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Theorem grpoinvid2 29782
Description: The inverse of a group element expressed in terms of the identity element. (Contributed by NM, 27-Oct-2006.) (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
grpoinvid2 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ ((π‘β€˜π΄) = 𝐡 ↔ (𝐡𝐺𝐴) = π‘ˆ))
Proof of Theorem grpoinvid2
StepHypRef Expression
1 oveq1 7416 . . . 4 ((π‘β€˜π΄) = 𝐡 β†’ ((π‘β€˜π΄)𝐺𝐴) = (𝐡𝐺𝐴))
21adantl 483 . . 3 (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ (π‘β€˜π΄) = 𝐡) β†’ ((π‘β€˜π΄)𝐺𝐴) = (𝐡𝐺𝐴))
3 grpinv.1 . . . . . 6 𝑋 = ran 𝐺
4 grpinv.2 . . . . . 6 π‘ˆ = (GIdβ€˜πΊ)
5 grpinv.3 . . . . . 6 𝑁 = (invβ€˜πΊ)
63, 4, 5grpolinv 29779 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) β†’ ((π‘β€˜π΄)𝐺𝐴) = π‘ˆ)
763adant3 1133 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ ((π‘β€˜π΄)𝐺𝐴) = π‘ˆ)
87adantr 482 . . 3 (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ (π‘β€˜π΄) = 𝐡) β†’ ((π‘β€˜π΄)𝐺𝐴) = π‘ˆ)
92, 8eqtr3d 2775 . 2 (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ (π‘β€˜π΄) = 𝐡) β†’ (𝐡𝐺𝐴) = π‘ˆ)
103, 5grpoinvcl 29777 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) β†’ (π‘β€˜π΄) ∈ 𝑋)
113, 4grpolid 29769 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ (π‘β€˜π΄) ∈ 𝑋) β†’ (π‘ˆπΊ(π‘β€˜π΄)) = (π‘β€˜π΄))
1210, 11syldan 592 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) β†’ (π‘ˆπΊ(π‘β€˜π΄)) = (π‘β€˜π΄))
13123adant3 1133 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (π‘ˆπΊ(π‘β€˜π΄)) = (π‘β€˜π΄))
1413eqcomd 2739 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (π‘β€˜π΄) = (π‘ˆπΊ(π‘β€˜π΄)))
1514adantr 482 . . 3 (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ (𝐡𝐺𝐴) = π‘ˆ) β†’ (π‘β€˜π΄) = (π‘ˆπΊ(π‘β€˜π΄)))
16 oveq1 7416 . . . 4 ((𝐡𝐺𝐴) = π‘ˆ β†’ ((𝐡𝐺𝐴)𝐺(π‘β€˜π΄)) = (π‘ˆπΊ(π‘β€˜π΄)))
1716adantl 483 . . 3 (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ (𝐡𝐺𝐴) = π‘ˆ) β†’ ((𝐡𝐺𝐴)𝐺(π‘β€˜π΄)) = (π‘ˆπΊ(π‘β€˜π΄)))
18 simprr 772 . . . . . . . 8 ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ 𝐡 ∈ 𝑋)
19 simprl 770 . . . . . . . 8 ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ 𝐴 ∈ 𝑋)
2010adantrr 716 . . . . . . . 8 ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ (π‘β€˜π΄) ∈ 𝑋)
2118, 19, 203jca 1129 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ (𝐡 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ (π‘β€˜π΄) ∈ 𝑋))
223grpoass 29756 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ (𝐡 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ (π‘β€˜π΄) ∈ 𝑋)) β†’ ((𝐡𝐺𝐴)𝐺(π‘β€˜π΄)) = (𝐡𝐺(𝐴𝐺(π‘β€˜π΄))))
2321, 22syldan 592 . . . . . 6 ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ ((𝐡𝐺𝐴)𝐺(π‘β€˜π΄)) = (𝐡𝐺(𝐴𝐺(π‘β€˜π΄))))
24233impb 1116 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ ((𝐡𝐺𝐴)𝐺(π‘β€˜π΄)) = (𝐡𝐺(𝐴𝐺(π‘β€˜π΄))))
253, 4, 5grporinv 29780 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) β†’ (𝐴𝐺(π‘β€˜π΄)) = π‘ˆ)
2625oveq2d 7425 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) β†’ (𝐡𝐺(𝐴𝐺(π‘β€˜π΄))) = (π΅πΊπ‘ˆ))
27263adant3 1133 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (𝐡𝐺(𝐴𝐺(π‘β€˜π΄))) = (π΅πΊπ‘ˆ))
283, 4grporid 29770 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐡 ∈ 𝑋) β†’ (π΅πΊπ‘ˆ) = 𝐡)
29283adant2 1132 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (π΅πΊπ‘ˆ) = 𝐡)
3024, 27, 293eqtrd 2777 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ ((𝐡𝐺𝐴)𝐺(π‘β€˜π΄)) = 𝐡)
3130adantr 482 . . 3 (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ (𝐡𝐺𝐴) = π‘ˆ) β†’ ((𝐡𝐺𝐴)𝐺(π‘β€˜π΄)) = 𝐡)
3215, 17, 313eqtr2d 2779 . 2 (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ (𝐡𝐺𝐴) = π‘ˆ) β†’ (π‘β€˜π΄) = 𝐡)
339, 32impbida 800 1 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ ((π‘β€˜π΄) = 𝐡 ↔ (𝐡𝐺𝐴) = π‘ˆ))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  ran crn 5678  β€˜cfv 6544  (class class class)co 7409  GrpOpcgr 29742  GIdcgi 29743  invcgn 29744
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 29746  df-gid 29747  df-ginv 29748
This theorem is referenced by:  rngonegmn1r  36810
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