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Theorem grpomuldivass 30338
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 1192 . . . 4 ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ 𝐴 ∈ 𝑋)
2 simpr2 1193 . . . 4 ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ 𝐡 ∈ 𝑋)
3 grpdivf.1 . . . . . 6 𝑋 = ran 𝐺
4 eqid 2727 . . . . . 6 (invβ€˜πΊ) = (invβ€˜πΊ)
53, 4grpoinvcl 30321 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐢 ∈ 𝑋) β†’ ((invβ€˜πΊ)β€˜πΆ) ∈ 𝑋)
653ad2antr3 1188 . . . 4 ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ ((invβ€˜πΊ)β€˜πΆ) ∈ 𝑋)
71, 2, 63jca 1126 . . 3 ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ ((invβ€˜πΊ)β€˜πΆ) ∈ 𝑋))
83grpoass 30300 . . 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 590 . 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 482 . . 3 ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ 𝐺 ∈ GrpOp)
113grpocl 30297 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (𝐴𝐺𝐡) ∈ 𝑋)
12113adant3r3 1182 . . 3 ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ (𝐴𝐺𝐡) ∈ 𝑋)
13 simpr3 1194 . . 3 ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ 𝐢 ∈ 𝑋)
14 grpdivf.3 . . . 4 𝐷 = ( /𝑔 β€˜πΊ)
153, 4, 14grpodivval 30332 . . 3 ((𝐺 ∈ GrpOp ∧ (𝐴𝐺𝐡) ∈ 𝑋 ∧ 𝐢 ∈ 𝑋) β†’ ((𝐴𝐺𝐡)𝐷𝐢) = ((𝐴𝐺𝐡)𝐺((invβ€˜πΊ)β€˜πΆ)))
1610, 12, 13, 15syl3anc 1369 . 2 ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ ((𝐴𝐺𝐡)𝐷𝐢) = ((𝐴𝐺𝐡)𝐺((invβ€˜πΊ)β€˜πΆ)))
173, 4, 14grpodivval 30332 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋) β†’ (𝐡𝐷𝐢) = (𝐡𝐺((invβ€˜πΊ)β€˜πΆ)))
18173adant3r1 1180 . . 3 ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ (𝐡𝐷𝐢) = (𝐡𝐺((invβ€˜πΊ)β€˜πΆ)))
1918oveq2d 7430 . 2 ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ (𝐴𝐺(𝐡𝐷𝐢)) = (𝐴𝐺(𝐡𝐺((invβ€˜πΊ)β€˜πΆ))))
209, 16, 193eqtr4d 2777 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 395   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099  ran crn 5673  β€˜cfv 6542  (class class class)co 7414  GrpOpcgr 30286  invcgn 30288   /𝑔 cgs 30289
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-1st 7987  df-2nd 7988  df-grpo 30290  df-gid 30291  df-ginv 30292  df-gdiv 30293
This theorem is referenced by:  ablomuldiv  30349  ablodivdiv  30350  ablo4pnp  37288
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