Theorem ablomuldiv 28323

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 28319	. . . 4 ⊢ ((𝐺 ∈ AbelOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺𝐵) = (𝐵𝐺𝐴))
3	2	3adant3r3 1180	. . 3 ⊢ ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐺𝐵) = (𝐵𝐺𝐴))
4	3	oveq1d 7165	. 2 ⊢ ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐺𝐵)𝐷𝐶) = ((𝐵𝐺𝐴)𝐷𝐶))
5		3ancoma 1094	. . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) ↔ (𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋))
6		ablogrpo 28318	. . . 4 ⊢ (𝐺 ∈ AbelOp → 𝐺 ∈ GrpOp)
7		abldiv.3	. . . . 5 ⊢ 𝐷 = ( /𝑔 ‘𝐺)
8	1, 7	grpomuldivass 28312	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐵𝐺𝐴)𝐷𝐶) = (𝐵𝐺(𝐴𝐷𝐶)))
9	6, 8	sylan 582	. . 3 ⊢ ((𝐺 ∈ AbelOp ∧ (𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐵𝐺𝐴)𝐷𝐶) = (𝐵𝐺(𝐴𝐷𝐶)))
10	5, 9	sylan2b 595	. 2 ⊢ ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐵𝐺𝐴)𝐷𝐶) = (𝐵𝐺(𝐴𝐷𝐶)))
11		simpr2 1191	. . . 4 ⊢ ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → 𝐵 ∈ 𝑋)
12	1, 7	grpodivcl 28310	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐴𝐷𝐶) ∈ 𝑋)
13	6, 12	syl3an1 1159	. . . . 5 ⊢ ((𝐺 ∈ AbelOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐴𝐷𝐶) ∈ 𝑋)
14	13	3adant3r2 1179	. . . 4 ⊢ ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐷𝐶) ∈ 𝑋)
15	11, 14	jca 514	. . 3 ⊢ ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐵 ∈ 𝑋 ∧ (𝐴𝐷𝐶) ∈ 𝑋))
16	1	ablocom 28319	. . . 4 ⊢ ((𝐺 ∈ AbelOp ∧ 𝐵 ∈ 𝑋 ∧ (𝐴𝐷𝐶) ∈ 𝑋) → (𝐵𝐺(𝐴𝐷𝐶)) = ((𝐴𝐷𝐶)𝐺𝐵))
17	16	3expb 1116	. . 3 ⊢ ((𝐺 ∈ AbelOp ∧ (𝐵 ∈ 𝑋 ∧ (𝐴𝐷𝐶) ∈ 𝑋)) → (𝐵𝐺(𝐴𝐷𝐶)) = ((𝐴𝐷𝐶)𝐺𝐵))
18	15, 17	syldan 593	. 2 ⊢ ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐵𝐺(𝐴𝐷𝐶)) = ((𝐴𝐷𝐶)𝐺𝐵))
19	4, 10, 18	3eqtrd 2860	1 ⊢ ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐺𝐵)𝐷𝐶) = ((𝐴𝐷𝐶)𝐺𝐵))

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110  ran crn 5550  ‘cfv 6349  (class class class)co 7150  GrpOpcgr 28260   /_𝑔 cgs 28263  AbelOpcablo 28315

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-1st 7683  df-2nd 7684  df-grpo 28264  df-gid 28265  df-ginv 28266  df-gdiv 28267  df-ablo 28316

This theorem is referenced by:  ablodivdiv  28324  nvaddsub  28426  ablo4pnp  35152