MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  grpoinvop Structured version   Visualization version   GIF version

Theorem grpoinvop 27617
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 1128 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → 𝐺 ∈ GrpOp)
2 simp2 1129 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → 𝐴𝑋)
3 simp3 1130 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → 𝐵𝑋)
4 grpasscan1.1 . . . . . . 7 𝑋 = ran 𝐺
5 grpasscan1.2 . . . . . . 7 𝑁 = (inv‘𝐺)
64, 5grpoinvcl 27608 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐵𝑋) → (𝑁𝐵) ∈ 𝑋)
763adant2 1123 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝑁𝐵) ∈ 𝑋)
84, 5grpoinvcl 27608 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝑁𝐴) ∈ 𝑋)
983adant3 1124 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝑁𝐴) ∈ 𝑋)
104grpocl 27584 . . . . 5 ((𝐺 ∈ GrpOp ∧ (𝑁𝐵) ∈ 𝑋 ∧ (𝑁𝐴) ∈ 𝑋) → ((𝑁𝐵)𝐺(𝑁𝐴)) ∈ 𝑋)
111, 7, 9, 10syl3anc 1439 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((𝑁𝐵)𝐺(𝑁𝐴)) ∈ 𝑋)
124grpoass 27587 . . . 4 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋 ∧ ((𝑁𝐵)𝐺(𝑁𝐴)) ∈ 𝑋)) → ((𝐴𝐺𝐵)𝐺((𝑁𝐵)𝐺(𝑁𝐴))) = (𝐴𝐺(𝐵𝐺((𝑁𝐵)𝐺(𝑁𝐴)))))
131, 2, 3, 11, 12syl13anc 1441 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((𝐴𝐺𝐵)𝐺((𝑁𝐵)𝐺(𝑁𝐴))) = (𝐴𝐺(𝐵𝐺((𝑁𝐵)𝐺(𝑁𝐴)))))
14 eqid 2724 . . . . . . . 8 (GId‘𝐺) = (GId‘𝐺)
154, 14, 5grporinv 27611 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ 𝐵𝑋) → (𝐵𝐺(𝑁𝐵)) = (GId‘𝐺))
16153adant2 1123 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐵𝐺(𝑁𝐵)) = (GId‘𝐺))
1716oveq1d 6780 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((𝐵𝐺(𝑁𝐵))𝐺(𝑁𝐴)) = ((GId‘𝐺)𝐺(𝑁𝐴)))
184grpoass 27587 . . . . . 6 ((𝐺 ∈ GrpOp ∧ (𝐵𝑋 ∧ (𝑁𝐵) ∈ 𝑋 ∧ (𝑁𝐴) ∈ 𝑋)) → ((𝐵𝐺(𝑁𝐵))𝐺(𝑁𝐴)) = (𝐵𝐺((𝑁𝐵)𝐺(𝑁𝐴))))
191, 3, 7, 9, 18syl13anc 1441 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((𝐵𝐺(𝑁𝐵))𝐺(𝑁𝐴)) = (𝐵𝐺((𝑁𝐵)𝐺(𝑁𝐴))))
204, 14grpolid 27600 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ (𝑁𝐴) ∈ 𝑋) → ((GId‘𝐺)𝐺(𝑁𝐴)) = (𝑁𝐴))
218, 20syldan 488 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → ((GId‘𝐺)𝐺(𝑁𝐴)) = (𝑁𝐴))
22213adant3 1124 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((GId‘𝐺)𝐺(𝑁𝐴)) = (𝑁𝐴))
2317, 19, 223eqtr3d 2766 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐵𝐺((𝑁𝐵)𝐺(𝑁𝐴))) = (𝑁𝐴))
2423oveq2d 6781 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐺(𝐵𝐺((𝑁𝐵)𝐺(𝑁𝐴)))) = (𝐴𝐺(𝑁𝐴)))
254, 14, 5grporinv 27611 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝐴𝐺(𝑁𝐴)) = (GId‘𝐺))
26253adant3 1124 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐺(𝑁𝐴)) = (GId‘𝐺))
2713, 24, 263eqtrd 2762 . 2 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((𝐴𝐺𝐵)𝐺((𝑁𝐵)𝐺(𝑁𝐴))) = (GId‘𝐺))
284grpocl 27584 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐺𝐵) ∈ 𝑋)
294, 14, 5grpoinvid1 27612 . . 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 1439 . 2 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((𝑁‘(𝐴𝐺𝐵)) = ((𝑁𝐵)𝐺(𝑁𝐴)) ↔ ((𝐴𝐺𝐵)𝐺((𝑁𝐵)𝐺(𝑁𝐴))) = (GId‘𝐺)))
3127, 30mpbird 247 1 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝑁‘(𝐴𝐺𝐵)) = ((𝑁𝐵)𝐺(𝑁𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  w3a 1072   = wceq 1596  wcel 2103  ran crn 5219  cfv 6001  (class class class)co 6765  GrpOpcgr 27573  GIdcgi 27574  invcgn 27575
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-rep 4879  ax-sep 4889  ax-nul 4897  ax-pr 5011  ax-un 7066
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-reu 3021  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-iun 4630  df-br 4761  df-opab 4821  df-mpt 4838  df-id 5128  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-riota 6726  df-ov 6768  df-grpo 27577  df-gid 27578  df-ginv 27579
This theorem is referenced by:  grpoinvdiv  27621
  Copyright terms: Public domain W3C validator