Theorem hvmapvalvalN 41096

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 41095	. . 3 ⊢ (𝜑 → (𝑀‘𝑋) = (𝑦 ∈ 𝑉 ↦ (℩𝑗 ∈ 𝑅 ∃𝑡 ∈ (𝑂‘{𝑋})𝑦 = (𝑡 + (𝑗 · 𝑋)))))
14	13	fveq1d 6893	. 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 7372	. . 3 ⊢ (℩𝑗 ∈ 𝑅 ∃𝑡 ∈ (𝑂‘{𝑋})𝑌 = (𝑡 + (𝑗 · 𝑋))) ∈ V
17		eqeq1 2735	. . . . . 6 ⊢ (𝑦 = 𝑌 → (𝑦 = (𝑡 + (𝑗 · 𝑋)) ↔ 𝑌 = (𝑡 + (𝑗 · 𝑋))))
18	17	rexbidv 3177	. . . . 5 ⊢ (𝑦 = 𝑌 → (∃𝑡 ∈ (𝑂‘{𝑋})𝑦 = (𝑡 + (𝑗 · 𝑋)) ↔ ∃𝑡 ∈ (𝑂‘{𝑋})𝑌 = (𝑡 + (𝑗 · 𝑋))))
19	18	riotabidv 7370	. . . 4 ⊢ (𝑦 = 𝑌 → (℩𝑗 ∈ 𝑅 ∃𝑡 ∈ (𝑂‘{𝑋})𝑦 = (𝑡 + (𝑗 · 𝑋))) = (℩𝑗 ∈ 𝑅 ∃𝑡 ∈ (𝑂‘{𝑋})𝑌 = (𝑡 + (𝑗 · 𝑋))))
20		eqid 2731	. . . 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 6996	. . 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 2771	1 ⊢ (𝜑 → ((𝑀‘𝑋)‘𝑌) = (℩𝑗 ∈ 𝑅 ∃𝑡 ∈ (𝑂‘{𝑋})𝑌 = (𝑡 + (𝑗 · 𝑋))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2105  ∃wrex 3069  Vcvv 3473   ∖ cdif 3945  {csn 4628   ↦ cmpt 5231  ‘cfv 6543  ℩crio 7367  (class class class)co 7412  Basecbs 17151  +_gcplusg 17204  Scalarcsca 17207   ·_𝑠 cvsca 17208  0_gc0g 17392  LHypclh 39319  DVecHcdvh 40413  ocHcoch 40682  HVMapchvm 41091

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-hvmap 41092

This theorem is referenced by: (None)