Theorem grpinvfng 13778

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 = (invg‘𝐺)

Assertion

Ref	Expression
grpinvfng	⊢ (𝐺 ∈ 𝑉 → 𝑁 Fn 𝐵)

Proof of Theorem grpinvfng

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		grpinvfn.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐺)
2		basfn 13292	. . . . . . 7 ⊢ Base Fn V
3		elex 2827	. . . . . . 7 ⊢ (𝐺 ∈ 𝑉 → 𝐺 ∈ V)
4		funfvex 5689	. . . . . . . 8 ⊢ ((Fun Base ∧ 𝐺 ∈ dom Base) → (Base‘𝐺) ∈ V)
5	4	funfni 5460	. . . . . . 7 ⊢ ((Base Fn V ∧ 𝐺 ∈ V) → (Base‘𝐺) ∈ V)
6	2, 3, 5	sylancr 414	. . . . . 6 ⊢ (𝐺 ∈ 𝑉 → (Base‘𝐺) ∈ V)
7	1, 6	eqeltrid 2321	. . . . 5 ⊢ (𝐺 ∈ 𝑉 → 𝐵 ∈ V)
8		riotaexg 6009	. . . . 5 ⊢ (𝐵 ∈ V → (℩𝑦 ∈ 𝐵 (𝑦(+g‘𝐺)𝑥) = (0g‘𝐺)) ∈ V)
9	7, 8	syl 14	. . . 4 ⊢ (𝐺 ∈ 𝑉 → (℩𝑦 ∈ 𝐵 (𝑦(+g‘𝐺)𝑥) = (0g‘𝐺)) ∈ V)
10	9	ralrimivw 2618	. . 3 ⊢ (𝐺 ∈ 𝑉 → ∀𝑥 ∈ 𝐵 (℩𝑦 ∈ 𝐵 (𝑦(+g‘𝐺)𝑥) = (0g‘𝐺)) ∈ V)
11		eqid 2234	. . . 4 ⊢ (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦(+g‘𝐺)𝑥) = (0g‘𝐺))) = (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦(+g‘𝐺)𝑥) = (0g‘𝐺)))
12	11	fnmpt 5487	. . 3 ⊢ (∀𝑥 ∈ 𝐵 (℩𝑦 ∈ 𝐵 (𝑦(+g‘𝐺)𝑥) = (0g‘𝐺)) ∈ V → (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦(+g‘𝐺)𝑥) = (0g‘𝐺))) Fn 𝐵)
13	10, 12	syl 14	. 2 ⊢ (𝐺 ∈ 𝑉 → (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦(+g‘𝐺)𝑥) = (0g‘𝐺))) Fn 𝐵)
14		eqid 2234	. . . 4 ⊢ (+g‘𝐺) = (+g‘𝐺)
15		eqid 2234	. . . 4 ⊢ (0g‘𝐺) = (0g‘𝐺)
16		grpinvfn.n	. . . 4 ⊢ 𝑁 = (invg‘𝐺)
17	1, 14, 15, 16	grpinvfvalg 13776	. . 3 ⊢ (𝐺 ∈ 𝑉 → 𝑁 = (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦(+g‘𝐺)𝑥) = (0g‘𝐺))))
18	17	fneq1d 5448	. 2 ⊢ (𝐺 ∈ 𝑉 → (𝑁 Fn 𝐵 ↔ (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦(+g‘𝐺)𝑥) = (0g‘𝐺))) Fn 𝐵))
19	13, 18	mpbird 167	1 ⊢ (𝐺 ∈ 𝑉 → 𝑁 Fn 𝐵)

Syntax hints:   → wi 4   = wceq 1398   ∈ wcel 2205  ∀wral 2522  Vcvv 2815   ↦ cmpt 4173   Fn wfn 5349  ‘cfv 5354  ℩crio 6004  (class class class)co 6052  Basecbs 13233  +_gcplusg 13311  0_gc0g 13490  inv_gcminusg 13735

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-coll 4227  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-cnex 8223  ax-resscn 8224  ax-1re 8226  ax-addrcl 8229

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-reu 2529  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-riota 6005  df-ov 6055  df-inn 9243  df-ndx 13236  df-slot 13237  df-base 13239  df-minusg 13738

This theorem is referenced by:  isgrpinv  13788  mulgval  13860  mulgfng  13862  invrfvald  14289