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Theorem odadd1 18967

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 18910	. . . . . . . . 9 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
2		odadd1.2	. . . . . . . . . 10 ⊢ 𝑋 = (Base‘𝐺)
3		odadd1.3	. . . . . . . . . 10 ⊢ + = (+_g‘𝐺)
4	2, 3	grpcl 18110	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 + 𝐵) ∈ 𝑋)
5	1, 4	syl3an1 1159	. . . . . . . 8 ⊢ ((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 + 𝐵) ∈ 𝑋)
6		odadd1.1	. . . . . . . . 9 ⊢ 𝑂 = (od‘𝐺)
7	2, 6	odcl 18663	. . . . . . . 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 12084	. . . . . 6 ⊢ ((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑂‘(𝐴 + 𝐵)) ∈ ℤ)
10	2, 6	odcl 18663	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑋 → (𝑂‘𝐴) ∈ ℕ₀)
11	10	3ad2ant2 1130	. . . . . . . . 9 ⊢ ((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑂‘𝐴) ∈ ℕ₀)
12	11	nn0zd 12084	. . . . . . . 8 ⊢ ((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑂‘𝐴) ∈ ℤ)
13	2, 6	odcl 18663	. . . . . . . . . 10 ⊢ (𝐵 ∈ 𝑋 → (𝑂‘𝐵) ∈ ℕ₀)
14	13	3ad2ant3 1131	. . . . . . . . 9 ⊢ ((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑂‘𝐵) ∈ ℕ₀)
15	14	nn0zd 12084	. . . . . . . 8 ⊢ ((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑂‘𝐵) ∈ ℤ)
16	12, 15	gcdcld 15856	. . . . . . 7 ⊢ ((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ∈ ℕ₀)
17	16	nn0zd 12084	. . . . . 6 ⊢ ((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ∈ ℤ)
18	9, 17	zmulcld 12092	. . . . 5 ⊢ ((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑂‘(𝐴 + 𝐵)) · ((𝑂‘𝐴) gcd (𝑂‘𝐵))) ∈ ℤ)
19	18	adantr 483	. . . 4 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) = 0) → ((𝑂‘(𝐴 + 𝐵)) · ((𝑂‘𝐴) gcd (𝑂‘𝐵))) ∈ ℤ)
20		dvds0 15624	. . . 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 15864	. . . . . 6 ⊢ (((𝑂‘𝐴) ∈ ℤ ∧ (𝑂‘𝐵) ∈ ℤ) → (((𝑂‘𝐴) gcd (𝑂‘𝐵)) = 0 ↔ ((𝑂‘𝐴) = 0 ∧ (𝑂‘𝐵) = 0)))
23	12, 15, 22	syl2anc 586	. . . . 5 ⊢ ((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((𝑂‘𝐴) gcd (𝑂‘𝐵)) = 0 ↔ ((𝑂‘𝐴) = 0 ∧ (𝑂‘𝐵) = 0)))
24	23	biimpa 479	. . . 4 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) = 0) → ((𝑂‘𝐴) = 0 ∧ (𝑂‘𝐵) = 0))
25		oveq12 7164	. . . . 5 ⊢ (((𝑂‘𝐴) = 0 ∧ (𝑂‘𝐵) = 0) → ((𝑂‘𝐴) · (𝑂‘𝐵)) = (0 · 0))
26		0cn 10632	. . . . . 6 ⊢ 0 ∈ ℂ
27	26	mul01i 10829	. . . . 5 ⊢ (0 · 0) = 0
28	25, 27	syl6eq 2872	. . . 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 5093	. 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 1187	. . . . . . 7 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → 𝐺 ∈ Abel)
32	17	adantr 483	. . . . . . . . 9 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ∈ ℤ)
33	12	adantr 483	. . . . . . . . 9 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (𝑂‘𝐴) ∈ ℤ)
34	15	adantr 483	. . . . . . . . 9 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (𝑂‘𝐵) ∈ ℤ)
35		gcddvds 15851	. . . . . . . . . . 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 586	. . . . . . . . . 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 497	. . . . . . . . 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 15647	. . . . . . . 8 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ∥ ((𝑂‘𝐴) · (𝑂‘𝐵)))
39		simpr 487	. . . . . . . . 9 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0)
40	33, 34	zmulcld 12092	. . . . . . . . 9 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((𝑂‘𝐴) · (𝑂‘𝐵)) ∈ ℤ)
41		dvdsval2 15609	. . . . . . . . 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 1367	. . . . . . . 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 234	. . . . . . 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 1188	. . . . . . 7 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → 𝐴 ∈ 𝑋)
45		simpl3 1189	. . . . . . 7 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → 𝐵 ∈ 𝑋)
46		eqid 2821	. . . . . . . 8 ⊢ (._g‘𝐺) = (._g‘𝐺)
47	2, 46, 3	mulgdi 18946	. . . . . . 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 1368	. . . . . 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 498	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ∥ (𝑂‘𝐵))
50		dvdsval2 15609	. . . . . . . . . . . 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 1367	. . . . . . . . . . 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 234	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((𝑂‘𝐵) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) ∈ ℤ)
