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Theorem hashgcdlem 16758

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 7394	. . . . 5 ⊢ (𝑦 = 𝑥 → (𝑦 gcd (𝑀 / 𝑁)) = (𝑥 gcd (𝑀 / 𝑁)))
3	2	eqeq1d 2731	. . . 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 3662	. . 3 ⊢ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ (𝑥 gcd (𝑀 / 𝑁)) = 1))
6		elfzonn0 13668	. . . . . . 7 ⊢ (𝑥 ∈ (0..^(𝑀 / 𝑁)) → 𝑥 ∈ ℕ₀)
7	6	ad2antrl 728	. . . . . 6 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ (𝑥 gcd (𝑀 / 𝑁)) = 1)) → 𝑥 ∈ ℕ₀)
8		nnnn0 12449	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ₀)
9	8	3ad2ant2 1134	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) → 𝑁 ∈ ℕ₀)
10	9	adantr 480	. . . . . 6 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ (𝑥 gcd (𝑀 / 𝑁)) = 1)) → 𝑁 ∈ ℕ₀)
11	7, 10	nn0mulcld 12508	. . . . 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 13629	. . . . . . 7 ⊢ (𝑥 ∈ (0..^(𝑀 / 𝑁)) → 𝑥 < (𝑀 / 𝑁))
14	13	ad2antrl 728	. . . . . 6 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ (𝑥 gcd (𝑀 / 𝑁)) = 1)) → 𝑥 < (𝑀 / 𝑁))
15		elfzoelz 13620	. . . . . . . . 9 ⊢ (𝑥 ∈ (0..^(𝑀 / 𝑁)) → 𝑥 ∈ ℤ)
16	15	ad2antrl 728	. . . . . . . 8 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ (𝑥 gcd (𝑀 / 𝑁)) = 1)) → 𝑥 ∈ ℤ)
17	16	zred 12638	. . . . . . 7 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ (𝑥 gcd (𝑀 / 𝑁)) = 1)) → 𝑥 ∈ ℝ)
18		nnre 12193	. . . . . . . . 9 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℝ)
19	18	3ad2ant1 1133	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) → 𝑀 ∈ ℝ)
20	19	adantr 480	. . . . . . 7 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ (𝑥 gcd (𝑀 / 𝑁)) = 1)) → 𝑀 ∈ ℝ)
21		nnre 12193	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ)
22		nngt0 12217	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → 0 < 𝑁)
23	21, 22	jca 511	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → (𝑁 ∈ ℝ ∧ 0 < 𝑁))
24	23	3ad2ant2 1134	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) → (𝑁 ∈ ℝ ∧ 0 < 𝑁))
25	24	adantr 480	. . . . . . 7 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ (𝑥 gcd (𝑀 / 𝑁)) = 1)) → (𝑁 ∈ ℝ ∧ 0 < 𝑁))
26		ltmuldiv 12056	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ ∧ 𝑀 ∈ ℝ ∧ (𝑁 ∈ ℝ ∧ 0 < 𝑁)) → ((𝑥 · 𝑁) < 𝑀 ↔ 𝑥 < (𝑀 / 𝑁)))
27	17, 20, 25, 26	syl3anc 1373	. . . . . 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 13661	. . . . 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 12194	. . . . . . . . . 10 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℂ)
32	31	3ad2ant1 1133	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) → 𝑀 ∈ ℂ)
33		nncn 12194	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ)
34	33	3ad2ant2 1134	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) → 𝑁 ∈ ℂ)
35		nnne0 12220	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → 𝑁 ≠ 0)
36	35	3ad2ant2 1134	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) → 𝑁 ≠ 0)
37	32, 34, 36	divcan1d 11959	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) → ((𝑀 / 𝑁) · 𝑁) = 𝑀)
38	37	adantr 480	. . . . . . 7 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ (𝑥 gcd (𝑀 / 𝑁)) = 1)) → ((𝑀 / 𝑁) · 𝑁) = 𝑀)
