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Theorem grpomuldivass 27235
Description: Associative-type law for multiplication and division. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)
Hypotheses
Ref Expression
grpdivf.1 𝑋 = ran 𝐺
grpdivf.3 𝐷 = ( /𝑔𝐺)
Assertion
Ref Expression
grpomuldivass ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐺𝐵)𝐷𝐶) = (𝐴𝐺(𝐵𝐷𝐶)))

Proof of Theorem grpomuldivass
StepHypRef Expression
1 simpr1 1065 . . . 4 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → 𝐴𝑋)
2 simpr2 1066 . . . 4 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → 𝐵𝑋)
3 grpdivf.1 . . . . . 6 𝑋 = ran 𝐺
4 eqid 2626 . . . . . 6 (inv‘𝐺) = (inv‘𝐺)
53, 4grpoinvcl 27218 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐶𝑋) → ((inv‘𝐺)‘𝐶) ∈ 𝑋)
653ad2antr3 1226 . . . 4 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((inv‘𝐺)‘𝐶) ∈ 𝑋)
71, 2, 63jca 1240 . . 3 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝑋𝐵𝑋 ∧ ((inv‘𝐺)‘𝐶) ∈ 𝑋))
83grpoass 27197 . . 3 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋 ∧ ((inv‘𝐺)‘𝐶) ∈ 𝑋)) → ((𝐴𝐺𝐵)𝐺((inv‘𝐺)‘𝐶)) = (𝐴𝐺(𝐵𝐺((inv‘𝐺)‘𝐶))))
97, 8syldan 487 . 2 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐺𝐵)𝐺((inv‘𝐺)‘𝐶)) = (𝐴𝐺(𝐵𝐺((inv‘𝐺)‘𝐶))))
10 simpl 473 . . 3 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → 𝐺 ∈ GrpOp)
113grpocl 27194 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐺𝐵) ∈ 𝑋)
12113adant3r3 1273 . . 3 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝐺𝐵) ∈ 𝑋)
13 simpr3 1067 . . 3 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → 𝐶𝑋)
14 grpdivf.3 . . . 4 𝐷 = ( /𝑔𝐺)
153, 4, 14grpodivval 27229 . . 3 ((𝐺 ∈ GrpOp ∧ (𝐴𝐺𝐵) ∈ 𝑋𝐶𝑋) → ((𝐴𝐺𝐵)𝐷𝐶) = ((𝐴𝐺𝐵)𝐺((inv‘𝐺)‘𝐶)))
1610, 12, 13, 15syl3anc 1323 . 2 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐺𝐵)𝐷𝐶) = ((𝐴𝐺𝐵)𝐺((inv‘𝐺)‘𝐶)))
173, 4, 14grpodivval 27229 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐵𝑋𝐶𝑋) → (𝐵𝐷𝐶) = (𝐵𝐺((inv‘𝐺)‘𝐶)))
18173adant3r1 1271 . . 3 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐵𝐷𝐶) = (𝐵𝐺((inv‘𝐺)‘𝐶)))
1918oveq2d 6621 . 2 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝐺(𝐵𝐷𝐶)) = (𝐴𝐺(𝐵𝐺((inv‘𝐺)‘𝐶))))
209, 16, 193eqtr4d 2670 1 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐺𝐵)𝐷𝐶) = (𝐴𝐺(𝐵𝐷𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1992  ran crn 5080  cfv 5850  (class class class)co 6605  GrpOpcgr 27183  invcgn 27185   /𝑔 cgs 27186
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-1st 7116  df-2nd 7117  df-grpo 27187  df-gid 27188  df-ginv 27189  df-gdiv 27190
This theorem is referenced by:  ablomuldiv  27246  ablodivdiv  27247  ablo4pnp  33297
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