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Theorem grpolcan 30559
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 7439 . . . . . 6 ((𝐶𝐺𝐴) = (𝐶𝐺𝐵) → (((inv‘𝐺)‘𝐶)𝐺(𝐶𝐺𝐴)) = (((inv‘𝐺)‘𝐶)𝐺(𝐶𝐺𝐵)))
21adantl 481 . . . . 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 2735 . . . . . . . . . . 11 (GId‘𝐺) = (GId‘𝐺)
5 eqid 2735 . . . . . . . . . . 11 (inv‘𝐺) = (inv‘𝐺)
63, 4, 5grpolinv 30555 . . . . . . . . . 10 ((𝐺 ∈ GrpOp ∧ 𝐶𝑋) → (((inv‘𝐺)‘𝐶)𝐺𝐶) = (GId‘𝐺))
76adantlr 715 . . . . . . . . 9 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ 𝐶𝑋) → (((inv‘𝐺)‘𝐶)𝐺𝐶) = (GId‘𝐺))
87oveq1d 7446 . . . . . . . 8 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ 𝐶𝑋) → ((((inv‘𝐺)‘𝐶)𝐺𝐶)𝐺𝐴) = ((GId‘𝐺)𝐺𝐴))
93, 5grpoinvcl 30553 . . . . . . . . . . . 12 ((𝐺 ∈ GrpOp ∧ 𝐶𝑋) → ((inv‘𝐺)‘𝐶) ∈ 𝑋)
109adantrl 716 . . . . . . . . . . 11 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐶𝑋)) → ((inv‘𝐺)‘𝐶) ∈ 𝑋)
11 simprr 773 . . . . . . . . . . 11 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐶𝑋)) → 𝐶𝑋)
12 simprl 771 . . . . . . . . . . 11 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐶𝑋)) → 𝐴𝑋)
1310, 11, 123jca 1127 . . . . . . . . . 10 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐶𝑋)) → (((inv‘𝐺)‘𝐶) ∈ 𝑋𝐶𝑋𝐴𝑋))
143grpoass 30532 . . . . . . . . . 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 467 . . . . . . . 8 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ 𝐶𝑋) → ((((inv‘𝐺)‘𝐶)𝐺𝐶)𝐺𝐴) = (((inv‘𝐺)‘𝐶)𝐺(𝐶𝐺𝐴)))
173, 4grpolid 30545 . . . . . . . . 9 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → ((GId‘𝐺)𝐺𝐴) = 𝐴)
1817adantr 480 . . . . . . . 8 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ 𝐶𝑋) → ((GId‘𝐺)𝐺𝐴) = 𝐴)
198, 16, 183eqtr3d 2783 . . . . . . 7 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ 𝐶𝑋) → (((inv‘𝐺)‘𝐶)𝐺(𝐶𝐺𝐴)) = 𝐴)
2019adantrl 716 . . . . . 6 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ (𝐵𝑋𝐶𝑋)) → (((inv‘𝐺)‘𝐶)𝐺(𝐶𝐺𝐴)) = 𝐴)
2120adantr 480 . . . . 5 ((((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ (𝐵𝑋𝐶𝑋)) ∧ (𝐶𝐺𝐴) = (𝐶𝐺𝐵)) → (((inv‘𝐺)‘𝐶)𝐺(𝐶𝐺𝐴)) = 𝐴)
226adantrl 716 . . . . . . . . 9 ((𝐺 ∈ GrpOp ∧ (𝐵𝑋𝐶𝑋)) → (((inv‘𝐺)‘𝐶)𝐺𝐶) = (GId‘𝐺))
2322oveq1d 7446 . . . . . . . 8 ((𝐺 ∈ GrpOp ∧ (𝐵𝑋𝐶𝑋)) → ((((inv‘𝐺)‘𝐶)𝐺𝐶)𝐺𝐵) = ((GId‘𝐺)𝐺𝐵))
249adantrl 716 . . . . . . . . . 10 ((𝐺 ∈ GrpOp ∧ (𝐵𝑋𝐶𝑋)) → ((inv‘𝐺)‘𝐶) ∈ 𝑋)
25 simprr 773 . . . . . . . . . 10 ((𝐺 ∈ GrpOp ∧ (𝐵𝑋𝐶𝑋)) → 𝐶𝑋)
26 simprl 771 . . . . . . . . . 10 ((𝐺 ∈ GrpOp ∧ (𝐵𝑋𝐶𝑋)) → 𝐵𝑋)
2724, 25, 263jca 1127 . . . . . . . . 9 ((𝐺 ∈ GrpOp ∧ (𝐵𝑋𝐶𝑋)) → (((inv‘𝐺)‘𝐶) ∈ 𝑋𝐶𝑋𝐵𝑋))
283grpoass 30532 . . . . . . . . 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 30545 . . . . . . . . 9 ((𝐺 ∈ GrpOp ∧ 𝐵𝑋) → ((GId‘𝐺)𝐺𝐵) = 𝐵)
3130adantrr 717 . . . . . . . 8 ((𝐺 ∈ GrpOp ∧ (𝐵𝑋𝐶𝑋)) → ((GId‘𝐺)𝐺𝐵) = 𝐵)
3223, 29, 313eqtr3d 2783 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ (𝐵𝑋𝐶𝑋)) → (((inv‘𝐺)‘𝐶)𝐺(𝐶𝐺𝐵)) = 𝐵)
3332adantlr 715 . . . . . 6 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ (𝐵𝑋𝐶𝑋)) → (((inv‘𝐺)‘𝐶)𝐺(𝐶𝐺𝐵)) = 𝐵)
3433adantr 480 . . . . 5 ((((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ (𝐵𝑋𝐶𝑋)) ∧ (𝐶𝐺𝐴) = (𝐶𝐺𝐵)) → (((inv‘𝐺)‘𝐶)𝐺(𝐶𝐺𝐵)) = 𝐵)
352, 21, 343eqtr3d 2783 . . . 4 ((((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ (𝐵𝑋𝐶𝑋)) ∧ (𝐶𝐺𝐴) = (𝐶𝐺𝐵)) → 𝐴 = 𝐵)
3635exp53 447 . . 3 (𝐺 ∈ GrpOp → (𝐴𝑋 → (𝐵𝑋 → (𝐶𝑋 → ((𝐶𝐺𝐴) = (𝐶𝐺𝐵) → 𝐴 = 𝐵)))))
37363imp2 1348 . 2 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐶𝐺𝐴) = (𝐶𝐺𝐵) → 𝐴 = 𝐵))
38 oveq2 7439 . 2 (𝐴 = 𝐵 → (𝐶𝐺𝐴) = (𝐶𝐺𝐵))
3937, 38impbid1 225 1 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐶𝐺𝐴) = (𝐶𝐺𝐵) ↔ 𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  ran crn 5690  cfv 6563  (class class class)co 7431  GrpOpcgr 30518  GIdcgi 30519  invcgn 30520
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-grpo 30522  df-gid 30523  df-ginv 30524
This theorem is referenced by:  grpo2inv  30560  vclcan  30600  rngolcan  37905  rngolz  37909
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