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Theorem grpolcan 28302
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 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐶𝐺𝐴) = (𝐶𝐺𝐵) ↔ 𝐴 = 𝐵))

Proof of Theorem grpolcan
StepHypRef Expression
1 oveq2 7146 . . . . . 6 ((𝐶𝐺𝐴) = (𝐶𝐺𝐵) → (((inv‘𝐺)‘𝐶)𝐺(𝐶𝐺𝐴)) = (((inv‘𝐺)‘𝐶)𝐺(𝐶𝐺𝐵)))
21adantl 485 . . . . 5 ((((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ (𝐵𝑋𝐶𝑋)) ∧ (𝐶𝐺𝐴) = (𝐶𝐺𝐵)) → (((inv‘𝐺)‘𝐶)𝐺(𝐶𝐺𝐴)) = (((inv‘𝐺)‘𝐶)𝐺(𝐶𝐺𝐵)))
3 grplcan.1 . . . . . . . . . . 11 𝑋 = ran 𝐺
4 eqid 2824 . . . . . . . . . . 11 (GId‘𝐺) = (GId‘𝐺)
5 eqid 2824 . . . . . . . . . . 11 (inv‘𝐺) = (inv‘𝐺)
63, 4, 5grpolinv 28298 . . . . . . . . . 10 ((𝐺 ∈ GrpOp ∧ 𝐶𝑋) → (((inv‘𝐺)‘𝐶)𝐺𝐶) = (GId‘𝐺))
76adantlr 714 . . . . . . . . 9 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ 𝐶𝑋) → (((inv‘𝐺)‘𝐶)𝐺𝐶) = (GId‘𝐺))
87oveq1d 7153 . . . . . . . 8 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ 𝐶𝑋) → ((((inv‘𝐺)‘𝐶)𝐺𝐶)𝐺𝐴) = ((GId‘𝐺)𝐺𝐴))
93, 5grpoinvcl 28296 . . . . . . . . . . . 12 ((𝐺 ∈ GrpOp ∧ 𝐶𝑋) → ((inv‘𝐺)‘𝐶) ∈ 𝑋)
109adantrl 715 . . . . . . . . . . 11 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐶𝑋)) → ((inv‘𝐺)‘𝐶) ∈ 𝑋)
11 simprr 772 . . . . . . . . . . 11 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐶𝑋)) → 𝐶𝑋)
12 simprl 770 . . . . . . . . . . 11 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐶𝑋)) → 𝐴𝑋)
1310, 11, 123jca 1125 . . . . . . . . . 10 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐶𝑋)) → (((inv‘𝐺)‘𝐶) ∈ 𝑋𝐶𝑋𝐴𝑋))
143grpoass 28275 . . . . . . . . . 10 ((𝐺 ∈ GrpOp ∧ (((inv‘𝐺)‘𝐶) ∈ 𝑋𝐶𝑋𝐴𝑋)) → ((((inv‘𝐺)‘𝐶)𝐺𝐶)𝐺𝐴) = (((inv‘𝐺)‘𝐶)𝐺(𝐶𝐺𝐴)))
1513, 14syldan 594 . . . . . . . . 9 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐶𝑋)) → ((((inv‘𝐺)‘𝐶)𝐺𝐶)𝐺𝐴) = (((inv‘𝐺)‘𝐶)𝐺(𝐶𝐺𝐴)))
1615anassrs 471 . . . . . . . 8 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ 𝐶𝑋) → ((((inv‘𝐺)‘𝐶)𝐺𝐶)𝐺𝐴) = (((inv‘𝐺)‘𝐶)𝐺(𝐶𝐺𝐴)))
173, 4grpolid 28288 . . . . . . . . 9 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → ((GId‘𝐺)𝐺𝐴) = 𝐴)
1817adantr 484 . . . . . . . 8 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ 𝐶𝑋) → ((GId‘𝐺)𝐺𝐴) = 𝐴)
198, 16, 183eqtr3d 2867 . . . . . . 7 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ 𝐶𝑋) → (((inv‘𝐺)‘𝐶)𝐺(𝐶𝐺𝐴)) = 𝐴)
2019adantrl 715 . . . . . 6 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ (𝐵𝑋𝐶𝑋)) → (((inv‘𝐺)‘𝐶)𝐺(𝐶𝐺𝐴)) = 𝐴)
2120adantr 484 . . . . 5 ((((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ (𝐵𝑋𝐶𝑋)) ∧ (𝐶𝐺𝐴) = (𝐶𝐺𝐵)) → (((inv‘𝐺)‘𝐶)𝐺(𝐶𝐺𝐴)) = 𝐴)
