odadd1 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > odadd1		Structured version Visualization version GIF version

Theorem odadd1 19976

Description: The order of a product in an abelian group divides the LCM of the orders of the factors. (Contributed by Mario Carneiro, 20-Oct-2015.)

**Hypotheses**
Ref	Expression
odadd1.1	⊢ 𝑂 = (od‘𝐺)
odadd1.2	⊢ 𝑋 = (Base‘𝐺)
odadd1.3	⊢ + = (+_g‘𝐺)

**Assertion**
Ref	Expression
odadd1	⊢ ((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑂‘(𝐴 + 𝐵)) · ((𝑂‘𝐴) gcd (𝑂‘𝐵))) ∥ ((𝑂‘𝐴) · (𝑂‘𝐵)))

**Proof of Theorem odadd1**
Step	Hyp	Ref	Expression
1		ablgrp 19913	. . . . . . . . 9 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
2		odadd1.2	. . . . . . . . . 10 ⊢ 𝑋 = (Base‘𝐺)
3		odadd1.3	. . . . . . . . . 10 ⊢ + = (+_g‘𝐺)
4	2, 3	grpcl 19066	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 + 𝐵) ∈ 𝑋)
5	1, 4	syl3an1 1181	. . . . . . . 8 ⊢ ((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 + 𝐵) ∈ 𝑋)
6		odadd1.1	. . . . . . . . 9 ⊢ 𝑂 = (od‘𝐺)
7	2, 6	odcl 19664	. . . . . . . 8 ⊢ ((𝐴 + 𝐵) ∈ 𝑋 → (𝑂‘(𝐴 + 𝐵)) ∈ ℕ₀)
8	5, 7	syl 18	. . . . . . 7 ⊢ ((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑂‘(𝐴 + 𝐵)) ∈ ℕ₀)
9	8	nn0zd 12641	. . . . . 6 ⊢ ((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑂‘(𝐴 + 𝐵)) ∈ ℤ)
10	2, 6	odcl 19664	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑋 → (𝑂‘𝐴) ∈ ℕ₀)
11	10	3ad2ant2 1152	. . . . . . . . 9 ⊢ ((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑂‘𝐴) ∈ ℕ₀)
12	11	nn0zd 12641	. . . . . . . 8 ⊢ ((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑂‘𝐴) ∈ ℤ)
13	2, 6	odcl 19664	. . . . . . . . . 10 ⊢ (𝐵 ∈ 𝑋 → (𝑂‘𝐵) ∈ ℕ₀)
14	13	3ad2ant3 1153	. . . . . . . . 9 ⊢ ((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑂‘𝐵) ∈ ℕ₀)
15	14	nn0zd 12641	. . . . . . . 8 ⊢ ((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑂‘𝐵) ∈ ℤ)
16	12, 15	gcdcld 16599	. . . . . . 7 ⊢ ((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ∈ ℕ₀)
17	16	nn0zd 12641	. . . . . 6 ⊢ ((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ∈ ℤ)
18	9, 17	zmulcld 12732	. . . . 5 ⊢ ((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑂‘(𝐴 + 𝐵)) · ((𝑂‘𝐴) gcd (𝑂‘𝐵))) ∈ ℤ)
19	18	adantr 486	. . . 4 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) = 0) → ((𝑂‘(𝐴 + 𝐵)) · ((𝑂‘𝐴) gcd (𝑂‘𝐵))) ∈ ℤ)
20		dvds0 16362	. . . 4 ⊢ (((𝑂‘(𝐴 + 𝐵)) · ((𝑂‘𝐴) gcd (𝑂‘𝐵))) ∈ ℤ → ((𝑂‘(𝐴 + 𝐵)) · ((𝑂‘𝐴) gcd (𝑂‘𝐵))) ∥ 0)
21	19, 20	syl 18	. . 3 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) = 0) → ((𝑂‘(𝐴 + 𝐵)) · ((𝑂‘𝐴) gcd (𝑂‘𝐵))) ∥ 0)
22		gcdeq0 16608	. . . . . 6 ⊢ (((𝑂‘𝐴) ∈ ℤ ∧ (𝑂‘𝐵) ∈ ℤ) → (((𝑂‘𝐴) gcd (𝑂‘𝐵)) = 0 ↔ ((𝑂‘𝐴) = 0 ∧ (𝑂‘𝐵) = 0)))
