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Theorem grpoinv 29809
Description: The properties of a group element's inverse. (Contributed by NM, 27-Oct-2006.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
grpinv.1 𝑋 = ran 𝐺
grpinv.2 π‘ˆ = (GIdβ€˜πΊ)
grpinv.3 𝑁 = (invβ€˜πΊ)
Assertion
Ref Expression
grpoinv ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) β†’ (((π‘β€˜π΄)𝐺𝐴) = π‘ˆ ∧ (𝐴𝐺(π‘β€˜π΄)) = π‘ˆ))
Proof of Theorem grpoinv
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 grpinv.1 . . . . . 6 𝑋 = ran 𝐺
2 grpinv.2 . . . . . 6 π‘ˆ = (GIdβ€˜πΊ)
3 grpinv.3 . . . . . 6 𝑁 = (invβ€˜πΊ)
41, 2, 3grpoinvval 29807 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) β†’ (π‘β€˜π΄) = (℩𝑦 ∈ 𝑋 (𝑦𝐺𝐴) = π‘ˆ))
51, 2grpoinveu 29803 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) β†’ βˆƒ!𝑦 ∈ 𝑋 (𝑦𝐺𝐴) = π‘ˆ)
6 riotacl2 7382 . . . . . 6 (βˆƒ!𝑦 ∈ 𝑋 (𝑦𝐺𝐴) = π‘ˆ β†’ (℩𝑦 ∈ 𝑋 (𝑦𝐺𝐴) = π‘ˆ) ∈ {𝑦 ∈ 𝑋 ∣ (𝑦𝐺𝐴) = π‘ˆ})
75, 6syl 17 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) β†’ (℩𝑦 ∈ 𝑋 (𝑦𝐺𝐴) = π‘ˆ) ∈ {𝑦 ∈ 𝑋 ∣ (𝑦𝐺𝐴) = π‘ˆ})
84, 7eqeltrd 2834 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) β†’ (π‘β€˜π΄) ∈ {𝑦 ∈ 𝑋 ∣ (𝑦𝐺𝐴) = π‘ˆ})
9 simpl 484 . . . . . . . . 9 (((𝑦𝐺𝐴) = π‘ˆ ∧ (𝐴𝐺𝑦) = π‘ˆ) β†’ (𝑦𝐺𝐴) = π‘ˆ)
109rgenw 3066 . . . . . . . 8 βˆ€π‘¦ ∈ 𝑋 (((𝑦𝐺𝐴) = π‘ˆ ∧ (𝐴𝐺𝑦) = π‘ˆ) β†’ (𝑦𝐺𝐴) = π‘ˆ)
1110a1i 11 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) β†’ βˆ€π‘¦ ∈ 𝑋 (((𝑦𝐺𝐴) = π‘ˆ ∧ (𝐴𝐺𝑦) = π‘ˆ) β†’ (𝑦𝐺𝐴) = π‘ˆ))
121, 2grpoidinv2 29799 . . . . . . . 8 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) β†’ (((π‘ˆπΊπ΄) = 𝐴 ∧ (π΄πΊπ‘ˆ) = 𝐴) ∧ βˆƒπ‘¦ ∈ 𝑋 ((𝑦𝐺𝐴) = π‘ˆ ∧ (𝐴𝐺𝑦) = π‘ˆ)))
1312simprd 497 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) β†’ βˆƒπ‘¦ ∈ 𝑋 ((𝑦𝐺𝐴) = π‘ˆ ∧ (𝐴𝐺𝑦) = π‘ˆ))
1411, 13, 53jca 1129 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) β†’ (βˆ€π‘¦ ∈ 𝑋 (((𝑦𝐺𝐴) = π‘ˆ ∧ (𝐴𝐺𝑦) = π‘ˆ) β†’ (𝑦𝐺𝐴) = π‘ˆ) ∧ βˆƒπ‘¦ ∈ 𝑋 ((𝑦𝐺𝐴) = π‘ˆ ∧ (𝐴𝐺𝑦) = π‘ˆ) ∧ βˆƒ!𝑦 ∈ 𝑋 (𝑦𝐺𝐴) = π‘ˆ))
