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Theorem odadd2 18075

Description: The order of a product in an abelian group is divisible by the LCM of the orders of the factors divided by the GCD. (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
odadd2	⊢ ((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑂‘𝐴) · (𝑂‘𝐵)) ∥ ((𝑂‘(𝐴 + 𝐵)) · (((𝑂‘𝐴) gcd (𝑂‘𝐵))↑2)))

**Proof of Theorem odadd2**
Step	Hyp	Ref	Expression
1		odadd1.2	. . . . . . . . 9 ⊢ 𝑋 = (Base‘𝐺)
2		odadd1.1	. . . . . . . . 9 ⊢ 𝑂 = (od‘𝐺)
3	1, 2	odcl 17778	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑋 → (𝑂‘𝐴) ∈ ℕ₀)
4	3	3ad2ant2 1076	. . . . . . 7 ⊢ ((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑂‘𝐴) ∈ ℕ₀)
5	4	nn0zd 11356	. . . . . 6 ⊢ ((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑂‘𝐴) ∈ ℤ)
6	1, 2	odcl 17778	. . . . . . . 8 ⊢ (𝐵 ∈ 𝑋 → (𝑂‘𝐵) ∈ ℕ₀)
7	6	3ad2ant3 1077	. . . . . . 7 ⊢ ((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑂‘𝐵) ∈ ℕ₀)
8	7	nn0zd 11356	. . . . . 6 ⊢ ((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑂‘𝐵) ∈ ℤ)
9	5, 8	zmulcld 11364	. . . . 5 ⊢ ((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑂‘𝐴) · (𝑂‘𝐵)) ∈ ℤ)
10	9	adantr 480	. . . 4 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) = 0) → ((𝑂‘𝐴) · (𝑂‘𝐵)) ∈ ℤ)
11		dvds0 14835	. . . 4 ⊢ (((𝑂‘𝐴) · (𝑂‘𝐵)) ∈ ℤ → ((𝑂‘𝐴) · (𝑂‘𝐵)) ∥ 0)
12	10, 11	syl 17	. . 3 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) = 0) → ((𝑂‘𝐴) · (𝑂‘𝐵)) ∥ 0)
13		simpr 476	. . . . . 6 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) = 0) → ((𝑂‘𝐴) gcd (𝑂‘𝐵)) = 0)
14	13	sq0id 12819	. . . . 5 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) = 0) → (((𝑂‘𝐴) gcd (𝑂‘𝐵))↑2) = 0)
15	14	oveq2d 6565	. . . 4 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) = 0) → ((𝑂‘(𝐴 + 𝐵)) · (((𝑂‘𝐴) gcd (𝑂‘𝐵))↑2)) = ((𝑂‘(𝐴 + 𝐵)) · 0))
16		ablgrp 18021	. . . . . . . . . 10 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
17		odadd1.3	. . . . . . . . . . 11 ⊢ + = (+_g‘𝐺)
18	1, 17	grpcl 17253	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 + 𝐵) ∈ 𝑋)
19	16, 18	syl3an1 1351	. . . . . . . . 9 ⊢ ((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 + 𝐵) ∈ 𝑋)
20	1, 2	odcl 17778	. . . . . . . . 9 ⊢ ((𝐴 + 𝐵) ∈ 𝑋 → (𝑂‘(𝐴 + 𝐵)) ∈ ℕ₀)
21	19, 20	syl 17	. . . . . . . 8 ⊢ ((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑂‘(𝐴 + 𝐵)) ∈ ℕ₀)
22	21	nn0zd 11356	. . . . . . 7 ⊢ ((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑂‘(𝐴 + 𝐵)) ∈ ℤ)
23	22	adantr 480	. . . . . 6 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) = 0) → (𝑂‘(𝐴 + 𝐵)) ∈ ℤ)
24	23	zcnd 11359	. . . . 5 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) = 0) → (𝑂‘(𝐴 + 𝐵)) ∈ ℂ)
25	24	mul01d 10114	. . . 4 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) = 0) → ((𝑂‘(𝐴 + 𝐵)) · 0) = 0)
26	15, 25	eqtrd 2644	. . 3 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) = 0) → ((𝑂‘(𝐴 + 𝐵)) · (((𝑂‘𝐴) gcd (𝑂‘𝐵))↑2)) = 0)
27	12, 26	breqtrrd 4611	. 2 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) = 0) → ((𝑂‘𝐴) · (𝑂‘𝐵)) ∥ ((𝑂‘(𝐴 + 𝐵)) · (((𝑂‘𝐴) gcd (𝑂‘𝐵))↑2)))
28	5	adantr 480	. . . . . . . 8 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (𝑂‘𝐴) ∈ ℤ)
29	8	adantr 480	. . . . . . . 8 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (𝑂‘𝐵) ∈ ℤ)
30	28, 29	gcdcld 15068	. . . . . . 7 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ∈ ℕ₀)
31	30	nn0cnd 11230	. . . . . 6 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ∈ ℂ)
32	31	sqvald 12867	. . . . 5 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (((𝑂‘𝐴) gcd (𝑂‘𝐵))↑2) = (((𝑂‘𝐴) gcd (𝑂‘𝐵)) · ((𝑂‘𝐴) gcd (𝑂‘𝐵))))
