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Theorem grpinvfn 18083
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
grpinvfn 𝑁 Fn 𝐵

Proof of Theorem grpinvfn
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 riotaex 7107 . 2 (𝑦𝐵 (𝑦(+g𝐺)𝑥) = (0g𝐺)) ∈ V
2 grpinvfn.b . . 3 𝐵 = (Base‘𝐺)
3 eqid 2818 . . 3 (+g𝐺) = (+g𝐺)
4 eqid 2818 . . 3 (0g𝐺) = (0g𝐺)
5 grpinvfn.n . . 3 𝑁 = (invg𝐺)
62, 3, 4, 5grpinvfval 18080 . 2 𝑁 = (𝑥𝐵 ↦ (𝑦𝐵 (𝑦(+g𝐺)𝑥) = (0g𝐺)))
71, 6fnmpti 6484 1 𝑁 Fn 𝐵
Colors of variables: wff setvar class
Syntax hints:   = wceq 1528   Fn wfn 6343  cfv 6348  crio 7102  (class class class)co 7145  Basecbs 16471  +gcplusg 16553  0gc0g 16701  invgcminusg 18042
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-riota 7103  df-ov 7148  df-minusg 18045
This theorem is referenced by:  grpinvfvi  18084  isgrpinv  18094  invrfval  19352
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