226adantrl 715 . . . . . . . . 9 ((𝐺 ∈ GrpOp ∧ (𝐵𝑋𝐶𝑋)) → (((inv‘𝐺)‘𝐶)𝐺𝐶) = (GId‘𝐺))
2322oveq1d 7153 . . . . . . . 8 ((𝐺 ∈ GrpOp ∧ (𝐵𝑋𝐶𝑋)) → ((((inv‘𝐺)‘𝐶)𝐺𝐶)𝐺𝐵) = ((GId‘𝐺)𝐺𝐵))
249adantrl 715 . . . . . . . . . 10 ((𝐺 ∈ GrpOp ∧ (𝐵𝑋𝐶𝑋)) → ((inv‘𝐺)‘𝐶) ∈ 𝑋)
25 simprr 772 . . . . . . . . . 10 ((𝐺 ∈ GrpOp ∧ (𝐵𝑋𝐶𝑋)) → 𝐶𝑋)
26 simprl 770 . . . . . . . . . 10 ((𝐺 ∈ GrpOp ∧ (𝐵𝑋𝐶𝑋)) → 𝐵𝑋)
2724, 25, 263jca 1125 . . . . . . . . 9 ((𝐺 ∈ GrpOp ∧ (𝐵𝑋𝐶𝑋)) → (((inv‘𝐺)‘𝐶) ∈ 𝑋𝐶𝑋𝐵𝑋))
283grpoass 28275 . . . . . . . . 9 ((𝐺 ∈ GrpOp ∧ (((inv‘𝐺)‘𝐶) ∈ 𝑋𝐶𝑋𝐵𝑋)) → ((((inv‘𝐺)‘𝐶)𝐺𝐶)𝐺𝐵) = (((inv‘𝐺)‘𝐶)𝐺(𝐶𝐺𝐵)))
2927, 28syldan 594 . . . . . . . 8 ((𝐺 ∈ GrpOp ∧ (𝐵𝑋𝐶𝑋)) → ((((inv‘𝐺)‘𝐶)𝐺𝐶)𝐺𝐵) = (((inv‘𝐺)‘𝐶)𝐺(𝐶𝐺𝐵)))
303, 4grpolid 28288 . . . . . . . . 9 ((𝐺 ∈ GrpOp ∧ 𝐵𝑋) → ((GId‘𝐺)𝐺𝐵) = 𝐵)
3130adantrr 716 . . . . . . . 8 ((𝐺 ∈ GrpOp ∧ (𝐵𝑋𝐶𝑋)) → ((GId‘𝐺)𝐺𝐵) = 𝐵)
3223, 29, 313eqtr3d 2867 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ (𝐵𝑋𝐶𝑋)) → (((inv‘𝐺)‘𝐶)𝐺(𝐶𝐺𝐵)) = 𝐵)
3332adantlr 714 . . . . . 6 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ (𝐵𝑋𝐶𝑋)) → (((inv‘𝐺)‘𝐶)𝐺(𝐶𝐺𝐵)) = 𝐵)
3433adantr 484 . . . . 5 ((((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ (𝐵𝑋𝐶𝑋)) ∧ (𝐶𝐺𝐴) = (𝐶𝐺𝐵)) → (((inv‘𝐺)‘𝐶)𝐺(𝐶𝐺𝐵)) = 𝐵)
352, 21, 343eqtr3d 2867 . . . 4 ((((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ (𝐵𝑋𝐶𝑋)) ∧ (𝐶𝐺𝐴) = (𝐶𝐺𝐵)) → 𝐴 = 𝐵)
3635exp53 451 . . 3 (𝐺 ∈ GrpOp → (𝐴𝑋 → (𝐵𝑋 → (𝐶𝑋 → ((𝐶𝐺𝐴) = (𝐶𝐺𝐵) → 𝐴 = 𝐵)))))
37363imp2 1346 . 2 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐶𝐺𝐴) = (𝐶𝐺𝐵) → 𝐴 = 𝐵))
38 oveq2 7146 . 2 (𝐴 = 𝐵 → (𝐶𝐺𝐴) = (𝐶𝐺𝐵))
3937, 38impbid1 228 1 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐶𝐺𝐴) = (𝐶𝐺𝐵) ↔ 𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2115  ran crn 5537  cfv 6336  (class class class)co 7138  GrpOpcgr 28261  GIdcgi 28262  invcgn 28263
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5171  ax-sep 5184  ax-nul 5191  ax-pr 5311  ax-un 7444
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3014  df-ral 3137  df-rex 3138  df-reu 3139  df-rab 3141  df-v 3481  df-sbc 3758  df-csb 3866  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4820  df-iun 4902  df-br 5048  df-opab 5110  df-mpt 5128  df-id 5441  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-riota 7096  df-ov 7141  df-grpo 28265  df-gid 28266  df-ginv 28267
This theorem is referenced by:  grpo2inv  28303  vclcan  28343  rngolcan  35256  rngolz  35260
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