33	32	oveq2d 6565	. . . 4 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((((𝑂‘𝐴) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) · ((𝑂‘𝐵) / ((𝑂‘𝐴) gcd (𝑂‘𝐵)))) · (((𝑂‘𝐴) gcd (𝑂‘𝐵))↑2)) = ((((𝑂‘𝐴) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) · ((𝑂‘𝐵) / ((𝑂‘𝐴) gcd (𝑂‘𝐵)))) · (((𝑂‘𝐴) gcd (𝑂‘𝐵)) · ((𝑂‘𝐴) gcd (𝑂‘𝐵)))))
34		gcddvds 15063	. . . . . . . . 9 ⊢ (((𝑂‘𝐴) ∈ ℤ ∧ (𝑂‘𝐵) ∈ ℤ) → (((𝑂‘𝐴) gcd (𝑂‘𝐵)) ∥ (𝑂‘𝐴) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ∥ (𝑂‘𝐵)))
35	28, 29, 34	syl2anc 691	. . . . . . . 8 ⊢ (((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 474	. . . . . . 7 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ∥ (𝑂‘𝐴))
37	30	nn0zd 11356	. . . . . . . 8 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ∈ ℤ)
38		simpr 476	. . . . . . . 8 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0)
39		dvdsval2 14824	. . . . . . . 8 ⊢ ((((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))
40	37, 38, 28, 39	syl3anc 1318	. . . . . . 7 ⊢ (((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))
41	36, 40	mpbid 221	. . . . . 6 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((𝑂‘𝐴) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) ∈ ℤ)
42	41	zcnd 11359	. . . . 5 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((𝑂‘𝐴) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) ∈ ℂ)
43	35	simprd 478	. . . . . . 7 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ∥ (𝑂‘𝐵))
44		dvdsval2 14824	. . . . . . . 8 ⊢ ((((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))
45	37, 38, 29, 44	syl3anc 1318	. . . . . . 7 ⊢ (((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))
46	43, 45	mpbid 221	. . . . . 6 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((𝑂‘𝐵) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) ∈ ℤ)
47	46	zcnd 11359	. . . . 5 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((𝑂‘𝐵) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) ∈ ℂ)
48	42, 31, 47, 31	mul4d 10127	. . . 4 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((((𝑂‘𝐴) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) · ((𝑂‘𝐴) gcd (𝑂‘𝐵))) · (((𝑂‘𝐵) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) · ((𝑂‘𝐴) gcd (𝑂‘𝐵)))) = ((((𝑂‘𝐴) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) · ((𝑂‘𝐵) / ((𝑂‘𝐴) gcd (𝑂‘𝐵)))) · (((𝑂‘𝐴) gcd (𝑂‘𝐵)) · ((𝑂‘𝐴) gcd (𝑂‘𝐵)))))
49	28	zcnd 11359	. . . . . 6 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (𝑂‘𝐴) ∈ ℂ)
50	49, 31, 38	divcan1d 10681	. . . . 5 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (((𝑂‘𝐴) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) · ((𝑂‘𝐴) gcd (𝑂‘𝐵))) = (𝑂‘𝐴))
51	29	zcnd 11359	. . . . . 6 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (𝑂‘𝐵) ∈ ℂ)
52	51, 31, 38	divcan1d 10681	. . . . 5 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (((𝑂‘𝐵) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) · ((𝑂‘𝐴) gcd (𝑂‘𝐵))) = (𝑂‘𝐵))
53	50, 52	oveq12d 6567	. . . 4 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((((𝑂‘𝐴) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) · ((𝑂‘𝐴) gcd (𝑂‘𝐵))) · (((𝑂‘𝐵) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) · ((𝑂‘𝐴) gcd (𝑂‘𝐵)))) = ((𝑂‘𝐴) · (𝑂‘𝐵)))
54	33, 48, 53	3eqtr2d 2650	. . 3 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((((𝑂‘𝐴) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) · ((𝑂‘𝐵) / ((𝑂‘𝐴) gcd (𝑂‘𝐵)))) · (((𝑂‘𝐴) gcd (𝑂‘𝐵))↑2)) = ((𝑂‘𝐴) · (𝑂‘𝐵)))
55	22	adantr 480	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (𝑂‘(𝐴 + 𝐵)) ∈ ℤ)
