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Theorem grpolcan 27957
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 6930 . . . . . 6 ((𝐶𝐺𝐴) = (𝐶𝐺𝐵) → (((inv‘𝐺)‘𝐶)𝐺(𝐶𝐺𝐴)) = (((inv‘𝐺)‘𝐶)𝐺(𝐶𝐺𝐵)))
21adantl 475 . . . . 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 2778 . . . . . . . . . . 11 (GId‘𝐺) = (GId‘𝐺)
5 eqid 2778 . . . . . . . . . . 11 (inv‘𝐺) = (inv‘𝐺)
63, 4, 5grpolinv 27953 . . . . . . . . . 10 ((𝐺 ∈ GrpOp ∧ 𝐶𝑋) → (((inv‘𝐺)‘𝐶)𝐺𝐶) = (GId‘𝐺))
76adantlr 705 . . . . . . . . 9 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ 𝐶𝑋) → (((inv‘𝐺)‘𝐶)𝐺𝐶) = (GId‘𝐺))
87oveq1d 6937 . . . . . . . 8 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ 𝐶𝑋) → ((((inv‘𝐺)‘𝐶)𝐺𝐶)𝐺𝐴) = ((GId‘𝐺)𝐺𝐴))
93, 5grpoinvcl 27951 . . . . . . . . . . . 12 ((𝐺 ∈ GrpOp ∧ 𝐶𝑋) → ((inv‘𝐺)‘𝐶) ∈ 𝑋)
109adantrl 706 . . . . . . . . . . 11 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐶𝑋)) → ((inv‘𝐺)‘𝐶) ∈ 𝑋)
11 simprr 763 . . . . . . . . . . 11 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐶𝑋)) → 𝐶𝑋)
12 simprl 761 . . . . . . . . . . 11 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐶𝑋)) → 𝐴𝑋)
1310, 11, 123jca 1119 . . . . . . . . . 10 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐶𝑋)) → (((inv‘𝐺)‘𝐶) ∈ 𝑋𝐶𝑋𝐴𝑋))
143grpoass 27930 . . . . . . . . . 10 ((𝐺 ∈ GrpOp ∧ (((inv‘𝐺)‘𝐶) ∈ 𝑋𝐶𝑋𝐴𝑋)) → ((((inv‘𝐺)‘𝐶)𝐺𝐶)𝐺𝐴) = (((inv‘𝐺)‘𝐶)𝐺(𝐶𝐺𝐴)))
1513, 14syldan 585 . . . . . . . . 9 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐶𝑋)) → ((((inv‘𝐺)‘𝐶)𝐺𝐶)𝐺𝐴) = (((inv‘𝐺)‘𝐶)𝐺(𝐶𝐺𝐴)))
1615anassrs 461 . . . . . . . 8 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ 𝐶𝑋) → ((((inv‘𝐺)‘𝐶)𝐺𝐶)𝐺𝐴) = (((inv‘𝐺)‘𝐶)𝐺(𝐶𝐺𝐴)))
173, 4grpolid 27943 . . . . . . . . 9 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → ((GId‘𝐺)𝐺𝐴) = 𝐴)
1817adantr 474 . . . . . . . 8 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ 𝐶𝑋) → ((GId‘𝐺)𝐺𝐴) = 𝐴)
198, 16, 183eqtr3d 2822 . . . . . . 7 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ 𝐶𝑋) → (((inv‘𝐺)‘𝐶)𝐺(𝐶𝐺𝐴)) = 𝐴)
2019adantrl 706 . . . . . 6 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ (𝐵𝑋𝐶𝑋)) → (((inv‘𝐺)‘𝐶)𝐺(𝐶𝐺𝐴)) = 𝐴)
2120adantr 474 . . . . 5 ((((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ (𝐵𝑋𝐶𝑋)) ∧ (𝐶𝐺𝐴) = (𝐶𝐺𝐵)) → (((inv‘𝐺)‘𝐶)𝐺(𝐶𝐺𝐴)) = 𝐴)
