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Theorem grpolcan 30549
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 2737 . . . . . . . . . . 11 (GId‘𝐺) = (GId‘𝐺)
5 eqid 2737 . . . . . . . . . . 11 (inv‘𝐺) = (inv‘𝐺)
63, 4, 5grpolinv 30545 . . . . . . . . . 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 30543 . . . . . . . . . . . 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 1129 . . . . . . . . . 10 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐶𝑋)) → (((inv‘𝐺)‘𝐶) ∈ 𝑋𝐶𝑋𝐴𝑋))
143grpoass 30522 . . . . . . . . . 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 30535 . . . . . . . . 9 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → ((GId‘𝐺)𝐺𝐴) = 𝐴)
1817adantr 480 . . . . . . . 8 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ 𝐶𝑋) → ((GId‘𝐺)𝐺𝐴) = 𝐴)
198, 16, 183eqtr3d 2785 . . . . . . 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 1129 . . . . . . . . 9 ((𝐺 ∈ GrpOp ∧ (𝐵𝑋𝐶𝑋)) → (((inv‘𝐺)‘𝐶) ∈ 𝑋𝐶𝑋𝐵𝑋))
283grpoass 30522 . . . . . . . . 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 30535 . . . . . . . . 9 ((𝐺 ∈ GrpOp ∧ 𝐵𝑋) → ((GId‘𝐺)𝐺𝐵) = 𝐵)
3130adantrr 717 . . . . . . . 8 ((𝐺 ∈ GrpOp ∧ (𝐵𝑋𝐶𝑋)) → ((GId‘𝐺)𝐺𝐵) = 𝐵)
3223, 29, 313eqtr3d 2785 . . . . . . 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 2785 . . . 4 ((((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ (𝐵𝑋𝐶𝑋)) ∧ (𝐶𝐺𝐴) = (𝐶𝐺𝐵)) → 𝐴 = 𝐵)
3635exp53 447 . . 3 (𝐺 ∈ GrpOp → (𝐴𝑋 → (𝐵𝑋 → (𝐶𝑋 → ((𝐶𝐺𝐴) = (𝐶𝐺𝐵) → 𝐴 = 𝐵)))))
37363imp2 1350 . 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 1087   = wceq 1540  wcel 2108  ran crn 5686  cfv 6561  (class class class)co 7431  GrpOpcgr 30508  GIdcgi 30509  invcgn 30510
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-grpo 30512  df-gid 30513  df-ginv 30514
This theorem is referenced by:  grpo2inv  30550  vclcan  30590  rngolcan  37925  rngolz  37929
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