56		dvdsmul2 14842	. . . . . . . . . 10 ⊢ (((𝑂‘(𝐴 + 𝐵)) ∈ ℤ ∧ (𝑂‘𝐴) ∈ ℤ) → (𝑂‘𝐴) ∥ ((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐴)))
57	55, 28, 56	syl2anc 691	. . . . . . . . 9 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (𝑂‘𝐴) ∥ ((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐴)))
58		simpl1 1057	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → 𝐺 ∈ Abel)
59	55, 29	zmulcld 11364	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐵)) ∈ ℤ)
60		simpl2 1058	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → 𝐴 ∈ 𝑋)
61		simpl3 1059	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → 𝐵 ∈ 𝑋)
62		eqid 2610	. . . . . . . . . . . . . 14 ⊢ (._g‘𝐺) = (._g‘𝐺)
63	1, 62, 17	mulgdi 18055	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ Abel ∧ (((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐵)) ∈ ℤ ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐵))(._g‘𝐺)(𝐴 + 𝐵)) = ((((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐵))(._g‘𝐺)𝐴) + (((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐵))(._g‘𝐺)𝐵)))
64	58, 59, 60, 61, 63	syl13anc 1320	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐵))(._g‘𝐺)(𝐴 + 𝐵)) = ((((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐵))(._g‘𝐺)𝐴) + (((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐵))(._g‘𝐺)𝐵)))
65		dvdsmul2 14842	. . . . . . . . . . . . . . 15 ⊢ (((𝑂‘(𝐴 + 𝐵)) ∈ ℤ ∧ (𝑂‘𝐵) ∈ ℤ) → (𝑂‘𝐵) ∥ ((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐵)))
66	55, 29, 65	syl2anc 691	. . . . . . . . . . . . . 14 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (𝑂‘𝐵) ∥ ((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐵)))
67	58, 16	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → 𝐺 ∈ Grp)
68		eqid 2610	. . . . . . . . . . . . . . . 16 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
69	1, 2, 62, 68	oddvds 17789	. . . . . . . . . . . . . . 15 ⊢ ((𝐺 ∈ Grp ∧ 𝐵 ∈ 𝑋 ∧ ((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐵)) ∈ ℤ) → ((𝑂‘𝐵) ∥ ((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐵)) ↔ (((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐵))(._g‘𝐺)𝐵) = (0_g‘𝐺)))
70	67, 61, 59, 69	syl3anc 1318	. . . . . . . . . . . . . 14 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((𝑂‘𝐵) ∥ ((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐵)) ↔ (((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐵))(._g‘𝐺)𝐵) = (0_g‘𝐺)))
71	66, 70	mpbid 221	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐵))(._g‘𝐺)𝐵) = (0_g‘𝐺))
72	71	oveq2d 6565	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐵))(._g‘𝐺)𝐴) + (((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐵))(._g‘𝐺)𝐵)) = ((((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐵))(._g‘𝐺)𝐴) + (0_g‘𝐺)))
73	64, 72	eqtrd 2644	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐵))(._g‘𝐺)(𝐴 + 𝐵)) = ((((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐵))(._g‘𝐺)𝐴) + (0_g‘𝐺)))
74		dvdsmul1 14841	. . . . . . . . . . . . 13 ⊢ (((𝑂‘(𝐴 + 𝐵)) ∈ ℤ ∧ (𝑂‘𝐵) ∈ ℤ) → (𝑂‘(𝐴 + 𝐵)) ∥ ((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐵)))
75	55, 29, 74	syl2anc 691	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (𝑂‘(𝐴 + 𝐵)) ∥ ((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐵)))
76	19	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (𝐴 + 𝐵) ∈ 𝑋)
77	1, 2, 62, 68	oddvds 17789	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ Grp ∧ (𝐴 + 𝐵) ∈ 𝑋 ∧ ((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐵)) ∈ ℤ) → ((𝑂‘(𝐴 + 𝐵)) ∥ ((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐵)) ↔ (((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐵))(._g‘𝐺)(𝐴 + 𝐵)) = (0_g‘𝐺)))
