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Theorem odadd2 20069

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 19756	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑋 → (𝑂‘𝐴) ∈ ℕ₀)
4	3	3ad2ant2 1152	. . . . . . 7 ⊢ ((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑂‘𝐴) ∈ ℕ₀)
5	4	nn0zd 12722	. . . . . 6 ⊢ ((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑂‘𝐴) ∈ ℤ)
6	1, 2	odcl 19756	. . . . . . . 8 ⊢ (𝐵 ∈ 𝑋 → (𝑂‘𝐵) ∈ ℕ₀)
7	6	3ad2ant3 1153	. . . . . . 7 ⊢ ((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑂‘𝐵) ∈ ℕ₀)
8	7	nn0zd 12722	. . . . . 6 ⊢ ((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑂‘𝐵) ∈ ℤ)
9	5, 8	zmulcld 12813	. . . . 5 ⊢ ((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑂‘𝐴) · (𝑂‘𝐵)) ∈ ℤ)
10	9	adantr 486	. . . 4 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) = 0) → ((𝑂‘𝐴) · (𝑂‘𝐵)) ∈ ℤ)
11		dvds0 16447	. . . 4 ⊢ (((𝑂‘𝐴) · (𝑂‘𝐵)) ∈ ℤ → ((𝑂‘𝐴) · (𝑂‘𝐵)) ∥ 0)
12	10, 11	syl 18	. . 3 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) = 0) → ((𝑂‘𝐴) · (𝑂‘𝐵)) ∥ 0)
13		simpr 490	. . . . . 6 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) = 0) → ((𝑂‘𝐴) gcd (𝑂‘𝐵)) = 0)
14	13	sq0id 14341	. . . . 5 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) = 0) → (((𝑂‘𝐴) gcd (𝑂‘𝐵))↑2) = 0)
15	14	oveq2d 7437	. . . 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 20005	. . . . . . . . . 10 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
17		odadd1.3	. . . . . . . . . . 11 ⊢ + = (+_g‘𝐺)
18	1, 17	grpcl 19158	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 + 𝐵) ∈ 𝑋)
19	16, 18	syl3an1 1181	. . . . . . . . 9 ⊢ ((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 + 𝐵) ∈ 𝑋)
20	1, 2	odcl 19756	. . . . . . . . 9 ⊢ ((𝐴 + 𝐵) ∈ 𝑋 → (𝑂‘(𝐴 + 𝐵)) ∈ ℕ₀)
21	19, 20	syl 18	. . . . . . . 8 ⊢ ((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑂‘(𝐴 + 𝐵)) ∈ ℕ₀)
22	21	nn0zd 12722	. . . . . . 7 ⊢ ((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑂‘(𝐴 + 𝐵)) ∈ ℤ)
23	22	adantr 486	. . . . . 6 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) = 0) → (𝑂‘(𝐴 + 𝐵)) ∈ ℤ)
24	23	zcnd 12808	. . . . 5 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) = 0) → (𝑂‘(𝐴 + 𝐵)) ∈ ℂ)
25	24	mul01d 11513	. . . 4 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) = 0) → ((𝑂‘(𝐴 + 𝐵)) · 0) = 0)
26	15, 25	eqtrd 2795	. . 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 5132	. 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 486	. . . . . . . 8 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (𝑂‘𝐴) ∈ ℤ)
29	8	adantr 486	. . . . . . . 8 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (𝑂‘𝐵) ∈ ℤ)
30	28, 29	gcdcld 16684	. . . . . . 7 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ∈ ℕ₀)
31	30	nn0cnd 12673	. . . . . 6 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ∈ ℂ)
32	31	sqvald 14290	. . . . 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 7437	. . . 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 16679	. . . . . . . . 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 596	. . . . . . . 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 500	. . . . . . 7 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ∥ (𝑂‘𝐴))
37	30	nn0zd 12722	. . . . . . . 8 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ∈ ℤ)
38		simpr 490	. . . . . . . 8 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0)
39		dvdsval2 16431	. . . . . . . 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 1398	. . . . . . 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 235	. . . . . 6 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((𝑂‘𝐴) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) ∈ ℤ)
42	41	zcnd 12808	. . . . 5 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((𝑂‘𝐴) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) ∈ ℂ)
43	35	simprd 501	. . . . . . 7 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ∥ (𝑂‘𝐵))
44		dvdsval2 16431	. . . . . . . 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 1398	. . . . . . 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 235	. . . . . 6 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((𝑂‘𝐵) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) ∈ ℤ)
47	46	zcnd 12808	. . . . 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 11526	. . . 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 12808	. . . . . 6 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (𝑂‘𝐴) ∈ ℂ)
50	49, 31, 38	divcan1d 12098	. . . . 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 12808	. . . . . 6 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (𝑂‘𝐵) ∈ ℂ)
52	51, 31, 38	divcan1d 12098	. . . . 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 7439	. . . 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 2801	. . 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 486	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (𝑂‘(𝐴 + 𝐵)) ∈ ℤ)
56		dvdsmul2 16454	. . . . . . . . . 10 ⊢ (((𝑂‘(𝐴 + 𝐵)) ∈ ℤ ∧ (𝑂‘𝐴) ∈ ℤ) → (𝑂‘𝐴) ∥ ((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐴)))
57	55, 28, 56	syl2anc 596	. . . . . . . . 9 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (𝑂‘𝐴) ∥ ((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐴)))
58		simpl1 1210	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → 𝐺 ∈ Abel)
59	55, 29	zmulcld 12813	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐵)) ∈ ℤ)
60		simpl2 1211	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → 𝐴 ∈ 𝑋)
61		simpl3 1212	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → 𝐵 ∈ 𝑋)
62		eqid 2760	. . . . . . . . . . . . . 14 ⊢ (._g‘𝐺) = (._g‘𝐺)
63	1, 62, 17	mulgdi 20046	. . . . . . . . . . . . 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 1399	. . . . . . . . . . . 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 16454	. . . . . . . . . . . . . . 15 ⊢ (((𝑂‘(𝐴 + 𝐵)) ∈ ℤ ∧ (𝑂‘𝐵) ∈ ℤ) → (𝑂‘𝐵) ∥ ((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐵)))
66	55, 29, 65	syl2anc 596	. . . . . . . . . . . . . 14 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (𝑂‘𝐵) ∥ ((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐵)))
67	58, 16	syl 18	. . . . . . . . . . . . . . 15 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → 𝐺 ∈ Grp)
68		eqid 2760	. . . . . . . . . . . . . . . 16 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
