Theorem ablomuldiv 31033

Description: Law for group multiplication and division. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)

Hypotheses

Ref	Expression
abldiv.1	⊢ 𝑋 = ran 𝐺
abldiv.3	⊢ 𝐷 = ( /𝑔 ‘𝐺)

Assertion

Ref	Expression
ablomuldiv	⊢ ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐺𝐵)𝐷𝐶) = ((𝐴𝐷𝐶)𝐺𝐵))

Proof of Theorem ablomuldiv

Step	Hyp	Ref	Expression
1		abldiv.1	. . . . 5 ⊢ 𝑋 = ran 𝐺
2	1	ablocom 31029	. . . 4 ⊢ ((𝐺 ∈ AbelOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺𝐵) = (𝐵𝐺𝐴))
3	2	3adant3r3 1203	. . 3 ⊢ ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐺𝐵) = (𝐵𝐺𝐴))
4	3	oveq1d 7428	. 2 ⊢ ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐺𝐵)𝐷𝐶) = ((𝐵𝐺𝐴)𝐷𝐶))
5		3ancoma 1115	. . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) ↔ (𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋))
6		ablogrpo 31028	. . . 4 ⊢ (𝐺 ∈ AbelOp → 𝐺 ∈ GrpOp)
7		abldiv.3	. . . . 5 ⊢ 𝐷 = ( /𝑔 ‘𝐺)
8	1, 7	grpomuldivass 31022	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐵𝐺𝐴)𝐷𝐶) = (𝐵𝐺(𝐴𝐷𝐶)))
9	6, 8	sylan 592	. . 3 ⊢ ((𝐺 ∈ AbelOp ∧ (𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐵𝐺𝐴)𝐷𝐶) = (𝐵𝐺(𝐴𝐷𝐶)))
10	5, 9	sylan2b 606	. 2 ⊢ ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐵𝐺𝐴)𝐷𝐶) = (𝐵𝐺(𝐴𝐷𝐶)))
11		simpr2 1214	. . . 4 ⊢ ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → 𝐵 ∈ 𝑋)
12	1, 7	grpodivcl 31020	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐴𝐷𝐶) ∈ 𝑋)
13	6, 12	syl3an1 1181	. . . . 5 ⊢ ((𝐺 ∈ AbelOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐴𝐷𝐶) ∈ 𝑋)
14	13	3adant3r2 1202	. . . 4 ⊢ ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐷𝐶) ∈ 𝑋)
15	11, 14	jca 521	. . 3 ⊢ ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐵 ∈ 𝑋 ∧ (𝐴𝐷𝐶) ∈ 𝑋))
16	1	ablocom 31029	. . . 4 ⊢ ((𝐺 ∈ AbelOp ∧ 𝐵 ∈ 𝑋 ∧ (𝐴𝐷𝐶) ∈ 𝑋) → (𝐵𝐺(𝐴𝐷𝐶)) = ((𝐴𝐷𝐶)𝐺𝐵))
17	16	3expb 1138	. . 3 ⊢ ((𝐺 ∈ AbelOp ∧ (𝐵 ∈ 𝑋 ∧ (𝐴𝐷𝐶) ∈ 𝑋)) → (𝐵𝐺(𝐴𝐷𝐶)) = ((𝐴𝐷𝐶)𝐺𝐵))
18	15, 17	syldan 603	. 2 ⊢ ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐵𝐺(𝐴𝐷𝐶)) = ((𝐴𝐷𝐶)𝐺𝐵))
19	4, 10, 18	3eqtrd 2799	1 ⊢ ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐺𝐵)𝐷𝐶) = ((𝐴𝐷𝐶)𝐺𝐵))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ran crn 5656  ‘cfv 6533  (class class class)co 7413  GrpOpcgr 30970   /_𝑔 cgs 30973  AbelOpcablo 31025

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7370  df-ov 7416  df-oprab 7417  df-mpo 7418  df-1st 7986  df-2nd 7987  df-grpo 30974  df-gid 30975  df-ginv 30976  df-gdiv 30977  df-ablo 31026

This theorem is used by:  ablodivdiv  31034  nvaddsub  31136  ablo4pnp  38630