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Theorem odadd1 18251

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 18198	. . . . . . . . 9 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
2		odadd1.2	. . . . . . . . . 10 ⊢ 𝑋 = (Base‘𝐺)
3		odadd1.3	. . . . . . . . . 10 ⊢ + = (+_g‘𝐺)
4	2, 3	grpcl 17430	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 + 𝐵) ∈ 𝑋)
5	1, 4	syl3an1 1359	. . . . . . . 8 ⊢ ((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 + 𝐵) ∈ 𝑋)
6		odadd1.1	. . . . . . . . 9 ⊢ 𝑂 = (od‘𝐺)
7	2, 6	odcl 17955	. . . . . . . 8 ⊢ ((𝐴 + 𝐵) ∈ 𝑋 → (𝑂‘(𝐴 + 𝐵)) ∈ ℕ₀)
8	5, 7	syl 17	. . . . . . 7 ⊢ ((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑂‘(𝐴 + 𝐵)) ∈ ℕ₀)
9	8	nn0zd 11480	. . . . . 6 ⊢ ((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑂‘(𝐴 + 𝐵)) ∈ ℤ)
10	2, 6	odcl 17955	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑋 → (𝑂‘𝐴) ∈ ℕ₀)
11	10	3ad2ant2 1083	. . . . . . . . 9 ⊢ ((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑂‘𝐴) ∈ ℕ₀)
12	11	nn0zd 11480	. . . . . . . 8 ⊢ ((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑂‘𝐴) ∈ ℤ)
13	2, 6	odcl 17955	. . . . . . . . . 10 ⊢ (𝐵 ∈ 𝑋 → (𝑂‘𝐵) ∈ ℕ₀)
14	13	3ad2ant3 1084	. . . . . . . . 9 ⊢ ((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑂‘𝐵) ∈ ℕ₀)
15	14	nn0zd 11480	. . . . . . . 8 ⊢ ((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑂‘𝐵) ∈ ℤ)
16	12, 15	gcdcld 15230	. . . . . . 7 ⊢ ((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ∈ ℕ₀)
17	16	nn0zd 11480	. . . . . 6 ⊢ ((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ∈ ℤ)
18	9, 17	zmulcld 11488	. . . . 5 ⊢ ((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑂‘(𝐴 + 𝐵)) · ((𝑂‘𝐴) gcd (𝑂‘𝐵))) ∈ ℤ)
19	18	adantr 481	. . . 4 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) = 0) → ((𝑂‘(𝐴 + 𝐵)) · ((𝑂‘𝐴) gcd (𝑂‘𝐵))) ∈ ℤ)
20		dvds0 14997	. . . 4 ⊢ (((𝑂‘(𝐴 + 𝐵)) · ((𝑂‘𝐴) gcd (𝑂‘𝐵))) ∈ ℤ → ((𝑂‘(𝐴 + 𝐵)) · ((𝑂‘𝐴) gcd (𝑂‘𝐵))) ∥ 0)
21	19, 20	syl 17	. . 3 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) = 0) → ((𝑂‘(𝐴 + 𝐵)) · ((𝑂‘𝐴) gcd (𝑂‘𝐵))) ∥ 0)
22		gcdeq0 15238	. . . . . 6 ⊢ (((𝑂‘𝐴) ∈ ℤ ∧ (𝑂‘𝐵) ∈ ℤ) → (((𝑂‘𝐴) gcd (𝑂‘𝐵)) = 0 ↔ ((𝑂‘𝐴) = 0 ∧ (𝑂‘𝐵) = 0)))
23	12, 15, 22	syl2anc 693	. . . . 5 ⊢ ((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((𝑂‘𝐴) gcd (𝑂‘𝐵)) = 0 ↔ ((𝑂‘𝐴) = 0 ∧ (𝑂‘𝐵) = 0)))
24	23	biimpa 501	. . . 4 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) = 0) → ((𝑂‘𝐴) = 0 ∧ (𝑂‘𝐵) = 0))
25		oveq12 6659	. . . . 5 ⊢ (((𝑂‘𝐴) = 0 ∧ (𝑂‘𝐵) = 0) → ((𝑂‘𝐴) · (𝑂‘𝐵)) = (0 · 0))
26		0cn 10032	. . . . . 6 ⊢ 0 ∈ ℂ
27	26	mul01i 10226	. . . . 5 ⊢ (0 · 0) = 0
28	25, 27	syl6eq 2672	. . . 4 ⊢ (((𝑂‘𝐴) = 0 ∧ (𝑂‘𝐵) = 0) → ((𝑂‘𝐴) · (𝑂‘𝐵)) = 0)
29	24, 28	syl 17	. . 3 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) = 0) → ((𝑂‘𝐴) · (𝑂‘𝐵)) = 0)
30	21, 29	breqtrrd 4681	. 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 1064	. . . . . . 7 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → 𝐺 ∈ Abel)
32	12	adantr 481	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (𝑂‘𝐴) ∈ ℤ)
33	15	adantr 481	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (𝑂‘𝐵) ∈ ℤ)
34		gcddvds 15225	. . . . . . . . . . 11 ⊢ (((𝑂‘𝐴) ∈ ℤ ∧ (𝑂‘𝐵) ∈ ℤ) → (((𝑂‘𝐴) gcd (𝑂‘𝐵)) ∥ (𝑂‘𝐴) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ∥ (𝑂‘𝐵)))
35	32, 33, 34	syl2anc 693	. . . . . . . . . 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)))
36	35	simpld 475	. . . . . . . . 9 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ∥ (𝑂‘𝐴))
37	17	adantr 481	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ∈ ℤ)
