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Theorem grpoinvid2 30561
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 𝑈 = (GId‘𝐺)
grpinv.3 𝑁 = (inv‘𝐺)
Assertion
Ref Expression
grpoinvid2 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((𝑁𝐴) = 𝐵 ↔ (𝐵𝐺𝐴) = 𝑈))

Proof of Theorem grpoinvid2
StepHypRef Expression
1 oveq1 7455 . . . 4 ((𝑁𝐴) = 𝐵 → ((𝑁𝐴)𝐺𝐴) = (𝐵𝐺𝐴))
21adantl 481 . . 3 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) ∧ (𝑁𝐴) = 𝐵) → ((𝑁𝐴)𝐺𝐴) = (𝐵𝐺𝐴))
3 grpinv.1 . . . . . 6 𝑋 = ran 𝐺
4 grpinv.2 . . . . . 6 𝑈 = (GId‘𝐺)
5 grpinv.3 . . . . . 6 𝑁 = (inv‘𝐺)
63, 4, 5grpolinv 30558 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → ((𝑁𝐴)𝐺𝐴) = 𝑈)
763adant3 1132 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((𝑁𝐴)𝐺𝐴) = 𝑈)
87adantr 480 . . 3 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) ∧ (𝑁𝐴) = 𝐵) → ((𝑁𝐴)𝐺𝐴) = 𝑈)
92, 8eqtr3d 2782 . 2 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) ∧ (𝑁𝐴) = 𝐵) → (𝐵𝐺𝐴) = 𝑈)
103, 5grpoinvcl 30556 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝑁𝐴) ∈ 𝑋)
113, 4grpolid 30548 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ (𝑁𝐴) ∈ 𝑋) → (𝑈𝐺(𝑁𝐴)) = (𝑁𝐴))
1210, 11syldan 590 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝑈𝐺(𝑁𝐴)) = (𝑁𝐴))
13123adant3 1132 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝑈𝐺(𝑁𝐴)) = (𝑁𝐴))
1413eqcomd 2746 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝑁𝐴) = (𝑈𝐺(𝑁𝐴)))
1514adantr 480 . . 3 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐵𝐺𝐴) = 𝑈) → (𝑁𝐴) = (𝑈𝐺(𝑁𝐴)))
16 oveq1 7455 . . . 4 ((𝐵𝐺𝐴) = 𝑈 → ((𝐵𝐺𝐴)𝐺(𝑁𝐴)) = (𝑈𝐺(𝑁𝐴)))
1716adantl 481 . . 3 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐵𝐺𝐴) = 𝑈) → ((𝐵𝐺𝐴)𝐺(𝑁𝐴)) = (𝑈𝐺(𝑁𝐴)))
18 simprr 772 . . . . . . . 8 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋)) → 𝐵𝑋)
19 simprl 770 . . . . . . . 8 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋)) → 𝐴𝑋)
2010adantrr 716 . . . . . . . 8 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋)) → (𝑁𝐴) ∈ 𝑋)
2118, 19, 203jca 1128 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋)) → (𝐵𝑋𝐴𝑋 ∧ (𝑁𝐴) ∈ 𝑋))
223grpoass 30535 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ (𝐵𝑋𝐴𝑋 ∧ (𝑁𝐴) ∈ 𝑋)) → ((𝐵𝐺𝐴)𝐺(𝑁𝐴)) = (𝐵𝐺(𝐴𝐺(𝑁𝐴))))
2321, 22syldan 590 . . . . . 6 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋)) → ((𝐵𝐺𝐴)𝐺(𝑁𝐴)) = (𝐵𝐺(𝐴𝐺(𝑁𝐴))))
24233impb 1115 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((𝐵𝐺𝐴)𝐺(𝑁𝐴)) = (𝐵𝐺(𝐴𝐺(𝑁𝐴))))
253, 4, 5grporinv 30559 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝐴𝐺(𝑁𝐴)) = 𝑈)
2625oveq2d 7464 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝐵𝐺(𝐴𝐺(𝑁𝐴))) = (𝐵𝐺𝑈))
27263adant3 1132 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐵𝐺(𝐴𝐺(𝑁𝐴))) = (𝐵𝐺𝑈))
283, 4grporid 30549 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐵𝑋) → (𝐵𝐺𝑈) = 𝐵)
29283adant2 1131 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐵𝐺𝑈) = 𝐵)
3024, 27, 293eqtrd 2784 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((𝐵𝐺𝐴)𝐺(𝑁𝐴)) = 𝐵)
3130adantr 480 . . 3 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐵𝐺𝐴) = 𝑈) → ((𝐵𝐺𝐴)𝐺(𝑁𝐴)) = 𝐵)
3215, 17, 313eqtr2d 2786 . 2 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐵𝐺𝐴) = 𝑈) → (𝑁𝐴) = 𝐵)
339, 32impbida 800 1 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((𝑁𝐴) = 𝐵 ↔ (𝐵𝐺𝐴) = 𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1537  wcel 2108  ran crn 5701  cfv 6573  (class class class)co 7448  GrpOpcgr 30521  GIdcgi 30522  invcgn 30523
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-grpo 30525  df-gid 30526  df-ginv 30527
This theorem is referenced by:  rngonegmn1r  37902
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