39	38	eqcomd 2735	. . . . . 6 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ (𝑥 gcd (𝑀 / 𝑁)) = 1)) → 𝑀 = ((𝑀 / 𝑁) · 𝑁))
40	39	oveq2d 7403	. . . . 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 16231	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑁 ∥ 𝑀 ↔ (𝑀 / 𝑁) ∈ ℕ))
42	41	biimp3a 1471	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) → (𝑀 / 𝑁) ∈ ℕ)
43	42	nnzd 12556	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) → (𝑀 / 𝑁) ∈ ℤ)
44	43	adantr 480	. . . . . 6 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ (𝑥 gcd (𝑀 / 𝑁)) = 1)) → (𝑀 / 𝑁) ∈ ℤ)
45		mulgcdr 16520	. . . . . 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 1373	. . . . 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 7402	. . . . . 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 11192	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) → (1 · 𝑁) = 𝑁)
50	49	adantr 480	. . . . . 6 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ (𝑥 gcd (𝑀 / 𝑁)) = 1)) → (1 · 𝑁) = 𝑁)
51	48, 50	eqtrd 2764	. . . . 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 2768	. . . 4 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ (𝑥 gcd (𝑀 / 𝑁)) = 1)) → ((𝑥 · 𝑁) gcd 𝑀) = 𝑁)
53		oveq1 7394	. . . . . 6 ⊢ (𝑧 = (𝑥 · 𝑁) → (𝑧 gcd 𝑀) = ((𝑥 · 𝑁) gcd 𝑀))
54	53	eqeq1d 2731	. . . . 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 3662	. . . 4 ⊢ ((𝑥 · 𝑁) ∈ 𝐵 ↔ ((𝑥 · 𝑁) ∈ (0..^𝑀) ∧ ((𝑥 · 𝑁) gcd 𝑀) = 𝑁))
57	30, 52, 56	sylanbrc 583	. . 3 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ (𝑥 gcd (𝑀 / 𝑁)) = 1)) → (𝑥 · 𝑁) ∈ 𝐵)
58	5, 57	sylan2b 594	. 2 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ 𝑥 ∈ 𝐴) → (𝑥 · 𝑁) ∈ 𝐵)
59		oveq1 7394	. . . . 5 ⊢ (𝑧 = 𝑤 → (𝑧 gcd 𝑀) = (𝑤 gcd 𝑀))
60	59	eqeq1d 2731	. . . 4 ⊢ (𝑧 = 𝑤 → ((𝑧 gcd 𝑀) = 𝑁 ↔ (𝑤 gcd 𝑀) = 𝑁))
61	60, 55	elrab2 3662	. . 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 13620	. . . . . . . . . . 11 ⊢ (𝑤 ∈ (0..^𝑀) → 𝑤 ∈ ℤ)
64	63	ad2antrl 728	. . . . . . . . . 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 12556	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → 𝑀 ∈ ℤ)
67		gcddvds 16473	. . . . . . . . . 10 ⊢ ((𝑤 ∈ ℤ ∧ 𝑀 ∈ ℤ) → ((𝑤 gcd 𝑀) ∥ 𝑤 ∧ (𝑤 gcd 𝑀) ∥ 𝑀))
68	64, 66, 67	syl2anc 584	. . . . . . . . 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 494	. . . . . . . 8 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → (𝑤 gcd 𝑀) ∥ 𝑤)
70	62, 69	eqbrtrrd 5131	. . . . . . 7 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → 𝑁 ∥ 𝑤)
71		nnz 12550	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ)
72	71	3ad2ant2 1134	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) → 𝑁 ∈ ℤ)
73	72	adantr 480	. . . . . . . 8 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → 𝑁 ∈ ℤ)
74	36	adantr 480	. . . . . . . 8 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → 𝑁 ≠ 0)
75		dvdsval2 16225	. . . . . . . 8 ⊢ ((𝑁 ∈ ℤ ∧ 𝑁 ≠ 0 ∧ 𝑤 ∈ ℤ) → (𝑁 ∥ 𝑤 ↔ (𝑤 / 𝑁) ∈ ℤ))
76	73, 74, 64, 75	syl3anc 1373	. . . . . . 7 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → (𝑁 ∥ 𝑤 ↔ (𝑤 / 𝑁) ∈ ℤ))
