Theorem hvmapfval 37048

Description: Map from nonzero vectors to nonzero functionals in the closed kernel dual space. (Contributed by NM, 23-Mar-2015.)

Hypotheses

Ref	Expression
hvmapval.h	⊢ 𝐻 = (LHyp‘𝐾)
hvmapval.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
hvmapval.o	⊢ 𝑂 = ((ocH‘𝐾)‘𝑊)
hvmapval.v	⊢ 𝑉 = (Base‘𝑈)
hvmapval.p	⊢ + = (+g‘𝑈)
hvmapval.t	⊢ · = ( ·𝑠 ‘𝑈)
hvmapval.z	⊢ 0 = (0g‘𝑈)
hvmapval.s	⊢ 𝑆 = (Scalar‘𝑈)
hvmapval.r	⊢ 𝑅 = (Base‘𝑆)
hvmapval.m	⊢ 𝑀 = ((HVMap‘𝐾)‘𝑊)
hvmapval.k	⊢ (𝜑 → (𝐾 ∈ 𝐴 ∧ 𝑊 ∈ 𝐻))

Assertion

Ref	Expression
hvmapfval	⊢ (𝜑 → 𝑀 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑗 ∈ 𝑅 ∃𝑡 ∈ (𝑂‘{𝑥})𝑣 = (𝑡 + (𝑗 · 𝑥))))))

Distinct variable groups:   𝑡,𝑗,𝑣,𝑥,𝐾   𝑡,𝑊   𝑡,𝑂   𝑅,𝑗   𝑥,𝑉   𝑗,𝑊,𝑣,𝑥   𝑥, 0

Allowed substitution hints:   𝜑(𝑥,𝑣,𝑡,𝑗)   𝐴(𝑥,𝑣,𝑡,𝑗)   + (𝑥,𝑣,𝑡,𝑗)   𝑅(𝑥,𝑣,𝑡)   𝑆(𝑥,𝑣,𝑡,𝑗)   · (𝑥,𝑣,𝑡,𝑗)   𝑈(𝑥,𝑣,𝑡,𝑗)   𝐻(𝑥,𝑣,𝑡,𝑗)   𝑀(𝑥,𝑣,𝑡,𝑗)   𝑂(𝑥,𝑣,𝑗)   𝑉(𝑣,𝑡,𝑗)   0 (𝑣,𝑡,𝑗)

