3ad2ant2 1083	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) → (𝑁 ∈ ℝ ∧ 0 < 𝑁))
25	24	adantr 481	. . . . . . 7 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ (𝑥 gcd (𝑀 / 𝑁)) = 1)) → (𝑁 ∈ ℝ ∧ 0 < 𝑁))
26		ltmuldiv 10896	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ ∧ 𝑀 ∈ ℝ ∧ (𝑁 ∈ ℝ ∧ 0 < 𝑁)) → ((𝑥 · 𝑁) < 𝑀 ↔ 𝑥 < (𝑀 / 𝑁)))
27	17, 20, 25, 26	syl3anc 1326	. . . . . 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 247	. . . . 5 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ (𝑥 gcd (𝑀 / 𝑁)) = 1)) → (𝑥 · 𝑁) < 𝑀)
29		elfzo0 12508	. . . . 5 ⊢ ((𝑥 · 𝑁) ∈ (0..^𝑀) ↔ ((𝑥 · 𝑁) ∈ ℕ₀ ∧ 𝑀 ∈ ℕ ∧ (𝑥 · 𝑁) < 𝑀))
30	11, 12, 28, 29	syl3anbrc 1246	. . . 4 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ (𝑥 gcd (𝑀 / 𝑁)) = 1)) → (𝑥 · 𝑁) ∈ (0..^𝑀))
31		nncn 11028	. . . . . . . . . 10 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℂ)
32	31	3ad2ant1 1082	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) → 𝑀 ∈ ℂ)
33		nncn 11028	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ)
34	33	3ad2ant2 1083	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) → 𝑁 ∈ ℂ)
35		nnne0 11053	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → 𝑁 ≠ 0)
36	35	3ad2ant2 1083	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) → 𝑁 ≠ 0)
37	32, 34, 36	divcan1d 10802	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) → ((𝑀 / 𝑁) · 𝑁) = 𝑀)
38	37	adantr 481	. . . . . . 7 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ (𝑥 gcd (𝑀 / 𝑁)) = 1)) → ((𝑀 / 𝑁) · 𝑁) = 𝑀)
39	38	eqcomd 2628	. . . . . 6 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ (𝑥 gcd (𝑀 / 𝑁)) = 1)) → 𝑀 = ((𝑀 / 𝑁) · 𝑁))
40	39	oveq2d 6666	. . . . 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 14989	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑁 ∥ 𝑀 ↔ (𝑀 / 𝑁) ∈ ℕ))
42	41	biimp3a 1432	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) → (𝑀 / 𝑁) ∈ ℕ)
43	42	nnzd 11481	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) → (𝑀 / 𝑁) ∈ ℤ)
44	43	adantr 481	. . . . . 6 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ (𝑥 gcd (𝑀 / 𝑁)) = 1)) → (𝑀 / 𝑁) ∈ ℤ)
45		mulgcdr 15267	. . . . . 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 1326	. . . . 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 796	. . . . . . 7 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ (𝑥 gcd (𝑀 / 𝑁)) = 1)) → (𝑥 gcd (𝑀 / 𝑁)) = 1)
48	47	oveq1d 6665	. . . . . 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	mulid2d 10058	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) → (1 · 𝑁) = 𝑁)
50	49	adantr 481	. . . . . 6 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ (𝑥 gcd (𝑀 / 𝑁)) = 1)) → (1 · 𝑁) = 𝑁)
51	48, 50	eqtrd 2656	. . . . 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 2660	. . . 4 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ (𝑥 gcd (𝑀 / 𝑁)) = 1)) → ((𝑥 · 𝑁) gcd 𝑀) = 𝑁)