77	70, 76	mpbid 232	. . . . . 6 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → (𝑤 / 𝑁) ∈ ℤ)
78		elfzofz 13636	. . . . . . . . 9 ⊢ (𝑤 ∈ (0..^𝑀) → 𝑤 ∈ (0...𝑀))
79	78	ad2antrl 728	. . . . . . . 8 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → 𝑤 ∈ (0...𝑀))
80		elfznn0 13581	. . . . . . . 8 ⊢ (𝑤 ∈ (0...𝑀) → 𝑤 ∈ ℕ₀)
81		nn0re 12451	. . . . . . . . 9 ⊢ (𝑤 ∈ ℕ₀ → 𝑤 ∈ ℝ)
82		nn0ge0 12467	. . . . . . . . 9 ⊢ (𝑤 ∈ ℕ₀ → 0 ≤ 𝑤)
83	81, 82	jca 511	. . . . . . . 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 480	. . . . . . 7 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → (𝑁 ∈ ℝ ∧ 0 < 𝑁))
86		divge0 12052	. . . . . . 7 ⊢ (((𝑤 ∈ ℝ ∧ 0 ≤ 𝑤) ∧ (𝑁 ∈ ℝ ∧ 0 < 𝑁)) → 0 ≤ (𝑤 / 𝑁))
87	84, 85, 86	syl2anc 584	. . . . . 6 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → 0 ≤ (𝑤 / 𝑁))
88		elnn0z 12542	. . . . . 6 ⊢ ((𝑤 / 𝑁) ∈ ℕ₀ ↔ ((𝑤 / 𝑁) ∈ ℤ ∧ 0 ≤ (𝑤 / 𝑁)))
89	77, 87, 88	sylanbrc 583	. . . . 5 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → (𝑤 / 𝑁) ∈ ℕ₀)
90	42	adantr 480	. . . . 5 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → (𝑀 / 𝑁) ∈ ℕ)
91		elfzolt2 13629	. . . . . . 7 ⊢ (𝑤 ∈ (0..^𝑀) → 𝑤 < 𝑀)
92	91	ad2antrl 728	. . . . . 6 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → 𝑤 < 𝑀)
93	64	zred 12638	. . . . . . 7 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → 𝑤 ∈ ℝ)
94	19	adantr 480	. . . . . . 7 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → 𝑀 ∈ ℝ)
95		ltdiv1 12047	. . . . . . 7 ⊢ ((𝑤 ∈ ℝ ∧ 𝑀 ∈ ℝ ∧ (𝑁 ∈ ℝ ∧ 0 < 𝑁)) → (𝑤 < 𝑀 ↔ (𝑤 / 𝑁) < (𝑀 / 𝑁)))
96	93, 94, 85, 95	syl3anc 1373	. . . . . 6 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → (𝑤 < 𝑀 ↔ (𝑤 / 𝑁) < (𝑀 / 𝑁)))
97	92, 96	mpbid 232	. . . . 5 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → (𝑤 / 𝑁) < (𝑀 / 𝑁))
98		elfzo0 13661	. . . . 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 7402	. . . . 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 16521	. . . . . 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 1380	. . . . 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 11956	. . . . . 6 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) → (𝑁 / 𝑁) = 1)
106	105	adantr 480	. . . . 5 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → (𝑁 / 𝑁) = 1)
107	100, 104, 106	3eqtr3d 2772	. . . 4 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → ((𝑤 / 𝑁) gcd (𝑀 / 𝑁)) = 1)
108		oveq1 7394	. . . . . 6 ⊢ (𝑦 = (𝑤 / 𝑁) → (𝑦 gcd (𝑀 / 𝑁)) = ((𝑤 / 𝑁) gcd (𝑀 / 𝑁)))
109	108	eqeq1d 2731	. . . . 5 ⊢ (𝑦 = (𝑤 / 𝑁) → ((𝑦 gcd (𝑀 / 𝑁)) = 1 ↔ ((𝑤 / 𝑁) gcd (𝑀 / 𝑁)) = 1))
110	109, 4	elrab2 3662	. . . 4 ⊢ ((𝑤 / 𝑁) ∈ 𝐴 ↔ ((𝑤 / 𝑁) ∈ (0..^(𝑀 / 𝑁)) ∧ ((𝑤 / 𝑁) gcd (𝑀 / 𝑁)) = 1))
111	99, 107, 110	sylanbrc 583	. . 3 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → (𝑤 / 𝑁) ∈ 𝐴)
112	61, 111	sylan2b 594	. 2 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ 𝑤 ∈ 𝐵) → (𝑤 / 𝑁) ∈ 𝐴)
113	5	simplbi 497	. . . 4 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ (0..^(𝑀 / 𝑁)))
114	61	simplbi 497	. . . 4 ⊢ (𝑤 ∈ 𝐵 → 𝑤 ∈ (0..^𝑀))
115	113, 114	anim12i 613	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) → (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ 𝑤 ∈ (0..^𝑀)))
