Theorem hvmapval 42130

Description: Value of 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))
hvmapval.x	⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))

Assertion

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

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

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

Proof of Theorem hvmapval

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		hvmapval.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
2		hvmapval.u	. . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
3		hvmapval.o	. . . 4 ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊)
4		hvmapval.v	. . . 4 ⊢ 𝑉 = (Base‘𝑈)
5		hvmapval.p	. . . 4 ⊢ + = (+g‘𝑈)
6		hvmapval.t	. . . 4 ⊢ · = ( ·𝑠 ‘𝑈)
7		hvmapval.z	. . . 4 ⊢ 0 = (0g‘𝑈)
8		hvmapval.s	. . . 4 ⊢ 𝑆 = (Scalar‘𝑈)
9		hvmapval.r	. . . 4 ⊢ 𝑅 = (Base‘𝑆)
10		hvmapval.m	. . . 4 ⊢ 𝑀 = ((HVMap‘𝐾)‘𝑊)
11		hvmapval.k	. . . 4 ⊢ (𝜑 → (𝐾 ∈ 𝐴 ∧ 𝑊 ∈ 𝐻))
12	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11	hvmapfval 42129	. . 3 ⊢ (𝜑 → 𝑀 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑗 ∈ 𝑅 ∃𝑡 ∈ (𝑂‘{𝑥})𝑣 = (𝑡 + (𝑗 · 𝑥))))))
13	12	fveq1d 6844	. 2 ⊢ (𝜑 → (𝑀‘𝑋) = ((𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑗 ∈ 𝑅 ∃𝑡 ∈ (𝑂‘{𝑥})𝑣 = (𝑡 + (𝑗 · 𝑥)))))‘𝑋))
14		hvmapval.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
15	4	fvexi 6856	. . . 4 ⊢ 𝑉 ∈ V
16	15	mptex 7179	. . 3 ⊢ (𝑣 ∈ 𝑉 ↦ (℩𝑗 ∈ 𝑅 ∃𝑡 ∈ (𝑂‘{𝑋})𝑣 = (𝑡 + (𝑗 · 𝑋)))) ∈ V
17		sneq 4592	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → {𝑥} = {𝑋})
18	17	fveq2d 6846	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑂‘{𝑥}) = (𝑂‘{𝑋}))
19		oveq2 7376	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (𝑗 · 𝑥) = (𝑗 · 𝑋))
20	19	oveq2d 7384	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝑡 + (𝑗 · 𝑥)) = (𝑡 + (𝑗 · 𝑋)))
21	20	eqeq2d 2748	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑣 = (𝑡 + (𝑗 · 𝑥)) ↔ 𝑣 = (𝑡 + (𝑗 · 𝑋))))
22	18, 21	rexeqbidv 3319	. . . . . 6 ⊢ (𝑥 = 𝑋 → (∃𝑡 ∈ (𝑂‘{𝑥})𝑣 = (𝑡 + (𝑗 · 𝑥)) ↔ ∃𝑡 ∈ (𝑂‘{𝑋})𝑣 = (𝑡 + (𝑗 · 𝑋))))
23	22	riotabidv 7327	. . . . 5 ⊢ (𝑥 = 𝑋 → (℩𝑗 ∈ 𝑅 ∃𝑡 ∈ (𝑂‘{𝑥})𝑣 = (𝑡 + (𝑗 · 𝑥))) = (℩𝑗 ∈ 𝑅 ∃𝑡 ∈ (𝑂‘{𝑋})𝑣 = (𝑡 + (𝑗 · 𝑋))))
24	23	mpteq2dv 5194	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑣 ∈ 𝑉 ↦ (℩𝑗 ∈ 𝑅 ∃𝑡 ∈ (𝑂‘{𝑥})𝑣 = (𝑡 + (𝑗 · 𝑥)))) = (𝑣 ∈ 𝑉 ↦ (℩𝑗 ∈ 𝑅 ∃𝑡 ∈ (𝑂‘{𝑋})𝑣 = (𝑡 + (𝑗 · 𝑋)))))
25		eqid 2737	. . . 4 ⊢ (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑗 ∈ 𝑅 ∃𝑡 ∈ (𝑂‘{𝑥})𝑣 = (𝑡 + (𝑗 · 𝑥))))) = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑗 ∈ 𝑅 ∃𝑡 ∈ (𝑂‘{𝑥})𝑣 = (𝑡 + (𝑗 · 𝑥)))))
26	24, 25	fvmptg 6947	. . 3 ⊢ ((𝑋 ∈ (𝑉 ∖ { 0 }) ∧ (𝑣 ∈ 𝑉 ↦ (℩𝑗 ∈ 𝑅 ∃𝑡 ∈ (𝑂‘{𝑋})𝑣 = (𝑡 + (𝑗 · 𝑋)))) ∈ V) → ((𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑗 ∈ 𝑅 ∃𝑡 ∈ (𝑂‘{𝑥})𝑣 = (𝑡 + (𝑗 · 𝑥)))))‘𝑋) = (𝑣 ∈ 𝑉 ↦ (℩𝑗 ∈ 𝑅 ∃𝑡 ∈ (𝑂‘{𝑋})𝑣 = (𝑡 + (𝑗 · 𝑋)))))
27	14, 16, 26	sylancl 587	. 2 ⊢ (𝜑 → ((𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑗 ∈ 𝑅 ∃𝑡 ∈ (𝑂‘{𝑥})𝑣 = (𝑡 + (𝑗 · 𝑥)))))‘𝑋) = (𝑣 ∈ 𝑉 ↦ (℩𝑗 ∈ 𝑅 ∃𝑡 ∈ (𝑂‘{𝑋})𝑣 = (𝑡 + (𝑗 · 𝑋)))))
28	13, 27	eqtrd 2772	1 ⊢ (𝜑 → (𝑀‘𝑋) = (𝑣 ∈ 𝑉 ↦ (℩𝑗 ∈ 𝑅 ∃𝑡 ∈ (𝑂‘{𝑋})𝑣 = (𝑡 + (𝑗 · 𝑋)))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∃wrex 3062  Vcvv 3442   ∖ cdif 3900  {csn 4582   ↦ cmpt 5181  ‘cfv 6500  ℩crio 7324  (class class class)co 7368  Basecbs 17148  +_gcplusg 17189  Scalarcsca 17192   ·_𝑠 cvsca 17193  0_gc0g 17371  LHypclh 40354  DVecHcdvh 41448  ocHcoch 41717  HVMapchvm 42126

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-hvmap 42127

This theorem is referenced by:  hvmapvalvalN  42131  hvmapidN  42132  hdmapevec2  42206