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Theorem grpoinvid2 30049
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 7418 . . . 4 ((π‘β€˜π΄) = 𝐡 β†’ ((π‘β€˜π΄)𝐺𝐴) = (𝐡𝐺𝐴))
21adantl 480 . . 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 30046 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) β†’ ((π‘β€˜π΄)𝐺𝐴) = π‘ˆ)
763adant3 1130 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ ((π‘β€˜π΄)𝐺𝐴) = π‘ˆ)
87adantr 479 . . 3 (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ (π‘β€˜π΄) = 𝐡) β†’ ((π‘β€˜π΄)𝐺𝐴) = π‘ˆ)
92, 8eqtr3d 2772 . 2 (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ (π‘β€˜π΄) = 𝐡) β†’ (𝐡𝐺𝐴) = π‘ˆ)
103, 5grpoinvcl 30044 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) β†’ (π‘β€˜π΄) ∈ 𝑋)
113, 4grpolid 30036 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ (π‘β€˜π΄) ∈ 𝑋) β†’ (π‘ˆπΊ(π‘β€˜π΄)) = (π‘β€˜π΄))
1210, 11syldan 589 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) β†’ (π‘ˆπΊ(π‘β€˜π΄)) = (π‘β€˜π΄))
13123adant3 1130 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (π‘ˆπΊ(π‘β€˜π΄)) = (π‘β€˜π΄))
1413eqcomd 2736 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (π‘β€˜π΄) = (π‘ˆπΊ(π‘β€˜π΄)))
1514adantr 479 . . 3 (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ (𝐡𝐺𝐴) = π‘ˆ) β†’ (π‘β€˜π΄) = (π‘ˆπΊ(π‘β€˜π΄)))
16 oveq1 7418 . . . 4 ((𝐡𝐺𝐴) = π‘ˆ β†’ ((𝐡𝐺𝐴)𝐺(π‘β€˜π΄)) = (π‘ˆπΊ(π‘β€˜π΄)))
1716adantl 480 . . 3 (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ (𝐡𝐺𝐴) = π‘ˆ) β†’ ((𝐡𝐺𝐴)𝐺(π‘β€˜π΄)) = (π‘ˆπΊ(π‘β€˜π΄)))
18 simprr 769 . . . . . . . 8 ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ 𝐡 ∈ 𝑋)
19 simprl 767 . . . . . . . 8 ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ 𝐴 ∈ 𝑋)
2010adantrr 713 . . . . . . . 8 ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ (π‘β€˜π΄) ∈ 𝑋)
2118, 19, 203jca 1126 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ (𝐡 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ (π‘β€˜π΄) ∈ 𝑋))
223grpoass 30023 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ (𝐡 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ (π‘β€˜π΄) ∈ 𝑋)) β†’ ((𝐡𝐺𝐴)𝐺(π‘β€˜π΄)) = (𝐡𝐺(𝐴𝐺(π‘β€˜π΄))))
2321, 22syldan 589 . . . . . 6 ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ ((𝐡𝐺𝐴)𝐺(π‘β€˜π΄)) = (𝐡𝐺(𝐴𝐺(π‘β€˜π΄))))
24233impb 1113 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ ((𝐡𝐺𝐴)𝐺(π‘β€˜π΄)) = (𝐡𝐺(𝐴𝐺(π‘β€˜π΄))))
253, 4, 5grporinv 30047 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) β†’ (𝐴𝐺(π‘β€˜π΄)) = π‘ˆ)
2625oveq2d 7427 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) β†’ (𝐡𝐺(𝐴𝐺(π‘β€˜π΄))) = (π΅πΊπ‘ˆ))
27263adant3 1130 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (𝐡𝐺(𝐴𝐺(π‘β€˜π΄))) = (π΅πΊπ‘ˆ))
283, 4grporid 30037 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐡 ∈ 𝑋) β†’ (π΅πΊπ‘ˆ) = 𝐡)
29283adant2 1129 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (π΅πΊπ‘ˆ) = 𝐡)
3024, 27, 293eqtrd 2774 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ ((𝐡𝐺𝐴)𝐺(π‘β€˜π΄)) = 𝐡)
3130adantr 479 . . 3 (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ (𝐡𝐺𝐴) = π‘ˆ) β†’ ((𝐡𝐺𝐴)𝐺(π‘β€˜π΄)) = 𝐡)
3215, 17, 313eqtr2d 2776 . 2 (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ (𝐡𝐺𝐴) = π‘ˆ) β†’ (π‘β€˜π΄) = 𝐡)
339, 32impbida 797 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 394   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104  ran crn 5676  β€˜cfv 6542  (class class class)co 7411  GrpOpcgr 30009  GIdcgi 30010  invcgn 30011
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7367  df-ov 7414  df-grpo 30013  df-gid 30014  df-ginv 30015
This theorem is referenced by:  rngonegmn1r  37113
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