MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  grpolcan Structured version   Visualization version   GIF version

Theorem grpolcan 29258
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 7358 . . . . . 6 ((𝐢𝐺𝐴) = (𝐢𝐺𝐡) β†’ (((invβ€˜πΊ)β€˜πΆ)𝐺(𝐢𝐺𝐴)) = (((invβ€˜πΊ)β€˜πΆ)𝐺(𝐢𝐺𝐡)))
21adantl 483 . . . . 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 2738 . . . . . . . . . . 11 (GIdβ€˜πΊ) = (GIdβ€˜πΊ)
5 eqid 2738 . . . . . . . . . . 11 (invβ€˜πΊ) = (invβ€˜πΊ)
63, 4, 5grpolinv 29254 . . . . . . . . . 10 ((𝐺 ∈ GrpOp ∧ 𝐢 ∈ 𝑋) β†’ (((invβ€˜πΊ)β€˜πΆ)𝐺𝐢) = (GIdβ€˜πΊ))
76adantlr 714 . . . . . . . . 9 (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ 𝐢 ∈ 𝑋) β†’ (((invβ€˜πΊ)β€˜πΆ)𝐺𝐢) = (GIdβ€˜πΊ))
87oveq1d 7365 . . . . . . . 8 (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ 𝐢 ∈ 𝑋) β†’ ((((invβ€˜πΊ)β€˜πΆ)𝐺𝐢)𝐺𝐴) = ((GIdβ€˜πΊ)𝐺𝐴))
93, 5grpoinvcl 29252 . . . . . . . . . . . 12 ((𝐺 ∈ GrpOp ∧ 𝐢 ∈ 𝑋) β†’ ((invβ€˜πΊ)β€˜πΆ) ∈ 𝑋)
109adantrl 715 . . . . . . . . . . 11 ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ ((invβ€˜πΊ)β€˜πΆ) ∈ 𝑋)
11 simprr 772 . . . . . . . . . . 11 ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ 𝐢 ∈ 𝑋)
12 simprl 770 . . . . . . . . . . 11 ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ 𝐴 ∈ 𝑋)
1310, 11, 123jca 1129 . . . . . . . . . 10 ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ (((invβ€˜πΊ)β€˜πΆ) ∈ 𝑋 ∧ 𝐢 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋))
143grpoass 29231 . . . . . . . . . 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 592 . . . . . . . . 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 469 . . . . . . . 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 29244 . . . . . . . . 9 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) β†’ ((GIdβ€˜πΊ)𝐺𝐴) = 𝐴)
1817adantr 482 . . . . . . . 8 (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ 𝐢 ∈ 𝑋) β†’ ((GIdβ€˜πΊ)𝐺𝐴) = 𝐴)
198, 16, 183eqtr3d 2786 . . . . . . 7 (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ 𝐢 ∈ 𝑋) β†’ (((invβ€˜πΊ)β€˜πΆ)𝐺(𝐢𝐺𝐴)) = 𝐴)
2019adantrl 715 . . . . . 6 (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ (𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ (((invβ€˜πΊ)β€˜πΆ)𝐺(𝐢𝐺𝐴)) = 𝐴)
2120adantr 482 . . . . 5 ((((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ (𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) ∧ (𝐢𝐺𝐴) = (𝐢𝐺𝐡)) β†’ (((invβ€˜πΊ)β€˜πΆ)𝐺(𝐢𝐺𝐴)) = 𝐴)
226adantrl 715 . . . . . . . . 9 ((𝐺 ∈ GrpOp ∧ (𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ (((invβ€˜πΊ)β€˜πΆ)𝐺𝐢) = (GIdβ€˜πΊ))
2322oveq1d 7365 . . . . . . . 8 ((𝐺 ∈ GrpOp ∧ (𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ ((((invβ€˜πΊ)β€˜πΆ)𝐺𝐢)𝐺𝐡) = ((GIdβ€˜πΊ)𝐺𝐡))
249adantrl 715 . . . . . . . . . 10 ((𝐺 ∈ GrpOp ∧ (𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ ((invβ€˜πΊ)β€˜πΆ) ∈ 𝑋)
25 simprr 772 . . . . . . . . . 10 ((𝐺 ∈ GrpOp ∧ (𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ 𝐢 ∈ 𝑋)
26 simprl 770 . . . . . . . . . 10 ((𝐺 ∈ GrpOp ∧ (𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ 𝐡 ∈ 𝑋)
2724, 25, 263jca 1129 . . . . . . . . 9 ((𝐺 ∈ GrpOp ∧ (𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ (((invβ€˜πΊ)β€˜πΆ) ∈ 𝑋 ∧ 𝐢 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋))
283grpoass 29231 . . . . . . . . 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 592 . . . . . . . 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 29244 . . . . . . . . 9 ((𝐺 ∈ GrpOp ∧ 𝐡 ∈ 𝑋) β†’ ((GIdβ€˜πΊ)𝐺𝐡) = 𝐡)
3130adantrr 716 . . . . . . . 8 ((𝐺 ∈ GrpOp ∧ (𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ ((GIdβ€˜πΊ)𝐺𝐡) = 𝐡)
3223, 29, 313eqtr3d 2786 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ (𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ (((invβ€˜πΊ)β€˜πΆ)𝐺(𝐢𝐺𝐡)) = 𝐡)
3332adantlr 714 . . . . . 6 (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ (𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ (((invβ€˜πΊ)β€˜πΆ)𝐺(𝐢𝐺𝐡)) = 𝐡)
3433adantr 482 . . . . 5 ((((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ (𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) ∧ (𝐢𝐺𝐴) = (𝐢𝐺𝐡)) β†’ (((invβ€˜πΊ)β€˜πΆ)𝐺(𝐢𝐺𝐡)) = 𝐡)
352, 21, 343eqtr3d 2786 . . . 4 ((((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ (𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) ∧ (𝐢𝐺𝐴) = (𝐢𝐺𝐡)) β†’ 𝐴 = 𝐡)
3635exp53 449 . . 3 (𝐺 ∈ GrpOp β†’ (𝐴 ∈ 𝑋 β†’ (𝐡 ∈ 𝑋 β†’ (𝐢 ∈ 𝑋 β†’ ((𝐢𝐺𝐴) = (𝐢𝐺𝐡) β†’ 𝐴 = 𝐡)))))
37363imp2 1350 . 2 ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ ((𝐢𝐺𝐴) = (𝐢𝐺𝐡) β†’ 𝐴 = 𝐡))
38 oveq2 7358 . 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  ran crn 5632  β€˜cfv 6492  (class class class)co 7350  GrpOpcgr 29217  GIdcgi 29218  invcgn 29219
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2709  ax-rep 5241  ax-sep 5255  ax-nul 5262  ax-pr 5383  ax-un 7663
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2888  df-ne 2943  df-ral 3064  df-rex 3073  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3739  df-csb 3855  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-iun 4955  df-br 5105  df-opab 5167  df-mpt 5188  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6444  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7306  df-ov 7353  df-grpo 29221  df-gid 29222  df-ginv 29223
This theorem is referenced by:  grpo2inv  29259  vclcan  29299  rngolcan  36263  rngolz  36267
  Copyright terms: Public domain W3C validator