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Theorem grpomuldivass 30390
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 1191 . . . 4 ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ 𝐴 ∈ 𝑋)
2 simpr2 1192 . . . 4 ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ 𝐡 ∈ 𝑋)
3 grpdivf.1 . . . . . 6 𝑋 = ran 𝐺
4 eqid 2725 . . . . . 6 (invβ€˜πΊ) = (invβ€˜πΊ)
53, 4grpoinvcl 30373 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐢 ∈ 𝑋) β†’ ((invβ€˜πΊ)β€˜πΆ) ∈ 𝑋)
653ad2antr3 1187 . . . 4 ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ ((invβ€˜πΊ)β€˜πΆ) ∈ 𝑋)
71, 2, 63jca 1125 . . 3 ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ ((invβ€˜πΊ)β€˜πΆ) ∈ 𝑋))
83grpoass 30352 . . 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 589 . 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 481 . . 3 ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ 𝐺 ∈ GrpOp)
113grpocl 30349 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (𝐴𝐺𝐡) ∈ 𝑋)
12113adant3r3 1181 . . 3 ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ (𝐴𝐺𝐡) ∈ 𝑋)
13 simpr3 1193 . . 3 ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ 𝐢 ∈ 𝑋)
14 grpdivf.3 . . . 4 𝐷 = ( /𝑔 β€˜πΊ)
153, 4, 14grpodivval 30384 . . 3 ((𝐺 ∈ GrpOp ∧ (𝐴𝐺𝐡) ∈ 𝑋 ∧ 𝐢 ∈ 𝑋) β†’ ((𝐴𝐺𝐡)𝐷𝐢) = ((𝐴𝐺𝐡)𝐺((invβ€˜πΊ)β€˜πΆ)))
1610, 12, 13, 15syl3anc 1368 . 2 ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ ((𝐴𝐺𝐡)𝐷𝐢) = ((𝐴𝐺𝐡)𝐺((invβ€˜πΊ)β€˜πΆ)))
173, 4, 14grpodivval 30384 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋) β†’ (𝐡𝐷𝐢) = (𝐡𝐺((invβ€˜πΊ)β€˜πΆ)))
18173adant3r1 1179 . . 3 ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ (𝐡𝐷𝐢) = (𝐡𝐺((invβ€˜πΊ)β€˜πΆ)))
1918oveq2d 7429 . 2 ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ (𝐴𝐺(𝐡𝐷𝐢)) = (𝐴𝐺(𝐡𝐺((invβ€˜πΊ)β€˜πΆ))))
209, 16, 193eqtr4d 2775 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 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  ran crn 5674  β€˜cfv 6543  (class class class)co 7413  GrpOpcgr 30338  invcgn 30340   /𝑔 cgs 30341
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 2166  ax-ext 2696  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pow 5360  ax-pr 5424  ax-un 7735
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  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 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-iun 4994  df-br 5145  df-opab 5207  df-mpt 5228  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7369  df-ov 7416  df-oprab 7417  df-mpo 7418  df-1st 7987  df-2nd 7988  df-grpo 30342  df-gid 30343  df-ginv 30344  df-gdiv 30345
This theorem is referenced by:  ablomuldiv  30401  ablodivdiv  30402  ablo4pnp  37406
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