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Theorem hashgcdlem 16970

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 7427	. . . . 5 ⊢ (𝑦 = 𝑥 → (𝑦 gcd (𝑀 / 𝑁)) = (𝑥 gcd (𝑀 / 𝑁)))
3	2	eqeq1d 2762	. . . 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 3648	. . 3 ⊢ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ (𝑥 gcd (𝑀 / 𝑁)) = 1))
6		elfzonn0 13845	. . . . . . 7 ⊢ (𝑥 ∈ (0..^(𝑀 / 𝑁)) → 𝑥 ∈ ℕ₀)
7	6	ad2antrl 741	. . . . . 6 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ (𝑥 gcd (𝑀 / 𝑁)) = 1)) → 𝑥 ∈ ℕ₀)
8		nnnn0 12616	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ₀)
9	8	3ad2ant2 1152	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) → 𝑁 ∈ ℕ₀)
10	9	adantr 486	. . . . . 6 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ (𝑥 gcd (𝑀 / 𝑁)) = 1)) → 𝑁 ∈ ℕ₀)
11	7, 10	nn0mulcld 12675	. . . . 5 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ (𝑥 gcd (𝑀 / 𝑁)) = 1)) → (𝑥 · 𝑁) ∈ ℕ₀)
12		simpl1 1210	. . . . 5 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ (𝑥 gcd (𝑀 / 𝑁)) = 1)) → 𝑀 ∈ ℕ)
13		elfzolt2 13806	. . . . . . 7 ⊢ (𝑥 ∈ (0..^(𝑀 / 𝑁)) → 𝑥 < (𝑀 / 𝑁))
14	13	ad2antrl 741	. . . . . 6 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ (𝑥 gcd (𝑀 / 𝑁)) = 1)) → 𝑥 < (𝑀 / 𝑁))
15		elfzoelz 13796	. . . . . . . . 9 ⊢ (𝑥 ∈ (0..^(𝑀 / 𝑁)) → 𝑥 ∈ ℤ)
16	15	ad2antrl 741	. . . . . . . 8 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ (𝑥 gcd (𝑀 / 𝑁)) = 1)) → 𝑥 ∈ ℤ)
17	16	zred 12806	. . . . . . 7 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ (𝑥 gcd (𝑀 / 𝑁)) = 1)) → 𝑥 ∈ ℝ)
18		nnre 12345	. . . . . . . . 9 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℝ)
19	18	3ad2ant1 1151	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) → 𝑀 ∈ ℝ)
20	19	adantr 486	. . . . . . 7 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ (𝑥 gcd (𝑀 / 𝑁)) = 1)) → 𝑀 ∈ ℝ)
21		nnre 12345	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ)
22		nngt0 12372	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → 0 < 𝑁)
23	21, 22	jca 521	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → (𝑁 ∈ ℝ ∧ 0 < 𝑁))
24	23	3ad2ant2 1152	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) → (𝑁 ∈ ℝ ∧ 0 < 𝑁))
25	24	adantr 486	. . . . . . 7 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ (𝑥 gcd (𝑀 / 𝑁)) = 1)) → (𝑁 ∈ ℝ ∧ 0 < 𝑁))
26		ltmuldiv 12193	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ ∧ 𝑀 ∈ ℝ ∧ (𝑁 ∈ ℝ ∧ 0 < 𝑁)) → ((𝑥 · 𝑁) < 𝑀 ↔ 𝑥 < (𝑀 / 𝑁)))
27	17, 20, 25, 26	syl3anc 1398	. . . . . 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 260	. . . . 5 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ (𝑥 gcd (𝑀 / 𝑁)) = 1)) → (𝑥 · 𝑁) < 𝑀)
29		elfzo0 13838	. . . . 5 ⊢ ((𝑥 · 𝑁) ∈ (0..^𝑀) ↔ ((𝑥 · 𝑁) ∈ ℕ₀ ∧ 𝑀 ∈ ℕ ∧ (𝑥 · 𝑁) < 𝑀))
30	11, 12, 28, 29	syl3anbrc 1362	. . . 4 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ (𝑥 gcd (𝑀 / 𝑁)) = 1)) → (𝑥 · 𝑁) ∈ (0..^𝑀))