53		oveq1 6657	. . . . . 6 ⊢ (𝑧 = (𝑥 · 𝑁) → (𝑧 gcd 𝑀) = ((𝑥 · 𝑁) gcd 𝑀))
54	53	eqeq1d 2624	. . . . 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 3366	. . . 4 ⊢ ((𝑥 · 𝑁) ∈ 𝐵 ↔ ((𝑥 · 𝑁) ∈ (0..^𝑀) ∧ ((𝑥 · 𝑁) gcd 𝑀) = 𝑁))
57	30, 52, 56	sylanbrc 698	. . 3 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ (𝑥 gcd (𝑀 / 𝑁)) = 1)) → (𝑥 · 𝑁) ∈ 𝐵)
58	5, 57	sylan2b 492	. 2 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ 𝑥 ∈ 𝐴) → (𝑥 · 𝑁) ∈ 𝐵)
59		oveq1 6657	. . . . 5 ⊢ (𝑧 = 𝑤 → (𝑧 gcd 𝑀) = (𝑤 gcd 𝑀))
60	59	eqeq1d 2624	. . . 4 ⊢ (𝑧 = 𝑤 → ((𝑧 gcd 𝑀) = 𝑁 ↔ (𝑤 gcd 𝑀) = 𝑁))
61	60, 55	elrab2 3366	. . 3 ⊢ (𝑤 ∈ 𝐵 ↔ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁))
62		simprr 796	. . . . . . . 8 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → (𝑤 gcd 𝑀) = 𝑁)
63		elfzoelz 12470	. . . . . . . . . . 11 ⊢ (𝑤 ∈ (0..^𝑀) → 𝑤 ∈ ℤ)
64	63	ad2antrl 764	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → 𝑤 ∈ ℤ)
65		simpl1 1064	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → 𝑀 ∈ ℕ)
66	65	nnzd 11481	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → 𝑀 ∈ ℤ)
67		gcddvds 15225	. . . . . . . . . 10 ⊢ ((𝑤 ∈ ℤ ∧ 𝑀 ∈ ℤ) → ((𝑤 gcd 𝑀) ∥ 𝑤 ∧ (𝑤 gcd 𝑀) ∥ 𝑀))
68	64, 66, 67	syl2anc 693	. . . . . . . . 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 475	. . . . . . . 8 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → (𝑤 gcd 𝑀) ∥ 𝑤)
70	62, 69	eqbrtrrd 4677	. . . . . . 7 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → 𝑁 ∥ 𝑤)
71		nnz 11399	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ)
72	71	3ad2ant2 1083	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) → 𝑁 ∈ ℤ)
73	72	adantr 481	. . . . . . . 8 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → 𝑁 ∈ ℤ)
74	36	adantr 481	. . . . . . . 8 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → 𝑁 ≠ 0)
75		dvdsval2 14986	. . . . . . . 8 ⊢ ((𝑁 ∈ ℤ ∧ 𝑁 ≠ 0 ∧ 𝑤 ∈ ℤ) → (𝑁 ∥ 𝑤 ↔ (𝑤 / 𝑁) ∈ ℤ))
76	73, 74, 64, 75	syl3anc 1326	. . . . . . 7 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → (𝑁 ∥ 𝑤 ↔ (𝑤 / 𝑁) ∈ ℤ))
77	70, 76	mpbid 222	. . . . . 6 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → (𝑤 / 𝑁) ∈ ℤ)
78		elfzofz 12485	. . . . . . . . 9 ⊢ (𝑤 ∈ (0..^𝑀) → 𝑤 ∈ (0...𝑀))
79	78	ad2antrl 764	. . . . . . . 8 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → 𝑤 ∈ (0...𝑀))
80		elfznn0 12433	. . . . . . . 8 ⊢ (𝑤 ∈ (0...𝑀) → 𝑤 ∈ ℕ₀)
81		nn0re 11301	. . . . . . . . 9 ⊢ (𝑤 ∈ ℕ₀ → 𝑤 ∈ ℝ)
82		nn0ge0 11318	. . . . . . . . 9 ⊢ (𝑤 ∈ ℕ₀ → 0 ≤ 𝑤)
83	81, 82	jca 554	. . . . . . . 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 481	. . . . . . 7 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → (𝑁 ∈ ℝ ∧ 0 < 𝑁))
86		divge0 10892	. . . . . . 7 ⊢ (((𝑤 ∈ ℝ ∧ 0 ≤ 𝑤) ∧ (𝑁 ∈ ℝ ∧ 0 < 𝑁)) → 0 ≤ (𝑤 / 𝑁))
