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Theorem grpoinvop 30562
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 1135 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → 𝐺 ∈ GrpOp)
2 simp2 1136 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → 𝐴𝑋)
3 simp3 1137 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → 𝐵𝑋)
4 grpasscan1.1 . . . . . . 7 𝑋 = ran 𝐺
5 grpasscan1.2 . . . . . . 7 𝑁 = (inv‘𝐺)
64, 5grpoinvcl 30553 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐵𝑋) → (𝑁𝐵) ∈ 𝑋)
763adant2 1130 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝑁𝐵) ∈ 𝑋)
84, 5grpoinvcl 30553 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝑁𝐴) ∈ 𝑋)
983adant3 1131 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝑁𝐴) ∈ 𝑋)
104grpocl 30529 . . . . 5 ((𝐺 ∈ GrpOp ∧ (𝑁𝐵) ∈ 𝑋 ∧ (𝑁𝐴) ∈ 𝑋) → ((𝑁𝐵)𝐺(𝑁𝐴)) ∈ 𝑋)
111, 7, 9, 10syl3anc 1370 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((𝑁𝐵)𝐺(𝑁𝐴)) ∈ 𝑋)
124grpoass 30532 . . . 4 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋 ∧ ((𝑁𝐵)𝐺(𝑁𝐴)) ∈ 𝑋)) → ((𝐴𝐺𝐵)𝐺((𝑁𝐵)𝐺(𝑁𝐴))) = (𝐴𝐺(𝐵𝐺((𝑁𝐵)𝐺(𝑁𝐴)))))
131, 2, 3, 11, 12syl13anc 1371 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((𝐴𝐺𝐵)𝐺((𝑁𝐵)𝐺(𝑁𝐴))) = (𝐴𝐺(𝐵𝐺((𝑁𝐵)𝐺(𝑁𝐴)))))
14 eqid 2735 . . . . . . . 8 (GId‘𝐺) = (GId‘𝐺)
154, 14, 5grporinv 30556 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ 𝐵𝑋) → (𝐵𝐺(𝑁𝐵)) = (GId‘𝐺))
16153adant2 1130 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐵𝐺(𝑁𝐵)) = (GId‘𝐺))
1716oveq1d 7446 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((𝐵𝐺(𝑁𝐵))𝐺(𝑁𝐴)) = ((GId‘𝐺)𝐺(𝑁𝐴)))
184grpoass 30532 . . . . . 6 ((𝐺 ∈ GrpOp ∧ (𝐵𝑋 ∧ (𝑁𝐵) ∈ 𝑋 ∧ (𝑁𝐴) ∈ 𝑋)) → ((𝐵𝐺(𝑁𝐵))𝐺(𝑁𝐴)) = (𝐵𝐺((𝑁𝐵)𝐺(𝑁𝐴))))
191, 3, 7, 9, 18syl13anc 1371 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((𝐵𝐺(𝑁𝐵))𝐺(𝑁𝐴)) = (𝐵𝐺((𝑁𝐵)𝐺(𝑁𝐴))))
204, 14grpolid 30545 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ (𝑁𝐴) ∈ 𝑋) → ((GId‘𝐺)𝐺(𝑁𝐴)) = (𝑁𝐴))
218, 20syldan 591 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → ((GId‘𝐺)𝐺(𝑁𝐴)) = (𝑁𝐴))
22213adant3 1131 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((GId‘𝐺)𝐺(𝑁𝐴)) = (𝑁𝐴))
2317, 19, 223eqtr3d 2783 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐵𝐺((𝑁𝐵)𝐺(𝑁𝐴))) = (𝑁𝐴))
2423oveq2d 7447 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐺(𝐵𝐺((𝑁𝐵)𝐺(𝑁𝐴)))) = (𝐴𝐺(𝑁𝐴)))
254, 14, 5grporinv 30556 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝐴𝐺(𝑁𝐴)) = (GId‘𝐺))
26253adant3 1131 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐺(𝑁𝐴)) = (GId‘𝐺))
2713, 24, 263eqtrd 2779 . 2 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((𝐴𝐺𝐵)𝐺((𝑁𝐵)𝐺(𝑁𝐴))) = (GId‘𝐺))
284grpocl 30529 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐺𝐵) ∈ 𝑋)
294, 14, 5grpoinvid1 30557 . . 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 1370 . 2 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((𝑁‘(𝐴𝐺𝐵)) = ((𝑁𝐵)𝐺(𝑁𝐴)) ↔ ((𝐴𝐺𝐵)𝐺((𝑁𝐵)𝐺(𝑁𝐴))) = (GId‘𝐺)))
3127, 30mpbird 257 1 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝑁‘(𝐴𝐺𝐵)) = ((𝑁𝐵)𝐺(𝑁𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  w3a 1086   = wceq 1537  wcel 2106  ran crn 5690  cfv 6563  (class class class)co 7431  GrpOpcgr 30518  GIdcgi 30519  invcgn 30520
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-grpo 30522  df-gid 30523  df-ginv 30524
This theorem is referenced by:  grpoinvdiv  30566
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