Proof of Theorem hvmapfval

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		hvmapval.k	. 2 ⊢ (𝜑 → (𝐾 ∈ 𝐴 ∧ 𝑊 ∈ 𝐻))
2		hvmapval.m	. . . 4 ⊢ 𝑀 = ((HVMap‘𝐾)‘𝑊)
3		hvmapval.h	. . . . . 6 ⊢ 𝐻 = (LHyp‘𝐾)
4	3	hvmapffval 37047	. . . . 5 ⊢ (𝐾 ∈ 𝐴 → (HVMap‘𝐾) = (𝑤 ∈ 𝐻 ↦ (𝑥 ∈ ((Base‘((DVecH‘𝐾)‘𝑤)) ∖ {(0g‘((DVecH‘𝐾)‘𝑤))}) ↦ (𝑣 ∈ (Base‘((DVecH‘𝐾)‘𝑤)) ↦ (℩𝑗 ∈ (Base‘(Scalar‘((DVecH‘𝐾)‘𝑤)))∃𝑡 ∈ (((ocH‘𝐾)‘𝑤)‘{𝑥})𝑣 = (𝑡(+g‘((DVecH‘𝐾)‘𝑤))(𝑗( ·𝑠 ‘((DVecH‘𝐾)‘𝑤))𝑥)))))))
5	4	fveq1d 6193	. . . 4 ⊢ (𝐾 ∈ 𝐴 → ((HVMap‘𝐾)‘𝑊) = ((𝑤 ∈ 𝐻 ↦ (𝑥 ∈ ((Base‘((DVecH‘𝐾)‘𝑤)) ∖ {(0g‘((DVecH‘𝐾)‘𝑤))}) ↦ (𝑣 ∈ (Base‘((DVecH‘𝐾)‘𝑤)) ↦ (℩𝑗 ∈ (Base‘(Scalar‘((DVecH‘𝐾)‘𝑤)))∃𝑡 ∈ (((ocH‘𝐾)‘𝑤)‘{𝑥})𝑣 = (𝑡(+g‘((DVecH‘𝐾)‘𝑤))(𝑗( ·𝑠 ‘((DVecH‘𝐾)‘𝑤))𝑥))))))‘𝑊))
6	2, 5	syl5eq 2668	. . 3 ⊢ (𝐾 ∈ 𝐴 → 𝑀 = ((𝑤 ∈ 𝐻 ↦ (𝑥 ∈ ((Base‘((DVecH‘𝐾)‘𝑤)) ∖ {(0g‘((DVecH‘𝐾)‘𝑤))}) ↦ (𝑣 ∈ (Base‘((DVecH‘𝐾)‘𝑤)) ↦ (℩𝑗 ∈ (Base‘(Scalar‘((DVecH‘𝐾)‘𝑤)))∃𝑡 ∈ (((ocH‘𝐾)‘𝑤)‘{𝑥})𝑣 = (𝑡(+g‘((DVecH‘𝐾)‘𝑤))(𝑗( ·𝑠 ‘((DVecH‘𝐾)‘𝑤))𝑥))))))‘𝑊))
7		fveq2 6191	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → ((DVecH‘𝐾)‘𝑤) = ((DVecH‘𝐾)‘𝑊))
8		hvmapval.u	. . . . . . . . 9 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
9	7, 8	syl6eqr 2674	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → ((DVecH‘𝐾)‘𝑤) = 𝑈)
10	9	fveq2d 6195	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (Base‘((DVecH‘𝐾)‘𝑤)) = (Base‘𝑈))
11		hvmapval.v	. . . . . . 7 ⊢ 𝑉 = (Base‘𝑈)
12	10, 11	syl6eqr 2674	. . . . . 6 ⊢ (𝑤 = 𝑊 → (Base‘((DVecH‘𝐾)‘𝑤)) = 𝑉)
13	9	fveq2d 6195	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (0g‘((DVecH‘𝐾)‘𝑤)) = (0g‘𝑈))
14		hvmapval.z	. . . . . . . 8 ⊢ 0 = (0g‘𝑈)
15	13, 14	syl6eqr 2674	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (0g‘((DVecH‘𝐾)‘𝑤)) = 0 )
16	15	sneqd 4189	. . . . . 6 ⊢ (𝑤 = 𝑊 → {(0g‘((DVecH‘𝐾)‘𝑤))} = { 0 })
17	12, 16	difeq12d 3729	. . . . 5 ⊢ (𝑤 = 𝑊 → ((Base‘((DVecH‘𝐾)‘𝑤)) ∖ {(0g‘((DVecH‘𝐾)‘𝑤))}) = (𝑉 ∖ { 0 }))
18	9	fveq2d 6195	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → (Scalar‘((DVecH‘𝐾)‘𝑤)) = (Scalar‘𝑈))
19		hvmapval.s	. . . . . . . . . 10 ⊢ 𝑆 = (Scalar‘𝑈)
20	18, 19	syl6eqr 2674	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (Scalar‘((DVecH‘𝐾)‘𝑤)) = 𝑆)