78	67, 76, 59, 77	syl3anc 1318	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((𝑂‘(𝐴 + 𝐵)) ∥ ((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐵)) ↔ (((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐵))(._g‘𝐺)(𝐴 + 𝐵)) = (0_g‘𝐺)))
79	75, 78	mpbid 221	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐵))(._g‘𝐺)(𝐴 + 𝐵)) = (0_g‘𝐺))
80	1, 62	mulgcl 17382	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ Grp ∧ ((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐵)) ∈ ℤ ∧ 𝐴 ∈ 𝑋) → (((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐵))(._g‘𝐺)𝐴) ∈ 𝑋)
81	67, 59, 60, 80	syl3anc 1318	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐵))(._g‘𝐺)𝐴) ∈ 𝑋)
82	1, 17, 68	grprid 17276	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ Grp ∧ (((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐵))(._g‘𝐺)𝐴) ∈ 𝑋) → ((((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐵))(._g‘𝐺)𝐴) + (0_g‘𝐺)) = (((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐵))(._g‘𝐺)𝐴))
83	67, 81, 82	syl2anc 691	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐵))(._g‘𝐺)𝐴) + (0_g‘𝐺)) = (((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐵))(._g‘𝐺)𝐴))
84	73, 79, 83	3eqtr3rd 2653	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐵))(._g‘𝐺)𝐴) = (0_g‘𝐺))
85	1, 2, 62, 68	oddvds 17789	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ ((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐵)) ∈ ℤ) → ((𝑂‘𝐴) ∥ ((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐵)) ↔ (((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐵))(._g‘𝐺)𝐴) = (0_g‘𝐺)))
86	67, 60, 59, 85	syl3anc 1318	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((𝑂‘𝐴) ∥ ((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐵)) ↔ (((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐵))(._g‘𝐺)𝐴) = (0_g‘𝐺)))
87	84, 86	mpbird 246	. . . . . . . . 9 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (𝑂‘𝐴) ∥ ((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐵)))
88	55, 28	zmulcld 11364	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐴)) ∈ ℤ)
89		dvdsgcd 15099	. . . . . . . . . 10 ⊢ (((𝑂‘𝐴) ∈ ℤ ∧ ((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐴)) ∈ ℤ ∧ ((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐵)) ∈ ℤ) → (((𝑂‘𝐴) ∥ ((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐴)) ∧ (𝑂‘𝐴) ∥ ((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐵))) → (𝑂‘𝐴) ∥ (((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐴)) gcd ((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐵)))))
90	28, 88, 59, 89	syl3anc 1318	. . . . . . . . 9 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (((𝑂‘𝐴) ∥ ((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐴)) ∧ (𝑂‘𝐴) ∥ ((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐵))) → (𝑂‘𝐴) ∥ (((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐴)) gcd ((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐵)))))
91	57, 87, 90	mp2and 711	. . . . . . . 8 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (𝑂‘𝐴) ∥ (((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐴)) gcd ((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐵))))
92	21	adantr 480	. . . . . . . . 9 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (𝑂‘(𝐴 + 𝐵)) ∈ ℕ₀)
93		mulgcd 15103	. . . . . . . . 9 ⊢ (((𝑂‘(𝐴 + 𝐵)) ∈ ℕ₀ ∧ (𝑂‘𝐴) ∈ ℤ ∧ (𝑂‘𝐵) ∈ ℤ) → (((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐴)) gcd ((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐵))) = ((𝑂‘(𝐴 + 𝐵)) · ((𝑂‘𝐴) gcd (𝑂‘𝐵))))
