Theorem hvmapvalvalN 41718

Description: Value of value of map (i.e. functional value) from nonzero vectors to nonzero functionals in the closed kernel dual space. (Contributed by NM, 23-Mar-2015.) (New usage is discouraged.)

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 }))
hvmapval.y	⊢ (𝜑 → 𝑌 ∈ 𝑉)

Assertion

Ref	Expression
hvmapvalvalN	⊢ (𝜑 → ((𝑀‘𝑋)‘𝑌) = (℩𝑗 ∈ 𝑅 ∃𝑡 ∈ (𝑂‘{𝑋})𝑌 = (𝑡 + (𝑗 · 𝑋))))

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

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

Proof of Theorem hvmapvalvalN

Dummy variable 𝑦 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		hvmapval.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
13	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12	hvmapval 41717	. . 3 ⊢ (𝜑 → (𝑀‘𝑋) = (𝑦 ∈ 𝑉 ↦ (℩𝑗 ∈ 𝑅 ∃𝑡 ∈ (𝑂‘{𝑋})𝑦 = (𝑡 + (𝑗 · 𝑋)))))
14	13	fveq1d 6922	. 2 ⊢ (𝜑 → ((𝑀‘𝑋)‘𝑌) = ((𝑦 ∈ 𝑉 ↦ (℩𝑗 ∈ 𝑅 ∃𝑡 ∈ (𝑂‘{𝑋})𝑦 = (𝑡 + (𝑗 · 𝑋))))‘𝑌))
15		hvmapval.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
16		riotaex 7408	. . 3 ⊢ (℩𝑗 ∈ 𝑅 ∃𝑡 ∈ (𝑂‘{𝑋})𝑌 = (𝑡 + (𝑗 · 𝑋))) ∈ V
17		eqeq1 2744	. . . . . 6 ⊢ (𝑦 = 𝑌 → (𝑦 = (𝑡 + (𝑗 · 𝑋)) ↔ 𝑌 = (𝑡 + (𝑗 · 𝑋))))
18	17	rexbidv 3185	. . . . 5 ⊢ (𝑦 = 𝑌 → (∃𝑡 ∈ (𝑂‘{𝑋})𝑦 = (𝑡 + (𝑗 · 𝑋)) ↔ ∃𝑡 ∈ (𝑂‘{𝑋})𝑌 = (𝑡 + (𝑗 · 𝑋))))
19	18	riotabidv 7406	. . . 4 ⊢ (𝑦 = 𝑌 → (℩𝑗 ∈ 𝑅 ∃𝑡 ∈ (𝑂‘{𝑋})𝑦 = (𝑡 + (𝑗 · 𝑋))) = (℩𝑗 ∈ 𝑅 ∃𝑡 ∈ (𝑂‘{𝑋})𝑌 = (𝑡 + (𝑗 · 𝑋))))
20		eqid 2740	. . . 4 ⊢ (𝑦 ∈ 𝑉 ↦ (℩𝑗 ∈ 𝑅 ∃𝑡 ∈ (𝑂‘{𝑋})𝑦 = (𝑡 + (𝑗 · 𝑋)))) = (𝑦 ∈ 𝑉 ↦ (℩𝑗 ∈ 𝑅 ∃𝑡 ∈ (𝑂‘{𝑋})𝑦 = (𝑡 + (𝑗 · 𝑋))))
21	19, 20	fvmptg 7027	. . 3 ⊢ ((𝑌 ∈ 𝑉 ∧ (℩𝑗 ∈ 𝑅 ∃𝑡 ∈ (𝑂‘{𝑋})𝑌 = (𝑡 + (𝑗 · 𝑋))) ∈ V) → ((𝑦 ∈ 𝑉 ↦ (℩𝑗 ∈ 𝑅 ∃𝑡 ∈ (𝑂‘{𝑋})𝑦 = (𝑡 + (𝑗 · 𝑋))))‘𝑌) = (℩𝑗 ∈ 𝑅 ∃𝑡 ∈ (𝑂‘{𝑋})𝑌 = (𝑡 + (𝑗 · 𝑋))))
22	15, 16, 21	sylancl 585	. 2 ⊢ (𝜑 → ((𝑦 ∈ 𝑉 ↦ (℩𝑗 ∈ 𝑅 ∃𝑡 ∈ (𝑂‘{𝑋})𝑦 = (𝑡 + (𝑗 · 𝑋))))‘𝑌) = (℩𝑗 ∈ 𝑅 ∃𝑡 ∈ (𝑂‘{𝑋})𝑌 = (𝑡 + (𝑗 · 𝑋))))
23	14, 22	eqtrd 2780	1 ⊢ (𝜑 → ((𝑀‘𝑋)‘𝑌) = (℩𝑗 ∈ 𝑅 ∃𝑡 ∈ (𝑂‘{𝑋})𝑌 = (𝑡 + (𝑗 · 𝑋))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ∃wrex 3076  Vcvv 3488   ∖ cdif 3973  {csn 4648   ↦ cmpt 5249  ‘cfv 6573  ℩crio 7403  (class class class)co 7448  Basecbs 17258  +_gcplusg 17311  Scalarcsca 17314   ·_𝑠 cvsca 17315  0_gc0g 17499  LHypclh 39941  DVecHcdvh 41035  ocHcoch 41304  HVMapchvm 41713

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-hvmap 41714

This theorem is referenced by: (None)