15 reupick2 4321 . . . . . 6 (((βˆ€π‘¦ ∈ 𝑋 (((𝑦𝐺𝐴) = π‘ˆ ∧ (𝐴𝐺𝑦) = π‘ˆ) β†’ (𝑦𝐺𝐴) = π‘ˆ) ∧ βˆƒπ‘¦ ∈ 𝑋 ((𝑦𝐺𝐴) = π‘ˆ ∧ (𝐴𝐺𝑦) = π‘ˆ) ∧ βˆƒ!𝑦 ∈ 𝑋 (𝑦𝐺𝐴) = π‘ˆ) ∧ 𝑦 ∈ 𝑋) β†’ ((𝑦𝐺𝐴) = π‘ˆ ↔ ((𝑦𝐺𝐴) = π‘ˆ ∧ (𝐴𝐺𝑦) = π‘ˆ)))
1614, 15sylan 581 . . . . 5 (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) β†’ ((𝑦𝐺𝐴) = π‘ˆ ↔ ((𝑦𝐺𝐴) = π‘ˆ ∧ (𝐴𝐺𝑦) = π‘ˆ)))
1716rabbidva 3440 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) β†’ {𝑦 ∈ 𝑋 ∣ (𝑦𝐺𝐴) = π‘ˆ} = {𝑦 ∈ 𝑋 ∣ ((𝑦𝐺𝐴) = π‘ˆ ∧ (𝐴𝐺𝑦) = π‘ˆ)})
188, 17eleqtrd 2836 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) β†’ (π‘β€˜π΄) ∈ {𝑦 ∈ 𝑋 ∣ ((𝑦𝐺𝐴) = π‘ˆ ∧ (𝐴𝐺𝑦) = π‘ˆ)})
19 oveq1 7416 . . . . . 6 (𝑦 = (π‘β€˜π΄) β†’ (𝑦𝐺𝐴) = ((π‘β€˜π΄)𝐺𝐴))
2019eqeq1d 2735 . . . . 5 (𝑦 = (π‘β€˜π΄) β†’ ((𝑦𝐺𝐴) = π‘ˆ ↔ ((π‘β€˜π΄)𝐺𝐴) = π‘ˆ))
21 oveq2 7417 . . . . . 6 (𝑦 = (π‘β€˜π΄) β†’ (𝐴𝐺𝑦) = (𝐴𝐺(π‘β€˜π΄)))
2221eqeq1d 2735 . . . . 5 (𝑦 = (π‘β€˜π΄) β†’ ((𝐴𝐺𝑦) = π‘ˆ ↔ (𝐴𝐺(π‘β€˜π΄)) = π‘ˆ))
2320, 22anbi12d 632 . . . 4 (𝑦 = (π‘β€˜π΄) β†’ (((𝑦𝐺𝐴) = π‘ˆ ∧ (𝐴𝐺𝑦) = π‘ˆ) ↔ (((π‘β€˜π΄)𝐺𝐴) = π‘ˆ ∧ (𝐴𝐺(π‘β€˜π΄)) = π‘ˆ)))
2423elrab 3684 . . 3 ((π‘β€˜π΄) ∈ {𝑦 ∈ 𝑋 ∣ ((𝑦𝐺𝐴) = π‘ˆ ∧ (𝐴𝐺𝑦) = π‘ˆ)} ↔ ((π‘β€˜π΄) ∈ 𝑋 ∧ (((π‘β€˜π΄)𝐺𝐴) = π‘ˆ ∧ (𝐴𝐺(π‘β€˜π΄)) = π‘ˆ)))
2518, 24sylib 217 . 2 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) β†’ ((π‘β€˜π΄) ∈ 𝑋 ∧ (((π‘β€˜π΄)𝐺𝐴) = π‘ˆ ∧ (𝐴𝐺(π‘β€˜π΄)) = π‘ˆ)))
2625simprd 497 1 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) β†’ (((π‘β€˜π΄)𝐺𝐴) = π‘ˆ ∧ (𝐴𝐺(π‘β€˜π΄)) = π‘ˆ))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3062  βˆƒwrex 3071  βˆƒ!wreu 3375  {crab 3433  ran crn 5678  β€˜cfv 6544  β„©crio 7364  (class class class)co 7409  GrpOpcgr 29773  GIdcgi 29774  invcgn 29775
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 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725
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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-grpo 29777  df-gid 29778  df-ginv 29779
This theorem is referenced by:  grpolinv  29810  grporinv  29811
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