94	92, 28, 29, 93	syl3anc 1318	. . . . . . . 8 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐴)) gcd ((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐵))) = ((𝑂‘(𝐴 + 𝐵)) · ((𝑂‘𝐴) gcd (𝑂‘𝐵))))
95	91, 94	breqtrd 4609	. . . . . . 7 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (𝑂‘𝐴) ∥ ((𝑂‘(𝐴 + 𝐵)) · ((𝑂‘𝐴) gcd (𝑂‘𝐵))))
96	50, 95	eqbrtrd 4605	. . . . . 6 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (((𝑂‘𝐴) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) · ((𝑂‘𝐴) gcd (𝑂‘𝐵))) ∥ ((𝑂‘(𝐴 + 𝐵)) · ((𝑂‘𝐴) gcd (𝑂‘𝐵))))
97		dvdsmulcr 14849	. . . . . . 7 ⊢ ((((𝑂‘𝐴) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) ∈ ℤ ∧ (𝑂‘(𝐴 + 𝐵)) ∈ ℤ ∧ (((𝑂‘𝐴) gcd (𝑂‘𝐵)) ∈ ℤ ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0)) → ((((𝑂‘𝐴) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) · ((𝑂‘𝐴) gcd (𝑂‘𝐵))) ∥ ((𝑂‘(𝐴 + 𝐵)) · ((𝑂‘𝐴) gcd (𝑂‘𝐵))) ↔ ((𝑂‘𝐴) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) ∥ (𝑂‘(𝐴 + 𝐵))))
98	41, 55, 37, 38, 97	syl112anc 1322	. . . . . 6 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((((𝑂‘𝐴) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) · ((𝑂‘𝐴) gcd (𝑂‘𝐵))) ∥ ((𝑂‘(𝐴 + 𝐵)) · ((𝑂‘𝐴) gcd (𝑂‘𝐵))) ↔ ((𝑂‘𝐴) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) ∥ (𝑂‘(𝐴 + 𝐵))))
99	96, 98	mpbid 221	. . . . 5 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((𝑂‘𝐴) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) ∥ (𝑂‘(𝐴 + 𝐵)))
100	1, 62, 17	mulgdi 18055	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ Abel ∧ (((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐴)) ∈ ℤ ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐴))(._g‘𝐺)(𝐴 + 𝐵)) = ((((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐴))(._g‘𝐺)𝐴) + (((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐴))(._g‘𝐺)𝐵)))
101	58, 88, 60, 61, 100	syl13anc 1320	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐴))(._g‘𝐺)(𝐴 + 𝐵)) = ((((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐴))(._g‘𝐺)𝐴) + (((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐴))(._g‘𝐺)𝐵)))
102	1, 2, 62, 68	oddvds 17789	. . . . . . . . . . . . . . 15 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ ((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐴)) ∈ ℤ) → ((𝑂‘𝐴) ∥ ((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐴)) ↔ (((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐴))(._g‘𝐺)𝐴) = (0_g‘𝐺)))
103	67, 60, 88, 102	syl3anc 1318	. . . . . . . . . . . . . 14 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((𝑂‘𝐴) ∥ ((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐴)) ↔ (((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐴))(._g‘𝐺)𝐴) = (0_g‘𝐺)))
104	57, 103	mpbid 221	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐴))(._g‘𝐺)𝐴) = (0_g‘𝐺))
105	104	oveq1d 6564	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐴))(._g‘𝐺)𝐴) + (((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐴))(._g‘𝐺)𝐵)) = ((0_g‘𝐺) + (((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐴))(._g‘𝐺)𝐵)))
106	101, 105	eqtrd 2644	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐴))(._g‘𝐺)(𝐴 + 𝐵)) = ((0_g‘𝐺) + (((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐴))(._g‘𝐺)𝐵)))
107		dvdsmul1 14841	. . . . . . . . . . . . 13 ⊢ (((𝑂‘(𝐴 + 𝐵)) ∈ ℤ ∧ (𝑂‘𝐴) ∈ ℤ) → (𝑂‘(𝐴 + 𝐵)) ∥ ((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐴)))
108	55, 28, 107	syl2anc 691	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (𝑂‘(𝐴 + 𝐵)) ∥ ((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐴)))