87	84, 85, 86	syl2anc 693	. . . . . 6 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → 0 ≤ (𝑤 / 𝑁))
88		elnn0z 11390	. . . . . 6 ⊢ ((𝑤 / 𝑁) ∈ ℕ₀ ↔ ((𝑤 / 𝑁) ∈ ℤ ∧ 0 ≤ (𝑤 / 𝑁)))
89	77, 87, 88	sylanbrc 698	. . . . 5 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → (𝑤 / 𝑁) ∈ ℕ₀)
90	42	adantr 481	. . . . 5 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → (𝑀 / 𝑁) ∈ ℕ)
91		elfzolt2 12479	. . . . . . 7 ⊢ (𝑤 ∈ (0..^𝑀) → 𝑤 < 𝑀)
92	91	ad2antrl 764	. . . . . 6 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → 𝑤 < 𝑀)
93	64	zred 11482	. . . . . . 7 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → 𝑤 ∈ ℝ)
94	19	adantr 481	. . . . . . 7 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → 𝑀 ∈ ℝ)
95		ltdiv1 10887	. . . . . . 7 ⊢ ((𝑤 ∈ ℝ ∧ 𝑀 ∈ ℝ ∧ (𝑁 ∈ ℝ ∧ 0 < 𝑁)) → (𝑤 < 𝑀 ↔ (𝑤 / 𝑁) < (𝑀 / 𝑁)))
96	93, 94, 85, 95	syl3anc 1326	. . . . . 6 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → (𝑤 < 𝑀 ↔ (𝑤 / 𝑁) < (𝑀 / 𝑁)))
97	92, 96	mpbid 222	. . . . 5 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → (𝑤 / 𝑁) < (𝑀 / 𝑁))
98		elfzo0 12508	. . . . 5 ⊢ ((𝑤 / 𝑁) ∈ (0..^(𝑀 / 𝑁)) ↔ ((𝑤 / 𝑁) ∈ ℕ₀ ∧ (𝑀 / 𝑁) ∈ ℕ ∧ (𝑤 / 𝑁) < (𝑀 / 𝑁)))
99	89, 90, 97, 98	syl3anbrc 1246	. . . 4 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → (𝑤 / 𝑁) ∈ (0..^(𝑀 / 𝑁)))
100	62	oveq1d 6665	. . . . 5 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → ((𝑤 gcd 𝑀) / 𝑁) = (𝑁 / 𝑁))
101		simpl2 1065	. . . . . 6 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → 𝑁 ∈ ℕ)
102		simpl3 1066	. . . . . 6 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → 𝑁 ∥ 𝑀)
103		gcddiv 15268	. . . . . 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 1334	. . . . 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 10799	. . . . . 6 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) → (𝑁 / 𝑁) = 1)
106	105	adantr 481	. . . . 5 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → (𝑁 / 𝑁) = 1)
107	100, 104, 106	3eqtr3d 2664	. . . 4 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → ((𝑤 / 𝑁) gcd (𝑀 / 𝑁)) = 1)
108		oveq1 6657	. . . . . 6 ⊢ (𝑦 = (𝑤 / 𝑁) → (𝑦 gcd (𝑀 / 𝑁)) = ((𝑤 / 𝑁) gcd (𝑀 / 𝑁)))
109	108	eqeq1d 2624	. . . . 5 ⊢ (𝑦 = (𝑤 / 𝑁) → ((𝑦 gcd (𝑀 / 𝑁)) = 1 ↔ ((𝑤 / 𝑁) gcd (𝑀 / 𝑁)) = 1))
110	109, 4	elrab2 3366	. . . 4 ⊢ ((𝑤 / 𝑁) ∈ 𝐴 ↔ ((𝑤 / 𝑁) ∈ (0..^(𝑀 / 𝑁)) ∧ ((𝑤 / 𝑁) gcd (𝑀 / 𝑁)) = 1))
111	99, 107, 110	sylanbrc 698	. . 3 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → (𝑤 / 𝑁) ∈ 𝐴)
112	61, 111	sylan2b 492	. 2 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ 𝑤 ∈ 𝐵) → (𝑤 / 𝑁) ∈ 𝐴)
113	5	simplbi 476	. . . 4 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ (0..^(𝑀 / 𝑁)))
114	61	simplbi 476	. . . 4 ⊢ (𝑤 ∈ 𝐵 → 𝑤 ∈ (0..^𝑀))
115	113, 114	anim12i 590	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) → (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ 𝑤 ∈ (0..^𝑀)))
