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Theorem odadd2 19769

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 19456	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑋 → (𝑂‘𝐴) ∈ ℕ₀)
4	3	3ad2ant2 1134	. . . . . . 7 ⊢ ((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑂‘𝐴) ∈ ℕ₀)
5	4	nn0zd 12504	. . . . . 6 ⊢ ((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑂‘𝐴) ∈ ℤ)
6	1, 2	odcl 19456	. . . . . . . 8 ⊢ (𝐵 ∈ 𝑋 → (𝑂‘𝐵) ∈ ℕ₀)
7	6	3ad2ant3 1135	. . . . . . 7 ⊢ ((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑂‘𝐵) ∈ ℕ₀)
8	7	nn0zd 12504	. . . . . 6 ⊢ ((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑂‘𝐵) ∈ ℤ)
9	5, 8	zmulcld 12593	. . . . 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 16189	. . . 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 484	. . . . . 6 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) = 0) → ((𝑂‘𝐴) gcd (𝑂‘𝐵)) = 0)
14	13	sq0id 14108	. . . . 5 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) = 0) → (((𝑂‘𝐴) gcd (𝑂‘𝐵))↑2) = 0)
15	14	oveq2d 7371	. . . 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 19705	. . . . . . . . . 10 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
17		odadd1.3	. . . . . . . . . . 11 ⊢ + = (+_g‘𝐺)
18	1, 17	grpcl 18862	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 + 𝐵) ∈ 𝑋)
19	16, 18	syl3an1 1163	. . . . . . . . 9 ⊢ ((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 + 𝐵) ∈ 𝑋)
20	1, 2	odcl 19456	. . . . . . . . 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 12504	. . . . . . 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 12588	. . . . 5 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) = 0) → (𝑂‘(𝐴 + 𝐵)) ∈ ℂ)
25	24	mul01d 11323	. . . 4 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) = 0) → ((𝑂‘(𝐴 + 𝐵)) · 0) = 0)
26	15, 25	eqtrd 2768	. . 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 5123	. 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 16426	. . . . . . 7 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ∈ ℕ₀)
31	30	nn0cnd 12455	. . . . . 6 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ∈ ℂ)
32	31	sqvald 14057	. . . . 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 7371	. . . 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 16421	. . . . . . . . 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 584	. . . . . . . 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 494	. . . . . . 7 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ∥ (𝑂‘𝐴))
37	30	nn0zd 12504	. . . . . . . 8 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ∈ ℤ)
38		simpr 484	. . . . . . . 8 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0)
39		dvdsval2 16173	. . . . . . . 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 1373	. . . . . . 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 232	. . . . . 6 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((𝑂‘𝐴) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) ∈ ℤ)
42	41	zcnd 12588	. . . . 5 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((𝑂‘𝐴) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) ∈ ℂ)
43	35	simprd 495	. . . . . . 7 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ∥ (𝑂‘𝐵))
44		dvdsval2 16173	. . . . . . . 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 1373	. . . . . . 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 232	. . . . . 6 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((𝑂‘𝐵) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) ∈ ℤ)
47	46	zcnd 12588	. . . . 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 11336	. . . 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 12588	. . . . . 6 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (𝑂‘𝐴) ∈ ℂ)
50	49, 31, 38	divcan1d 11909	. . . . 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 12588	. . . . . 6 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (𝑂‘𝐵) ∈ ℂ)
52	51, 31, 38	divcan1d 11909	. . . . 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 7373	. . . 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 2774	. . 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 16196	. . . . . . . . . 10 ⊢ (((𝑂‘(𝐴 + 𝐵)) ∈ ℤ ∧ (𝑂‘𝐴) ∈ ℤ) → (𝑂‘𝐴) ∥ ((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐴)))
57	55, 28, 56	syl2anc 584	. . . . . . . . 9 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (𝑂‘𝐴) ∥ ((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐴)))
58		simpl1 1192	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → 𝐺 ∈ Abel)
59	55, 29	zmulcld 12593	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐵)) ∈ ℤ)
60		simpl2 1193	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → 𝐴 ∈ 𝑋)
61		simpl3 1194	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → 𝐵 ∈ 𝑋)
62		eqid 2733	. . . . . . . . . . . . . 14 ⊢ (._g‘𝐺) = (._g‘𝐺)
63	1, 62, 17	mulgdi 19746	. . . . . . . . . . . . 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 1374	. . . . . . . . . . . 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 16196	. . . . . . . . . . . . . . 15 ⊢ (((𝑂‘(𝐴 + 𝐵)) ∈ ℤ ∧ (𝑂‘𝐵) ∈ ℤ) → (𝑂‘𝐵) ∥ ((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐵)))
66	55, 29, 65	syl2anc 584	. . . . . . . . . . . . . 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 2733	. . . . . . . . . . . . . . . 16 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
69	1, 2, 62, 68	oddvds 19467	. . . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . . 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 232	. . . . . . . . . . . . 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 7371	. . . . . . . . . . . 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 2768	. . . . . . . . . . 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 16195	. . . . . . . . . . . . 13 ⊢ (((𝑂‘(𝐴 + 𝐵)) ∈ ℤ ∧ (𝑂‘𝐵) ∈ ℤ) → (𝑂‘(𝐴 + 𝐵)) ∥ ((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐵)))
75	55, 29, 74	syl2anc 584	. . . . . . . . . . . 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 19467	. . . . . . . . . . . . 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 1373	. . . . . . . . . . . 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 232	. . . . . . . . . . 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 19012	. . . . . . . . . . . . 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 1373	. . . . . . . . . . . 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 18889	. . . . . . . . . . . 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 584	. . . . . . . . . . 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 2777	. . . . . . . . . 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 19467	. . . . . . . . . . 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 1373	. . . . . . . . . 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 257	. . . . . . . . 9 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (𝑂‘𝐴) ∥ ((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐵)))
88	55, 28	zmulcld 12593	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐴)) ∈ ℤ)
89		dvdsgcd 16462	. . . . . . . . . 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 1373	. . . . . . . . 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 699	. . . . . . . 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 16466	. . . . . . . . 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 1373	. . . . . . . 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 5121	. . . . . . 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 5117	. . . . . 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 16203	. . . . . . 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 1376	. . . . . 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 232	. . . . 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 19746	. . . . . . . . . . . . 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 1374	. . . . . . . . . . . 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 19467	. . . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . . 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 232	. . . . . . . . . . . . 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 7370	. . . . . . . . . . . 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 2768	. . . . . . . . . . 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 16195	. . . . . . . . . . . . 13 ⊢ (((𝑂‘(𝐴 + 𝐵)) ∈ ℤ ∧ (𝑂‘𝐴) ∈ ℤ) → (𝑂‘(𝐴 + 𝐵)) ∥ ((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐴)))
108	55, 28, 107	syl2anc 584	. . . . . . . . . . . 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 19467	. . . . . . . . . . . . 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 1373	. . . . . . . . . . . 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 232	. . . . . . . . . . 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 19012	. . . . . . . . . . . . 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 1373	. . . . . . . . . . . 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 18888	. . . . . . . . . . . 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 584	. . . . . . . . . . 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 2777	. . . . . . . . . 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 19467	. . . . . . . . . . 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 1373	. . . . . . . . . 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 257	. . . . . . . . 9 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (𝑂‘𝐵) ∥ ((𝑂‘(𝐴 + 𝐵)) · (𝑂‘𝐴)))
120		dvdsgcd 16462	. . . . . . . . . 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 1373	. . . . . . . . 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 699	. . . . . . . 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 5121	. . . . . . 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 5117	. . . . . 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 16203	. . . . . . 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 1376	. . . . . 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 232	. . . . 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 16426	. . . . . . . 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 12455	. . . . . . 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 11118	. . . . . . 7 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → 1 ∈ ℂ)
131	31	mullidd 11141	. . . . . . . 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 7373	. . . . . . . 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 16468	. . . . . . . . 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 1373	. . . . . . . 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 2775	. . . . . . 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 11764	. . . . . 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 16572	. . . . . 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 1375	. . . . 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 699	. . . 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 12593	. . . . 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 14043	. . . . . 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 16201	. . . . 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 1373	. . . 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 5119	. 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 3016	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 206 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2113 ≠ wne 2929 class class class wbr 5095 ‘cfv 6489 (class class class)co 7355 0cc0 11017 1c1 11018 · cmul 11022 / cdiv 11785 2c2 12191 ℕ₀cn0 12392 ℤcz 12479 ↑cexp 13975 ∥ cdvds 16170 gcd cgcd 16412 Basecbs 17127 +_gcplusg 17168 0_gc0g 17350 Grpcgrp 18854 ._gcmg 18988 odcod 19444 Abelcabl 19701