109	1, 2, 62, 68	oddvds 17789	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ Grp ∧ (𝐴 + 𝐵) ∈ 𝑋 ∧ ((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐴)) ∈ ℤ) → ((𝑂‘(𝐴 + 𝐵)) ∥ ((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐴)) ↔ (((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐴))(._g‘𝐺)(𝐴 + 𝐵)) = (0_g‘𝐺)))
110	67, 76, 88, 109	syl3anc 1318	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((𝑂‘(𝐴 + 𝐵)) ∥ ((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐴)) ↔ (((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐴))(._g‘𝐺)(𝐴 + 𝐵)) = (0_g‘𝐺)))
111	108, 110	mpbid 221	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐴))(._g‘𝐺)(𝐴 + 𝐵)) = (0_g‘𝐺))
112	1, 62	mulgcl 17382	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ Grp ∧ ((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐴)) ∈ ℤ ∧ 𝐵 ∈ 𝑋) → (((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐴))(._g‘𝐺)𝐵) ∈ 𝑋)
113	67, 88, 61, 112	syl3anc 1318	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐴))(._g‘𝐺)𝐵) ∈ 𝑋)
114	1, 17, 68	grplid 17275	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ Grp ∧ (((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐴))(._g‘𝐺)𝐵) ∈ 𝑋) → ((0_g‘𝐺) + (((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐴))(._g‘𝐺)𝐵)) = (((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐴))(._g‘𝐺)𝐵))
115	67, 113, 114	syl2anc 691	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((0_g‘𝐺) + (((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐴))(._g‘𝐺)𝐵)) = (((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐴))(._g‘𝐺)𝐵))
116	106, 111, 115	3eqtr3rd 2653	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐴))(._g‘𝐺)𝐵) = (0_g‘𝐺))
117	1, 2, 62, 68	oddvds 17789	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Grp ∧ 𝐵 ∈ 𝑋 ∧ ((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐴)) ∈ ℤ) → ((𝑂‘𝐵) ∥ ((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐴)) ↔ (((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐴))(._g‘𝐺)𝐵) = (0_g‘𝐺)))
118	67, 61, 88, 117	syl3anc 1318	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((𝑂‘𝐵) ∥ ((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐴)) ↔ (((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐴))(._g‘𝐺)𝐵) = (0_g‘𝐺)))
119	116, 118	mpbird 246	. . . . . . . . 9 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (𝑂‘𝐵) ∥ ((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐴)))
120		dvdsgcd 15099	. . . . . . . . . 10 ⊢ (((𝑂‘𝐵) ∈ ℤ ∧ ((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐴)) ∈ ℤ ∧ ((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐵)) ∈ ℤ) → (((𝑂‘𝐵) ∥ ((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐴)) ∧ (𝑂‘𝐵) ∥ ((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐵))) → (𝑂‘𝐵) ∥ (((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐴)) gcd ((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐵)))))
121	29, 88, 59, 120	syl3anc 1318	. . . . . . . . 9 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (((𝑂‘𝐵) ∥ ((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐴)) ∧ (𝑂‘𝐵) ∥ ((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐵))) → (𝑂‘𝐵) ∥ (((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐴)) gcd ((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐵)))))
122	119, 66, 121	mp2and 711	. . . . . . . 8 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (𝑂‘𝐵) ∥ (((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐴)) gcd ((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐵))))
