Theorem hgmapfnN 41912

Description: Functionality of scalar sigma map. (Contributed by NM, 7-Jun-2015.) (New usage is discouraged.)

Hypotheses

Ref	Expression
hgmapfn.h	⊢ 𝐻 = (LHyp‘𝐾)
hgmapfn.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
hgmapfn.r	⊢ 𝑅 = (Scalar‘𝑈)
hgmapfn.b	⊢ 𝐵 = (Base‘𝑅)
hgmapfn.g	⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊)
hgmapfn.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))

Assertion

Ref	Expression
hgmapfnN	⊢ (𝜑 → 𝐺 Fn 𝐵)

Proof of Theorem hgmapfnN

Dummy variables 𝑗 𝑘 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		riotaex 7371	. . 3 ⊢ (℩𝑗 ∈ 𝐵 ∀𝑥 ∈ (Base‘𝑈)(((HDMap‘𝐾)‘𝑊)‘(𝑘( ·𝑠 ‘𝑈)𝑥)) = (𝑗( ·𝑠 ‘((LCDual‘𝐾)‘𝑊))(((HDMap‘𝐾)‘𝑊)‘𝑥))) ∈ V
2		eqid 2736	. . 3 ⊢ (𝑘 ∈ 𝐵 ↦ (℩𝑗 ∈ 𝐵 ∀𝑥 ∈ (Base‘𝑈)(((HDMap‘𝐾)‘𝑊)‘(𝑘( ·𝑠 ‘𝑈)𝑥)) = (𝑗( ·𝑠 ‘((LCDual‘𝐾)‘𝑊))(((HDMap‘𝐾)‘𝑊)‘𝑥)))) = (𝑘 ∈ 𝐵 ↦ (℩𝑗 ∈ 𝐵 ∀𝑥 ∈ (Base‘𝑈)(((HDMap‘𝐾)‘𝑊)‘(𝑘( ·𝑠 ‘𝑈)𝑥)) = (𝑗( ·𝑠 ‘((LCDual‘𝐾)‘𝑊))(((HDMap‘𝐾)‘𝑊)‘𝑥))))
3	1, 2	fnmpti 6686	. 2 ⊢ (𝑘 ∈ 𝐵 ↦ (℩𝑗 ∈ 𝐵 ∀𝑥 ∈ (Base‘𝑈)(((HDMap‘𝐾)‘𝑊)‘(𝑘( ·𝑠 ‘𝑈)𝑥)) = (𝑗( ·𝑠 ‘((LCDual‘𝐾)‘𝑊))(((HDMap‘𝐾)‘𝑊)‘𝑥)))) Fn 𝐵
4		hgmapfn.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
5		hgmapfn.u	. . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
6		eqid 2736	. . . 4 ⊢ (Base‘𝑈) = (Base‘𝑈)
7		eqid 2736	. . . 4 ⊢ ( ·𝑠 ‘𝑈) = ( ·𝑠 ‘𝑈)
8		hgmapfn.r	. . . 4 ⊢ 𝑅 = (Scalar‘𝑈)
9		hgmapfn.b	. . . 4 ⊢ 𝐵 = (Base‘𝑅)
10		eqid 2736	. . . 4 ⊢ ((LCDual‘𝐾)‘𝑊) = ((LCDual‘𝐾)‘𝑊)
11		eqid 2736	. . . 4 ⊢ ( ·𝑠 ‘((LCDual‘𝐾)‘𝑊)) = ( ·𝑠 ‘((LCDual‘𝐾)‘𝑊))
12		eqid 2736	. . . 4 ⊢ ((HDMap‘𝐾)‘𝑊) = ((HDMap‘𝐾)‘𝑊)
13		hgmapfn.g	. . . 4 ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊)
14		hgmapfn.k	. . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
15	4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14	hgmapfval 41910	. . 3 ⊢ (𝜑 → 𝐺 = (𝑘 ∈ 𝐵 ↦ (℩𝑗 ∈ 𝐵 ∀𝑥 ∈ (Base‘𝑈)(((HDMap‘𝐾)‘𝑊)‘(𝑘( ·𝑠 ‘𝑈)𝑥)) = (𝑗( ·𝑠 ‘((LCDual‘𝐾)‘𝑊))(((HDMap‘𝐾)‘𝑊)‘𝑥)))))
16	15	fneq1d 6636	. 2 ⊢ (𝜑 → (𝐺 Fn 𝐵 ↔ (𝑘 ∈ 𝐵 ↦ (℩𝑗 ∈ 𝐵 ∀𝑥 ∈ (Base‘𝑈)(((HDMap‘𝐾)‘𝑊)‘(𝑘( ·𝑠 ‘𝑈)𝑥)) = (𝑗( ·𝑠 ‘((LCDual‘𝐾)‘𝑊))(((HDMap‘𝐾)‘𝑊)‘𝑥)))) Fn 𝐵))
17	3, 16	mpbiri 258	1 ⊢ (𝜑 → 𝐺 Fn 𝐵)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3052   ↦ cmpt 5206   Fn wfn 6531  ‘cfv 6536  ℩crio 7366  (class class class)co 7410  Basecbs 17233  Scalarcsca 17279   ·_𝑠 cvsca 17280  HLchlt 39373  LHypclh 40008  DVecHcdvh 41102  LCDualclcd 41610  HDMapchdma 41816  HGMapchg 41907

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pr 5407

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-hgmap 41908

This theorem is referenced by:  hgmaprnlem1N  41920  hgmaprnN  41925  hgmapf1oN  41927