31		nncn 12346	. . . . . . . . . 10 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℂ)
32	31	3ad2ant1 1151	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) → 𝑀 ∈ ℂ)
33		nncn 12346	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ)
34	33	3ad2ant2 1152	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) → 𝑁 ∈ ℂ)
35		nnne0 12375	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → 𝑁 ≠ 0)
36	35	3ad2ant2 1152	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) → 𝑁 ≠ 0)
37	32, 34, 36	divcan1d 12097	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) → ((𝑀 / 𝑁) · 𝑁) = 𝑀)
38	37	adantr 486	. . . . . . 7 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ (𝑥 gcd (𝑀 / 𝑁)) = 1)) → ((𝑀 / 𝑁) · 𝑁) = 𝑀)
39	38	eqcomd 2766	. . . . . 6 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ (𝑥 gcd (𝑀 / 𝑁)) = 1)) → 𝑀 = ((𝑀 / 𝑁) · 𝑁))
40	39	oveq2d 7436	. . . . 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 16436	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑁 ∥ 𝑀 ↔ (𝑀 / 𝑁) ∈ ℕ))
42	41	biimp3a 1498	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) → (𝑀 / 𝑁) ∈ ℕ)
43	42	nnzd 12722	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) → (𝑀 / 𝑁) ∈ ℤ)
44	43	adantr 486	. . . . . 6 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ (𝑥 gcd (𝑀 / 𝑁)) = 1)) → (𝑀 / 𝑁) ∈ ℤ)
45		mulgcdr 16728	. . . . . 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 1398	. . . . 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 785	. . . . . . 7 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ (𝑥 gcd (𝑀 / 𝑁)) = 1)) → (𝑥 gcd (𝑀 / 𝑁)) = 1)
48	47	oveq1d 7435	. . . . . 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 11330	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) → (1 · 𝑁) = 𝑁)
50	49	adantr 486	. . . . . 6 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ (𝑥 gcd (𝑀 / 𝑁)) = 1)) → (1 · 𝑁) = 𝑁)
51	48, 50	eqtrd 2795	. . . . 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 2799	. . . 4 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ (𝑥 gcd (𝑀 / 𝑁)) = 1)) → ((𝑥 · 𝑁) gcd 𝑀) = 𝑁)
53		oveq1 7427	. . . . . 6 ⊢ (𝑧 = (𝑥 · 𝑁) → (𝑧 gcd 𝑀) = ((𝑥 · 𝑁) gcd 𝑀))
54	53	eqeq1d 2762	. . . . 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 3648	. . . 4 ⊢ ((𝑥 · 𝑁) ∈ 𝐵 ↔ ((𝑥 · 𝑁) ∈ (0..^𝑀) ∧ ((𝑥 · 𝑁) gcd 𝑀) = 𝑁))
57	30, 52, 56	sylanbrc 595	. . 3 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ (𝑥 gcd (𝑀 / 𝑁)) = 1)) → (𝑥 · 𝑁) ∈ 𝐵)
58	5, 57	sylan2b 606	. 2 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ 𝑥 ∈ 𝐴) → (𝑥 · 𝑁) ∈ 𝐵)
59		oveq1 7427	. . . . 5 ⊢ (𝑧 = 𝑤 → (𝑧 gcd 𝑀) = (𝑤 gcd 𝑀))
60	59	eqeq1d 2762	. . . 4 ⊢ (𝑧 = 𝑤 → ((𝑧 gcd 𝑀) = 𝑁 ↔ (𝑤 gcd 𝑀) = 𝑁))
61	60, 55	elrab2 3648	. . 3 ⊢ (𝑤 ∈ 𝐵 ↔ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁))
62		simprr 785	. . . . . . . 8 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → (𝑤 gcd 𝑀) = 𝑁)
