Theorem grpinvfng 13543

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 13057	. . . . . . 7 ⊢ Base Fn V
3		elex 2791	. . . . . . 7 ⊢ (𝐺 ∈ 𝑉 → 𝐺 ∈ V)
4		funfvex 5620	. . . . . . . 8 ⊢ ((Fun Base ∧ 𝐺 ∈ dom Base) → (Base‘𝐺) ∈ V)
5	4	funfni 5399	. . . . . . 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 2296	. . . . 5 ⊢ (𝐺 ∈ 𝑉 → 𝐵 ∈ V)
8		riotaexg 5931	. . . . 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 2584	. . 3 ⊢ (𝐺 ∈ 𝑉 → ∀𝑥 ∈ 𝐵 (℩𝑦 ∈ 𝐵 (𝑦(+g‘𝐺)𝑥) = (0g‘𝐺)) ∈ V)
11		eqid 2209	. . . 4 ⊢ (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦(+g‘𝐺)𝑥) = (0g‘𝐺))) = (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦(+g‘𝐺)𝑥) = (0g‘𝐺)))
12	11	fnmpt 5426	. . 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 2209	. . . 4 ⊢ (+g‘𝐺) = (+g‘𝐺)
15		eqid 2209	. . . 4 ⊢ (0g‘𝐺) = (0g‘𝐺)
16		grpinvfn.n	. . . 4 ⊢ 𝑁 = (invg‘𝐺)
17	1, 14, 15, 16	grpinvfvalg 13541	. . 3 ⊢ (𝐺 ∈ 𝑉 → 𝑁 = (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦(+g‘𝐺)𝑥) = (0g‘𝐺))))
18	17	fneq1d 5387	. 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 1375   ∈ wcel 2180  ∀wral 2488  Vcvv 2779   ↦ cmpt 4124   Fn wfn 5289  ‘cfv 5294  ℩crio 5926  (class class class)co 5974  Basecbs 12998  +_gcplusg 13076  0_gc0g 13255  inv_gcminusg 13500

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 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-13 2182  ax-14 2183  ax-ext 2191  ax-coll 4178  ax-sep 4181  ax-pow 4237  ax-pr 4272  ax-un 4501  ax-cnex 8058  ax-resscn 8059  ax-1re 8061  ax-addrcl 8064

This theorem depends on definitions:  df-bi 117  df-3an 985  df-tru 1378  df-nf 1487  df-sb 1789  df-eu 2060  df-mo 2061  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ral 2493  df-rex 2494  df-reu 2495  df-rab 2497  df-v 2781  df-sbc 3009  df-csb 3105  df-un 3181  df-in 3183  df-ss 3190  df-pw 3631  df-sn 3652  df-pr 3653  df-op 3655  df-uni 3868  df-int 3903  df-iun 3946  df-br 4063  df-opab 4125  df-mpt 4126  df-id 4361  df-xp 4702  df-rel 4703  df-cnv 4704  df-co 4705  df-dm 4706  df-rn 4707  df-res 4708  df-ima 4709  df-iota 5254  df-fun 5296  df-fn 5297  df-f 5298  df-f1 5299  df-fo 5300  df-f1o 5301  df-fv 5302  df-riota 5927  df-ov 5977  df-inn 9079  df-ndx 13001  df-slot 13002  df-base 13004  df-minusg 13503

This theorem is referenced by:  isgrpinv  13553  mulgval  13625  mulgfng  13627  invrfvald  14051