23	12, 15, 22	syl2anc 596	. . . . 5 ⊢ ((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((𝑂‘𝐴) gcd (𝑂‘𝐵)) = 0 ↔ ((𝑂‘𝐴) = 0 ∧ (𝑂‘𝐵) = 0)))
24	23	biimpa 482	. . . 4 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) = 0) → ((𝑂‘𝐴) = 0 ∧ (𝑂‘𝐵) = 0))
25		oveq12 7423	. . . . 5 ⊢ (((𝑂‘𝐴) = 0 ∧ (𝑂‘𝐵) = 0) → ((𝑂‘𝐴) · (𝑂‘𝐵)) = (0 · 0))
26		0cn 11223	. . . . . 6 ⊢ 0 ∈ ℂ
27	26	mul01i 11425	. . . . 5 ⊢ (0 · 0) = 0
28	25, 27	eqtrdi 2811	. . . 4 ⊢ (((𝑂‘𝐴) = 0 ∧ (𝑂‘𝐵) = 0) → ((𝑂‘𝐴) · (𝑂‘𝐵)) = 0)
29	24, 28	syl 18	. . 3 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) = 0) → ((𝑂‘𝐴) · (𝑂‘𝐵)) = 0)
30	21, 29	breqtrrd 5133	. 2 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) = 0) → ((𝑂‘(𝐴 + 𝐵)) · ((𝑂‘𝐴) gcd (𝑂‘𝐵))) ∥ ((𝑂‘𝐴) · (𝑂‘𝐵)))
31		simpl1 1210	. . . . . . 7 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → 𝐺 ∈ Abel)
32	17	adantr 486	. . . . . . . . 9 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ∈ ℤ)
33	12	adantr 486	. . . . . . . . 9 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (𝑂‘𝐴) ∈ ℤ)
34	15	adantr 486	. . . . . . . . 9 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (𝑂‘𝐵) ∈ ℤ)
35		gcddvds 16594	. . . . . . . . . . 11 ⊢ (((𝑂‘𝐴) ∈ ℤ ∧ (𝑂‘𝐵) ∈ ℤ) → (((𝑂‘𝐴) gcd (𝑂‘𝐵)) ∥ (𝑂‘𝐴) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ∥ (𝑂‘𝐵)))
36	33, 34, 35	syl2anc 596	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (((𝑂‘𝐴) gcd (𝑂‘𝐵)) ∥ (𝑂‘𝐴) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ∥ (𝑂‘𝐵)))
37	36	simpld 500	. . . . . . . . 9 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ∥ (𝑂‘𝐴))
38	32, 33, 34, 37	dvdsmultr1d 16388	. . . . . . . 8 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ∥ ((𝑂‘𝐴) · (𝑂‘𝐵)))
39		simpr 490	. . . . . . . . 9 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0)
40	33, 34	zmulcld 12732	. . . . . . . . 9 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((𝑂‘𝐴) · (𝑂‘𝐵)) ∈ ℤ)
41		dvdsval2 16346	. . . . . . . . 9 ⊢ ((((𝑂‘𝐴) gcd (𝑂‘𝐵)) ∈ ℤ ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0 ∧ ((𝑂‘𝐴) · (𝑂‘𝐵)) ∈ ℤ) → (((𝑂‘𝐴) gcd (𝑂‘𝐵)) ∥ ((𝑂‘𝐴) · (𝑂‘𝐵)) ↔ (((𝑂‘𝐴) · (𝑂‘𝐵)) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) ∈ ℤ))
42	32, 39, 40, 41	syl3anc 1398	. . . . . . . 8 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (((𝑂‘𝐴) gcd (𝑂‘𝐵)) ∥ ((𝑂‘𝐴) · (𝑂‘𝐵)) ↔ (((𝑂‘𝐴) · (𝑂‘𝐵)) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) ∈ ℤ))
43	38, 42	mpbid 235	. . . . . . 7 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (((𝑂‘𝐴) · (𝑂‘𝐵)) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) ∈ ℤ)
44		simpl2 1211	. . . . . . 7 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → 𝐴 ∈ 𝑋)