69	1, 2, 62, 68	oddvds 19767	. . . . . . . . . . . . . . 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 1398	. . . . . . . . . . . . . 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 235	. . . . . . . . . . . . 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 7437	. . . . . . . . . . . 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 2795	. . . . . . . . . . 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 16453	. . . . . . . . . . . . 13 ⊢ (((𝑂‘(𝐴 + 𝐵)) ∈ ℤ ∧ (𝑂‘𝐵) ∈ ℤ) → (𝑂‘(𝐴 + 𝐵)) ∥ ((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐵)))
75	55, 29, 74	syl2anc 596	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (𝑂‘(𝐴 + 𝐵)) ∥ ((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐵)))
76	19	adantr 486	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (𝐴 + 𝐵) ∈ 𝑋)
77	1, 2, 62, 68	oddvds 19767	. . . . . . . . . . . . 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 1398	. . . . . . . . . . . 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 235	. . . . . . . . . . 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 19307	. . . . . . . . . . . . 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 1398	. . . . . . . . . . . 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 19185	. . . . . . . . . . . 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 596	. . . . . . . . . . 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 2804	. . . . . . . . . 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 19767	. . . . . . . . . . 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 1398	. . . . . . . . . 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 260	. . . . . . . . 9 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (𝑂‘𝐴) ∥ ((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐵)))
88	55, 28	zmulcld 12813	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐴)) ∈ ℤ)
89		dvdsgcd 16723	. . . . . . . . . 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 1398	. . . . . . . . 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 712	. . . . . . . 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 486	. . . . . . . . 9 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (𝑂‘(𝐴 + 𝐵)) ∈ ℕ₀)
93		mulgcd 16727	. . . . . . . . 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 1398	. . . . . . . 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 5130	. . . . . . 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 5126	. . . . . 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 16461	. . . . . . 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 1401	. . . . . 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 235	. . . . 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 20046	. . . . . . . . . . . . 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 1399	. . . . . . . . . . . 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 19767	. . . . . . . . . . . . . . 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 1398	. . . . . . . . . . . . . 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 235	. . . . . . . . . . . . 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 7436	. . . . . . . . . . . 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 2795	. . . . . . . . . . 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 16453	. . . . . . . . . . . . 13 ⊢ (((𝑂‘(𝐴 + 𝐵)) ∈ ℤ ∧ (𝑂‘𝐴) ∈ ℤ) → (𝑂‘(𝐴 + 𝐵)) ∥ ((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐴)))
108	55, 28, 107	syl2anc 596	. . . . . . . . . . . 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 19767	. . . . . . . . . . . . 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 1398	. . . . . . . . . . . 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 235	. . . . . . . . . . 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 19307	. . . . . . . . . . . . 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 1398	. . . . . . . . . . . 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 19184	. . . . . . . . . . . 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 596	. . . . . . . . . . 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 2804	. . . . . . . . . 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 19767	. . . . . . . . . . 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 1398	. . . . . . . . . 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 260	. . . . . . . . 9 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (𝑂‘𝐵) ∥ ((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐴)))
120		dvdsgcd 16723	. . . . . . . . . 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 1398	. . . . . . . . 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 712	. . . . . . . 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 5130	. . . . . . 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 5126	. . . . . 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 16461	. . . . . . 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 1401	. . . . . 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 235	. . . . 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 16684	. . . . . . . 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 12673	. . . . . . 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 11306	. . . . . . 7 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → 1 ∈ ℂ)
131	31	mullidd 11331	. . . . . . . 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 7439	. . . . . . . 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 16729	. . . . . . . . 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 1398	. . . . . . . 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 2802	. . . . . . 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 11956	. . . . . 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 16835	. . . . . 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 1400	. . . . 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 712	. . . 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 12813	. . . . 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 14276	. . . . . 6 ⊢ (((𝑂‘𝐴) gcd (𝑂‘𝐵)) ∈ ℤ → (((𝑂‘𝐴) gcd (𝑂‘𝐵))↑2) ∈ ℤ)
142	37, 141	syl 18	. . . . 5 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (((𝑂‘𝐴) gcd (𝑂‘𝐵))↑2) ∈ ℤ)
143		dvdsmulc 16459	. . . . 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 1398	. . . 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 16	. . 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 5128	. 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 3042	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

