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Theorem grpolcan 29270
Description: Left cancellation law for groups. (Contributed by NM, 27-Oct-2006.) (New usage is discouraged.)
Hypothesis
Ref Expression
grplcan.1 𝑋 = ran 𝐺
Assertion
Ref Expression
grpolcan ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ ((𝐢𝐺𝐴) = (𝐢𝐺𝐡) ↔ 𝐴 = 𝐡))

Proof of Theorem grpolcan
StepHypRef Expression
1 oveq2 7357 . . . . . 6 ((𝐢𝐺𝐴) = (𝐢𝐺𝐡) β†’ (((invβ€˜πΊ)β€˜πΆ)𝐺(𝐢𝐺𝐴)) = (((invβ€˜πΊ)β€˜πΆ)𝐺(𝐢𝐺𝐡)))
21adantl 482 . . . . 5 ((((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ (𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) ∧ (𝐢𝐺𝐴) = (𝐢𝐺𝐡)) β†’ (((invβ€˜πΊ)β€˜πΆ)𝐺(𝐢𝐺𝐴)) = (((invβ€˜πΊ)β€˜πΆ)𝐺(𝐢𝐺𝐡)))
3 grplcan.1 . . . . . . . . . . 11 𝑋 = ran 𝐺
4 eqid 2737 . . . . . . . . . . 11 (GIdβ€˜πΊ) = (GIdβ€˜πΊ)
5 eqid 2737 . . . . . . . . . . 11 (invβ€˜πΊ) = (invβ€˜πΊ)
63, 4, 5grpolinv 29266 . . . . . . . . . 10 ((𝐺 ∈ GrpOp ∧ 𝐢 ∈ 𝑋) β†’ (((invβ€˜πΊ)β€˜πΆ)𝐺𝐢) = (GIdβ€˜πΊ))
76adantlr 713 . . . . . . . . 9 (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ 𝐢 ∈ 𝑋) β†’ (((invβ€˜πΊ)β€˜πΆ)𝐺𝐢) = (GIdβ€˜πΊ))
87oveq1d 7364 . . . . . . . 8 (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ 𝐢 ∈ 𝑋) β†’ ((((invβ€˜πΊ)β€˜πΆ)𝐺𝐢)𝐺𝐴) = ((GIdβ€˜πΊ)𝐺𝐴))
93, 5grpoinvcl 29264 . . . . . . . . . . . 12 ((𝐺 ∈ GrpOp ∧ 𝐢 ∈ 𝑋) β†’ ((invβ€˜πΊ)β€˜πΆ) ∈ 𝑋)
109adantrl 714 . . . . . . . . . . 11 ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ ((invβ€˜πΊ)β€˜πΆ) ∈ 𝑋)
11 simprr 771 . . . . . . . . . . 11 ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ 𝐢 ∈ 𝑋)
12 simprl 769 . . . . . . . . . . 11 ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ 𝐴 ∈ 𝑋)
1310, 11, 123jca 1128 . . . . . . . . . 10 ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ (((invβ€˜πΊ)β€˜πΆ) ∈ 𝑋 ∧ 𝐢 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋))
143grpoass 29243 . . . . . . . . . 10 ((𝐺 ∈ GrpOp ∧ (((invβ€˜πΊ)β€˜πΆ) ∈ 𝑋 ∧ 𝐢 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) β†’ ((((invβ€˜πΊ)β€˜πΆ)𝐺𝐢)𝐺𝐴) = (((invβ€˜πΊ)β€˜πΆ)𝐺(𝐢𝐺𝐴)))
1513, 14syldan 591 . . . . . . . . 9 ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ ((((invβ€˜πΊ)β€˜πΆ)𝐺𝐢)𝐺𝐴) = (((invβ€˜πΊ)β€˜πΆ)𝐺(𝐢𝐺𝐴)))
1615anassrs 468 . . . . . . . 8 (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ 𝐢 ∈ 𝑋) β†’ ((((invβ€˜πΊ)β€˜πΆ)𝐺𝐢)𝐺𝐴) = (((invβ€˜πΊ)β€˜πΆ)𝐺(𝐢𝐺𝐴)))
173, 4grpolid 29256 . . . . . . . . 9 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) β†’ ((GIdβ€˜πΊ)𝐺𝐴) = 𝐴)
1817adantr 481 . . . . . . . 8 (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ 𝐢 ∈ 𝑋) β†’ ((GIdβ€˜πΊ)𝐺𝐴) = 𝐴)
198, 16, 183eqtr3d 2785 . . . . . . 7 (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ 𝐢 ∈ 𝑋) β†’ (((invβ€˜πΊ)β€˜πΆ)𝐺(𝐢𝐺𝐴)) = 𝐴)
2019adantrl 714 . . . . . 6 (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ (𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ (((invβ€˜πΊ)β€˜πΆ)𝐺(𝐢𝐺𝐴)) = 𝐴)
2120adantr 481 . . . . 5 ((((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ (𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) ∧ (𝐢𝐺𝐴) = (𝐢𝐺𝐡)) β†’ (((invβ€˜πΊ)β€˜πΆ)𝐺(𝐢𝐺𝐴)) = 𝐴)