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 2705 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7677 ax-cnex 11073 ax-resscn 11074 ax-1cn 11075 ax-icn 11076 ax-addcl 11077 ax-addrcl 11078 ax-mulcl 11079 ax-mulrcl 11080 ax-mulcom 11081 ax-addass 11082 ax-mulass 11083 ax-distr 11084 ax-i2m1 11085 ax-1ne0 11086 ax-1rid 11087 ax-rnegex 11088 ax-rrecex 11089 ax-cnre 11090 ax-pre-lttri 11091 ax-pre-lttrn 11092 ax-pre-ltadd 11093 ax-pre-mulgt0 11094 ax-pre-sup 11095

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 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-nel 3034 df-ral 3049 df-rex 3058 df-rmo 3347 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4283 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4861 df-iun 4945 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6256 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-riota 7312 df-ov 7358 df-oprab 7359 df-mpo 7360 df-om 7806 df-1st 7930 df-2nd 7931 df-frecs 8220 df-wrecs 8251 df-recs 8300 df-rdg 8338 df-er 8631 df-en 8880 df-dom 8881 df-sdom 8882 df-sup 9337 df-inf 9338 df-pnf 11159 df-mnf 11160 df-xr 11161 df-ltxr 11162 df-le 11163 df-sub 11357 df-neg 11358 df-div 11786 df-nn 12137 df-2 12199 df-3 12200 df-n0 12393 df-z 12480 df-uz 12743 df-rp 12897 df-fz 13415 df-fzo 13562 df-fl 13703 df-mod 13781 df-seq 13916 df-exp 13976 df-cj 15013 df-re 15014 df-im 15015 df-sqrt 15149 df-abs 15150 df-dvds 16171 df-gcd 16413 df-0g 17352 df-mgm 18556 df-sgrp 18635 df-mnd 18651 df-grp 18857 df-minusg 18858 df-sbg 18859 df-mulg 18989 df-od 19448 df-cmn 19702 df-abl 19703

This theorem is referenced by: odadd 19770

W3C validator