21	20	fveq2d 6195	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (Base‘(Scalar‘((DVecH‘𝐾)‘𝑤))) = (Base‘𝑆))
22		hvmapval.r	. . . . . . . 8 ⊢ 𝑅 = (Base‘𝑆)
23	21, 22	syl6eqr 2674	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (Base‘(Scalar‘((DVecH‘𝐾)‘𝑤))) = 𝑅)
24		fveq2 6191	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → ((ocH‘𝐾)‘𝑤) = ((ocH‘𝐾)‘𝑊))
25		hvmapval.o	. . . . . . . . . 10 ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊)
26	24, 25	syl6eqr 2674	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → ((ocH‘𝐾)‘𝑤) = 𝑂)
27	26	fveq1d 6193	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (((ocH‘𝐾)‘𝑤)‘{𝑥}) = (𝑂‘{𝑥}))
28	9	fveq2d 6195	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑊 → (+g‘((DVecH‘𝐾)‘𝑤)) = (+g‘𝑈))
29		hvmapval.p	. . . . . . . . . . 11 ⊢ + = (+g‘𝑈)
30	28, 29	syl6eqr 2674	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → (+g‘((DVecH‘𝐾)‘𝑤)) = + )
31		eqidd 2623	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → 𝑡 = 𝑡)
32	9	fveq2d 6195	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑊 → ( ·𝑠 ‘((DVecH‘𝐾)‘𝑤)) = ( ·𝑠 ‘𝑈))
33		hvmapval.t	. . . . . . . . . . . 12 ⊢ · = ( ·𝑠 ‘𝑈)
34	32, 33	syl6eqr 2674	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑊 → ( ·𝑠 ‘((DVecH‘𝐾)‘𝑤)) = · )
35	34	oveqd 6667	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → (𝑗( ·𝑠 ‘((DVecH‘𝐾)‘𝑤))𝑥) = (𝑗 · 𝑥))
36	30, 31, 35	oveq123d 6671	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (𝑡(+g‘((DVecH‘𝐾)‘𝑤))(𝑗( ·𝑠 ‘((DVecH‘𝐾)‘𝑤))𝑥)) = (𝑡 + (𝑗 · 𝑥)))
37	36	eqeq2d 2632	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (𝑣 = (𝑡(+g‘((DVecH‘𝐾)‘𝑤))(𝑗( ·𝑠 ‘((DVecH‘𝐾)‘𝑤))𝑥)) ↔ 𝑣 = (𝑡 + (𝑗 · 𝑥))))
38	27, 37	rexeqbidv 3153	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (∃𝑡 ∈ (((ocH‘𝐾)‘𝑤)‘{𝑥})𝑣 = (𝑡(+g‘((DVecH‘𝐾)‘𝑤))(𝑗( ·𝑠 ‘((DVecH‘𝐾)‘𝑤))𝑥)) ↔ ∃𝑡 ∈ (𝑂‘{𝑥})𝑣 = (𝑡 + (𝑗 · 𝑥))))
39	23, 38	riotaeqbidv 6614	. . . . . 6 ⊢ (𝑤 = 𝑊 → (℩𝑗 ∈ (Base‘(Scalar‘((DVecH‘𝐾)‘𝑤)))∃𝑡 ∈ (((ocH‘𝐾)‘𝑤)‘{𝑥})𝑣 = (𝑡(+g‘((DVecH‘𝐾)‘𝑤))(𝑗( ·𝑠 ‘((DVecH‘𝐾)‘𝑤))𝑥))) = (℩𝑗 ∈ 𝑅 ∃𝑡 ∈ (𝑂‘{𝑥})𝑣 = (𝑡 + (𝑗 · 𝑥))))
40	12, 39	mpteq12dv 4733	. . . . 5 ⊢ (𝑤 = 𝑊 → (𝑣 ∈ (Base‘((DVecH‘𝐾)‘𝑤)) ↦ (℩𝑗 ∈ (Base‘(Scalar‘((DVecH‘𝐾)‘𝑤)))∃𝑡 ∈ (((ocH‘𝐾)‘𝑤)‘{𝑥})𝑣 = (𝑡(+g‘((DVecH‘𝐾)‘𝑤))(𝑗( ·𝑠 ‘((DVecH‘𝐾)‘𝑤))𝑥)))) = (𝑣 ∈ 𝑉 ↦ (℩𝑗 ∈ 𝑅 ∃𝑡 ∈ (𝑂‘{𝑥})𝑣 = (𝑡 + (𝑗 · 𝑥)))))