38		dvdsmultr1 15019	. . . . . . . . . 10 ⊢ ((((𝑂‘𝐴) gcd (𝑂‘𝐵)) ∈ ℤ ∧ (𝑂‘𝐴) ∈ ℤ ∧ (𝑂‘𝐵) ∈ ℤ) → (((𝑂‘𝐴) gcd (𝑂‘𝐵)) ∥ (𝑂‘𝐴) → ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ∥ ((𝑂‘𝐴) · (𝑂‘𝐵))))
39	37, 32, 33, 38	syl3anc 1326	. . . . . . . . 9 ⊢ (((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‘𝐴) · (𝑂‘𝐵))))
40	36, 39	mpd 15	. . . . . . . 8 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ∥ ((𝑂‘𝐴) · (𝑂‘𝐵)))
41		simpr 477	. . . . . . . . 9 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0)
42	32, 33	zmulcld 11488	. . . . . . . . 9 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((𝑂‘𝐴) · (𝑂‘𝐵)) ∈ ℤ)
43		dvdsval2 14986	. . . . . . . . 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))
44	37, 41, 42, 43	syl3anc 1326	. . . . . . . 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))
45	40, 44	mpbid 222	. . . . . . 7 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (((𝑂‘𝐴) · (𝑂‘𝐵)) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) ∈ ℤ)
46		simpl2 1065	. . . . . . 7 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → 𝐴 ∈ 𝑋)
47		simpl3 1066	. . . . . . 7 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → 𝐵 ∈ 𝑋)
48		eqid 2622	. . . . . . . 8 ⊢ (._g‘𝐺) = (._g‘𝐺)
49	2, 48, 3	mulgdi 18232	. . . . . . 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)))
50	31, 45, 46, 47, 49	syl13anc 1328	. . . . . 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)))
51	35	simprd 479	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ∥ (𝑂‘𝐵))
52		dvdsval2 14986	. . . . . . . . . . . . 13 ⊢ ((((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))
53	37, 41, 33, 52	syl3anc 1326	. . . . . . . . . . . 12 ⊢ (((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))
54	51, 53	mpbid 222	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((𝑂‘𝐵) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) ∈ ℤ)
55		dvdsmul1 15003	. . . . . . . . . . 11 ⊢ (((𝑂‘𝐴) ∈ ℤ ∧ ((𝑂‘𝐵) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) ∈ ℤ) → (𝑂‘𝐴) ∥ ((𝑂‘𝐴) · ((𝑂‘𝐵) / ((𝑂‘𝐴) gcd (𝑂‘𝐵)))))
56	32, 54, 55	syl2anc 693	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (𝑂‘𝐴) ∥ ((𝑂‘𝐴) · ((𝑂‘𝐵) / ((𝑂‘𝐴) gcd (𝑂‘𝐵)))))
57	32	zcnd 11483	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (𝑂‘𝐴) ∈ ℂ)
58	33	zcnd 11483	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (𝑂‘𝐵) ∈ ℂ)
59	37	zcnd 11483	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ∈ ℂ)
60	57, 58, 59, 41	divassd 10836	. . . . . . . . . 10 ⊢ (((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)))))
61	56, 60	breqtrrd 4681	. . . . . . . . 9 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (𝑂‘𝐴) ∥ (((𝑂‘𝐴) · (𝑂‘𝐵)) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))))
62	31, 1	syl 17	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → 𝐺 ∈ Grp)
63		eqid 2622	. . . . . . . . . . 11 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
64	2, 6, 48, 63	oddvds 17966	. . . . . . . . . 10 ⊢ ((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)))
65	62, 46, 45, 64	syl3anc 1326	. . . . . . . . 9 ⊢ (((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)))
66	61, 65	mpbid 222	. . . . . . . 8 ⊢ (((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))
67		dvdsval2 14986	. . . . . . . . . . . . 13 ⊢ ((((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))
68	37, 41, 32, 67	syl3anc 1326	. . . . . . . . . . . 12 ⊢ (((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))
69	36, 68	mpbid 222	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((𝑂‘𝐴) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) ∈ ℤ)
70		dvdsmul1 15003	. . . . . . . . . . 11 ⊢ (((𝑂‘𝐵) ∈ ℤ ∧ ((𝑂‘𝐴) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) ∈ ℤ) → (𝑂‘𝐵) ∥ ((𝑂‘𝐵) · ((𝑂‘𝐴) / ((𝑂‘𝐴) gcd (𝑂‘𝐵)))))
