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Theorem grpolcan 28234
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 7153 . . . . . 6 ((𝐶𝐺𝐴) = (𝐶𝐺𝐵) → (((inv‘𝐺)‘𝐶)𝐺(𝐶𝐺𝐴)) = (((inv‘𝐺)‘𝐶)𝐺(𝐶𝐺𝐵)))
21adantl 482 . . . . 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 2818 . . . . . . . . . . 11 (GId‘𝐺) = (GId‘𝐺)
5 eqid 2818 . . . . . . . . . . 11 (inv‘𝐺) = (inv‘𝐺)
63, 4, 5grpolinv 28230 . . . . . . . . . 10 ((𝐺 ∈ GrpOp ∧ 𝐶𝑋) → (((inv‘𝐺)‘𝐶)𝐺𝐶) = (GId‘𝐺))
76adantlr 711 . . . . . . . . 9 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ 𝐶𝑋) → (((inv‘𝐺)‘𝐶)𝐺𝐶) = (GId‘𝐺))
87oveq1d 7160 . . . . . . . 8 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ 𝐶𝑋) → ((((inv‘𝐺)‘𝐶)𝐺𝐶)𝐺𝐴) = ((GId‘𝐺)𝐺𝐴))
93, 5grpoinvcl 28228 . . . . . . . . . . . 12 ((𝐺 ∈ GrpOp ∧ 𝐶𝑋) → ((inv‘𝐺)‘𝐶) ∈ 𝑋)
109adantrl 712 . . . . . . . . . . 11 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐶𝑋)) → ((inv‘𝐺)‘𝐶) ∈ 𝑋)
11 simprr 769 . . . . . . . . . . 11 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐶𝑋)) → 𝐶𝑋)
12 simprl 767 . . . . . . . . . . 11 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐶𝑋)) → 𝐴𝑋)
1310, 11, 123jca 1120 . . . . . . . . . 10 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐶𝑋)) → (((inv‘𝐺)‘𝐶) ∈ 𝑋𝐶𝑋𝐴𝑋))
143grpoass 28207 . . . . . . . . . 10 ((𝐺 ∈ GrpOp ∧ (((inv‘𝐺)‘𝐶) ∈ 𝑋𝐶𝑋𝐴𝑋)) → ((((inv‘𝐺)‘𝐶)𝐺𝐶)𝐺𝐴) = (((inv‘𝐺)‘𝐶)𝐺(𝐶𝐺𝐴)))
1513, 14syldan 591 . . . . . . . . 9 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐶𝑋)) → ((((inv‘𝐺)‘𝐶)𝐺𝐶)𝐺𝐴) = (((inv‘𝐺)‘𝐶)𝐺(𝐶𝐺𝐴)))
1615anassrs 468 . . . . . . . 8 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ 𝐶𝑋) → ((((inv‘𝐺)‘𝐶)𝐺𝐶)𝐺𝐴) = (((inv‘𝐺)‘𝐶)𝐺(𝐶𝐺𝐴)))
173, 4grpolid 28220 . . . . . . . . 9 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → ((GId‘𝐺)𝐺𝐴) = 𝐴)
1817adantr 481 . . . . . . . 8 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ 𝐶𝑋) → ((GId‘𝐺)𝐺𝐴) = 𝐴)
198, 16, 183eqtr3d 2861 . . . . . . 7 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ 𝐶𝑋) → (((inv‘𝐺)‘𝐶)𝐺(𝐶𝐺𝐴)) = 𝐴)
2019adantrl 712 . . . . . 6 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ (𝐵𝑋𝐶𝑋)) → (((inv‘𝐺)‘𝐶)𝐺(𝐶𝐺𝐴)) = 𝐴)
