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Theorem fn0g 12958
Description: The group zero extractor is a function. (Contributed by Stefan O'Rear, 10-Jan-2015.)
Assertion
Ref Expression
fn0g 0g Fn V

Proof of Theorem fn0g
Dummy variables 𝑒 𝑔 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-riota 5873 . . 3 (𝑒 ∈ (Base‘𝑔)∀𝑥 ∈ (Base‘𝑔)((𝑒(+g𝑔)𝑥) = 𝑥 ∧ (𝑥(+g𝑔)𝑒) = 𝑥)) = (℩𝑒(𝑒 ∈ (Base‘𝑔) ∧ ∀𝑥 ∈ (Base‘𝑔)((𝑒(+g𝑔)𝑥) = 𝑥 ∧ (𝑥(+g𝑔)𝑒) = 𝑥)))
2 basfn 12676 . . . . 5 Base Fn V
3 vex 2763 . . . . 5 𝑔 ∈ V
4 funfvex 5571 . . . . . 6 ((Fun Base ∧ 𝑔 ∈ dom Base) → (Base‘𝑔) ∈ V)
54funfni 5354 . . . . 5 ((Base Fn V ∧ 𝑔 ∈ V) → (Base‘𝑔) ∈ V)
62, 3, 5mp2an 426 . . . 4 (Base‘𝑔) ∈ V
7 riotaexg 5877 . . . 4 ((Base‘𝑔) ∈ V → (𝑒 ∈ (Base‘𝑔)∀𝑥 ∈ (Base‘𝑔)((𝑒(+g𝑔)𝑥) = 𝑥 ∧ (𝑥(+g𝑔)𝑒) = 𝑥)) ∈ V)
86, 7ax-mp 5 . . 3 (𝑒 ∈ (Base‘𝑔)∀𝑥 ∈ (Base‘𝑔)((𝑒(+g𝑔)𝑥) = 𝑥 ∧ (𝑥(+g𝑔)𝑒) = 𝑥)) ∈ V
91, 8eqeltrri 2267 . 2 (℩𝑒(𝑒 ∈ (Base‘𝑔) ∧ ∀𝑥 ∈ (Base‘𝑔)((𝑒(+g𝑔)𝑥) = 𝑥 ∧ (𝑥(+g𝑔)𝑒) = 𝑥))) ∈ V
10 df-0g 12869 . 2 0g = (𝑔 ∈ V ↦ (℩𝑒(𝑒 ∈ (Base‘𝑔) ∧ ∀𝑥 ∈ (Base‘𝑔)((𝑒(+g𝑔)𝑥) = 𝑥 ∧ (𝑥(+g𝑔)𝑒) = 𝑥))))
119, 10fnmpti 5382 1 0g Fn V
Colors of variables: wff set class
Syntax hints:  wa 104   = wceq 1364  wcel 2164  wral 2472  Vcvv 2760  cio 5213   Fn wfn 5249  cfv 5254  crio 5872  (class class class)co 5918  Basecbs 12618  +gcplusg 12695  0gc0g 12867
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-cnex 7963  ax-resscn 7964  ax-1re 7966  ax-addrcl 7969
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2986  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-iota 5215  df-fun 5256  df-fn 5257  df-fv 5262  df-riota 5873  df-inn 8983  df-ndx 12621  df-slot 12622  df-base 12624  df-0g 12869
This theorem is referenced by:  fngsum  12971  igsumvalx  12972  gsumfzval  12974  gsum0g  12979  0mhm  13058  mulgval  13192  mulgfng  13194  issrg  13461  isdomn  13765
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