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Theorem grpinvfn 18797
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 7318 . 2 (𝑦𝐵 (𝑦(+g𝐺)𝑥) = (0g𝐺)) ∈ V
2 grpinvfn.b . . 3 𝐵 = (Base‘𝐺)
3 eqid 2733 . . 3 (+g𝐺) = (+g𝐺)
4 eqid 2733 . . 3 (0g𝐺) = (0g𝐺)
5 grpinvfn.n . . 3 𝑁 = (invg𝐺)
62, 3, 4, 5grpinvfval 18794 . 2 𝑁 = (𝑥𝐵 ↦ (𝑦𝐵 (𝑦(+g𝐺)𝑥) = (0g𝐺)))
71, 6fnmpti 6645 1 𝑁 Fn 𝐵
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542   Fn wfn 6492  cfv 6497  crio 7313  (class class class)co 7358  Basecbs 17088  +gcplusg 17138  0gc0g 17326  invgcminusg 18754
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fv 6505  df-riota 7314  df-ov 7361  df-minusg 18757
This theorem is referenced by:  grpinvfvi  18798  isgrpinv  18809  invrfval  20107  mplsubglem  21421  mhpinvcl  21558
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