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Theorem grpoinvid1 30380
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
grpoinvid1 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ ((π‘β€˜π΄) = 𝐡 ↔ (𝐴𝐺𝐡) = π‘ˆ))

Proof of Theorem grpoinvid1
StepHypRef Expression
1 oveq2 7423 . . . 4 ((π‘β€˜π΄) = 𝐡 β†’ (𝐴𝐺(π‘β€˜π΄)) = (𝐴𝐺𝐡))
21adantl 480 . . 3 (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ (π‘β€˜π΄) = 𝐡) β†’ (𝐴𝐺(π‘β€˜π΄)) = (𝐴𝐺𝐡))
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, 5grporinv 30379 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) β†’ (𝐴𝐺(π‘β€˜π΄)) = π‘ˆ)
763adant3 1129 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (𝐴𝐺(π‘β€˜π΄)) = π‘ˆ)
87adantr 479 . . 3 (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ (π‘β€˜π΄) = 𝐡) β†’ (𝐴𝐺(π‘β€˜π΄)) = π‘ˆ)
92, 8eqtr3d 2767 . 2 (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ (π‘β€˜π΄) = 𝐡) β†’ (𝐴𝐺𝐡) = π‘ˆ)
10 oveq2 7423 . . . 4 ((𝐴𝐺𝐡) = π‘ˆ β†’ ((π‘β€˜π΄)𝐺(𝐴𝐺𝐡)) = ((π‘β€˜π΄)πΊπ‘ˆ))
1110adantl 480 . . 3 (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ (𝐴𝐺𝐡) = π‘ˆ) β†’ ((π‘β€˜π΄)𝐺(𝐴𝐺𝐡)) = ((π‘β€˜π΄)πΊπ‘ˆ))
123, 4, 5grpolinv 30378 . . . . . . . 8 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) β†’ ((π‘β€˜π΄)𝐺𝐴) = π‘ˆ)
1312oveq1d 7430 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) β†’ (((π‘β€˜π΄)𝐺𝐴)𝐺𝐡) = (π‘ˆπΊπ΅))
14133adant3 1129 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (((π‘β€˜π΄)𝐺𝐴)𝐺𝐡) = (π‘ˆπΊπ΅))
153, 5grpoinvcl 30376 . . . . . . . . . 10 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) β†’ (π‘β€˜π΄) ∈ 𝑋)
1615adantrr 715 . . . . . . . . 9 ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ (π‘β€˜π΄) ∈ 𝑋)
17 simprl 769 . . . . . . . . 9 ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ 𝐴 ∈ 𝑋)
18 simprr 771 . . . . . . . . 9 ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ 𝐡 ∈ 𝑋)
1916, 17, 183jca 1125 . . . . . . . 8 ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ ((π‘β€˜π΄) ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋))
203grpoass 30355 . . . . . . . 8 ((𝐺 ∈ GrpOp ∧ ((π‘β€˜π΄) ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ (((π‘β€˜π΄)𝐺𝐴)𝐺𝐡) = ((π‘β€˜π΄)𝐺(𝐴𝐺𝐡)))
2119, 20syldan 589 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ (((π‘β€˜π΄)𝐺𝐴)𝐺𝐡) = ((π‘β€˜π΄)𝐺(𝐴𝐺𝐡)))
22213impb 1112 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (((π‘β€˜π΄)𝐺𝐴)𝐺𝐡) = ((π‘β€˜π΄)𝐺(𝐴𝐺𝐡)))
2314, 22eqtr3d 2767 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (π‘ˆπΊπ΅) = ((π‘β€˜π΄)𝐺(𝐴𝐺𝐡)))
243, 4grpolid 30368 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐡 ∈ 𝑋) β†’ (π‘ˆπΊπ΅) = 𝐡)
25243adant2 1128 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (π‘ˆπΊπ΅) = 𝐡)
2623, 25eqtr3d 2767 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ ((π‘β€˜π΄)𝐺(𝐴𝐺𝐡)) = 𝐡)
2726adantr 479 . . 3 (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ (𝐴𝐺𝐡) = π‘ˆ) β†’ ((π‘β€˜π΄)𝐺(𝐴𝐺𝐡)) = 𝐡)
283, 4grporid 30369 . . . . . 6 ((𝐺 ∈ GrpOp ∧ (π‘β€˜π΄) ∈ 𝑋) β†’ ((π‘β€˜π΄)πΊπ‘ˆ) = (π‘β€˜π΄))
2915, 28syldan 589 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) β†’ ((π‘β€˜π΄)πΊπ‘ˆ) = (π‘β€˜π΄))
30293adant3 1129 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ ((π‘β€˜π΄)πΊπ‘ˆ) = (π‘β€˜π΄))
3130adantr 479 . . 3 (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ (𝐴𝐺𝐡) = π‘ˆ) β†’ ((π‘β€˜π΄)πΊπ‘ˆ) = (π‘β€˜π΄))
3211, 27, 313eqtr3rd 2774 . 2 (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ (𝐴𝐺𝐡) = π‘ˆ) β†’ (π‘β€˜π΄) = 𝐡)
339, 32impbida 799 1 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ ((π‘β€˜π΄) = 𝐡 ↔ (𝐴𝐺𝐡) = π‘ˆ))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  ran crn 5673  β€˜cfv 6542  (class class class)co 7415  GrpOpcgr 30341  GIdcgi 30342  invcgn 30343
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pr 5423  ax-un 7737
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  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 7371  df-ov 7418  df-grpo 30345  df-gid 30346  df-ginv 30347
This theorem is referenced by:  grpoinvop  30385  rngonegmn1l  37470
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