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Theorem ablomuldiv 30314
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 = ( /𝑔𝐺)
Assertion
Ref Expression
ablomuldiv ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐺𝐵)𝐷𝐶) = ((𝐴𝐷𝐶)𝐺𝐵))

Proof of Theorem ablomuldiv
StepHypRef Expression
1 abldiv.1 . . . . 5 𝑋 = ran 𝐺
21ablocom 30310 . . . 4 ((𝐺 ∈ AbelOp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐺𝐵) = (𝐵𝐺𝐴))
323adant3r3 1181 . . 3 ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝐺𝐵) = (𝐵𝐺𝐴))
43oveq1d 7420 . 2 ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐺𝐵)𝐷𝐶) = ((𝐵𝐺𝐴)𝐷𝐶))
5 3ancoma 1095 . . 3 ((𝐴𝑋𝐵𝑋𝐶𝑋) ↔ (𝐵𝑋𝐴𝑋𝐶𝑋))
6 ablogrpo 30309 . . . 4 (𝐺 ∈ AbelOp → 𝐺 ∈ GrpOp)
7 abldiv.3 . . . . 5 𝐷 = ( /𝑔𝐺)
81, 7grpomuldivass 30303 . . . 4 ((𝐺 ∈ GrpOp ∧ (𝐵𝑋𝐴𝑋𝐶𝑋)) → ((𝐵𝐺𝐴)𝐷𝐶) = (𝐵𝐺(𝐴𝐷𝐶)))
96, 8sylan 579 . . 3 ((𝐺 ∈ AbelOp ∧ (𝐵𝑋𝐴𝑋𝐶𝑋)) → ((𝐵𝐺𝐴)𝐷𝐶) = (𝐵𝐺(𝐴𝐷𝐶)))
105, 9sylan2b 593 . 2 ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐵𝐺𝐴)𝐷𝐶) = (𝐵𝐺(𝐴𝐷𝐶)))
11 simpr2 1192 . . . 4 ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → 𝐵𝑋)
121, 7grpodivcl 30301 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐶𝑋) → (𝐴𝐷𝐶) ∈ 𝑋)
136, 12syl3an1 1160 . . . . 5 ((𝐺 ∈ AbelOp ∧ 𝐴𝑋𝐶𝑋) → (𝐴𝐷𝐶) ∈ 𝑋)
14133adant3r2 1180 . . . 4 ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝐷𝐶) ∈ 𝑋)
1511, 14jca 511 . . 3 ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐵𝑋 ∧ (𝐴𝐷𝐶) ∈ 𝑋))
161ablocom 30310 . . . 4 ((𝐺 ∈ AbelOp ∧ 𝐵𝑋 ∧ (𝐴𝐷𝐶) ∈ 𝑋) → (𝐵𝐺(𝐴𝐷𝐶)) = ((𝐴𝐷𝐶)𝐺𝐵))
17163expb 1117 . . 3 ((𝐺 ∈ AbelOp ∧ (𝐵𝑋 ∧ (𝐴𝐷𝐶) ∈ 𝑋)) → (𝐵𝐺(𝐴𝐷𝐶)) = ((𝐴𝐷𝐶)𝐺𝐵))
1815, 17syldan 590 . 2 ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐵𝐺(𝐴𝐷𝐶)) = ((𝐴𝐷𝐶)𝐺𝐵))
194, 10, 183eqtrd 2770 1 ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐺𝐵)𝐷𝐶) = ((𝐴𝐷𝐶)𝐺𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1084   = wceq 1533  wcel 2098  ran crn 5670  cfv 6537  (class class class)co 7405  GrpOpcgr 30251   /𝑔 cgs 30254  AbelOpcablo 30306
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7974  df-2nd 7975  df-grpo 30255  df-gid 30256  df-ginv 30257  df-gdiv 30258  df-ablo 30307
This theorem is referenced by:  ablodivdiv  30315  nvaddsub  30417  ablo4pnp  37261
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