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Theorem grpoinvid2 29369
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 7361 . . . 4 ((𝑁𝐴) = 𝐵 → ((𝑁𝐴)𝐺𝐴) = (𝐵𝐺𝐴))
21adantl 482 . . 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 29366 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → ((𝑁𝐴)𝐺𝐴) = 𝑈)
763adant3 1132 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((𝑁𝐴)𝐺𝐴) = 𝑈)
87adantr 481 . . 3 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) ∧ (𝑁𝐴) = 𝐵) → ((𝑁𝐴)𝐺𝐴) = 𝑈)
92, 8eqtr3d 2778 . 2 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) ∧ (𝑁𝐴) = 𝐵) → (𝐵𝐺𝐴) = 𝑈)
103, 5grpoinvcl 29364 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝑁𝐴) ∈ 𝑋)
113, 4grpolid 29356 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ (𝑁𝐴) ∈ 𝑋) → (𝑈𝐺(𝑁𝐴)) = (𝑁𝐴))
1210, 11syldan 591 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝑈𝐺(𝑁𝐴)) = (𝑁𝐴))
13123adant3 1132 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝑈𝐺(𝑁𝐴)) = (𝑁𝐴))
1413eqcomd 2742 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝑁𝐴) = (𝑈𝐺(𝑁𝐴)))
1514adantr 481 . . 3 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐵𝐺𝐴) = 𝑈) → (𝑁𝐴) = (𝑈𝐺(𝑁𝐴)))
16 oveq1 7361 . . . 4 ((𝐵𝐺𝐴) = 𝑈 → ((𝐵𝐺𝐴)𝐺(𝑁𝐴)) = (𝑈𝐺(𝑁𝐴)))
1716adantl 482 . . 3 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐵𝐺𝐴) = 𝑈) → ((𝐵𝐺𝐴)𝐺(𝑁𝐴)) = (𝑈𝐺(𝑁𝐴)))
18 simprr 771 . . . . . . . 8 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋)) → 𝐵𝑋)
19 simprl 769 . . . . . . . 8 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋)) → 𝐴𝑋)
2010adantrr 715 . . . . . . . 8 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋)) → (𝑁𝐴) ∈ 𝑋)
2118, 19, 203jca 1128 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋)) → (𝐵𝑋𝐴𝑋 ∧ (𝑁𝐴) ∈ 𝑋))
223grpoass 29343 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ (𝐵𝑋𝐴𝑋 ∧ (𝑁𝐴) ∈ 𝑋)) → ((𝐵𝐺𝐴)𝐺(𝑁𝐴)) = (𝐵𝐺(𝐴𝐺(𝑁𝐴))))
2321, 22syldan 591 . . . . . 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 29367 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝐴𝐺(𝑁𝐴)) = 𝑈)
2625oveq2d 7370 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝐵𝐺(𝐴𝐺(𝑁𝐴))) = (𝐵𝐺𝑈))
27263adant3 1132 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐵𝐺(𝐴𝐺(𝑁𝐴))) = (𝐵𝐺𝑈))
283, 4grporid 29357 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐵𝑋) → (𝐵𝐺𝑈) = 𝐵)
29283adant2 1131 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐵𝐺𝑈) = 𝐵)
3024, 27, 293eqtrd 2780 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((𝐵𝐺𝐴)𝐺(𝑁𝐴)) = 𝐵)
3130adantr 481 . . 3 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐵𝐺𝐴) = 𝑈) → ((𝐵𝐺𝐴)𝐺(𝑁𝐴)) = 𝐵)
3215, 17, 313eqtr2d 2782 . 2 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐵𝐺𝐴) = 𝑈) → (𝑁𝐴) = 𝐵)
339, 32impbida 799 1 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((𝑁𝐴) = 𝐵 ↔ (𝐵𝐺𝐴) = 𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  ran crn 5633  cfv 6494  (class class class)co 7354  GrpOpcgr 29329  GIdcgi 29330  invcgn 29331
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5241  ax-sep 5255  ax-nul 5262  ax-pr 5383  ax-un 7669
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2888  df-ne 2943  df-ral 3064  df-rex 3073  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3739  df-csb 3855  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-iun 4955  df-br 5105  df-opab 5167  df-mpt 5188  df-id 5530  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6446  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-riota 7310  df-ov 7357  df-grpo 29333  df-gid 29334  df-ginv 29335
This theorem is referenced by:  rngonegmn1r  36390
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