226adantrl 714 . . . . . . . . 9 ((𝐺 ∈ GrpOp ∧ (𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ (((invβ€˜πΊ)β€˜πΆ)𝐺𝐢) = (GIdβ€˜πΊ))
2322oveq1d 7364 . . . . . . . 8 ((𝐺 ∈ GrpOp ∧ (𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ ((((invβ€˜πΊ)β€˜πΆ)𝐺𝐢)𝐺𝐡) = ((GIdβ€˜πΊ)𝐺𝐡))
249adantrl 714 . . . . . . . . . 10 ((𝐺 ∈ GrpOp ∧ (𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ ((invβ€˜πΊ)β€˜πΆ) ∈ 𝑋)
25 simprr 771 . . . . . . . . . 10 ((𝐺 ∈ GrpOp ∧ (𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ 𝐢 ∈ 𝑋)
26 simprl 769 . . . . . . . . . 10 ((𝐺 ∈ GrpOp ∧ (𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ 𝐡 ∈ 𝑋)
2724, 25, 263jca 1128 . . . . . . . . 9 ((𝐺 ∈ GrpOp ∧ (𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ (((invβ€˜πΊ)β€˜πΆ) ∈ 𝑋 ∧ 𝐢 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋))
283grpoass 29243 . . . . . . . . 9 ((𝐺 ∈ GrpOp ∧ (((invβ€˜πΊ)β€˜πΆ) ∈ 𝑋 ∧ 𝐢 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ ((((invβ€˜πΊ)β€˜πΆ)𝐺𝐢)𝐺𝐡) = (((invβ€˜πΊ)β€˜πΆ)𝐺(𝐢𝐺𝐡)))
2927, 28syldan 591 . . . . . . . 8 ((𝐺 ∈ GrpOp ∧ (𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ ((((invβ€˜πΊ)β€˜πΆ)𝐺𝐢)𝐺𝐡) = (((invβ€˜πΊ)β€˜πΆ)𝐺(𝐢𝐺𝐡)))
303, 4grpolid 29256 . . . . . . . . 9 ((𝐺 ∈ GrpOp ∧ 𝐡 ∈ 𝑋) β†’ ((GIdβ€˜πΊ)𝐺𝐡) = 𝐡)
3130adantrr 715 . . . . . . . 8 ((𝐺 ∈ GrpOp ∧ (𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ ((GIdβ€˜πΊ)𝐺𝐡) = 𝐡)
3223, 29, 313eqtr3d 2785 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ (𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ (((invβ€˜πΊ)β€˜πΆ)𝐺(𝐢𝐺𝐡)) = 𝐡)
3332adantlr 713 . . . . . 6 (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ (𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ (((invβ€˜πΊ)β€˜πΆ)𝐺(𝐢𝐺𝐡)) = 𝐡)
3433adantr 481 . . . . 5 ((((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ (𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) ∧ (𝐢𝐺𝐴) = (𝐢𝐺𝐡)) β†’ (((invβ€˜πΊ)β€˜πΆ)𝐺(𝐢𝐺𝐡)) = 𝐡)
352, 21, 343eqtr3d 2785 . . . 4 ((((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ (𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) ∧ (𝐢𝐺𝐴) = (𝐢𝐺𝐡)) β†’ 𝐴 = 𝐡)
3635exp53 448 . . 3 (𝐺 ∈ GrpOp β†’ (𝐴 ∈ 𝑋 β†’ (𝐡 ∈ 𝑋 β†’ (𝐢 ∈ 𝑋 β†’ ((𝐢𝐺𝐴) = (𝐢𝐺𝐡) β†’ 𝐴 = 𝐡)))))
37363imp2 1349 . 2 ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ ((𝐢𝐺𝐴) = (𝐢𝐺𝐡) β†’ 𝐴 = 𝐡))
38 oveq2 7357 . 2 (𝐴 = 𝐡 β†’ (𝐢𝐺𝐴) = (𝐢𝐺𝐡))
3937, 38impbid1 224 1 ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ ((𝐢𝐺𝐴) = (𝐢𝐺𝐡) ↔ 𝐴 = 𝐡))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ran crn 5631  β€˜cfv 6491  (class class class)co 7349  GrpOpcgr 29229  GIdcgi 29230  invcgn 29231
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 2708  ax-rep 5240  ax-sep 5254  ax-nul 5261  ax-pr 5382  ax-un 7662
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5528  df-xp 5636  df-rel 5637  df-cnv 5638  df-co 5639  df-dm 5640  df-rn 5641  df-res 5642  df-ima 5643  df-iota 6443  df-fun 6493  df-fn 6494  df-f 6495  df-f1 6496  df-fo 6497  df-f1o 6498  df-fv 6499  df-riota 7305  df-ov 7352  df-grpo 29233  df-gid 29234  df-ginv 29235
This theorem is referenced by:  grpo2inv  29271  vclcan  29311  rngolcan  36272  rngolz  36276
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