116	63	ad2antll 765	. . . . . . . 8 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ 𝑤 ∈ (0..^𝑀))) → 𝑤 ∈ ℤ)
117	116	zcnd 11483	. . . . . . 7 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ 𝑤 ∈ (0..^𝑀))) → 𝑤 ∈ ℂ)
118	34	adantr 481	. . . . . . 7 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ 𝑤 ∈ (0..^𝑀))) → 𝑁 ∈ ℂ)
119	36	adantr 481	. . . . . . 7 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ 𝑤 ∈ (0..^𝑀))) → 𝑁 ≠ 0)
120	117, 118, 119	divcan1d 10802	. . . . . 6 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ 𝑤 ∈ (0..^𝑀))) → ((𝑤 / 𝑁) · 𝑁) = 𝑤)
121	120	eqcomd 2628	. . . . 5 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ 𝑤 ∈ (0..^𝑀))) → 𝑤 = ((𝑤 / 𝑁) · 𝑁))
122		oveq1 6657	. . . . . 6 ⊢ (𝑥 = (𝑤 / 𝑁) → (𝑥 · 𝑁) = ((𝑤 / 𝑁) · 𝑁))
123	122	eqeq2d 2632	. . . . 5 ⊢ (𝑥 = (𝑤 / 𝑁) → (𝑤 = (𝑥 · 𝑁) ↔ 𝑤 = ((𝑤 / 𝑁) · 𝑁)))
124	121, 123	syl5ibrcom 237	. . . 4 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ 𝑤 ∈ (0..^𝑀))) → (𝑥 = (𝑤 / 𝑁) → 𝑤 = (𝑥 · 𝑁)))
125	15	ad2antrl 764	. . . . . . . 8 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ 𝑤 ∈ (0..^𝑀))) → 𝑥 ∈ ℤ)
126	125	zcnd 11483	. . . . . . 7 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ 𝑤 ∈ (0..^𝑀))) → 𝑥 ∈ ℂ)
127	126, 118, 119	divcan4d 10807	. . . . . 6 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ 𝑤 ∈ (0..^𝑀))) → ((𝑥 · 𝑁) / 𝑁) = 𝑥)
128	127	eqcomd 2628	. . . . 5 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ 𝑤 ∈ (0..^𝑀))) → 𝑥 = ((𝑥 · 𝑁) / 𝑁))
129		oveq1 6657	. . . . . 6 ⊢ (𝑤 = (𝑥 · 𝑁) → (𝑤 / 𝑁) = ((𝑥 · 𝑁) / 𝑁))
130	129	eqeq2d 2632	. . . . 5 ⊢ (𝑤 = (𝑥 · 𝑁) → (𝑥 = (𝑤 / 𝑁) ↔ 𝑥 = ((𝑥 · 𝑁) / 𝑁)))
131	128, 130	syl5ibrcom 237	. . . 4 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ 𝑤 ∈ (0..^𝑀))) → (𝑤 = (𝑥 · 𝑁) → 𝑥 = (𝑤 / 𝑁)))
132	124, 131	impbid 202	. . 3 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ 𝑤 ∈ (0..^𝑀))) → (𝑥 = (𝑤 / 𝑁) ↔ 𝑤 = (𝑥 · 𝑁)))
133	115, 132	sylan2 491	. 2 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵)) → (𝑥 = (𝑤 / 𝑁) ↔ 𝑤 = (𝑥 · 𝑁)))
134	1, 58, 112, 133	f1o2d 6887	1 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) → 𝐹:𝐴–1-1-onto→𝐵)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 {crab 2916 class class class wbr 4653 ↦ cmpt 4729 –1-1-onto→wf1o 5887 (class class class)co 6650 ℂcc 9934 ℝcr 9935 0cc0 9936 1c1 9937 · cmul 9941 < clt 10074 ≤ cle 10075 / cdiv 10684 ℕcn 11020 ℕ₀cn0 11292 ℤcz 11377 ...cfz 12326 ..^cfzo 12465 ∥ cdvds 14983 gcd cgcd 15216

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-inf 8349 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-fz 12327 df-fzo 12466 df-fl 12593 df-mod 12669 df-seq 12802 df-exp 12861 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-dvds 14984 df-gcd 15217

This theorem is referenced by: hashgcdeq 15494

W3C validator