45		simpl3 1212	. . . . . . 7 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → 𝐵 ∈ 𝑋)
46		eqid 2760	. . . . . . . 8 ⊢ (._g‘𝐺) = (._g‘𝐺)
47	2, 46, 3	mulgdi 19954	. . . . . . 7 ⊢ ((𝐺 ∈ Abel ∧ ((((𝑂‘𝐴) · (𝑂‘𝐵)) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) ∈ ℤ ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ((((𝑂‘𝐴) · (𝑂‘𝐵)) / ((𝑂‘𝐴) gcd (𝑂‘𝐵)))(._g‘𝐺)(𝐴 + 𝐵)) = (((((𝑂‘𝐴) · (𝑂‘𝐵)) / ((𝑂‘𝐴) gcd (𝑂‘𝐵)))(._g‘𝐺)𝐴) + ((((𝑂‘𝐴) · (𝑂‘𝐵)) / ((𝑂‘𝐴) gcd (𝑂‘𝐵)))(._g‘𝐺)𝐵)))
48	31, 43, 44, 45, 47	syl13anc 1399	. . . . . 6 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((((𝑂‘𝐴) · (𝑂‘𝐵)) / ((𝑂‘𝐴) gcd (𝑂‘𝐵)))(._g‘𝐺)(𝐴 + 𝐵)) = (((((𝑂‘𝐴) · (𝑂‘𝐵)) / ((𝑂‘𝐴) gcd (𝑂‘𝐵)))(._g‘𝐺)𝐴) + ((((𝑂‘𝐴) · (𝑂‘𝐵)) / ((𝑂‘𝐴) gcd (𝑂‘𝐵)))(._g‘𝐺)𝐵)))
49	36	simprd 501	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ∥ (𝑂‘𝐵))
50		dvdsval2 16346	. . . . . . . . . . . 12 ⊢ ((((𝑂‘𝐴) gcd (𝑂‘𝐵)) ∈ ℤ ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0 ∧ (𝑂‘𝐵) ∈ ℤ) → (((𝑂‘𝐴) gcd (𝑂‘𝐵)) ∥ (𝑂‘𝐵) ↔ ((𝑂‘𝐵) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) ∈ ℤ))
51	32, 39, 34, 50	syl3anc 1398	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (((𝑂‘𝐴) gcd (𝑂‘𝐵)) ∥ (𝑂‘𝐵) ↔ ((𝑂‘𝐵) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) ∈ ℤ))
52	49, 51	mpbid 235	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((𝑂‘𝐵) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) ∈ ℤ)
53		dvdsmul1 16368	. . . . . . . . . 10 ⊢ (((𝑂‘𝐴) ∈ ℤ ∧ ((𝑂‘𝐵) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) ∈ ℤ) → (𝑂‘𝐴) ∥ ((𝑂‘𝐴) · ((𝑂‘𝐵) / ((𝑂‘𝐴) gcd (𝑂‘𝐵)))))
54	33, 52, 53	syl2anc 596	. . . . . . . . 9 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (𝑂‘𝐴) ∥ ((𝑂‘𝐴) · ((𝑂‘𝐵) / ((𝑂‘𝐴) gcd (𝑂‘𝐵)))))
55	33	zcnd 12727	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (𝑂‘𝐴) ∈ ℂ)
56	34	zcnd 12727	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (𝑂‘𝐵) ∈ ℂ)
57	32	zcnd 12727	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ∈ ℂ)
58	55, 56, 57, 39	divassd 12051	. . . . . . . . 9 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (((𝑂‘𝐴) · (𝑂‘𝐵)) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) = ((𝑂‘𝐴) · ((𝑂‘𝐵) / ((𝑂‘𝐴) gcd (𝑂‘𝐵)))))
59	54, 58	breqtrrd 5133	. . . . . . . 8 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (𝑂‘𝐴) ∥ (((𝑂‘𝐴) · (𝑂‘𝐵)) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))))
60	31, 1	syl 18	. . . . . . . . 9 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → 𝐺 ∈ Grp)
61		eqid 2760	. . . . . . . . . 10 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
