Theorem hvmapfval 41738

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 41737	. . . . 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 6828	. . . 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	eqtrid 2776	. . 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 6826	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → ((DVecH‘𝐾)‘𝑤) = ((DVecH‘𝐾)‘𝑊))
8		hvmapval.u	. . . . . . . . 9 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
9	7, 8	eqtr4di 2782	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → ((DVecH‘𝐾)‘𝑤) = 𝑈)
10	9	fveq2d 6830	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (Base‘((DVecH‘𝐾)‘𝑤)) = (Base‘𝑈))
11		hvmapval.v	. . . . . . 7 ⊢ 𝑉 = (Base‘𝑈)
12	10, 11	eqtr4di 2782	. . . . . 6 ⊢ (𝑤 = 𝑊 → (Base‘((DVecH‘𝐾)‘𝑤)) = 𝑉)
13	9	fveq2d 6830	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (0g‘((DVecH‘𝐾)‘𝑤)) = (0g‘𝑈))
14		hvmapval.z	. . . . . . . 8 ⊢ 0 = (0g‘𝑈)
15	13, 14	eqtr4di 2782	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (0g‘((DVecH‘𝐾)‘𝑤)) = 0 )
16	15	sneqd 4591	. . . . . 6 ⊢ (𝑤 = 𝑊 → {(0g‘((DVecH‘𝐾)‘𝑤))} = { 0 })
17	12, 16	difeq12d 4080	. . . . 5 ⊢ (𝑤 = 𝑊 → ((Base‘((DVecH‘𝐾)‘𝑤)) ∖ {(0g‘((DVecH‘𝐾)‘𝑤))}) = (𝑉 ∖ { 0 }))
18	9	fveq2d 6830	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → (Scalar‘((DVecH‘𝐾)‘𝑤)) = (Scalar‘𝑈))
19		hvmapval.s	. . . . . . . . . 10 ⊢ 𝑆 = (Scalar‘𝑈)
20	18, 19	eqtr4di 2782	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (Scalar‘((DVecH‘𝐾)‘𝑤)) = 𝑆)
21	20	fveq2d 6830	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (Base‘(Scalar‘((DVecH‘𝐾)‘𝑤))) = (Base‘𝑆))
22		hvmapval.r	. . . . . . . 8 ⊢ 𝑅 = (Base‘𝑆)
23	21, 22	eqtr4di 2782	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (Base‘(Scalar‘((DVecH‘𝐾)‘𝑤))) = 𝑅)
24		fveq2 6826	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → ((ocH‘𝐾)‘𝑤) = ((ocH‘𝐾)‘𝑊))
25		hvmapval.o	. . . . . . . . . 10 ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊)
26	24, 25	eqtr4di 2782	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → ((ocH‘𝐾)‘𝑤) = 𝑂)
27	26	fveq1d 6828	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (((ocH‘𝐾)‘𝑤)‘{𝑥}) = (𝑂‘{𝑥}))
28	9	fveq2d 6830	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑊 → (+g‘((DVecH‘𝐾)‘𝑤)) = (+g‘𝑈))
29		hvmapval.p	. . . . . . . . . . 11 ⊢ + = (+g‘𝑈)
30	28, 29	eqtr4di 2782	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → (+g‘((DVecH‘𝐾)‘𝑤)) = + )
31		eqidd 2730	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → 𝑡 = 𝑡)
32	9	fveq2d 6830	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑊 → ( ·𝑠 ‘((DVecH‘𝐾)‘𝑤)) = ( ·𝑠 ‘𝑈))
33		hvmapval.t	. . . . . . . . . . . 12 ⊢ · = ( ·𝑠 ‘𝑈)
34	32, 33	eqtr4di 2782	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑊 → ( ·𝑠 ‘((DVecH‘𝐾)‘𝑤)) = · )
35	34	oveqd 7370	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → (𝑗( ·𝑠 ‘((DVecH‘𝐾)‘𝑤))𝑥) = (𝑗 · 𝑥))
36	30, 31, 35	oveq123d 7374	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (𝑡(+g‘((DVecH‘𝐾)‘𝑤))(𝑗( ·𝑠 ‘((DVecH‘𝐾)‘𝑤))𝑥)) = (𝑡 + (𝑗 · 𝑥)))
37	36	eqeq2d 2740	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (𝑣 = (𝑡(+g‘((DVecH‘𝐾)‘𝑤))(𝑗( ·𝑠 ‘((DVecH‘𝐾)‘𝑤))𝑥)) ↔ 𝑣 = (𝑡 + (𝑗 · 𝑥))))
38	27, 37	rexeqbidv 3311	. . . . . . 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 7313	. . . . . 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 5182	. . . . 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 5182	. . . 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 2729	. . . 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	11	fvexi 6840	. . . . . 6 ⊢ 𝑉 ∈ V
44	43	difexi 5272	. . . . 5 ⊢ (𝑉 ∖ { 0 }) ∈ V
45	44	mptex 7163	. . . 4 ⊢ (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑗 ∈ 𝑅 ∃𝑡 ∈ (𝑂‘{𝑥})𝑣 = (𝑡 + (𝑗 · 𝑥))))) ∈ V
46	41, 42, 45	fvmpt 6934	. . 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))))))
47	6, 46	sylan9eq 2784	. 2 ⊢ ((𝐾 ∈ 𝐴 ∧ 𝑊 ∈ 𝐻) → 𝑀 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑗 ∈ 𝑅 ∃𝑡 ∈ (𝑂‘{𝑥})𝑣 = (𝑡 + (𝑗 · 𝑥))))))
48	1, 47	syl 17	1 ⊢ (𝜑 → 𝑀 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑗 ∈ 𝑅 ∃𝑡 ∈ (𝑂‘{𝑥})𝑣 = (𝑡 + (𝑗 · 𝑥))))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∃wrex 3053   ∖ cdif 3902  {csn 4579   ↦ cmpt 5176  ‘cfv 6486  ℩crio 7309  (class class class)co 7353  Basecbs 17138  +_gcplusg 17179  Scalarcsca 17182   ·_𝑠 cvsca 17183  0_gc0g 17361  LHypclh 39963  DVecHcdvh 41057  ocHcoch 41326  HVMapchvm 41735

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 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pr 5374

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7310  df-ov 7356  df-hvmap 41736

This theorem is referenced by:  hvmapval  41739  hvmap1o  41742  hvmaplkr  41747