63		elfzoelz 13796	. . . . . . . . . . 11 ⊢ (𝑤 ∈ (0..^𝑀) → 𝑤 ∈ ℤ)
64	63	ad2antrl 741	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → 𝑤 ∈ ℤ)
65		simpl1 1210	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → 𝑀 ∈ ℕ)
66	65	nnzd 12722	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → 𝑀 ∈ ℤ)
67		gcddvds 16678	. . . . . . . . . 10 ⊢ ((𝑤 ∈ ℤ ∧ 𝑀 ∈ ℤ) → ((𝑤 gcd 𝑀) ∥ 𝑤 ∧ (𝑤 gcd 𝑀) ∥ 𝑀))
68	64, 66, 67	syl2anc 596	. . . . . . . . 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 500	. . . . . . . 8 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → (𝑤 gcd 𝑀) ∥ 𝑤)
70	62, 69	eqbrtrrd 5128	. . . . . . 7 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → 𝑁 ∥ 𝑤)
71		nnz 12717	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ)
72	71	3ad2ant2 1152	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) → 𝑁 ∈ ℤ)
73	72	adantr 486	. . . . . . . 8 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → 𝑁 ∈ ℤ)
74	36	adantr 486	. . . . . . . 8 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → 𝑁 ≠ 0)
75		dvdsval2 16430	. . . . . . . 8 ⊢ ((𝑁 ∈ ℤ ∧ 𝑁 ≠ 0 ∧ 𝑤 ∈ ℤ) → (𝑁 ∥ 𝑤 ↔ (𝑤 / 𝑁) ∈ ℤ))
76	73, 74, 64, 75	syl3anc 1398	. . . . . . 7 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → (𝑁 ∥ 𝑤 ↔ (𝑤 / 𝑁) ∈ ℤ))
77	70, 76	mpbid 235	. . . . . 6 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → (𝑤 / 𝑁) ∈ ℤ)
78		elfzofz 13813	. . . . . . . . 9 ⊢ (𝑤 ∈ (0..^𝑀) → 𝑤 ∈ (0...𝑀))
79	78	ad2antrl 741	. . . . . . . 8 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → 𝑤 ∈ (0...𝑀))
80		elfznn0 13757	. . . . . . . 8 ⊢ (𝑤 ∈ (0...𝑀) → 𝑤 ∈ ℕ₀)
81		nn0re 12618	. . . . . . . . 9 ⊢ (𝑤 ∈ ℕ₀ → 𝑤 ∈ ℝ)
82		nn0ge0 12634	. . . . . . . . 9 ⊢ (𝑤 ∈ ℕ₀ → 0 ≤ 𝑤)
83	81, 82	jca 521	. . . . . . . 8 ⊢ (𝑤 ∈ ℕ₀ → (𝑤 ∈ ℝ ∧ 0 ≤ 𝑤))
84	79, 80, 83	3syl 19	. . . . . . 7 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → (𝑤 ∈ ℝ ∧ 0 ≤ 𝑤))
85	24	adantr 486	. . . . . . 7 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → (𝑁 ∈ ℝ ∧ 0 < 𝑁))
86		divge0 12189	. . . . . . 7 ⊢ (((𝑤 ∈ ℝ ∧ 0 ≤ 𝑤) ∧ (𝑁 ∈ ℝ ∧ 0 < 𝑁)) → 0 ≤ (𝑤 / 𝑁))
87	84, 85, 86	syl2anc 596	. . . . . 6 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → 0 ≤ (𝑤 / 𝑁))
88		elnn0z 12709	. . . . . 6 ⊢ ((𝑤 / 𝑁) ∈ ℕ₀ ↔ ((𝑤 / 𝑁) ∈ ℤ ∧ 0 ≤ (𝑤 / 𝑁)))
89	77, 87, 88	sylanbrc 595	. . . . 5 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → (𝑤 / 𝑁) ∈ ℕ₀)
90	42	adantr 486	. . . . 5 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → (𝑀 / 𝑁) ∈ ℕ)
91		elfzolt2 13806	. . . . . . 7 ⊢ (𝑤 ∈ (0..^𝑀) → 𝑤 < 𝑀)
92	91	ad2antrl 741	. . . . . 6 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → 𝑤 < 𝑀)
93	64	zred 12806	. . . . . . 7 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → 𝑤 ∈ ℝ)
94	19	adantr 486	. . . . . . 7 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → 𝑀 ∈ ℝ)
95		ltdiv1 12184	. . . . . . 7 ⊢ ((𝑤 ∈ ℝ ∧ 𝑀 ∈ ℝ ∧ (𝑁 ∈ ℝ ∧ 0 < 𝑁)) → (𝑤 < 𝑀 ↔ (𝑤 / 𝑁) < (𝑀 / 𝑁)))
