Theorem fn0g 13672

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.

Step	Hyp	Ref	Expression
1		df-riota 6011	. . 3 ⊢ (℩𝑒 ∈ (Base‘𝑔)∀𝑥 ∈ (Base‘𝑔)((𝑒(+g‘𝑔)𝑥) = 𝑥 ∧ (𝑥(+g‘𝑔)𝑒) = 𝑥)) = (℩𝑒(𝑒 ∈ (Base‘𝑔) ∧ ∀𝑥 ∈ (Base‘𝑔)((𝑒(+g‘𝑔)𝑥) = 𝑥 ∧ (𝑥(+g‘𝑔)𝑒) = 𝑥)))
2		basfn 13355	. . . . 5 ⊢ Base Fn V
3		vex 2818	. . . . 5 ⊢ 𝑔 ∈ V
4		funfvex 5692	. . . . . 6 ⊢ ((Fun Base ∧ 𝑔 ∈ dom Base) → (Base‘𝑔) ∈ V)
5	4	funfni 5463	. . . . 5 ⊢ ((Base Fn V ∧ 𝑔 ∈ V) → (Base‘𝑔) ∈ V)
6	2, 3, 5	mp2an 426	. . . 4 ⊢ (Base‘𝑔) ∈ V
7		riotaexg 6015	. . . 4 ⊢ ((Base‘𝑔) ∈ V → (℩𝑒 ∈ (Base‘𝑔)∀𝑥 ∈ (Base‘𝑔)((𝑒(+g‘𝑔)𝑥) = 𝑥 ∧ (𝑥(+g‘𝑔)𝑒) = 𝑥)) ∈ V)
8	6, 7	ax-mp 5	. . 3 ⊢ (℩𝑒 ∈ (Base‘𝑔)∀𝑥 ∈ (Base‘𝑔)((𝑒(+g‘𝑔)𝑥) = 𝑥 ∧ (𝑥(+g‘𝑔)𝑒) = 𝑥)) ∈ V
9	1, 8	eqeltrri 2308	. 2 ⊢ (℩𝑒(𝑒 ∈ (Base‘𝑔) ∧ ∀𝑥 ∈ (Base‘𝑔)((𝑒(+g‘𝑔)𝑥) = 𝑥 ∧ (𝑥(+g‘𝑔)𝑒) = 𝑥))) ∈ V
10		df-0g 13555	. 2 ⊢ 0g = (𝑔 ∈ V ↦ (℩𝑒(𝑒 ∈ (Base‘𝑔) ∧ ∀𝑥 ∈ (Base‘𝑔)((𝑒(+g‘𝑔)𝑥) = 𝑥 ∧ (𝑥(+g‘𝑔)𝑒) = 𝑥))))
11	9, 10	fnmpti 5492	1 ⊢ 0g Fn V

Syntax hints:   ∧ wa 104   = wceq 1398   ∈ wcel 2205  ∀wral 2522  Vcvv 2815  ℩cio 5315   Fn wfn 5352  ‘cfv 5357  ℩crio 6010  (class class class)co 6058  Basecbs 13296  +_gcplusg 13374  0_gc0g 13553

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-cnex 8234  ax-resscn 8235  ax-1re 8237  ax-addrcl 8240

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3046  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-iota 5317  df-fun 5359  df-fn 5360  df-fv 5365  df-riota 6011  df-inn 9255  df-ndx 13299  df-slot 13300  df-base 13302  df-0g 13555

This theorem is referenced by:  fngsum  13685  igsumvalx  13686  gsumfzval  13688  gsum0g  13693  prdsidlem  13744  pws0g  13748  0mhm  13783  prdsinvlem  13905  mulgval  13923  mulgfng  13925  issrg  14193  isdomn  14501