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Theorem grpomuldivass 29781
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 = ( /𝑔 β€˜πΊ)
Assertion
Ref Expression
grpomuldivass ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ ((𝐴𝐺𝐡)𝐷𝐢) = (𝐴𝐺(𝐡𝐷𝐢)))
Proof of Theorem grpomuldivass
StepHypRef Expression
1 simpr1 1194 . . . 4 ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ 𝐴 ∈ 𝑋)
2 simpr2 1195 . . . 4 ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ 𝐡 ∈ 𝑋)
3 grpdivf.1 . . . . . 6 𝑋 = ran 𝐺
4 eqid 2732 . . . . . 6 (invβ€˜πΊ) = (invβ€˜πΊ)
53, 4grpoinvcl 29764 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐢 ∈ 𝑋) β†’ ((invβ€˜πΊ)β€˜πΆ) ∈ 𝑋)
653ad2antr3 1190 . . . 4 ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ ((invβ€˜πΊ)β€˜πΆ) ∈ 𝑋)
71, 2, 63jca 1128 . . 3 ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ ((invβ€˜πΊ)β€˜πΆ) ∈ 𝑋))
83grpoass 29743 . . 3 ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ ((invβ€˜πΊ)β€˜πΆ) ∈ 𝑋)) β†’ ((𝐴𝐺𝐡)𝐺((invβ€˜πΊ)β€˜πΆ)) = (𝐴𝐺(𝐡𝐺((invβ€˜πΊ)β€˜πΆ))))
97, 8syldan 591 . 2 ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ ((𝐴𝐺𝐡)𝐺((invβ€˜πΊ)β€˜πΆ)) = (𝐴𝐺(𝐡𝐺((invβ€˜πΊ)β€˜πΆ))))
10 simpl 483 . . 3 ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ 𝐺 ∈ GrpOp)
113grpocl 29740 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (𝐴𝐺𝐡) ∈ 𝑋)
12113adant3r3 1184 . . 3 ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ (𝐴𝐺𝐡) ∈ 𝑋)
13 simpr3 1196 . . 3 ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ 𝐢 ∈ 𝑋)
14 grpdivf.3 . . . 4 𝐷 = ( /𝑔 β€˜πΊ)
153, 4, 14grpodivval 29775 . . 3 ((𝐺 ∈ GrpOp ∧ (𝐴𝐺𝐡) ∈ 𝑋 ∧ 𝐢 ∈ 𝑋) β†’ ((𝐴𝐺𝐡)𝐷𝐢) = ((𝐴𝐺𝐡)𝐺((invβ€˜πΊ)β€˜πΆ)))
1610, 12, 13, 15syl3anc 1371 . 2 ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ ((𝐴𝐺𝐡)𝐷𝐢) = ((𝐴𝐺𝐡)𝐺((invβ€˜πΊ)β€˜πΆ)))
173, 4, 14grpodivval 29775 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋) β†’ (𝐡𝐷𝐢) = (𝐡𝐺((invβ€˜πΊ)β€˜πΆ)))
18173adant3r1 1182 . . 3 ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ (𝐡𝐷𝐢) = (𝐡𝐺((invβ€˜πΊ)β€˜πΆ)))
1918oveq2d 7421 . 2 ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ (𝐴𝐺(𝐡𝐷𝐢)) = (𝐴𝐺(𝐡𝐺((invβ€˜πΊ)β€˜πΆ))))
209, 16, 193eqtr4d 2782 1 ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ ((𝐴𝐺𝐡)𝐷𝐢) = (𝐴𝐺(𝐡𝐷𝐢)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ran crn 5676  β€˜cfv 6540  (class class class)co 7405  GrpOpcgr 29729  invcgn 29731   /𝑔 cgs 29732
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7971  df-2nd 7972  df-grpo 29733  df-gid 29734  df-ginv 29735  df-gdiv 29736
This theorem is referenced by:  ablomuldiv  29792  ablodivdiv  29793  ablo4pnp  36736
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