53		dvdsmul1 15630	. . . . . . . . . 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 586	. . . . . . . . 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 12087	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (𝑂‘𝐴) ∈ ℂ)
56	34	zcnd 12087	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (𝑂‘𝐵) ∈ ℂ)
57	32	zcnd 12087	. . . . . . . . . 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 11450	. . . . . . . . 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 5093	. . . . . . . 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 17	. . . . . . . . 9 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → 𝐺 ∈ Grp)
61		eqid 2821	. . . . . . . . . 10 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
62	2, 6, 46, 61	oddvds 18674	. . . . . . . . 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 1367	. . . . . . . 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 234	. . . . . . 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 15609	. . . . . . . . . . . 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 1367	. . . . . . . . . . 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 234	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((𝑂‘𝐴) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) ∈ ℤ)
68		dvdsmul1 15630	. . . . . . . . . 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 586	. . . . . . . . 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 10661	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((𝑂‘𝐴) · (𝑂‘𝐵)) = ((𝑂‘𝐵) · (𝑂‘𝐴)))
71	70	oveq1d 7170	. . . . . . . . . 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 11450	. . . . . . . . . 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 2856	. . . . . . . . 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 5093	. . . . . . . 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 18674	. . . . . . . . 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 1367	. . . . . . . 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 234	. . . . . . 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 7173	. . . . . 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 18130	. . . . . . 7 ⊢ (𝐺 ∈ Grp → (0_g‘𝐺) ∈ 𝑋)
80	2, 3, 61	grplid 18132	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ (0_g‘𝐺) ∈ 𝑋) → ((0_g‘𝐺) + (0_g‘𝐺)) = (0_g‘𝐺))
81	60, 79, 80	syl2anc2 587	. . . . . 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 2860	. . . . 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 483	. . . . . 6 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (𝐴 + 𝐵) ∈ 𝑋)
84	2, 6, 46, 61	oddvds 18674	. . . . . 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 1367	. . . . 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 259	. . . 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 483	. . . . 5 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (𝑂‘(𝐴 + 𝐵)) ∈ ℤ)
88		dvdsmulcr 15638	. . . . 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 1370	. . . 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 259	. . 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 12087	. . . 4 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((𝑂‘𝐴) · (𝑂‘𝐵)) ∈ ℂ)
92	91, 57, 39	divcan1d 11416	. . 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 5091	. 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 3104	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 208 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 class class class wbr 5065 ‘cfv 6354 (class class class)co 7155 0cc0 10536 · cmul 10541 / cdiv 11296 ℕ₀cn0 11896 ℤcz 11980 ∥ cdvds 15606 gcd cgcd 15842 Basecbs 16482 +_gcplusg 16564 0_gc0g 16712 Grpcgrp 18102 ._gcmg 18223 odcod 18651 Abelcabl 18906

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 ax-pre-sup 10614

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-1st 7688 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-er 8288 df-en 8509 df-dom 8510 df-sdom 8511 df-sup 8905 df-inf 8906 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-div 11297 df-nn 11638 df-2 11699 df-3 11700 df-n0 11897 df-z 11981 df-uz 12243 df-rp 12389 df-fz 12892 df-fzo 13033 df-fl 13161 df-mod 13237 df-seq 13369 df-exp 13429 df-cj 14457 df-re 14458 df-im 14459 df-sqrt 14593 df-abs 14594 df-dvds 15607 df-gcd 15843 df-0g 16714 df-mgm 17851 df-sgrp 17900 df-mnd 17911 df-grp 18105 df-minusg 18106 df-sbg 18107 df-mulg 18224 df-od 18655 df-cmn 18907 df-abl 18908

This theorem is referenced by: odadd 18969 torsubg 18973

W3C validator