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Theorem grpinvfng 12923
Description: Functionality of the group inverse function. (Contributed by Stefan O'Rear, 21-Mar-2015.)
Hypotheses
Ref Expression
grpinvfn.b 𝐵 = (Base‘𝐺)
grpinvfn.n 𝑁 = (invg𝐺)
Assertion
Ref Expression
grpinvfng (𝐺𝑉𝑁 Fn 𝐵)

Proof of Theorem grpinvfng
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 grpinvfn.b . . . . . 6 𝐵 = (Base‘𝐺)
2 basfn 12523 . . . . . . 7 Base Fn V
3 elex 2750 . . . . . . 7 (𝐺𝑉𝐺 ∈ V)
4 funfvex 5534 . . . . . . . 8 ((Fun Base ∧ 𝐺 ∈ dom Base) → (Base‘𝐺) ∈ V)
54funfni 5318 . . . . . . 7 ((Base Fn V ∧ 𝐺 ∈ V) → (Base‘𝐺) ∈ V)
62, 3, 5sylancr 414 . . . . . 6 (𝐺𝑉 → (Base‘𝐺) ∈ V)
71, 6eqeltrid 2264 . . . . 5 (𝐺𝑉𝐵 ∈ V)
8 riotaexg 5838 . . . . 5 (𝐵 ∈ V → (𝑦𝐵 (𝑦(+g𝐺)𝑥) = (0g𝐺)) ∈ V)
97, 8syl 14 . . . 4 (𝐺𝑉 → (𝑦𝐵 (𝑦(+g𝐺)𝑥) = (0g𝐺)) ∈ V)
109ralrimivw 2551 . . 3 (𝐺𝑉 → ∀𝑥𝐵 (𝑦𝐵 (𝑦(+g𝐺)𝑥) = (0g𝐺)) ∈ V)
11 eqid 2177 . . . 4 (𝑥𝐵 ↦ (𝑦𝐵 (𝑦(+g𝐺)𝑥) = (0g𝐺))) = (𝑥𝐵 ↦ (𝑦𝐵 (𝑦(+g𝐺)𝑥) = (0g𝐺)))
1211fnmpt 5344 . . 3 (∀𝑥𝐵 (𝑦𝐵 (𝑦(+g𝐺)𝑥) = (0g𝐺)) ∈ V → (𝑥𝐵 ↦ (𝑦𝐵 (𝑦(+g𝐺)𝑥) = (0g𝐺))) Fn 𝐵)
1310, 12syl 14 . 2 (𝐺𝑉 → (𝑥𝐵 ↦ (𝑦𝐵 (𝑦(+g𝐺)𝑥) = (0g𝐺))) Fn 𝐵)
14 eqid 2177 . . . 4 (+g𝐺) = (+g𝐺)
15 eqid 2177 . . . 4 (0g𝐺) = (0g𝐺)
16 grpinvfn.n . . . 4 𝑁 = (invg𝐺)
171, 14, 15, 16grpinvfvalg 12921 . . 3 (𝐺𝑉𝑁 = (𝑥𝐵 ↦ (𝑦𝐵 (𝑦(+g𝐺)𝑥) = (0g𝐺))))
1817fneq1d 5308 . 2 (𝐺𝑉 → (𝑁 Fn 𝐵 ↔ (𝑥𝐵 ↦ (𝑦𝐵 (𝑦(+g𝐺)𝑥) = (0g𝐺))) Fn 𝐵))
1913, 18mpbird 167 1 (𝐺𝑉𝑁 Fn 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1353  wcel 2148  wral 2455  Vcvv 2739  cmpt 4066   Fn wfn 5213  cfv 5218  crio 5833  (class class class)co 5878  Basecbs 12465  +gcplusg 12539  0gc0g 12711  invgcminusg 12884
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-cnex 7905  ax-resscn 7906  ax-1re 7908  ax-addrcl 7911
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-riota 5834  df-ov 5881  df-inn 8923  df-ndx 12468  df-slot 12469  df-base 12471  df-minusg 12887
This theorem is referenced by:  isgrpinv  12932  mulgval  12992  mulgfng  12993  invrfvald  13297
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