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Theorem grpolcan 30292
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 ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ ((𝐢𝐺𝐴) = (𝐢𝐺𝐡) ↔ 𝐴 = 𝐡))

Proof of Theorem grpolcan
StepHypRef Expression
1 oveq2 7413 . . . . . 6 ((𝐢𝐺𝐴) = (𝐢𝐺𝐡) β†’ (((invβ€˜πΊ)β€˜πΆ)𝐺(𝐢𝐺𝐴)) = (((invβ€˜πΊ)β€˜πΆ)𝐺(𝐢𝐺𝐡)))
21adantl 481 . . . . 5 ((((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ (𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) ∧ (𝐢𝐺𝐴) = (𝐢𝐺𝐡)) β†’ (((invβ€˜πΊ)β€˜πΆ)𝐺(𝐢𝐺𝐴)) = (((invβ€˜πΊ)β€˜πΆ)𝐺(𝐢𝐺𝐡)))
3 grplcan.1 . . . . . . . . . . 11 𝑋 = ran 𝐺
4 eqid 2726 . . . . . . . . . . 11 (GIdβ€˜πΊ) = (GIdβ€˜πΊ)
5 eqid 2726 . . . . . . . . . . 11 (invβ€˜πΊ) = (invβ€˜πΊ)
63, 4, 5grpolinv 30288 . . . . . . . . . 10 ((𝐺 ∈ GrpOp ∧ 𝐢 ∈ 𝑋) β†’ (((invβ€˜πΊ)β€˜πΆ)𝐺𝐢) = (GIdβ€˜πΊ))
76adantlr 712 . . . . . . . . 9 (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ 𝐢 ∈ 𝑋) β†’ (((invβ€˜πΊ)β€˜πΆ)𝐺𝐢) = (GIdβ€˜πΊ))
87oveq1d 7420 . . . . . . . 8 (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ 𝐢 ∈ 𝑋) β†’ ((((invβ€˜πΊ)β€˜πΆ)𝐺𝐢)𝐺𝐴) = ((GIdβ€˜πΊ)𝐺𝐴))
93, 5grpoinvcl 30286 . . . . . . . . . . . 12 ((𝐺 ∈ GrpOp ∧ 𝐢 ∈ 𝑋) β†’ ((invβ€˜πΊ)β€˜πΆ) ∈ 𝑋)
109adantrl 713 . . . . . . . . . . 11 ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ ((invβ€˜πΊ)β€˜πΆ) ∈ 𝑋)
11 simprr 770 . . . . . . . . . . 11 ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ 𝐢 ∈ 𝑋)
12 simprl 768 . . . . . . . . . . 11 ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ 𝐴 ∈ 𝑋)
1310, 11, 123jca 1125 . . . . . . . . . 10 ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ (((invβ€˜πΊ)β€˜πΆ) ∈ 𝑋 ∧ 𝐢 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋))
143grpoass 30265 . . . . . . . . . 10 ((𝐺 ∈ GrpOp ∧ (((invβ€˜πΊ)β€˜πΆ) ∈ 𝑋 ∧ 𝐢 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) β†’ ((((invβ€˜πΊ)β€˜πΆ)𝐺𝐢)𝐺𝐴) = (((invβ€˜πΊ)β€˜πΆ)𝐺(𝐢𝐺𝐴)))
1513, 14syldan 590 . . . . . . . . 9 ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ ((((invβ€˜πΊ)β€˜πΆ)𝐺𝐢)𝐺𝐴) = (((invβ€˜πΊ)β€˜πΆ)𝐺(𝐢𝐺𝐴)))
1615anassrs 467 . . . . . . . 8 (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ 𝐢 ∈ 𝑋) β†’ ((((invβ€˜πΊ)β€˜πΆ)𝐺𝐢)𝐺𝐴) = (((invβ€˜πΊ)β€˜πΆ)𝐺(𝐢𝐺𝐴)))
173, 4grpolid 30278 . . . . . . . . 9 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) β†’ ((GIdβ€˜πΊ)𝐺𝐴) = 𝐴)
1817adantr 480 . . . . . . . 8 (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ 𝐢 ∈ 𝑋) β†’ ((GIdβ€˜πΊ)𝐺𝐴) = 𝐴)
198, 16, 183eqtr3d 2774 . . . . . . 7 (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ 𝐢 ∈ 𝑋) β†’ (((invβ€˜πΊ)β€˜πΆ)𝐺(𝐢𝐺𝐴)) = 𝐴)
2019adantrl 713 . . . . . 6 (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ (𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ (((invβ€˜πΊ)β€˜πΆ)𝐺(𝐢𝐺𝐴)) = 𝐴)
2120adantr 480 . . . . 5 ((((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ (𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) ∧ (𝐢𝐺𝐴) = (𝐢𝐺𝐡)) β†’ (((invβ€˜πΊ)β€˜πΆ)𝐺(𝐢𝐺𝐴)) = 𝐴)