96	93, 94, 85, 95	syl3anc 1398	. . . . . 6 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → (𝑤 < 𝑀 ↔ (𝑤 / 𝑁) < (𝑀 / 𝑁)))
97	92, 96	mpbid 235	. . . . 5 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → (𝑤 / 𝑁) < (𝑀 / 𝑁))
98		elfzo0 13838	. . . . 5 ⊢ ((𝑤 / 𝑁) ∈ (0..^(𝑀 / 𝑁)) ↔ ((𝑤 / 𝑁) ∈ ℕ₀ ∧ (𝑀 / 𝑁) ∈ ℕ ∧ (𝑤 / 𝑁) < (𝑀 / 𝑁)))
99	89, 90, 97, 98	syl3anbrc 1362	. . . 4 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → (𝑤 / 𝑁) ∈ (0..^(𝑀 / 𝑁)))
100	62	oveq1d 7435	. . . . 5 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → ((𝑤 gcd 𝑀) / 𝑁) = (𝑁 / 𝑁))
101		simpl2 1211	. . . . . 6 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → 𝑁 ∈ ℕ)
102		simpl3 1212	. . . . . 6 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → 𝑁 ∥ 𝑀)
103		gcddiv 16729	. . . . . 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 1405	. . . . 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 12094	. . . . . 6 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) → (𝑁 / 𝑁) = 1)
106	105	adantr 486	. . . . 5 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → (𝑁 / 𝑁) = 1)
107	100, 104, 106	3eqtr3d 2803	. . . 4 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → ((𝑤 / 𝑁) gcd (𝑀 / 𝑁)) = 1)
108		oveq1 7427	. . . . . 6 ⊢ (𝑦 = (𝑤 / 𝑁) → (𝑦 gcd (𝑀 / 𝑁)) = ((𝑤 / 𝑁) gcd (𝑀 / 𝑁)))
109	108	eqeq1d 2762	. . . . 5 ⊢ (𝑦 = (𝑤 / 𝑁) → ((𝑦 gcd (𝑀 / 𝑁)) = 1 ↔ ((𝑤 / 𝑁) gcd (𝑀 / 𝑁)) = 1))
110	109, 4	elrab2 3648	. . . 4 ⊢ ((𝑤 / 𝑁) ∈ 𝐴 ↔ ((𝑤 / 𝑁) ∈ (0..^(𝑀 / 𝑁)) ∧ ((𝑤 / 𝑁) gcd (𝑀 / 𝑁)) = 1))
111	99, 107, 110	sylanbrc 595	. . 3 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑤 ∈ (0..^𝑀) ∧ (𝑤 gcd 𝑀) = 𝑁)) → (𝑤 / 𝑁) ∈ 𝐴)
112	61, 111	sylan2b 606	. 2 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ 𝑤 ∈ 𝐵) → (𝑤 / 𝑁) ∈ 𝐴)
113	5	simplbi 502	. . . 4 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ (0..^(𝑀 / 𝑁)))
114	61	simplbi 502	. . . 4 ⊢ (𝑤 ∈ 𝐵 → 𝑤 ∈ (0..^𝑀))
115	113, 114	anim12i 625	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) → (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ 𝑤 ∈ (0..^𝑀)))
116	63	ad2antll 742	. . . . . . . 8 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ 𝑤 ∈ (0..^𝑀))) → 𝑤 ∈ ℤ)
117	116	zcnd 12807	. . . . . . 7 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ 𝑤 ∈ (0..^𝑀))) → 𝑤 ∈ ℂ)
118	34	adantr 486	. . . . . . 7 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ 𝑤 ∈ (0..^𝑀))) → 𝑁 ∈ ℂ)
119	36	adantr 486	. . . . . . 7 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ 𝑤 ∈ (0..^𝑀))) → 𝑁 ≠ 0)
120	117, 118, 119	divcan1d 12097	. . . . . 6 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ 𝑤 ∈ (0..^𝑀))) → ((𝑤 / 𝑁) · 𝑁) = 𝑤)
121	120	eqcomd 2766	. . . . 5 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ 𝑤 ∈ (0..^𝑀))) → 𝑤 = ((𝑤 / 𝑁) · 𝑁))
122		oveq1 7427	. . . . . 6 ⊢ (𝑥 = (𝑤 / 𝑁) → (𝑥 · 𝑁) = ((𝑤 / 𝑁) · 𝑁))
