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Theorem grpoinvop 28292
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 1132 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → 𝐺 ∈ GrpOp)
2 simp2 1133 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → 𝐴𝑋)
3 simp3 1134 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → 𝐵𝑋)
4 grpasscan1.1 . . . . . . 7 𝑋 = ran 𝐺
5 grpasscan1.2 . . . . . . 7 𝑁 = (inv‘𝐺)
64, 5grpoinvcl 28283 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐵𝑋) → (𝑁𝐵) ∈ 𝑋)
763adant2 1127 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝑁𝐵) ∈ 𝑋)
84, 5grpoinvcl 28283 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝑁𝐴) ∈ 𝑋)
983adant3 1128 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝑁𝐴) ∈ 𝑋)
104grpocl 28259 . . . . 5 ((𝐺 ∈ GrpOp ∧ (𝑁𝐵) ∈ 𝑋 ∧ (𝑁𝐴) ∈ 𝑋) → ((𝑁𝐵)𝐺(𝑁𝐴)) ∈ 𝑋)
111, 7, 9, 10syl3anc 1367 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((𝑁𝐵)𝐺(𝑁𝐴)) ∈ 𝑋)
124grpoass 28262 . . . 4 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋 ∧ ((𝑁𝐵)𝐺(𝑁𝐴)) ∈ 𝑋)) → ((𝐴𝐺𝐵)𝐺((𝑁𝐵)𝐺(𝑁𝐴))) = (𝐴𝐺(𝐵𝐺((𝑁𝐵)𝐺(𝑁𝐴)))))
131, 2, 3, 11, 12syl13anc 1368 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((𝐴𝐺𝐵)𝐺((𝑁𝐵)𝐺(𝑁𝐴))) = (𝐴𝐺(𝐵𝐺((𝑁𝐵)𝐺(𝑁𝐴)))))
14 eqid 2820 . . . . . . . 8 (GId‘𝐺) = (GId‘𝐺)
154, 14, 5grporinv 28286 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ 𝐵𝑋) → (𝐵𝐺(𝑁𝐵)) = (GId‘𝐺))
16153adant2 1127 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐵𝐺(𝑁𝐵)) = (GId‘𝐺))
1716oveq1d 7146 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((𝐵𝐺(𝑁𝐵))𝐺(𝑁𝐴)) = ((GId‘𝐺)𝐺(𝑁𝐴)))
184grpoass 28262 . . . . . 6 ((𝐺 ∈ GrpOp ∧ (𝐵𝑋 ∧ (𝑁𝐵) ∈ 𝑋 ∧ (𝑁𝐴) ∈ 𝑋)) → ((𝐵𝐺(𝑁𝐵))𝐺(𝑁𝐴)) = (𝐵𝐺((𝑁𝐵)𝐺(𝑁𝐴))))
191, 3, 7, 9, 18syl13anc 1368 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((𝐵𝐺(𝑁𝐵))𝐺(𝑁𝐴)) = (𝐵𝐺((𝑁𝐵)𝐺(𝑁𝐴))))
204, 14grpolid 28275 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ (𝑁𝐴) ∈ 𝑋) → ((GId‘𝐺)𝐺(𝑁𝐴)) = (𝑁𝐴))
218, 20syldan 593 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → ((GId‘𝐺)𝐺(𝑁𝐴)) = (𝑁𝐴))
22213adant3 1128 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((GId‘𝐺)𝐺(𝑁𝐴)) = (𝑁𝐴))
2317, 19, 223eqtr3d 2863 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐵𝐺((𝑁𝐵)𝐺(𝑁𝐴))) = (𝑁𝐴))
2423oveq2d 7147 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐺(𝐵𝐺((𝑁𝐵)𝐺(𝑁𝐴)))) = (𝐴𝐺(𝑁𝐴)))
254, 14, 5grporinv 28286 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝐴𝐺(𝑁𝐴)) = (GId‘𝐺))
26253adant3 1128 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐺(𝑁𝐴)) = (GId‘𝐺))
2713, 24, 263eqtrd 2859 . 2 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((𝐴𝐺𝐵)𝐺((𝑁𝐵)𝐺(𝑁𝐴))) = (GId‘𝐺))
284grpocl 28259 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐺𝐵) ∈ 𝑋)
294, 14, 5grpoinvid1 28287 . . 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 1367 . 2 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((𝑁‘(𝐴𝐺𝐵)) = ((𝑁𝐵)𝐺(𝑁𝐴)) ↔ ((𝐴𝐺𝐵)𝐺((𝑁𝐵)𝐺(𝑁𝐴))) = (GId‘𝐺)))
3127, 30mpbird 259 1 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝑁‘(𝐴𝐺𝐵)) = ((𝑁𝐵)𝐺(𝑁𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  w3a 1083   = wceq 1537  wcel 2114  ran crn 5530  cfv 6329  (class class class)co 7131  GrpOpcgr 28248  GIdcgi 28249  invcgn 28250
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-rep 5164  ax-sep 5177  ax-nul 5184  ax-pr 5304  ax-un 7437
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-reu 3132  df-rab 3134  df-v 3475  df-sbc 3752  df-csb 3860  df-dif 3915  df-un 3917  df-in 3919  df-ss 3928  df-nul 4268  df-if 4442  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4813  df-iun 4895  df-br 5041  df-opab 5103  df-mpt 5121  df-id 5434  df-xp 5535  df-rel 5536  df-cnv 5537  df-co 5538  df-dm 5539  df-rn 5540  df-res 5541  df-ima 5542  df-iota 6288  df-fun 6331  df-fn 6332  df-f 6333  df-f1 6334  df-fo 6335  df-f1o 6336  df-fv 6337  df-riota 7089  df-ov 7134  df-grpo 28252  df-gid 28253  df-ginv 28254
This theorem is referenced by:  grpoinvdiv  28296
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