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Theorem odadd1 19775

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 19712	. . . . . . . . 9 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
2		odadd1.2	. . . . . . . . . 10 ⊢ 𝑋 = (Base‘𝐺)
3		odadd1.3	. . . . . . . . . 10 ⊢ + = (+_g‘𝐺)
4	2, 3	grpcl 18869	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 + 𝐵) ∈ 𝑋)
5	1, 4	syl3an1 1163	. . . . . . . 8 ⊢ ((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 + 𝐵) ∈ 𝑋)
6		odadd1.1	. . . . . . . . 9 ⊢ 𝑂 = (od‘𝐺)
7	2, 6	odcl 19463	. . . . . . . 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 12511	. . . . . 6 ⊢ ((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑂‘(𝐴 + 𝐵)) ∈ ℤ)
10	2, 6	odcl 19463	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑋 → (𝑂‘𝐴) ∈ ℕ₀)
11	10	3ad2ant2 1134	. . . . . . . . 9 ⊢ ((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑂‘𝐴) ∈ ℕ₀)
12	11	nn0zd 12511	. . . . . . . 8 ⊢ ((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑂‘𝐴) ∈ ℤ)
13	2, 6	odcl 19463	. . . . . . . . . 10 ⊢ (𝐵 ∈ 𝑋 → (𝑂‘𝐵) ∈ ℕ₀)
14	13	3ad2ant3 1135	. . . . . . . . 9 ⊢ ((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑂‘𝐵) ∈ ℕ₀)
15	14	nn0zd 12511	. . . . . . . 8 ⊢ ((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑂‘𝐵) ∈ ℤ)
16	12, 15	gcdcld 16433	. . . . . . 7 ⊢ ((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ∈ ℕ₀)
17	16	nn0zd 12511	. . . . . 6 ⊢ ((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ∈ ℤ)
18	9, 17	zmulcld 12600	. . . . 5 ⊢ ((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑂‘(𝐴 + 𝐵)) · ((𝑂‘𝐴) gcd (𝑂‘𝐵))) ∈ ℤ)
19	18	adantr 480	. . . 4 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) = 0) → ((𝑂‘(𝐴 + 𝐵)) · ((𝑂‘𝐴) gcd (𝑂‘𝐵))) ∈ ℤ)
20		dvds0 16196	. . . 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 16442	. . . . . 6 ⊢ (((𝑂‘𝐴) ∈ ℤ ∧ (𝑂‘𝐵) ∈ ℤ) → (((𝑂‘𝐴) gcd (𝑂‘𝐵)) = 0 ↔ ((𝑂‘𝐴) = 0 ∧ (𝑂‘𝐵) = 0)))
23	12, 15, 22	syl2anc 584	. . . . 5 ⊢ ((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((𝑂‘𝐴) gcd (𝑂‘𝐵)) = 0 ↔ ((𝑂‘𝐴) = 0 ∧ (𝑂‘𝐵) = 0)))
24	23	biimpa 476	. . . 4 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) = 0) → ((𝑂‘𝐴) = 0 ∧ (𝑂‘𝐵) = 0))
25		oveq12 7365	. . . . 5 ⊢ (((𝑂‘𝐴) = 0 ∧ (𝑂‘𝐵) = 0) → ((𝑂‘𝐴) · (𝑂‘𝐵)) = (0 · 0))
26		0cn 11122	. . . . . 6 ⊢ 0 ∈ ℂ
27	26	mul01i 11321	. . . . 5 ⊢ (0 · 0) = 0
28	25, 27	eqtrdi 2785	. . . 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 5124	. 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 1192	. . . . . . 7 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → 𝐺 ∈ Abel)
32	17	adantr 480	. . . . . . . . 9 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ∈ ℤ)
33	12	adantr 480	. . . . . . . . 9 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (𝑂‘𝐴) ∈ ℤ)
34	15	adantr 480	. . . . . . . . 9 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (𝑂‘𝐵) ∈ ℤ)
35		gcddvds 16428	. . . . . . . . . . 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 584	. . . . . . . . . 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 494	. . . . . . . . 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 16222	. . . . . . . 8 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ∥ ((𝑂‘𝐴) · (𝑂‘𝐵)))
39		simpr 484	. . . . . . . . 9 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0)
40	33, 34	zmulcld 12600	. . . . . . . . 9 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((𝑂‘𝐴) · (𝑂‘𝐵)) ∈ ℤ)
41		dvdsval2 16180	. . . . . . . . 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 1373	. . . . . . . 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 232	. . . . . . 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 1193	. . . . . . 7 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → 𝐴 ∈ 𝑋)
45		simpl3 1194	. . . . . . 7 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → 𝐵 ∈ 𝑋)
46		eqid 2734	. . . . . . . 8 ⊢ (._g‘𝐺) = (._g‘𝐺)
47	2, 46, 3	mulgdi 19753	. . . . . . 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 1374	. . . . . 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 495	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ∥ (𝑂‘𝐵))
50		dvdsval2 16180	. . . . . . . . . . . 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 1373	. . . . . . . . . . 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 232	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((𝑂‘𝐵) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) ∈ ℤ)
53		dvdsmul1 16202	. . . . . . . . . 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 584	. . . . . . . . 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 12595	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (𝑂‘𝐴) ∈ ℂ)
56	34	zcnd 12595	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (𝑂‘𝐵) ∈ ℂ)
57	32	zcnd 12595	. . . . . . . . . 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 11950	. . . . . . . . 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 5124	. . . . . . . 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 2734	. . . . . . . . . 10 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
62	2, 6, 46, 61	oddvds 19474	. . . . . . . . 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 1373	. . . . . . . 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 232	. . . . . . 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 16180	. . . . . . . . . . . 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 1373	. . . . . . . . . . 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 232	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((𝑂‘𝐴) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) ∈ ℤ)
68		dvdsmul1 16202	. . . . . . . . . 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 584	. . . . . . . . 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 11151	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((𝑂‘𝐴) · (𝑂‘𝐵)) = ((𝑂‘𝐵) · (𝑂‘𝐴)))
71	70	oveq1d 7371	. . . . . . . . . 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 11950	. . . . . . . . . 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 2769	. . . . . . . . 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 5124	. . . . . . . 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 19474	. . . . . . . . 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 1373	. . . . . . . 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 232	. . . . . . 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 7374	. . . . . 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 18893	. . . . . . 7 ⊢ (𝐺 ∈ Grp → (0_g‘𝐺) ∈ 𝑋)
80	2, 3, 61	grplid 18895	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ (0_g‘𝐺) ∈ 𝑋) → ((0_g‘𝐺) + (0_g‘𝐺)) = (0_g‘𝐺))
81	60, 79, 80	syl2anc2 585	. . . . . 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 2773	. . . . 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 480	. . . . . 6 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (𝐴 + 𝐵) ∈ 𝑋)
84	2, 6, 46, 61	oddvds 19474	. . . . . 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 1373	. . . . 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 257	. . . 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 480	. . . . 5 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (𝑂‘(𝐴 + 𝐵)) ∈ ℤ)
88		dvdsmulcr 16210	. . . . 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 1376	. . . 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 257	. . 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 12595	. . . 4 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((𝑂‘𝐴) · (𝑂‘𝐵)) ∈ ℂ)
92	91, 57, 39	divcan1d 11916	. . 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 5122	. 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 3017	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 206 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2113 ≠ wne 2930 class class class wbr 5096 ‘cfv 6490 (class class class)co 7356 0cc0 11024 · cmul 11029 / cdiv 11792 ℕ₀cn0 12399 ℤcz 12486 ∥ cdvds 16177 gcd cgcd 16419 Basecbs 17134 +_gcplusg 17175 0_gc0g 17357 Grpcgrp 18861 ._gcmg 18995 odcod 19451 Abelcabl 19708