123	122	eqeq2d 2771	. . . . 5 ⊢ (𝑥 = (𝑤 / 𝑁) → (𝑤 = (𝑥 · 𝑁) ↔ 𝑤 = ((𝑤 / 𝑁) · 𝑁)))
124	121, 123	syl5ibrcom 250	. . . 4 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ 𝑤 ∈ (0..^𝑀))) → (𝑥 = (𝑤 / 𝑁) → 𝑤 = (𝑥 · 𝑁)))
125	15	ad2antrl 741	. . . . . . . 8 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ 𝑤 ∈ (0..^𝑀))) → 𝑥 ∈ ℤ)
126	125	zcnd 12807	. . . . . . 7 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ 𝑤 ∈ (0..^𝑀))) → 𝑥 ∈ ℂ)
127	126, 118, 119	divcan4d 12102	. . . . . 6 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ 𝑤 ∈ (0..^𝑀))) → ((𝑥 · 𝑁) / 𝑁) = 𝑥)
128	127	eqcomd 2766	. . . . 5 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ 𝑤 ∈ (0..^𝑀))) → 𝑥 = ((𝑥 · 𝑁) / 𝑁))
129		oveq1 7427	. . . . . 6 ⊢ (𝑤 = (𝑥 · 𝑁) → (𝑤 / 𝑁) = ((𝑥 · 𝑁) / 𝑁))
130	129	eqeq2d 2771	. . . . 5 ⊢ (𝑤 = (𝑥 · 𝑁) → (𝑥 = (𝑤 / 𝑁) ↔ 𝑥 = ((𝑥 · 𝑁) / 𝑁)))
131	128, 130	syl5ibrcom 250	. . . 4 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ 𝑤 ∈ (0..^𝑀))) → (𝑤 = (𝑥 · 𝑁) → 𝑥 = (𝑤 / 𝑁)))
132	124, 131	impbid 215	. . 3 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ (0..^(𝑀 / 𝑁)) ∧ 𝑤 ∈ (0..^𝑀))) → (𝑥 = (𝑤 / 𝑁) ↔ 𝑤 = (𝑥 · 𝑁)))
133	115, 132	sylan2 605	. 2 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) ∧ (𝑥 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵)) → (𝑥 = (𝑤 / 𝑁) ↔ 𝑤 = (𝑥 · 𝑁)))
134	1, 58, 112, 133	f1o2d 7675	1 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑀) → 𝐹:𝐴–1-1-onto→𝐵)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1569 ∈ wcel 2144 ≠ wne 2955 {crab 3412 class class class wbr 5102 ↦ cmpt 5185 –1-1-onto→wf1o 6537 (class class class)co 7420 ℂcc 11201 ℝcr 11202 0cc0 11203 1c1 11204 · cmul 11208 < clt 11346 ≤ cle 11347 / cdiv 11976 ℕcn 12338 ℕ₀cn0 12609 ℤcz 12696 ...cfz 13642 ..^cfzo 13791 ∥ cdvds 16427 gcd cgcd 16669

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7751 ax-cnex 11259 ax-resscn 11260 ax-1cn 11261 ax-icn 11262 ax-addcl 11263 ax-addrcl 11264 ax-mulcl 11265 ax-mulrcl 11266 ax-mulcom 11267 ax-addass 11268 ax-mulass 11269 ax-distr 11270 ax-i2m1 11271 ax-1ne0 11272 ax-1rid 11273 ax-rnegex 11274 ax-rrecex 11275 ax-cnre 11276 ax-pre-lttri 11277 ax-pre-lttrn 11278 ax-pre-ltadd 11279 ax-pre-mulgt0 11280 ax-pre-sup 11281

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-op 4835 df-uni 4867 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7878 df-1st 8001 df-2nd 8002 df-frecs 8302 df-wrecs 8333 df-recs 8382 df-rdg 8421 df-er 8720 df-en 8977 df-dom 8978 df-sdom 8979 df-sup 9437 df-inf 9438 df-pnf 11348 df-mnf 11349 df-xr 11350 df-ltxr 11351 df-le 11352 df-sub 11546 df-neg 11547 df-div 11977 df-nn 12339 df-2 12408 df-3 12409 df-n0 12610 df-z 12697 df-uz 12969 df-rp 13124 df-fz 13643 df-fzo 13792 df-fl 13935 df-mod 14013 df-seq 14148 df-exp 14208 df-cj 15271 df-re 15272 df-im 15273 df-sqrt 15407 df-abs 15408 df-dvds 16428 df-gcd 16670

This theorem is used by: hashgcdeq 16972

W3C validator