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Theorem grpoinvid2 30326
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 7421 . . . 4 ((π‘β€˜π΄) = 𝐡 β†’ ((π‘β€˜π΄)𝐺𝐴) = (𝐡𝐺𝐴))
21adantl 481 . . 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 30323 . . . . 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 480 . . 3 (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ (π‘β€˜π΄) = 𝐡) β†’ ((π‘β€˜π΄)𝐺𝐴) = π‘ˆ)
92, 8eqtr3d 2769 . 2 (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ (π‘β€˜π΄) = 𝐡) β†’ (𝐡𝐺𝐴) = π‘ˆ)
103, 5grpoinvcl 30321 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) β†’ (π‘β€˜π΄) ∈ 𝑋)
113, 4grpolid 30313 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ (π‘β€˜π΄) ∈ 𝑋) β†’ (π‘ˆπΊ(π‘β€˜π΄)) = (π‘β€˜π΄))
1210, 11syldan 590 . . . . . 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 2733 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (π‘β€˜π΄) = (π‘ˆπΊ(π‘β€˜π΄)))
1514adantr 480 . . 3 (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ (𝐡𝐺𝐴) = π‘ˆ) β†’ (π‘β€˜π΄) = (π‘ˆπΊ(π‘β€˜π΄)))
16 oveq1 7421 . . . 4 ((𝐡𝐺𝐴) = π‘ˆ β†’ ((𝐡𝐺𝐴)𝐺(π‘β€˜π΄)) = (π‘ˆπΊ(π‘β€˜π΄)))
1716adantl 481 . . 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 1126 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ (𝐡 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ (π‘β€˜π΄) ∈ 𝑋))
223grpoass 30300 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ (𝐡 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ (π‘β€˜π΄) ∈ 𝑋)) β†’ ((𝐡𝐺𝐴)𝐺(π‘β€˜π΄)) = (𝐡𝐺(𝐴𝐺(π‘β€˜π΄))))
2321, 22syldan 590 . . . . . 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 30324 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) β†’ (𝐴𝐺(π‘β€˜π΄)) = π‘ˆ)
2625oveq2d 7430 . . . . . 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 30314 . . . . . 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 2771 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ ((𝐡𝐺𝐴)𝐺(π‘β€˜π΄)) = 𝐡)
3130adantr 480 . . 3 (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ (𝐡𝐺𝐴) = π‘ˆ) β†’ ((𝐡𝐺𝐴)𝐺(π‘β€˜π΄)) = 𝐡)
3215, 17, 313eqtr2d 2773 . 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 395   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099  ran crn 5673  β€˜cfv 6542  (class class class)co 7414  GrpOpcgr 30286  GIdcgi 30287  invcgn 30288
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  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 7370  df-ov 7417  df-grpo 30290  df-gid 30291  df-ginv 30292
This theorem is referenced by:  rngonegmn1r  37350
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