71	33, 69, 70	syl2anc 693	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (𝑂‘𝐵) ∥ ((𝑂‘𝐵) · ((𝑂‘𝐴) / ((𝑂‘𝐴) gcd (𝑂‘𝐵)))))
72	57, 58	mulcomd 10061	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((𝑂‘𝐴) · (𝑂‘𝐵)) = ((𝑂‘𝐵) · (𝑂‘𝐴)))
73	72	oveq1d 6665	. . . . . . . . . . 11 ⊢ (((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	58, 57, 59, 41	divassd 10836	. . . . . . . . . . 11 ⊢ (((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)))))
75	73, 74	eqtrd 2656	. . . . . . . . . 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)))))
76	71, 75	breqtrrd 4681	. . . . . . . . 9 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (𝑂‘𝐵) ∥ (((𝑂‘𝐴) · (𝑂‘𝐵)) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))))
77	2, 6, 48, 63	oddvds 17966	. . . . . . . . . 10 ⊢ ((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)))
78	62, 47, 45, 77	syl3anc 1326	. . . . . . . . 9 ⊢ (((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)))
79	76, 78	mpbid 222	. . . . . . . 8 ⊢ (((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))
80	66, 79	oveq12d 6668	. . . . . . 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) + ((((𝑂‘𝐴) · (𝑂‘𝐵)) / ((𝑂‘𝐴) gcd (𝑂‘𝐵)))(._g‘𝐺)𝐵)) = ((0_g‘𝐺) + (0_g‘𝐺)))
81	2, 63	grpidcl 17450	. . . . . . . . 9 ⊢ (𝐺 ∈ Grp → (0_g‘𝐺) ∈ 𝑋)
82	62, 81	syl 17	. . . . . . . 8 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (0_g‘𝐺) ∈ 𝑋)
83	2, 3, 63	grplid 17452	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ (0_g‘𝐺) ∈ 𝑋) → ((0_g‘𝐺) + (0_g‘𝐺)) = (0_g‘𝐺))
84	62, 82, 83	syl2anc 693	. . . . . . 7 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((0_g‘𝐺) + (0_g‘𝐺)) = (0_g‘𝐺))
85	80, 84	eqtrd 2656	. . . . . 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))
86	50, 85	eqtrd 2656	. . . . 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))
87	5	adantr 481	. . . . . 6 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (𝐴 + 𝐵) ∈ 𝑋)
88	2, 6, 48, 63	oddvds 17966	. . . . . 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)))
89	62, 87, 45, 88	syl3anc 1326	. . . . 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)))
90	86, 89	mpbird 247	. . . 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))))
91	9	adantr 481	. . . . 5 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (𝑂‘(𝐴 + 𝐵)) ∈ ℤ)
92		dvdsmulcr 15011	. . . . 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)))))
93	91, 45, 37, 41, 92	syl112anc 1330	. . . 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)))))
94	90, 93	mpbird 247	. . 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))))
95	42	zcnd 11483	. . . 4 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((𝑂‘𝐴) · (𝑂‘𝐵)) ∈ ℂ)
96	95, 59, 41	divcan1d 10802	. . 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)))
97	94, 96	breqtrd 4679	. 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)))
98	30, 97	pm2.61dane 2881	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

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 class class class wbr 4653 ‘cfv 5888 (class class class)co 6650 0cc0 9936 · cmul 9941 / cdiv 10684 ℕ₀cn0 11292 ℤcz 11377 ∥ cdvds 14983 gcd cgcd 15216 Basecbs 15857 +_gcplusg 15941 0_gc0g 16100 Grpcgrp 17422 ._gcmg 17540 odcod 17944 Abelcabl 18194

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-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 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 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-minusg 17426 df-sbg 17427 df-mulg 17541 df-od 17948 df-cmn 18195 df-abl 18196

This theorem is referenced by: odadd 18253 torsubg 18257

W3C validator