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Theorem grpoinvid1 30290
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 7413 . . . 4 ((π‘β€˜π΄) = 𝐡 β†’ (𝐴𝐺(π‘β€˜π΄)) = (𝐴𝐺𝐡))
21adantl 481 . . 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 30289 . . . . 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 480 . . 3 (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ (π‘β€˜π΄) = 𝐡) β†’ (𝐴𝐺(π‘β€˜π΄)) = π‘ˆ)
92, 8eqtr3d 2768 . 2 (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ (π‘β€˜π΄) = 𝐡) β†’ (𝐴𝐺𝐡) = π‘ˆ)
10 oveq2 7413 . . . 4 ((𝐴𝐺𝐡) = π‘ˆ β†’ ((π‘β€˜π΄)𝐺(𝐴𝐺𝐡)) = ((π‘β€˜π΄)πΊπ‘ˆ))
1110adantl 481 . . 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 30288 . . . . . . . 8 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) β†’ ((π‘β€˜π΄)𝐺𝐴) = π‘ˆ)
1312oveq1d 7420 . . . . . . 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 30286 . . . . . . . . . 10 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) β†’ (π‘β€˜π΄) ∈ 𝑋)
1615adantrr 714 . . . . . . . . 9 ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ (π‘β€˜π΄) ∈ 𝑋)
17 simprl 768 . . . . . . . . 9 ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ 𝐴 ∈ 𝑋)
18 simprr 770 . . . . . . . . 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 30265 . . . . . . . 8 ((𝐺 ∈ GrpOp ∧ ((π‘β€˜π΄) ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ (((π‘β€˜π΄)𝐺𝐴)𝐺𝐡) = ((π‘β€˜π΄)𝐺(𝐴𝐺𝐡)))
2119, 20syldan 590 . . . . . . 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 2768 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (π‘ˆπΊπ΅) = ((π‘β€˜π΄)𝐺(𝐴𝐺𝐡)))
243, 4grpolid 30278 . . . . . 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 2768 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ ((π‘β€˜π΄)𝐺(𝐴𝐺𝐡)) = 𝐡)
2726adantr 480 . . 3 (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ (𝐴𝐺𝐡) = π‘ˆ) β†’ ((π‘β€˜π΄)𝐺(𝐴𝐺𝐡)) = 𝐡)
283, 4grporid 30279 . . . . . 6 ((𝐺 ∈ GrpOp ∧ (π‘β€˜π΄) ∈ 𝑋) β†’ ((π‘β€˜π΄)πΊπ‘ˆ) = (π‘β€˜π΄))
2915, 28syldan 590 . . . . 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 480 . . 3 (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ (𝐴𝐺𝐡) = π‘ˆ) β†’ ((π‘β€˜π΄)πΊπ‘ˆ) = (π‘β€˜π΄))
3211, 27, 313eqtr3rd 2775 . 2 (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ (𝐴𝐺𝐡) = π‘ˆ) β†’ (π‘β€˜π΄) = 𝐡)
339, 32impbida 798 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 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  ran crn 5670  β€˜cfv 6537  (class class class)co 7405  GrpOpcgr 30251  GIdcgi 30252  invcgn 30253
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-ov 7408  df-grpo 30255  df-gid 30256  df-ginv 30257
This theorem is referenced by:  grpoinvop  30295  rngonegmn1l  37322
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