226adantrl 713 . . . . . . . . 9 ((𝐺 ∈ GrpOp ∧ (𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ (((invβ€˜πΊ)β€˜πΆ)𝐺𝐢) = (GIdβ€˜πΊ))
2322oveq1d 7420 . . . . . . . 8 ((𝐺 ∈ GrpOp ∧ (𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ ((((invβ€˜πΊ)β€˜πΆ)𝐺𝐢)𝐺𝐡) = ((GIdβ€˜πΊ)𝐺𝐡))
249adantrl 713 . . . . . . . . . 10 ((𝐺 ∈ GrpOp ∧ (𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ ((invβ€˜πΊ)β€˜πΆ) ∈ 𝑋)
25 simprr 770 . . . . . . . . . 10 ((𝐺 ∈ GrpOp ∧ (𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ 𝐢 ∈ 𝑋)
26 simprl 768 . . . . . . . . . 10 ((𝐺 ∈ GrpOp ∧ (𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ 𝐡 ∈ 𝑋)
2724, 25, 263jca 1125 . . . . . . . . 9 ((𝐺 ∈ GrpOp ∧ (𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ (((invβ€˜πΊ)β€˜πΆ) ∈ 𝑋 ∧ 𝐢 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋))
283grpoass 30265 . . . . . . . . 9 ((𝐺 ∈ GrpOp ∧ (((invβ€˜πΊ)β€˜πΆ) ∈ 𝑋 ∧ 𝐢 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ ((((invβ€˜πΊ)β€˜πΆ)𝐺𝐢)𝐺𝐡) = (((invβ€˜πΊ)β€˜πΆ)𝐺(𝐢𝐺𝐡)))
2927, 28syldan 590 . . . . . . . 8 ((𝐺 ∈ GrpOp ∧ (𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ ((((invβ€˜πΊ)β€˜πΆ)𝐺𝐢)𝐺𝐡) = (((invβ€˜πΊ)β€˜πΆ)𝐺(𝐢𝐺𝐡)))
303, 4grpolid 30278 . . . . . . . . 9 ((𝐺 ∈ GrpOp ∧ 𝐡 ∈ 𝑋) β†’ ((GIdβ€˜πΊ)𝐺𝐡) = 𝐡)
3130adantrr 714 . . . . . . . 8 ((𝐺 ∈ GrpOp ∧ (𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ ((GIdβ€˜πΊ)𝐺𝐡) = 𝐡)
3223, 29, 313eqtr3d 2774 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ (𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ (((invβ€˜πΊ)β€˜πΆ)𝐺(𝐢𝐺𝐡)) = 𝐡)
3332adantlr 712 . . . . . 6 (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ (𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ (((invβ€˜πΊ)β€˜πΆ)𝐺(𝐢𝐺𝐡)) = 𝐡)
3433adantr 480 . . . . 5 ((((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ (𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) ∧ (𝐢𝐺𝐴) = (𝐢𝐺𝐡)) β†’ (((invβ€˜πΊ)β€˜πΆ)𝐺(𝐢𝐺𝐡)) = 𝐡)
352, 21, 343eqtr3d 2774 . . . 4 ((((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ (𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) ∧ (𝐢𝐺𝐴) = (𝐢𝐺𝐡)) β†’ 𝐴 = 𝐡)
3635exp53 447 . . 3 (𝐺 ∈ GrpOp β†’ (𝐴 ∈ 𝑋 β†’ (𝐡 ∈ 𝑋 β†’ (𝐢 ∈ 𝑋 β†’ ((𝐢𝐺𝐴) = (𝐢𝐺𝐡) β†’ 𝐴 = 𝐡)))))
37363imp2 1346 . 2 ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ ((𝐢𝐺𝐴) = (𝐢𝐺𝐡) β†’ 𝐴 = 𝐡))
38 oveq2 7413 . 2 (𝐴 = 𝐡 β†’ (𝐢𝐺𝐴) = (𝐢𝐺𝐡))
3937, 38impbid1 224 1 ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ ((𝐢𝐺𝐴) = (𝐢𝐺𝐡) ↔ 𝐴 = 𝐡))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  ran crn 5670  β€˜cfv 6537  (class class class)co 7405  GrpOpcgr 30251  GIdcgi 30252  invcgn 30253
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-ov 7408  df-grpo 30255  df-gid 30256  df-ginv 30257
This theorem is referenced by:  grpo2inv  30293  vclcan  30333  rngolcan  37299  rngolz  37303
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