Theorem grpinvfng 13650

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 13164	. . . . . . 7 ⊢ Base Fn V
3		elex 2813	. . . . . . 7 ⊢ (𝐺 ∈ 𝑉 → 𝐺 ∈ V)
4		funfvex 5659	. . . . . . . 8 ⊢ ((Fun Base ∧ 𝐺 ∈ dom Base) → (Base‘𝐺) ∈ V)
5	4	funfni 5434	. . . . . . 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 2317	. . . . 5 ⊢ (𝐺 ∈ 𝑉 → 𝐵 ∈ V)
8		riotaexg 5980	. . . . 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 2605	. . 3 ⊢ (𝐺 ∈ 𝑉 → ∀𝑥 ∈ 𝐵 (℩𝑦 ∈ 𝐵 (𝑦(+g‘𝐺)𝑥) = (0g‘𝐺)) ∈ V)
11		eqid 2230	. . . 4 ⊢ (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦(+g‘𝐺)𝑥) = (0g‘𝐺))) = (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦(+g‘𝐺)𝑥) = (0g‘𝐺)))
12	11	fnmpt 5461	. . 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 2230	. . . 4 ⊢ (+g‘𝐺) = (+g‘𝐺)
15		eqid 2230	. . . 4 ⊢ (0g‘𝐺) = (0g‘𝐺)
16		grpinvfn.n	. . . 4 ⊢ 𝑁 = (invg‘𝐺)
17	1, 14, 15, 16	grpinvfvalg 13648	. . 3 ⊢ (𝐺 ∈ 𝑉 → 𝑁 = (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦(+g‘𝐺)𝑥) = (0g‘𝐺))))
18	17	fneq1d 5422	. 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 1397   ∈ wcel 2201  ∀wral 2509  Vcvv 2801   ↦ cmpt 4151   Fn wfn 5323  ‘cfv 5328  ℩crio 5975  (class class class)co 6023  Basecbs 13105  +_gcplusg 13183  0_gc0g 13362  inv_gcminusg 13607

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2203  ax-14 2204  ax-ext 2212  ax-coll 4205  ax-sep 4208  ax-pow 4266  ax-pr 4301  ax-un 4532  ax-cnex 8128  ax-resscn 8129  ax-1re 8131  ax-addrcl 8134

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ral 2514  df-rex 2515  df-reu 2516  df-rab 2518  df-v 2803  df-sbc 3031  df-csb 3127  df-un 3203  df-in 3205  df-ss 3212  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-uni 3895  df-int 3930  df-iun 3973  df-br 4090  df-opab 4152  df-mpt 4153  df-id 4392  df-xp 4733  df-rel 4734  df-cnv 4735  df-co 4736  df-dm 4737  df-rn 4738  df-res 4739  df-ima 4740  df-iota 5288  df-fun 5330  df-fn 5331  df-f 5332  df-f1 5333  df-fo 5334  df-f1o 5335  df-fv 5336  df-riota 5976  df-ov 6026  df-inn 9149  df-ndx 13108  df-slot 13109  df-base 13111  df-minusg 13610

This theorem is referenced by:  isgrpinv  13660  mulgval  13732  mulgfng  13734  invrfvald  14160