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Theorem grpinvfn 17778
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 6843 . 2 (𝑦𝐵 (𝑦(+g𝐺)𝑥) = (0g𝐺)) ∈ V
2 grpinvfn.b . . 3 𝐵 = (Base‘𝐺)
3 eqid 2799 . . 3 (+g𝐺) = (+g𝐺)
4 eqid 2799 . . 3 (0g𝐺) = (0g𝐺)
5 grpinvfn.n . . 3 𝑁 = (invg𝐺)
62, 3, 4, 5grpinvfval 17776 . 2 𝑁 = (𝑥𝐵 ↦ (𝑦𝐵 (𝑦(+g𝐺)𝑥) = (0g𝐺)))
71, 6fnmpti 6233 1 𝑁 Fn 𝐵
Colors of variables: wff setvar class
Syntax hints:   = wceq 1653   Fn wfn 6096  cfv 6101  crio 6838  (class class class)co 6878  Basecbs 16184  +gcplusg 16267  0gc0g 16415  invgcminusg 17739
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-riota 6839  df-ov 6881  df-minusg 17742
This theorem is referenced by:  grpinvfvi  17779  isgrpinv  17788  invrfval  18989
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