62	2, 6, 46, 61	oddvds 19675	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ (((𝑂‘𝐴) · (𝑂‘𝐵)) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) ∈ ℤ) → ((𝑂‘𝐴) ∥ (((𝑂‘𝐴) · (𝑂‘𝐵)) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) ↔ ((((𝑂‘𝐴) · (𝑂‘𝐵)) / ((𝑂‘𝐴) gcd (𝑂‘𝐵)))(._g‘𝐺)𝐴) = (0_g‘𝐺)))
63	60, 44, 43, 62	syl3anc 1398	. . . . . . . 8 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((𝑂‘𝐴) ∥ (((𝑂‘𝐴) · (𝑂‘𝐵)) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) ↔ ((((𝑂‘𝐴) · (𝑂‘𝐵)) / ((𝑂‘𝐴) gcd (𝑂‘𝐵)))(._g‘𝐺)𝐴) = (0_g‘𝐺)))
64	59, 63	mpbid 235	. . . . . . 7 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((((𝑂‘𝐴) · (𝑂‘𝐵)) / ((𝑂‘𝐴) gcd (𝑂‘𝐵)))(._g‘𝐺)𝐴) = (0_g‘𝐺))
65		dvdsval2 16346	. . . . . . . . . . . 12 ⊢ ((((𝑂‘𝐴) gcd (𝑂‘𝐵)) ∈ ℤ ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0 ∧ (𝑂‘𝐴) ∈ ℤ) → (((𝑂‘𝐴) gcd (𝑂‘𝐵)) ∥ (𝑂‘𝐴) ↔ ((𝑂‘𝐴) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) ∈ ℤ))
66	32, 39, 33, 65	syl3anc 1398	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (((𝑂‘𝐴) gcd (𝑂‘𝐵)) ∥ (𝑂‘𝐴) ↔ ((𝑂‘𝐴) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) ∈ ℤ))
67	37, 66	mpbid 235	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((𝑂‘𝐴) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) ∈ ℤ)
68		dvdsmul1 16368	. . . . . . . . . 10 ⊢ (((𝑂‘𝐵) ∈ ℤ ∧ ((𝑂‘𝐴) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) ∈ ℤ) → (𝑂‘𝐵) ∥ ((𝑂‘𝐵) · ((𝑂‘𝐴) / ((𝑂‘𝐴) gcd (𝑂‘𝐵)))))
69	34, 67, 68	syl2anc 596	. . . . . . . . 9 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (𝑂‘𝐵) ∥ ((𝑂‘𝐵) · ((𝑂‘𝐴) / ((𝑂‘𝐴) gcd (𝑂‘𝐵)))))
70	55, 56	mulcomd 11255	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((𝑂‘𝐴) · (𝑂‘𝐵)) = ((𝑂‘𝐵) · (𝑂‘𝐴)))
71	70	oveq1d 7429	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (((𝑂‘𝐴) · (𝑂‘𝐵)) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) = (((𝑂‘𝐵) · (𝑂‘𝐴)) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))))
72	56, 55, 57, 39	divassd 12051	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (((𝑂‘𝐵) · (𝑂‘𝐴)) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) = ((𝑂‘𝐵) · ((𝑂‘𝐴) / ((𝑂‘𝐴) gcd (𝑂‘𝐵)))))
73	71, 72	eqtrd 2795	. . . . . . . . 9 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (((𝑂‘𝐴) · (𝑂‘𝐵)) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) = ((𝑂‘𝐵) · ((𝑂‘𝐴) / ((𝑂‘𝐴) gcd (𝑂‘𝐵)))))
74	69, 73	breqtrrd 5133	. . . . . . . 8 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (𝑂‘𝐵) ∥ (((𝑂‘𝐴) · (𝑂‘𝐵)) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))))
75	2, 6, 46, 61	oddvds 19675	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ 𝐵 ∈ 𝑋 ∧ (((𝑂‘𝐴) · (𝑂‘𝐵)) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) ∈ ℤ) → ((𝑂‘𝐵) ∥ (((𝑂‘𝐴) · (𝑂‘𝐵)) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) ↔ ((((𝑂‘𝐴) · (𝑂‘𝐵)) / ((𝑂‘𝐴) gcd (𝑂‘𝐵)))(._g‘𝐺)𝐵) = (0_g‘𝐺)))