123	122, 94	breqtrd 4609	. . . . . . 7 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (𝑂‘𝐵) ∥ ((𝑂‘(𝐴 + 𝐵)) · ((𝑂‘𝐴) gcd (𝑂‘𝐵))))
124	52, 123	eqbrtrd 4605	. . . . . 6 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (((𝑂‘𝐵) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) · ((𝑂‘𝐴) gcd (𝑂‘𝐵))) ∥ ((𝑂‘(𝐴 + 𝐵)) · ((𝑂‘𝐴) gcd (𝑂‘𝐵))))
125		dvdsmulcr 14849	. . . . . . 7 ⊢ ((((𝑂‘𝐵) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) ∈ ℤ ∧ (𝑂‘(𝐴 + 𝐵)) ∈ ℤ ∧ (((𝑂‘𝐴) gcd (𝑂‘𝐵)) ∈ ℤ ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0)) → ((((𝑂‘𝐵) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) · ((𝑂‘𝐴) gcd (𝑂‘𝐵))) ∥ ((𝑂‘(𝐴 + 𝐵)) · ((𝑂‘𝐴) gcd (𝑂‘𝐵))) ↔ ((𝑂‘𝐵) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) ∥ (𝑂‘(𝐴 + 𝐵))))
126	46, 55, 37, 38, 125	syl112anc 1322	. . . . . 6 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((((𝑂‘𝐵) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) · ((𝑂‘𝐴) gcd (𝑂‘𝐵))) ∥ ((𝑂‘(𝐴 + 𝐵)) · ((𝑂‘𝐴) gcd (𝑂‘𝐵))) ↔ ((𝑂‘𝐵) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) ∥ (𝑂‘(𝐴 + 𝐵))))
127	124, 126	mpbid 221	. . . . 5 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((𝑂‘𝐵) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) ∥ (𝑂‘(𝐴 + 𝐵)))
128	41, 46	gcdcld 15068	. . . . . . . 8 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (((𝑂‘𝐴) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) gcd ((𝑂‘𝐵) / ((𝑂‘𝐴) gcd (𝑂‘𝐵)))) ∈ ℕ₀)
129	128	nn0cnd 11230	. . . . . . 7 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (((𝑂‘𝐴) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) gcd ((𝑂‘𝐵) / ((𝑂‘𝐴) gcd (𝑂‘𝐵)))) ∈ ℂ)
130		1cnd 9935	. . . . . . 7 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → 1 ∈ ℂ)
131	31	mulid2d 9937	. . . . . . . 8 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (1 · ((𝑂‘𝐴) gcd (𝑂‘𝐵))) = ((𝑂‘𝐴) gcd (𝑂‘𝐵)))
132	50, 52	oveq12d 6567	. . . . . . . 8 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((((𝑂‘𝐴) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) · ((𝑂‘𝐴) gcd (𝑂‘𝐵))) gcd (((𝑂‘𝐵) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) · ((𝑂‘𝐴) gcd (𝑂‘𝐵)))) = ((𝑂‘𝐴) gcd (𝑂‘𝐵)))
133		mulgcdr 15105	. . . . . . . . 9 ⊢ ((((𝑂‘𝐴) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) ∈ ℤ ∧ ((𝑂‘𝐵) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) ∈ ℤ ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ∈ ℕ₀) → ((((𝑂‘𝐴) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) · ((𝑂‘𝐴) gcd (𝑂‘𝐵))) gcd (((𝑂‘𝐵) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) · ((𝑂‘𝐴) gcd (𝑂‘𝐵)))) = ((((𝑂‘𝐴) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) gcd ((𝑂‘𝐵) / ((𝑂‘𝐴) gcd (𝑂‘𝐵)))) · ((𝑂‘𝐴) gcd (𝑂‘𝐵))))
134	41, 46, 30, 133	syl3anc 1318	. . . . . . . 8 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((((𝑂‘𝐴) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) · ((𝑂‘𝐴) gcd (𝑂‘𝐵))) gcd (((𝑂‘𝐵) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) · ((𝑂‘𝐴) gcd (𝑂‘𝐵)))) = ((((𝑂‘𝐴) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) gcd ((𝑂‘𝐵) / ((𝑂‘𝐴) gcd (𝑂‘𝐵)))) · ((𝑂‘𝐴) gcd (𝑂‘𝐵))))
135	131, 132, 134	3eqtr2rd 2651	. . . . . . 7 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((((𝑂‘𝐴) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) gcd ((𝑂‘𝐵) / ((𝑂‘𝐴) gcd (𝑂‘𝐵)))) · ((𝑂‘𝐴) gcd (𝑂‘𝐵))) = (1 · ((𝑂‘𝐴) gcd (𝑂‘𝐵))))
136	129, 130, 31, 38, 135	mulcan2ad 10542	. . . . . 6 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (((𝑂‘𝐴) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) gcd ((𝑂‘𝐵) / ((𝑂‘𝐴) gcd (𝑂‘𝐵)))) = 1)
