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Theorem ablomuldiv 30581
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 30577 . . . 4 ((𝐺 ∈ AbelOp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐺𝐵) = (𝐵𝐺𝐴))
323adant3r3 1183 . . 3 ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝐺𝐵) = (𝐵𝐺𝐴))
43oveq1d 7446 . 2 ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐺𝐵)𝐷𝐶) = ((𝐵𝐺𝐴)𝐷𝐶))
5 3ancoma 1097 . . 3 ((𝐴𝑋𝐵𝑋𝐶𝑋) ↔ (𝐵𝑋𝐴𝑋𝐶𝑋))
6 ablogrpo 30576 . . . 4 (𝐺 ∈ AbelOp → 𝐺 ∈ GrpOp)
7 abldiv.3 . . . . 5 𝐷 = ( /𝑔𝐺)
81, 7grpomuldivass 30570 . . . 4 ((𝐺 ∈ GrpOp ∧ (𝐵𝑋𝐴𝑋𝐶𝑋)) → ((𝐵𝐺𝐴)𝐷𝐶) = (𝐵𝐺(𝐴𝐷𝐶)))
96, 8sylan 580 . . 3 ((𝐺 ∈ AbelOp ∧ (𝐵𝑋𝐴𝑋𝐶𝑋)) → ((𝐵𝐺𝐴)𝐷𝐶) = (𝐵𝐺(𝐴𝐷𝐶)))
105, 9sylan2b 594 . 2 ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐵𝐺𝐴)𝐷𝐶) = (𝐵𝐺(𝐴𝐷𝐶)))
11 simpr2 1194 . . . 4 ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → 𝐵𝑋)
121, 7grpodivcl 30568 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐶𝑋) → (𝐴𝐷𝐶) ∈ 𝑋)
136, 12syl3an1 1162 . . . . 5 ((𝐺 ∈ AbelOp ∧ 𝐴𝑋𝐶𝑋) → (𝐴𝐷𝐶) ∈ 𝑋)
14133adant3r2 1182 . . . 4 ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝐷𝐶) ∈ 𝑋)
1511, 14jca 511 . . 3 ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐵𝑋 ∧ (𝐴𝐷𝐶) ∈ 𝑋))
161ablocom 30577 . . . 4 ((𝐺 ∈ AbelOp ∧ 𝐵𝑋 ∧ (𝐴𝐷𝐶) ∈ 𝑋) → (𝐵𝐺(𝐴𝐷𝐶)) = ((𝐴𝐷𝐶)𝐺𝐵))
17163expb 1119 . . 3 ((𝐺 ∈ AbelOp ∧ (𝐵𝑋 ∧ (𝐴𝐷𝐶) ∈ 𝑋)) → (𝐵𝐺(𝐴𝐷𝐶)) = ((𝐴𝐷𝐶)𝐺𝐵))
1815, 17syldan 591 . 2 ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐵𝐺(𝐴𝐷𝐶)) = ((𝐴𝐷𝐶)𝐺𝐵))
194, 10, 183eqtrd 2779 1 ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐺𝐵)𝐷𝐶) = ((𝐴𝐷𝐶)𝐺𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1537  wcel 2106  ran crn 5690  cfv 6563  (class class class)co 7431  GrpOpcgr 30518   /𝑔 cgs 30521  AbelOpcablo 30573
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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-grpo 30522  df-gid 30523  df-ginv 30524  df-gdiv 30525  df-ablo 30574
This theorem is referenced by:  ablodivdiv  30582  nvaddsub  30684  ablo4pnp  37867
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