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2706 ax-sep 5239 ax-nul 5249 ax-pow 5308 ax-pr 5375 ax-un 7678 ax-cnex 11080 ax-resscn 11081 ax-1cn 11082 ax-icn 11083 ax-addcl 11084 ax-addrcl 11085 ax-mulcl 11086 ax-mulrcl 11087 ax-mulcom 11088 ax-addass 11089 ax-mulass 11090 ax-distr 11091 ax-i2m1 11092 ax-1ne0 11093 ax-1rid 11094 ax-rnegex 11095 ax-rrecex 11096 ax-cnre 11097 ax-pre-lttri 11098 ax-pre-lttrn 11099 ax-pre-ltadd 11100 ax-pre-mulgt0 11101 ax-pre-sup 11102

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2809 df-nfc 2883 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3059 df-rmo 3348 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4579 df-pr 4581 df-op 4585 df-uni 4862 df-iun 4946 df-br 5097 df-opab 5159 df-mpt 5178 df-tr 5204 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-er 8633 df-en 8882 df-dom 8883 df-sdom 8884 df-sup 9343 df-inf 9344 df-pnf 11166 df-mnf 11167 df-xr 11168 df-ltxr 11169 df-le 11170 df-sub 11364 df-neg 11365 df-div 11793 df-nn 12144 df-2 12206 df-3 12207 df-n0 12400 df-z 12487 df-uz 12750 df-rp 12904 df-fz 13422 df-fzo 13569 df-fl 13710 df-mod 13788 df-seq 13923 df-exp 13983 df-cj 15020 df-re 15021 df-im 15022 df-sqrt 15156 df-abs 15157 df-dvds 16178 df-gcd 16420 df-0g 17359 df-mgm 18563 df-sgrp 18642 df-mnd 18658 df-grp 18864 df-minusg 18865 df-sbg 18866 df-mulg 18996 df-od 19455 df-cmn 19709 df-abl 19710

This theorem is referenced by: odadd 19777 torsubg 19781

W3C validator