odadd1 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > odadd1		Structured version Visualization version GIF version

Theorem odadd1 19449

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 19391	. . . . . . . . 9 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
2		odadd1.2	. . . . . . . . . 10 ⊢ 𝑋 = (Base‘𝐺)
3		odadd1.3	. . . . . . . . . 10 ⊢ + = (+_g‘𝐺)
4	2, 3	grpcl 18585	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 + 𝐵) ∈ 𝑋)
5	1, 4	syl3an1 1162	. . . . . . . 8 ⊢ ((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 + 𝐵) ∈ 𝑋)
6		odadd1.1	. . . . . . . . 9 ⊢ 𝑂 = (od‘𝐺)
7	2, 6	odcl 19144	. . . . . . . 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 12424	. . . . . 6 ⊢ ((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑂‘(𝐴 + 𝐵)) ∈ ℤ)
10	2, 6	odcl 19144	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑋 → (𝑂‘𝐴) ∈ ℕ₀)
11	10	3ad2ant2 1133	. . . . . . . . 9 ⊢ ((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑂‘𝐴) ∈ ℕ₀)
12	11	nn0zd 12424	. . . . . . . 8 ⊢ ((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑂‘𝐴) ∈ ℤ)
13	2, 6	odcl 19144	. . . . . . . . . 10 ⊢ (𝐵 ∈ 𝑋 → (𝑂‘𝐵) ∈ ℕ₀)
14	13	3ad2ant3 1134	. . . . . . . . 9 ⊢ ((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑂‘𝐵) ∈ ℕ₀)
15	14	nn0zd 12424	. . . . . . . 8 ⊢ ((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑂‘𝐵) ∈ ℤ)
16	12, 15	gcdcld 16215	. . . . . . 7 ⊢ ((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ∈ ℕ₀)
17	16	nn0zd 12424	. . . . . 6 ⊢ ((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ∈ ℤ)
18	9, 17	zmulcld 12432	. . . . 5 ⊢ ((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑂‘(𝐴 + 𝐵)) · ((𝑂‘𝐴) gcd (𝑂‘𝐵))) ∈ ℤ)
19	18	adantr 481	. . . 4 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) = 0) → ((𝑂‘(𝐴 + 𝐵)) · ((𝑂‘𝐴) gcd (𝑂‘𝐵))) ∈ ℤ)
20		dvds0 15981	. . . 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 16224	. . . . . 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 477	. . . 4 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) = 0) → ((𝑂‘𝐴) = 0 ∧ (𝑂‘𝐵) = 0))
25		oveq12 7284	. . . . 5 ⊢ (((𝑂‘𝐴) = 0 ∧ (𝑂‘𝐵) = 0) → ((𝑂‘𝐴) · (𝑂‘𝐵)) = (0 · 0))
26		0cn 10967	. . . . . 6 ⊢ 0 ∈ ℂ
27	26	mul01i 11165	. . . . 5 ⊢ (0 · 0) = 0
28	25, 27	eqtrdi 2794	. . . 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 5102	. 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 1190	. . . . . . 7 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → 𝐺 ∈ Abel)
32	17	adantr 481	. . . . . . . . 9 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ∈ ℤ)
33	12	adantr 481	. . . . . . . . 9 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (𝑂‘𝐴) ∈ ℤ)
34	15	adantr 481	. . . . . . . . 9 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (𝑂‘𝐵) ∈ ℤ)
35		gcddvds 16210	. . . . . . . . . . 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 495	. . . . . . . . 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 16006	. . . . . . . 8 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ∥ ((𝑂‘𝐴) · (𝑂‘𝐵)))
39		simpr 485	. . . . . . . . 9 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0)
40	33, 34	zmulcld 12432	. . . . . . . . 9 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((𝑂‘𝐴) · (𝑂‘𝐵)) ∈ ℤ)
41		dvdsval2 15966	. . . . . . . . 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 1370	. . . . . . . 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 231	. . . . . . 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 1191	. . . . . . 7 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → 𝐴 ∈ 𝑋)
45		simpl3 1192	. . . . . . 7 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → 𝐵 ∈ 𝑋)
46		eqid 2738	. . . . . . . 8 ⊢ (._g‘𝐺) = (._g‘𝐺)
47	2, 46, 3	mulgdi 19428	. . . . . . 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 1371	. . . . . 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 496	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ∥ (𝑂‘𝐵))
50		dvdsval2 15966	. . . . . . . . . . . 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 1370	. . . . . . . . . . 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 231	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((𝑂‘𝐵) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) ∈ ℤ)
53		dvdsmul1 15987	. . . . . . . . . 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 12427	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (𝑂‘𝐴) ∈ ℂ)
56	34	zcnd 12427	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (𝑂‘𝐵) ∈ ℂ)
57	32	zcnd 12427	. . . . . . . . . 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 11786	. . . . . . . . 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 5102	. . . . . . . 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 2738	. . . . . . . . . 10 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
62	2, 6, 46, 61	oddvds 19155	. . . . . . . . 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 1370	. . . . . . . 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 231	. . . . . . 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 15966	. . . . . . . . . . . 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 1370	. . . . . . . . . . 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 231	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((𝑂‘𝐴) / ((𝑂‘𝐴) gcd (𝑂‘𝐵))) ∈ ℤ)
68		dvdsmul1 15987	. . . . . . . . . 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 10996	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((𝑂‘𝐴) · (𝑂‘𝐵)) = ((𝑂‘𝐵) · (𝑂‘𝐴)))
71	70	oveq1d 7290	. . . . . . . . . 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 11786	. . . . . . . . . 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 2778	. . . . . . . . 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 5102	. . . . . . . 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 19155	. . . . . . . . 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 1370	. . . . . . . 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 231	. . . . . . 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 7293	. . . . . 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 18607	. . . . . . 7 ⊢ (𝐺 ∈ Grp → (0_g‘𝐺) ∈ 𝑋)
80	2, 3, 61	grplid 18609	. . . . . . 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 2782	. . . . 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 481	. . . . . 6 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (𝐴 + 𝐵) ∈ 𝑋)
84	2, 6, 46, 61	oddvds 19155	. . . . . 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 1370	. . . . 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 256	. . . 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 481	. . . . 5 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → (𝑂‘(𝐴 + 𝐵)) ∈ ℤ)
88		dvdsmulcr 15995	. . . . 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 1373	. . . 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 256	. . 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 12427	. . . 4 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) ≠ 0) → ((𝑂‘𝐴) · (𝑂‘𝐵)) ∈ ℂ)
92	91, 57, 39	divcan1d 11752	. . 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 5100	. 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 3032	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 205 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 class class class wbr 5074 ‘cfv 6433 (class class class)co 7275 0cc0 10871 · cmul 10876 / cdiv 11632 ℕ₀cn0 12233 ℤcz 12319 ∥ cdvds 15963 gcd cgcd 16201 Basecbs 16912 +_gcplusg 16962 0_gc0g 17150 Grpcgrp 18577 ._gcmg 18700 odcod 19132 Abelcabl 19387

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-sup 9201 df-inf 9202 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-n0 12234 df-z 12320 df-uz 12583 df-rp 12731 df-fz 13240 df-fzo 13383 df-fl 13512 df-mod 13590 df-seq 13722 df-exp 13783 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-dvds 15964 df-gcd 16202 df-0g 17152 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-grp 18580 df-minusg 18581 df-sbg 18582 df-mulg 18701 df-od 19136 df-cmn 19388 df-abl 19389

This theorem is referenced by: odadd 19451 torsubg 19455

W3C validator