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Theorem grpoinvop 26771
 Description: The inverse of the group operation reverses the arguments. Lemma 2.2.1(d) of [Herstein] p. 55. (Contributed by NM, 27-Oct-2006.) (New usage is discouraged.)
Hypotheses
Ref Expression
grpasscan1.1 𝑋 = ran 𝐺
grpasscan1.2 𝑁 = (inv‘𝐺)
Assertion
Ref Expression
grpoinvop ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝑁‘(𝐴𝐺𝐵)) = ((𝑁𝐵)𝐺(𝑁𝐴)))

Proof of Theorem grpoinvop
StepHypRef Expression
1 simp1 1054 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → 𝐺 ∈ GrpOp)
2 simp2 1055 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → 𝐴𝑋)
3 simp3 1056 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → 𝐵𝑋)
4 grpasscan1.1 . . . . . . 7 𝑋 = ran 𝐺
5 grpasscan1.2 . . . . . . 7 𝑁 = (inv‘𝐺)
64, 5grpoinvcl 26762 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐵𝑋) → (𝑁𝐵) ∈ 𝑋)
763adant2 1073 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝑁𝐵) ∈ 𝑋)
84, 5grpoinvcl 26762 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝑁𝐴) ∈ 𝑋)
983adant3 1074 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝑁𝐴) ∈ 𝑋)
104grpocl 26738 . . . . 5 ((𝐺 ∈ GrpOp ∧ (𝑁𝐵) ∈ 𝑋 ∧ (𝑁𝐴) ∈ 𝑋) → ((𝑁𝐵)𝐺(𝑁𝐴)) ∈ 𝑋)
111, 7, 9, 10syl3anc 1318 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((𝑁𝐵)𝐺(𝑁𝐴)) ∈ 𝑋)
124grpoass 26741 . . . 4 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋 ∧ ((𝑁𝐵)𝐺(𝑁𝐴)) ∈ 𝑋)) → ((𝐴𝐺𝐵)𝐺((𝑁𝐵)𝐺(𝑁𝐴))) = (𝐴𝐺(𝐵𝐺((𝑁𝐵)𝐺(𝑁𝐴)))))
131, 2, 3, 11, 12syl13anc 1320 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((𝐴𝐺𝐵)𝐺((𝑁𝐵)𝐺(𝑁𝐴))) = (𝐴𝐺(𝐵𝐺((𝑁𝐵)𝐺(𝑁𝐴)))))
14 eqid 2610 . . . . . . . 8 (GId‘𝐺) = (GId‘𝐺)
154, 14, 5grporinv 26765 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ 𝐵𝑋) → (𝐵𝐺(𝑁𝐵)) = (GId‘𝐺))
16153adant2 1073 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐵𝐺(𝑁𝐵)) = (GId‘𝐺))
1716oveq1d 6564 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((𝐵𝐺(𝑁𝐵))𝐺(𝑁𝐴)) = ((GId‘𝐺)𝐺(𝑁𝐴)))
184grpoass 26741 . . . . . 6 ((𝐺 ∈ GrpOp ∧ (𝐵𝑋 ∧ (𝑁𝐵) ∈ 𝑋 ∧ (𝑁𝐴) ∈ 𝑋)) → ((𝐵𝐺(𝑁𝐵))𝐺(𝑁𝐴)) = (𝐵𝐺((𝑁𝐵)𝐺(𝑁𝐴))))
191, 3, 7, 9, 18syl13anc 1320 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((𝐵𝐺(𝑁𝐵))𝐺(𝑁𝐴)) = (𝐵𝐺((𝑁𝐵)𝐺(𝑁𝐴))))
204, 14grpolid 26754 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ (𝑁𝐴) ∈ 𝑋) → ((GId‘𝐺)𝐺(𝑁𝐴)) = (𝑁𝐴))
218, 20syldan 486 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → ((GId‘𝐺)𝐺(𝑁𝐴)) = (𝑁𝐴))
22213adant3 1074 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((GId‘𝐺)𝐺(𝑁𝐴)) = (𝑁𝐴))
2317, 19, 223eqtr3d 2652 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐵𝐺((𝑁𝐵)𝐺(𝑁𝐴))) = (𝑁𝐴))
2423oveq2d 6565 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐺(𝐵𝐺((𝑁𝐵)𝐺(𝑁𝐴)))) = (𝐴𝐺(𝑁𝐴)))
254, 14, 5grporinv 26765 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝐴𝐺(𝑁𝐴)) = (GId‘𝐺))
26253adant3 1074 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐺(𝑁𝐴)) = (GId‘𝐺))
2713, 24, 263eqtrd 2648 . 2 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((𝐴𝐺𝐵)𝐺((𝑁𝐵)𝐺(𝑁𝐴))) = (GId‘𝐺))
284grpocl 26738 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐺𝐵) ∈ 𝑋)
294, 14, 5grpoinvid1 26766 . . 3 ((𝐺 ∈ GrpOp ∧ (𝐴𝐺𝐵) ∈ 𝑋 ∧ ((𝑁𝐵)𝐺(𝑁𝐴)) ∈ 𝑋) → ((𝑁‘(𝐴𝐺𝐵)) = ((𝑁𝐵)𝐺(𝑁𝐴)) ↔ ((𝐴𝐺𝐵)𝐺((𝑁𝐵)𝐺(𝑁𝐴))) = (GId‘𝐺)))
301, 28, 11, 29syl3anc 1318 . 2 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((𝑁‘(𝐴𝐺𝐵)) = ((𝑁𝐵)𝐺(𝑁𝐴)) ↔ ((𝐴𝐺𝐵)𝐺((𝑁𝐵)𝐺(𝑁𝐴))) = (GId‘𝐺)))
3127, 30mpbird 246 1 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝑁‘(𝐴𝐺𝐵)) = ((𝑁𝐵)𝐺(𝑁𝐴)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ 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:  grpoinvdiv  26775
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