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Theorem grpoinvid2 29513
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 7365 . . . 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 29510 . . . . 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 29508 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) β†’ (π‘β€˜π΄) ∈ 𝑋)
113, 4grpolid 29500 . . . . . . 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 7365 . . . 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 29487 . . . . . . 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 29511 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) β†’ (𝐴𝐺(π‘β€˜π΄)) = π‘ˆ)
2625oveq2d 7374 . . . . . 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 29501 . . . . . 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 5635  β€˜cfv 6497  (class class class)co 7358  GrpOpcgr 29473  GIdcgi 29474  invcgn 29475
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 5243  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
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 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-grpo 29477  df-gid 29478  df-ginv 29479
This theorem is referenced by:  rngonegmn1r  36447
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