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Theorem grpoinvid2 26767
 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 6556 . . . 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 26764 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → ((𝑁𝐴)𝐺𝐴) = 𝑈)
763adant3 1074 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((𝑁𝐴)𝐺𝐴) = 𝑈)
87adantr 480 . . 3 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) ∧ (𝑁𝐴) = 𝐵) → ((𝑁𝐴)𝐺𝐴) = 𝑈)
92, 8eqtr3d 2646 . 2 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) ∧ (𝑁𝐴) = 𝐵) → (𝐵𝐺𝐴) = 𝑈)
103, 5grpoinvcl 26762 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝑁𝐴) ∈ 𝑋)
113, 4grpolid 26754 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ (𝑁𝐴) ∈ 𝑋) → (𝑈𝐺(𝑁𝐴)) = (𝑁𝐴))
1210, 11syldan 486 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝑈𝐺(𝑁𝐴)) = (𝑁𝐴))
13123adant3 1074 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝑈𝐺(𝑁𝐴)) = (𝑁𝐴))
1413eqcomd 2616 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝑁𝐴) = (𝑈𝐺(𝑁𝐴)))
1514adantr 480 . . 3 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐵𝐺𝐴) = 𝑈) → (𝑁𝐴) = (𝑈𝐺(𝑁𝐴)))
16 oveq1 6556 . . . 4 ((𝐵𝐺𝐴) = 𝑈 → ((𝐵𝐺𝐴)𝐺(𝑁𝐴)) = (𝑈𝐺(𝑁𝐴)))
1716adantl 481 . . 3 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐵𝐺𝐴) = 𝑈) → ((𝐵𝐺𝐴)𝐺(𝑁𝐴)) = (𝑈𝐺(𝑁𝐴)))
18 simprr 792 . . . . . . . 8 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋)) → 𝐵𝑋)
19 simprl 790 . . . . . . . 8 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋)) → 𝐴𝑋)
2010adantrr 749 . . . . . . . 8 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋)) → (𝑁𝐴) ∈ 𝑋)
2118, 19, 203jca 1235 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋)) → (𝐵𝑋𝐴𝑋 ∧ (𝑁𝐴) ∈ 𝑋))
223grpoass 26741 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ (𝐵𝑋𝐴𝑋 ∧ (𝑁𝐴) ∈ 𝑋)) → ((𝐵𝐺𝐴)𝐺(𝑁𝐴)) = (𝐵𝐺(𝐴𝐺(𝑁𝐴))))
2321, 22syldan 486 . . . . . 6 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋)) → ((𝐵𝐺𝐴)𝐺(𝑁𝐴)) = (𝐵𝐺(𝐴𝐺(𝑁𝐴))))
24233impb 1252 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((𝐵𝐺𝐴)𝐺(𝑁𝐴)) = (𝐵𝐺(𝐴𝐺(𝑁𝐴))))
253, 4, 5grporinv 26765 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝐴𝐺(𝑁𝐴)) = 𝑈)
2625oveq2d 6565 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝐵𝐺(𝐴𝐺(𝑁𝐴))) = (𝐵𝐺𝑈))
27263adant3 1074 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐵𝐺(𝐴𝐺(𝑁𝐴))) = (𝐵𝐺𝑈))
283, 4grporid 26755 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐵𝑋) → (𝐵𝐺𝑈) = 𝐵)
29283adant2 1073 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐵𝐺𝑈) = 𝐵)
3024, 27, 293eqtrd 2648 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((𝐵𝐺𝐴)𝐺(𝑁𝐴)) = 𝐵)
3130adantr 480 . . 3 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐵𝐺𝐴) = 𝑈) → ((𝐵𝐺𝐴)𝐺(𝑁𝐴)) = 𝐵)
3215, 17, 313eqtr2d 2650 . 2 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐵𝐺𝐴) = 𝑈) → (𝑁𝐴) = 𝐵)
339, 32impbida 873 1 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((𝑁𝐴) = 𝐵 ↔ (𝐵𝐺𝐴) = 𝑈))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ran crn 5039  ‘cfv 5804  (class class class)co 6549  GrpOpcgr 26727  GIdcgi 26728  invcgn 26729 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-grpo 26731  df-gid 26732  df-ginv 26733 This theorem is referenced by:  rngonegmn1r  32911
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