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1569 ∈ wcel 2144 ≠ wne 2955 class class class wbr 5102 ‘cfv 6539 (class class class)co 7421 0cc0 11204 1c1 11205 · cmul 11209 / cdiv 11977 2c2 12401 ℕ₀cn0 12610 ℤcz 12697 ↑cexp 14208 ∥ cdvds 16428 gcd cgcd 16670 Basecbs 17393 +_gcplusg 17434 0_gc0g 17616 Grpcgrp 19150 ._gcmg 19283 odcod 19744 Abelcabl 20001

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7752 ax-cnex 11260 ax-resscn 11261 ax-1cn 11262 ax-icn 11263 ax-addcl 11264 ax-addrcl 11265 ax-mulcl 11266 ax-mulrcl 11267 ax-mulcom 11268 ax-addass 11269 ax-mulass 11270 ax-distr 11271 ax-i2m1 11272 ax-1ne0 11273 ax-1rid 11274 ax-rnegex 11275 ax-rrecex 11276 ax-cnre 11277 ax-pre-lttri 11278 ax-pre-lttrn 11279 ax-pre-ltadd 11280 ax-pre-mulgt0 11281 ax-pre-sup 11282

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 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 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-op 4835 df-uni 4867 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-riota 7378 df-ov 7424 df-oprab 7425 df-mpo 7426 df-om 7879 df-1st 8002 df-2nd 8003 df-frecs 8303 df-wrecs 8334 df-recs 8383 df-rdg 8422 df-er 8721 df-en 8978 df-dom 8979 df-sdom 8980 df-sup 9438 df-inf 9439 df-pnf 11349 df-mnf 11350 df-xr 11351 df-ltxr 11352 df-le 11353 df-sub 11547 df-neg 11548 df-div 11978 df-nn 12340 df-2 12409 df-3 12410 df-n0 12611 df-z 12698 df-uz 12970 df-rp 13125 df-fz 13644 df-fzo 13793 df-fl 13936 df-mod 14014 df-seq 14149 df-exp 14209 df-cj 15272 df-re 15273 df-im 15274 df-sqrt 15408 df-abs 15409 df-dvds 16429 df-gcd 16671 df-0g 17618 df-mgm 18822 df-sgrp 18914 df-mnd 18930 df-grp 19153 df-minusg 19154 df-sbg 19155 df-mulg 19284 df-od 19748 df-cmn 20002 df-abl 20003

This theorem is used by: odadd 20070

W3C validator