41	17, 40	mpteq12dv 4733	. . . 4 ⊢ (𝑤 = 𝑊 → (𝑥 ∈ ((Base‘((DVecH‘𝐾)‘𝑤)) ∖ {(0g‘((DVecH‘𝐾)‘𝑤))}) ↦ (𝑣 ∈ (Base‘((DVecH‘𝐾)‘𝑤)) ↦ (℩𝑗 ∈ (Base‘(Scalar‘((DVecH‘𝐾)‘𝑤)))∃𝑡 ∈ (((ocH‘𝐾)‘𝑤)‘{𝑥})𝑣 = (𝑡(+g‘((DVecH‘𝐾)‘𝑤))(𝑗( ·𝑠 ‘((DVecH‘𝐾)‘𝑤))𝑥))))) = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑗 ∈ 𝑅 ∃𝑡 ∈ (𝑂‘{𝑥})𝑣 = (𝑡 + (𝑗 · 𝑥))))))
42		eqid 2622	. . . 4 ⊢ (𝑤 ∈ 𝐻 ↦ (𝑥 ∈ ((Base‘((DVecH‘𝐾)‘𝑤)) ∖ {(0g‘((DVecH‘𝐾)‘𝑤))}) ↦ (𝑣 ∈ (Base‘((DVecH‘𝐾)‘𝑤)) ↦ (℩𝑗 ∈ (Base‘(Scalar‘((DVecH‘𝐾)‘𝑤)))∃𝑡 ∈ (((ocH‘𝐾)‘𝑤)‘{𝑥})𝑣 = (𝑡(+g‘((DVecH‘𝐾)‘𝑤))(𝑗( ·𝑠 ‘((DVecH‘𝐾)‘𝑤))𝑥)))))) = (𝑤 ∈ 𝐻 ↦ (𝑥 ∈ ((Base‘((DVecH‘𝐾)‘𝑤)) ∖ {(0g‘((DVecH‘𝐾)‘𝑤))}) ↦ (𝑣 ∈ (Base‘((DVecH‘𝐾)‘𝑤)) ↦ (℩𝑗 ∈ (Base‘(Scalar‘((DVecH‘𝐾)‘𝑤)))∃𝑡 ∈ (((ocH‘𝐾)‘𝑤)‘{𝑥})𝑣 = (𝑡(+g‘((DVecH‘𝐾)‘𝑤))(𝑗( ·𝑠 ‘((DVecH‘𝐾)‘𝑤))𝑥))))))
43		fvex 6201	. . . . . . 7 ⊢ (Base‘𝑈) ∈ V
44	11, 43	eqeltri 2697	. . . . . 6 ⊢ 𝑉 ∈ V
45		difexg 4808	. . . . . 6 ⊢ (𝑉 ∈ V → (𝑉 ∖ { 0 }) ∈ V)
46	44, 45	ax-mp 5	. . . . 5 ⊢ (𝑉 ∖ { 0 }) ∈ V
47	46	mptex 6486	. . . 4 ⊢ (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑗 ∈ 𝑅 ∃𝑡 ∈ (𝑂‘{𝑥})𝑣 = (𝑡 + (𝑗 · 𝑥))))) ∈ V
48	41, 42, 47	fvmpt 6282	. . 3 ⊢ (𝑊 ∈ 𝐻 → ((𝑤 ∈ 𝐻 ↦ (𝑥 ∈ ((Base‘((DVecH‘𝐾)‘𝑤)) ∖ {(0g‘((DVecH‘𝐾)‘𝑤))}) ↦ (𝑣 ∈ (Base‘((DVecH‘𝐾)‘𝑤)) ↦ (℩𝑗 ∈ (Base‘(Scalar‘((DVecH‘𝐾)‘𝑤)))∃𝑡 ∈ (((ocH‘𝐾)‘𝑤)‘{𝑥})𝑣 = (𝑡(+g‘((DVecH‘𝐾)‘𝑤))(𝑗( ·𝑠 ‘((DVecH‘𝐾)‘𝑤))𝑥))))))‘𝑊) = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑗 ∈ 𝑅 ∃𝑡 ∈ (𝑂‘{𝑥})𝑣 = (𝑡 + (𝑗 · 𝑥))))))
49	6, 48	sylan9eq 2676	. 2 ⊢ ((𝐾 ∈ 𝐴 ∧ 𝑊 ∈ 𝐻) → 𝑀 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑗 ∈ 𝑅 ∃𝑡 ∈ (𝑂‘{𝑥})𝑣 = (𝑡 + (𝑗 · 𝑥))))))
50	1, 49	syl 17	1 ⊢ (𝜑 → 𝑀 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑗 ∈ 𝑅 ∃𝑡 ∈ (𝑂‘{𝑥})𝑣 = (𝑡 + (𝑗 · 𝑥))))))

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∃wrex 2913  Vcvv 3200   ∖ cdif 3571  {csn 4177   ↦ cmpt 4729  ‘cfv 5888  ℩crio 6610  (class class class)co 6650  Basecbs 15857  +_gcplusg 15941  Scalarcsca 15944   ·_𝑠 cvsca 15945  0_gc0g 16100  LHypclh 35270  DVecHcdvh 36367  ocHcoch 36636  HVMapchvm 37045

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-hvmap 37046

This theorem is referenced by:  hvmapval  37049  hvmap1o  37052  hvmaplkr  37057