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Theorem grpinvfn 18911
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 7365 . 2 (𝑦𝐵 (𝑦(+g𝐺)𝑥) = (0g𝐺)) ∈ V
2 grpinvfn.b . . 3 𝐵 = (Base‘𝐺)
3 eqid 2726 . . 3 (+g𝐺) = (+g𝐺)
4 eqid 2726 . . 3 (0g𝐺) = (0g𝐺)
5 grpinvfn.n . . 3 𝑁 = (invg𝐺)
62, 3, 4, 5grpinvfval 18908 . 2 𝑁 = (𝑥𝐵 ↦ (𝑦𝐵 (𝑦(+g𝐺)𝑥) = (0g𝐺)))
71, 6fnmpti 6687 1 𝑁 Fn 𝐵
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533   Fn wfn 6532  cfv 6537  crio 7360  (class class class)co 7405  Basecbs 17153  +gcplusg 17206  0gc0g 17394  invgcminusg 18864
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-riota 7361  df-ov 7408  df-minusg 18867
This theorem is referenced by:  grpinvfvi  18912  isgrpinv  18923  invrfval  20291  mplsubglem  21900  mhpinvcl  22035
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