226adantrl 706 . . . . . . . . 9 ((𝐺 ∈ GrpOp ∧ (𝐵𝑋𝐶𝑋)) → (((inv‘𝐺)‘𝐶)𝐺𝐶) = (GId‘𝐺))
2322oveq1d 6937 . . . . . . . 8 ((𝐺 ∈ GrpOp ∧ (𝐵𝑋𝐶𝑋)) → ((((inv‘𝐺)‘𝐶)𝐺𝐶)𝐺𝐵) = ((GId‘𝐺)𝐺𝐵))
249adantrl 706 . . . . . . . . . 10 ((𝐺 ∈ GrpOp ∧ (𝐵𝑋𝐶𝑋)) → ((inv‘𝐺)‘𝐶) ∈ 𝑋)
25 simprr 763 . . . . . . . . . 10 ((𝐺 ∈ GrpOp ∧ (𝐵𝑋𝐶𝑋)) → 𝐶𝑋)
26 simprl 761 . . . . . . . . . 10 ((𝐺 ∈ GrpOp ∧ (𝐵𝑋𝐶𝑋)) → 𝐵𝑋)
2724, 25, 263jca 1119 . . . . . . . . 9 ((𝐺 ∈ GrpOp ∧ (𝐵𝑋𝐶𝑋)) → (((inv‘𝐺)‘𝐶) ∈ 𝑋𝐶𝑋𝐵𝑋))
283grpoass 27930 . . . . . . . . 9 ((𝐺 ∈ GrpOp ∧ (((inv‘𝐺)‘𝐶) ∈ 𝑋𝐶𝑋𝐵𝑋)) → ((((inv‘𝐺)‘𝐶)𝐺𝐶)𝐺𝐵) = (((inv‘𝐺)‘𝐶)𝐺(𝐶𝐺𝐵)))
2927, 28syldan 585 . . . . . . . 8 ((𝐺 ∈ GrpOp ∧ (𝐵𝑋𝐶𝑋)) → ((((inv‘𝐺)‘𝐶)𝐺𝐶)𝐺𝐵) = (((inv‘𝐺)‘𝐶)𝐺(𝐶𝐺𝐵)))
303, 4grpolid 27943 . . . . . . . . 9 ((𝐺 ∈ GrpOp ∧ 𝐵𝑋) → ((GId‘𝐺)𝐺𝐵) = 𝐵)
3130adantrr 707 . . . . . . . 8 ((𝐺 ∈ GrpOp ∧ (𝐵𝑋𝐶𝑋)) → ((GId‘𝐺)𝐺𝐵) = 𝐵)
3223, 29, 313eqtr3d 2822 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ (𝐵𝑋𝐶𝑋)) → (((inv‘𝐺)‘𝐶)𝐺(𝐶𝐺𝐵)) = 𝐵)
3332adantlr 705 . . . . . 6 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ (𝐵𝑋𝐶𝑋)) → (((inv‘𝐺)‘𝐶)𝐺(𝐶𝐺𝐵)) = 𝐵)
3433adantr 474 . . . . 5 ((((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ (𝐵𝑋𝐶𝑋)) ∧ (𝐶𝐺𝐴) = (𝐶𝐺𝐵)) → (((inv‘𝐺)‘𝐶)𝐺(𝐶𝐺𝐵)) = 𝐵)
352, 21, 343eqtr3d 2822 . . . 4 ((((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ (𝐵𝑋𝐶𝑋)) ∧ (𝐶𝐺𝐴) = (𝐶𝐺𝐵)) → 𝐴 = 𝐵)
3635exp53 440 . . 3 (𝐺 ∈ GrpOp → (𝐴𝑋 → (𝐵𝑋 → (𝐶𝑋 → ((𝐶𝐺𝐴) = (𝐶𝐺𝐵) → 𝐴 = 𝐵)))))
37363imp2 1411 . 2 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐶𝐺𝐴) = (𝐶𝐺𝐵) → 𝐴 = 𝐵))
38 oveq2 6930 . 2 (𝐴 = 𝐵 → (𝐶𝐺𝐴) = (𝐶𝐺𝐵))
3937, 38impbid1 217 1 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐶𝐺𝐴) = (𝐶𝐺𝐵) ↔ 𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386  w3a 1071   = wceq 1601  wcel 2107  ran crn 5356  cfv 6135  (class class class)co 6922  GrpOpcgr 27916  GIdcgi 27917  invcgn 27918
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pr 5138  ax-un 7226
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-riota 6883  df-ov 6925  df-grpo 27920  df-gid 27921  df-ginv 27922
This theorem is referenced by:  grpo2inv  27958  vclcan  27998  rngolcan  34341  rngolz  34345
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