116	63	ad2antll 729	. . . . . . . 8 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ 𝑤 ∈ (0..^𝑀))) → 𝑤 ∈ ℤ)
117	116	zcnd 12639	. . . . . . 7 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ 𝑤 ∈ (0..^𝑀))) → 𝑤 ∈ ℂ)
118	34	adantr 480	. . . . . . 7 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ 𝑤 ∈ (0..^𝑀))) → 𝑁 ∈ ℂ)
119	36	adantr 480	. . . . . . 7 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ 𝑤 ∈ (0..^𝑀))) → 𝑁 ≠ 0)
120	117, 118, 119	divcan1d 11959	. . . . . 6 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ 𝑤 ∈ (0..^𝑀))) → ((𝑤 / 𝑁) · 𝑁) = 𝑤)
121	120	eqcomd 2735	. . . . 5 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ 𝑤 ∈ (0..^𝑀))) → 𝑤 = ((𝑤 / 𝑁) · 𝑁))
122		oveq1 7394	. . . . . 6 ⊢ (𝑥 = (𝑤 / 𝑁) → (𝑥 · 𝑁) = ((𝑤 / 𝑁) · 𝑁))
123	122	eqeq2d 2740	. . . . 5 ⊢ (𝑥 = (𝑤 / 𝑁) → (𝑤 = (𝑥 · 𝑁) ↔ 𝑤 = ((𝑤 / 𝑁) · 𝑁)))
124	121, 123	syl5ibrcom 247	. . . 4 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ 𝑤 ∈ (0..^𝑀))) → (𝑥 = (𝑤 / 𝑁) → 𝑤 = (𝑥 · 𝑁)))
125	15	ad2antrl 728	. . . . . . . 8 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ 𝑤 ∈ (0..^𝑀))) → 𝑥 ∈ ℤ)
126	125	zcnd 12639	. . . . . . 7 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ 𝑤 ∈ (0..^𝑀))) → 𝑥 ∈ ℂ)
127	126, 118, 119	divcan4d 11964	. . . . . 6 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ 𝑤 ∈ (0..^𝑀))) → ((𝑥 · 𝑁) / 𝑁) = 𝑥)
128	127	eqcomd 2735	. . . . 5 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ 𝑤 ∈ (0..^𝑀))) → 𝑥 = ((𝑥 · 𝑁) / 𝑁))
129		oveq1 7394	. . . . . 6 ⊢ (𝑤 = (𝑥 · 𝑁) → (𝑤 / 𝑁) = ((𝑥 · 𝑁) / 𝑁))
130	129	eqeq2d 2740	. . . . 5 ⊢ (𝑤 = (𝑥 · 𝑁) → (𝑥 = (𝑤 / 𝑁) ↔ 𝑥 = ((𝑥 · 𝑁) / 𝑁)))
131	128, 130	syl5ibrcom 247	. . . 4 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ 𝑤 ∈ (0..^𝑀))) → (𝑤 = (𝑥 · 𝑁) → 𝑥 = (𝑤 / 𝑁)))
132	124, 131	impbid 212	. . 3 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ 𝑤 ∈ (0..^𝑀))) → (𝑥 = (𝑤 / 𝑁) ↔ 𝑤 = (𝑥 · 𝑁)))
133	115, 132	sylan2 593	. 2 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵)) → (𝑥 = (𝑤 / 𝑁) ↔ 𝑤 = (𝑥 · 𝑁)))
134	1, 58, 112, 133	f1o2d 7643	1 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) → 𝐹:𝐴–1-1-onto→𝐵)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 {crab 3405 class class class wbr 5107 ↦ cmpt 5188 –1-1-onto→wf1o 6510 (class class class)co 7387 ℂcc 11066 ℝcr 11067 0cc0 11068 1c1 11069 · cmul 11073 < clt 11208 ≤ cle 11209 / cdiv 11835 ℕcn 12186 ℕ₀cn0 12442 ℤcz 12529 ...cfz 13468 ..^cfzo 13615 ∥ cdvds 16222 gcd cgcd 16464

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-pre-sup 11146

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-om 7843 df-1st 7968 df-2nd 7969 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-er 8671 df-en 8919 df-dom 8920 df-sdom 8921 df-sup 9393 df-inf 9394 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-div 11836 df-nn 12187 df-2 12249 df-3 12250 df-n0 12443 df-z 12530 df-uz 12794 df-rp 12952 df-fz 13469 df-fzo 13616 df-fl 13754 df-mod 13832 df-seq 13967 df-exp 14027 df-cj 15065 df-re 15066 df-im 15067 df-sqrt 15201 df-abs 15202 df-dvds 16223 df-gcd 16465

This theorem is referenced by: hashgcdeq 16760

W3C validator