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Theorem grpoinvop 30552
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 1137 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → 𝐺 ∈ GrpOp)
2 simp2 1138 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → 𝐴𝑋)
3 simp3 1139 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → 𝐵𝑋)
4 grpasscan1.1 . . . . . . 7 𝑋 = ran 𝐺
5 grpasscan1.2 . . . . . . 7 𝑁 = (inv‘𝐺)
64, 5grpoinvcl 30543 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐵𝑋) → (𝑁𝐵) ∈ 𝑋)
763adant2 1132 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝑁𝐵) ∈ 𝑋)
84, 5grpoinvcl 30543 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝑁𝐴) ∈ 𝑋)
983adant3 1133 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝑁𝐴) ∈ 𝑋)
104grpocl 30519 . . . . 5 ((𝐺 ∈ GrpOp ∧ (𝑁𝐵) ∈ 𝑋 ∧ (𝑁𝐴) ∈ 𝑋) → ((𝑁𝐵)𝐺(𝑁𝐴)) ∈ 𝑋)
111, 7, 9, 10syl3anc 1373 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((𝑁𝐵)𝐺(𝑁𝐴)) ∈ 𝑋)
124grpoass 30522 . . . 4 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋 ∧ ((𝑁𝐵)𝐺(𝑁𝐴)) ∈ 𝑋)) → ((𝐴𝐺𝐵)𝐺((𝑁𝐵)𝐺(𝑁𝐴))) = (𝐴𝐺(𝐵𝐺((𝑁𝐵)𝐺(𝑁𝐴)))))
131, 2, 3, 11, 12syl13anc 1374 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((𝐴𝐺𝐵)𝐺((𝑁𝐵)𝐺(𝑁𝐴))) = (𝐴𝐺(𝐵𝐺((𝑁𝐵)𝐺(𝑁𝐴)))))
14 eqid 2737 . . . . . . . 8 (GId‘𝐺) = (GId‘𝐺)
154, 14, 5grporinv 30546 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ 𝐵𝑋) → (𝐵𝐺(𝑁𝐵)) = (GId‘𝐺))
16153adant2 1132 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐵𝐺(𝑁𝐵)) = (GId‘𝐺))
1716oveq1d 7446 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((𝐵𝐺(𝑁𝐵))𝐺(𝑁𝐴)) = ((GId‘𝐺)𝐺(𝑁𝐴)))
184grpoass 30522 . . . . . 6 ((𝐺 ∈ GrpOp ∧ (𝐵𝑋 ∧ (𝑁𝐵) ∈ 𝑋 ∧ (𝑁𝐴) ∈ 𝑋)) → ((𝐵𝐺(𝑁𝐵))𝐺(𝑁𝐴)) = (𝐵𝐺((𝑁𝐵)𝐺(𝑁𝐴))))
191, 3, 7, 9, 18syl13anc 1374 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((𝐵𝐺(𝑁𝐵))𝐺(𝑁𝐴)) = (𝐵𝐺((𝑁𝐵)𝐺(𝑁𝐴))))
204, 14grpolid 30535 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ (𝑁𝐴) ∈ 𝑋) → ((GId‘𝐺)𝐺(𝑁𝐴)) = (𝑁𝐴))
218, 20syldan 591 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → ((GId‘𝐺)𝐺(𝑁𝐴)) = (𝑁𝐴))
22213adant3 1133 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((GId‘𝐺)𝐺(𝑁𝐴)) = (𝑁𝐴))
2317, 19, 223eqtr3d 2785 . . . 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 30546 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝐴𝐺(𝑁𝐴)) = (GId‘𝐺))
26253adant3 1133 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐺(𝑁𝐴)) = (GId‘𝐺))
2713, 24, 263eqtrd 2781 . 2 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((𝐴𝐺𝐵)𝐺((𝑁𝐵)𝐺(𝑁𝐴))) = (GId‘𝐺))
284grpocl 30519 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐺𝐵) ∈ 𝑋)
294, 14, 5grpoinvid1 30547 . . 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 1373 . 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 1087   = wceq 1540  wcel 2108  ran crn 5686  cfv 6561  (class class class)co 7431  GrpOpcgr 30508  GIdcgi 30509  invcgn 30510
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-grpo 30512  df-gid 30513  df-ginv 30514
This theorem is referenced by:  grpoinvdiv  30556
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