137		coprmdvds2 15206	. . . . . 6 ⊢ (((((𝑂‘𝐴) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) ∈ ℤ ∧ ((𝑂‘𝐵) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) ∈ ℤ ∧ (𝑂‘(𝐴 + 𝐵)) ∈ ℤ) ∧ (((𝑂‘𝐴) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) gcd ((𝑂‘𝐵) / ((𝑂‘𝐴) gcd (𝑂‘𝐵)))) = 1) → ((((𝑂‘𝐴) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) ∥ (𝑂‘(𝐴 + 𝐵)) ∧ ((𝑂‘𝐵) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) ∥ (𝑂‘(𝐴 + 𝐵))) → (((𝑂‘𝐴) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) · ((𝑂‘𝐵) / ((𝑂‘𝐴) gcd (𝑂‘𝐵)))) ∥ (𝑂‘(𝐴 + 𝐵))))
138	41, 46, 55, 136, 137	syl31anc 1321	. . . . 5 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((((𝑂‘𝐴) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) ∥ (𝑂‘(𝐴 + 𝐵)) ∧ ((𝑂‘𝐵) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) ∥ (𝑂‘(𝐴 + 𝐵))) → (((𝑂‘𝐴) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) · ((𝑂‘𝐵) / ((𝑂‘𝐴) gcd (𝑂‘𝐵)))) ∥ (𝑂‘(𝐴 + 𝐵))))
139	99, 127, 138	mp2and 711	. . . 4 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (((𝑂‘𝐴) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) · ((𝑂‘𝐵) / ((𝑂‘𝐴) gcd (𝑂‘𝐵)))) ∥ (𝑂‘(𝐴 + 𝐵)))
140	41, 46	zmulcld 11364	. . . . 5 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (((𝑂‘𝐴) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) · ((𝑂‘𝐵) / ((𝑂‘𝐴) gcd (𝑂‘𝐵)))) ∈ ℤ)
141		zsqcl 12796	. . . . . 6 ⊢ (((𝑂‘𝐴) gcd (𝑂‘𝐵)) ∈ ℤ → (((𝑂‘𝐴) gcd (𝑂‘𝐵))↑2) ∈ ℤ)
142	37, 141	syl 17	. . . . 5 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (((𝑂‘𝐴) gcd (𝑂‘𝐵))↑2) ∈ ℤ)
143		dvdsmulc 14847	. . . . 5 ⊢ (((((𝑂‘𝐴) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) · ((𝑂‘𝐵) / ((𝑂‘𝐴) gcd (𝑂‘𝐵)))) ∈ ℤ ∧ (𝑂‘(𝐴 + 𝐵)) ∈ ℤ ∧ (((𝑂‘𝐴) gcd (𝑂‘𝐵))↑2) ∈ ℤ) → ((((𝑂‘𝐴) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) · ((𝑂‘𝐵) / ((𝑂‘𝐴) gcd (𝑂‘𝐵)))) ∥ (𝑂‘(𝐴 + 𝐵)) → ((((𝑂‘𝐴) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) · ((𝑂‘𝐵) / ((𝑂‘𝐴) gcd (𝑂‘𝐵)))) · (((𝑂‘𝐴) gcd (𝑂‘𝐵))↑2)) ∥ ((𝑂‘(𝐴 + 𝐵)) · (((𝑂‘𝐴) gcd (𝑂‘𝐵))↑2))))
144	140, 55, 142, 143	syl3anc 1318	. . . 4 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((((𝑂‘𝐴) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) · ((𝑂‘𝐵) / ((𝑂‘𝐴) gcd (𝑂‘𝐵)))) ∥ (𝑂‘(𝐴 + 𝐵)) → ((((𝑂‘𝐴) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) · ((𝑂‘𝐵) / ((𝑂‘𝐴) gcd (𝑂‘𝐵)))) · (((𝑂‘𝐴) gcd (𝑂‘𝐵))↑2)) ∥ ((𝑂‘(𝐴 + 𝐵)) · (((𝑂‘𝐴) gcd (𝑂‘𝐵))↑2))))
145	139, 144	mpd 15	. . 3 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((((𝑂‘𝐴) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) · ((𝑂‘𝐵) / ((𝑂‘𝐴) gcd (𝑂‘𝐵)))) · (((𝑂‘𝐴) gcd (𝑂‘𝐵))↑2)) ∥ ((𝑂‘(𝐴 + 𝐵)) · (((𝑂‘𝐴) gcd (𝑂‘𝐵))↑2)))
146	54, 145	eqbrtrrd 4607	. 2 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((𝑂‘𝐴) · (𝑂‘𝐵)) ∥ ((𝑂‘(𝐴 + 𝐵)) · (((𝑂‘𝐴) gcd (𝑂‘𝐵))↑2)))
147	27, 146	pm2.61dane 2869	1 ⊢ ((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑂‘𝐴) · (𝑂‘𝐵)) ∥ ((𝑂‘(𝐴 + 𝐵)) · (((𝑂‘𝐴) gcd (𝑂‘𝐵))↑2)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 0cc0 9815 1c1 9816 · cmul 9820 / cdiv 10563 2c2 10947 ℕ₀cn0 11169 ℤcz 11254 ↑cexp 12722 ∥ cdvds 14821 gcd cgcd 15054 Basecbs 15695 +_gcplusg 15768 0_gc0g 15923 Grpcgrp 17245 ._gcmg 17363 odcod 17767 Abelcabl 18017

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-sup 8231 df-inf 8232 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-fz 12198 df-fzo 12335 df-fl 12455 df-mod 12531 df-seq 12664 df-exp 12723 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-dvds 14822 df-gcd 15055 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-grp 17248 df-minusg 17249 df-sbg 17250 df-mulg 17364 df-od 17771 df-cmn 18018 df-abl 18019

This theorem is referenced by: odadd 18076

W3C validator