76	60, 45, 43, 75	syl3anc 1398	. . . . . . . 8 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((𝑂‘𝐵) ∥ (((𝑂‘𝐴) · (𝑂‘𝐵)) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) ↔ ((((𝑂‘𝐴) · (𝑂‘𝐵)) / ((𝑂‘𝐴) gcd (𝑂‘𝐵)))(._g‘𝐺)𝐵) = (0_g‘𝐺)))
77	74, 76	mpbid 235	. . . . . . 7 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((((𝑂‘𝐴) · (𝑂‘𝐵)) / ((𝑂‘𝐴) gcd (𝑂‘𝐵)))(._g‘𝐺)𝐵) = (0_g‘𝐺))
78	64, 77	oveq12d 7432	. . . . . 6 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (((((𝑂‘𝐴) · (𝑂‘𝐵)) / ((𝑂‘𝐴) gcd (𝑂‘𝐵)))(._g‘𝐺)𝐴) + ((((𝑂‘𝐴) · (𝑂‘𝐵)) / ((𝑂‘𝐴) gcd (𝑂‘𝐵)))(._g‘𝐺)𝐵)) = ((0_g‘𝐺) + (0_g‘𝐺)))
79	2, 61	grpidcl 19090	. . . . . . 7 ⊢ (𝐺 ∈ Grp → (0_g‘𝐺) ∈ 𝑋)
80	2, 3, 61	grplid 19092	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ (0_g‘𝐺) ∈ 𝑋) → ((0_g‘𝐺) + (0_g‘𝐺)) = (0_g‘𝐺))
81	60, 79, 80	syl2anc2 597	. . . . . 6 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((0_g‘𝐺) + (0_g‘𝐺)) = (0_g‘𝐺))
82	48, 78, 81	3eqtrd 2799	. . . . 5 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((((𝑂‘𝐴) · (𝑂‘𝐵)) / ((𝑂‘𝐴) gcd (𝑂‘𝐵)))(._g‘𝐺)(𝐴 + 𝐵)) = (0_g‘𝐺))
83	5	adantr 486	. . . . . 6 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (𝐴 + 𝐵) ∈ 𝑋)
84	2, 6, 46, 61	oddvds 19675	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ (𝐴 + 𝐵) ∈ 𝑋 ∧ (((𝑂‘𝐴) · (𝑂‘𝐵)) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) ∈ ℤ) → ((𝑂‘(𝐴 + 𝐵)) ∥ (((𝑂‘𝐴) · (𝑂‘𝐵)) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) ↔ ((((𝑂‘𝐴) · (𝑂‘𝐵)) / ((𝑂‘𝐴) gcd (𝑂‘𝐵)))(._g‘𝐺)(𝐴 + 𝐵)) = (0_g‘𝐺)))
85	60, 83, 43, 84	syl3anc 1398	. . . . 5 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((𝑂‘(𝐴 + 𝐵)) ∥ (((𝑂‘𝐴) · (𝑂‘𝐵)) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) ↔ ((((𝑂‘𝐴) · (𝑂‘𝐵)) / ((𝑂‘𝐴) gcd (𝑂‘𝐵)))(._g‘𝐺)(𝐴 + 𝐵)) = (0_g‘𝐺)))
86	82, 85	mpbird 260	. . . 4 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (𝑂‘(𝐴 + 𝐵)) ∥ (((𝑂‘𝐴) · (𝑂‘𝐵)) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))))
87	9	adantr 486	. . . . 5 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (𝑂‘(𝐴 + 𝐵)) ∈ ℤ)
88		dvdsmulcr 16376	. . . . 5 ⊢ (((𝑂‘(𝐴 + 𝐵)) ∈ ℤ ∧ (((𝑂‘𝐴) · (𝑂‘𝐵)) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) ∈ ℤ ∧ (((𝑂‘𝐴) gcd (𝑂‘𝐵)) ∈ ℤ ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0)) → (((𝑂‘(𝐴 + 𝐵)) · ((𝑂‘𝐴) gcd (𝑂‘𝐵))) ∥ ((((𝑂‘𝐴) · (𝑂‘𝐵)) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) · ((𝑂‘𝐴) gcd (𝑂‘𝐵))) ↔ (𝑂‘(𝐴 + 𝐵)) ∥ (((𝑂‘𝐴) · (𝑂‘𝐵)) / ((𝑂‘𝐴) gcd (𝑂‘𝐵)))))
