Theorem hgmapfnN 43085

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 7388	. . 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 2763	. . 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 6690	. 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 2763	. . . 4 ⊢ (Base‘𝑈) = (Base‘𝑈)
7		eqid 2763	. . . 4 ⊢ ( ·𝑠 ‘𝑈) = ( ·𝑠 ‘𝑈)
8		hgmapfn.r	. . . 4 ⊢ 𝑅 = (Scalar‘𝑈)
9		hgmapfn.b	. . . 4 ⊢ 𝐵 = (Base‘𝑅)
10		eqid 2763	. . . 4 ⊢ ((LCDual‘𝐾)‘𝑊) = ((LCDual‘𝐾)‘𝑊)
11		eqid 2763	. . . 4 ⊢ ( ·𝑠 ‘((LCDual‘𝐾)‘𝑊)) = ( ·𝑠 ‘((LCDual‘𝐾)‘𝑊))
12		eqid 2763	. . . 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 43083	. . 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 6640	. 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 261	1 ⊢ (𝜑 → 𝐺 Fn 𝐵)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2146  ∀wral 3079   ↦ cmpt 5188   Fn wfn 6541  ‘cfv 6546  ℩crio 7383  (class class class)co 7427  Basecbs 17399  Scalarcsca 17443   ·_𝑠 cvsca 17444  HLchlt 40547  LHypclh 41181  DVecHcdvh 42275  LCDualclcd 42783  HDMapchdma 42989  HGMapchg 43080

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-rep 5234  ax-sep 5251  ax-nul 5262  ax-pr 5394

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3369  df-rab 3416  df-v 3455  df-sbc 3742  df-csb 3851  df-ss 3900  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-sn 4588  df-pr 4590  df-op 4838  df-uni 4870  df-iun 4955  df-br 5106  df-opab 5170  df-mpt 5189  df-id 5549  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6500  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-riota 7384  df-ov 7430  df-hgmap 43081

This theorem is used by:  hgmaprnlem1N  43093  hgmaprnN  43098  hgmapf1oN  43100