2120adantr 481 . . . . 5 ((((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ (𝐵𝑋𝐶𝑋)) ∧ (𝐶𝐺𝐴) = (𝐶𝐺𝐵)) → (((inv‘𝐺)‘𝐶)𝐺(𝐶𝐺𝐴)) = 𝐴)
226adantrl 712 . . . . . . . . 9 ((𝐺 ∈ GrpOp ∧ (𝐵𝑋𝐶𝑋)) → (((inv‘𝐺)‘𝐶)𝐺𝐶) = (GId‘𝐺))
2322oveq1d 7160 . . . . . . . 8 ((𝐺 ∈ GrpOp ∧ (𝐵𝑋𝐶𝑋)) → ((((inv‘𝐺)‘𝐶)𝐺𝐶)𝐺𝐵) = ((GId‘𝐺)𝐺𝐵))
249adantrl 712 . . . . . . . . . 10 ((𝐺 ∈ GrpOp ∧ (𝐵𝑋𝐶𝑋)) → ((inv‘𝐺)‘𝐶) ∈ 𝑋)
25 simprr 769 . . . . . . . . . 10 ((𝐺 ∈ GrpOp ∧ (𝐵𝑋𝐶𝑋)) → 𝐶𝑋)
26 simprl 767 . . . . . . . . . 10 ((𝐺 ∈ GrpOp ∧ (𝐵𝑋𝐶𝑋)) → 𝐵𝑋)
2724, 25, 263jca 1120 . . . . . . . . 9 ((𝐺 ∈ GrpOp ∧ (𝐵𝑋𝐶𝑋)) → (((inv‘𝐺)‘𝐶) ∈ 𝑋𝐶𝑋𝐵𝑋))
283grpoass 28207 . . . . . . . . 9 ((𝐺 ∈ GrpOp ∧ (((inv‘𝐺)‘𝐶) ∈ 𝑋𝐶𝑋𝐵𝑋)) → ((((inv‘𝐺)‘𝐶)𝐺𝐶)𝐺𝐵) = (((inv‘𝐺)‘𝐶)𝐺(𝐶𝐺𝐵)))
2927, 28syldan 591 . . . . . . . 8 ((𝐺 ∈ GrpOp ∧ (𝐵𝑋𝐶𝑋)) → ((((inv‘𝐺)‘𝐶)𝐺𝐶)𝐺𝐵) = (((inv‘𝐺)‘𝐶)𝐺(𝐶𝐺𝐵)))
303, 4grpolid 28220 . . . . . . . . 9 ((𝐺 ∈ GrpOp ∧ 𝐵𝑋) → ((GId‘𝐺)𝐺𝐵) = 𝐵)
3130adantrr 713 . . . . . . . 8 ((𝐺 ∈ GrpOp ∧ (𝐵𝑋𝐶𝑋)) → ((GId‘𝐺)𝐺𝐵) = 𝐵)
3223, 29, 313eqtr3d 2861 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ (𝐵𝑋𝐶𝑋)) → (((inv‘𝐺)‘𝐶)𝐺(𝐶𝐺𝐵)) = 𝐵)
3332adantlr 711 . . . . . 6 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ (𝐵𝑋𝐶𝑋)) → (((inv‘𝐺)‘𝐶)𝐺(𝐶𝐺𝐵)) = 𝐵)
3433adantr 481 . . . . 5 ((((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ (𝐵𝑋𝐶𝑋)) ∧ (𝐶𝐺𝐴) = (𝐶𝐺𝐵)) → (((inv‘𝐺)‘𝐶)𝐺(𝐶𝐺𝐵)) = 𝐵)
352, 21, 343eqtr3d 2861 . . . 4 ((((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ (𝐵𝑋𝐶𝑋)) ∧ (𝐶𝐺𝐴) = (𝐶𝐺𝐵)) → 𝐴 = 𝐵)
3635exp53 448 . . 3 (𝐺 ∈ GrpOp → (𝐴𝑋 → (𝐵𝑋 → (𝐶𝑋 → ((𝐶𝐺𝐴) = (𝐶𝐺𝐵) → 𝐴 = 𝐵)))))
37363imp2 1341 . 2 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐶𝐺𝐴) = (𝐶𝐺𝐵) → 𝐴 = 𝐵))
38 oveq2 7153 . 2 (𝐴 = 𝐵 → (𝐶𝐺𝐴) = (𝐶𝐺𝐵))
3937, 38impbid1 226 1 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐶𝐺𝐴) = (𝐶𝐺𝐵) ↔ 𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  ran crn 5549  cfv 6348  (class class class)co 7145  GrpOpcgr 28193  GIdcgi 28194  invcgn 28195
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-grpo 28197  df-gid 28198  df-ginv 28199
This theorem is referenced by:  grpo2inv  28235  vclcan  28275  rngolcan  35077  rngolz  35081
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