89	87, 43, 32, 39, 88	syl112anc 1401	. . . 4 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (((𝑂‘(𝐴 + 𝐵)) · ((𝑂‘𝐴) gcd (𝑂‘𝐵))) ∥ ((((𝑂‘𝐴) · (𝑂‘𝐵)) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) · ((𝑂‘𝐴) gcd (𝑂‘𝐵))) ↔ (𝑂‘(𝐴 + 𝐵)) ∥ (((𝑂‘𝐴) · (𝑂‘𝐵)) / ((𝑂‘𝐴) gcd (𝑂‘𝐵)))))
90	86, 89	mpbird 260	. . 3 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((𝑂‘(𝐴 + 𝐵)) · ((𝑂‘𝐴) gcd (𝑂‘𝐵))) ∥ ((((𝑂‘𝐴) · (𝑂‘𝐵)) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) · ((𝑂‘𝐴) gcd (𝑂‘𝐵))))
91	40	zcnd 12727	. . . 4 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((𝑂‘𝐴) · (𝑂‘𝐵)) ∈ ℂ)
92	91, 57, 39	divcan1d 12017	. . 3 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((((𝑂‘𝐴) · (𝑂‘𝐵)) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) · ((𝑂‘𝐴) gcd (𝑂‘𝐵))) = ((𝑂‘𝐴) · (𝑂‘𝐵)))
93	90, 92	breqtrd 5131	. 2 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((𝑂‘(𝐴 + 𝐵)) · ((𝑂‘𝐴) gcd (𝑂‘𝐵))) ∥ ((𝑂‘𝐴) · (𝑂‘𝐵)))
94	30, 93	pm2.61dane 3042	1 ⊢ ((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑂‘(𝐴 + 𝐵)) · ((𝑂‘𝐴) gcd (𝑂‘𝐵))) ∥ ((𝑂‘𝐴) · (𝑂‘𝐵)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ≠ wne 2955 class class class wbr 5103 ‘cfv 6533 (class class class)co 7414 0cc0 11125 · cmul 11130 / cdiv 11896 ℕ₀cn0 12529 ℤcz 12616 ∥ cdvds 16343 gcd cgcd 16585 Basecbs 17302 +_gcplusg 17343 0_gc0g 17525 Grpcgrp 19058 ._gcmg 19191 odcod 19652 Abelcabl 19909

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7737 ax-cnex 11181 ax-resscn 11182 ax-1cn 11183 ax-icn 11184 ax-addcl 11185 ax-addrcl 11186 ax-mulcl 11187 ax-mulrcl 11188 ax-mulcom 11189 ax-addass 11190 ax-mulass 11191 ax-distr 11192 ax-i2m1 11193 ax-1ne0 11194 ax-1rid 11195 ax-rnegex 11196 ax-rrecex 11197 ax-cnre 11198 ax-pre-lttri 11199 ax-pre-lttrn 11200 ax-pre-ltadd 11201 ax-pre-mulgt0 11202 ax-pre-sup 11203

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 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 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7371 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7864 df-1st 7987 df-2nd 7988 df-frecs 8281 df-wrecs 8312 df-recs 8361 df-rdg 8400 df-er 8697 df-en 8954 df-dom 8955 df-sdom 8956 df-sup 9413 df-inf 9414 df-pnf 11270 df-mnf 11271 df-xr 11272 df-ltxr 11273 df-le 11274 df-sub 11468 df-neg 11469 df-div 11897 df-nn 12259 df-2 12328 df-3 12329 df-n0 12530 df-z 12617 df-uz 12889 df-rp 13044 df-fz 13563 df-fzo 13711 df-fl 13854 df-mod 13932 df-seq 14067 df-exp 14127 df-cj 15187 df-re 15188 df-im 15189 df-sqrt 15323 df-abs 15324 df-dvds 16344 df-gcd 16586 df-0g 17527 df-mgm 18731 df-sgrp 18822 df-mnd 18838 df-grp 19061 df-minusg 19062 df-sbg 19063 df-mulg 19192 df-od 19656 df-cmn 19910 df-abl 19911

This theorem is used by: odadd 19978 torsubg 19982

W3C validator