Theorem grpinvfng 13420

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 12934	. . . . . . 7 ⊢ Base Fn V
3		elex 2784	. . . . . . 7 ⊢ (𝐺 ∈ 𝑉 → 𝐺 ∈ V)
4		funfvex 5600	. . . . . . . 8 ⊢ ((Fun Base ∧ 𝐺 ∈ dom Base) → (Base‘𝐺) ∈ V)
5	4	funfni 5381	. . . . . . 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 2293	. . . . 5 ⊢ (𝐺 ∈ 𝑉 → 𝐵 ∈ V)
8		riotaexg 5910	. . . . 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 2581	. . 3 ⊢ (𝐺 ∈ 𝑉 → ∀𝑥 ∈ 𝐵 (℩𝑦 ∈ 𝐵 (𝑦(+g‘𝐺)𝑥) = (0g‘𝐺)) ∈ V)
11		eqid 2206	. . . 4 ⊢ (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦(+g‘𝐺)𝑥) = (0g‘𝐺))) = (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦(+g‘𝐺)𝑥) = (0g‘𝐺)))
12	11	fnmpt 5408	. . 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 2206	. . . 4 ⊢ (+g‘𝐺) = (+g‘𝐺)
15		eqid 2206	. . . 4 ⊢ (0g‘𝐺) = (0g‘𝐺)
16		grpinvfn.n	. . . 4 ⊢ 𝑁 = (invg‘𝐺)
17	1, 14, 15, 16	grpinvfvalg 13418	. . 3 ⊢ (𝐺 ∈ 𝑉 → 𝑁 = (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦(+g‘𝐺)𝑥) = (0g‘𝐺))))
18	17	fneq1d 5369	. 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 1373   ∈ wcel 2177  ∀wral 2485  Vcvv 2773   ↦ cmpt 4109   Fn wfn 5271  ‘cfv 5276  ℩crio 5905  (class class class)co 5951  Basecbs 12876  +_gcplusg 12953  0_gc0g 13132  inv_gcminusg 13377

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4163  ax-sep 4166  ax-pow 4222  ax-pr 4257  ax-un 4484  ax-cnex 8023  ax-resscn 8024  ax-1re 8026  ax-addrcl 8029

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3000  df-csb 3095  df-un 3171  df-in 3173  df-ss 3180  df-pw 3619  df-sn 3640  df-pr 3641  df-op 3643  df-uni 3853  df-int 3888  df-iun 3931  df-br 4048  df-opab 4110  df-mpt 4111  df-id 4344  df-xp 4685  df-rel 4686  df-cnv 4687  df-co 4688  df-dm 4689  df-rn 4690  df-res 4691  df-ima 4692  df-iota 5237  df-fun 5278  df-fn 5279  df-f 5280  df-f1 5281  df-fo 5282  df-f1o 5283  df-fv 5284  df-riota 5906  df-ov 5954  df-inn 9044  df-ndx 12879  df-slot 12880  df-base 12882  df-minusg 13380

This theorem is referenced by:  isgrpinv  13430  mulgval  13502  mulgfng  13504  invrfvald  13928