Theorem hvmapval 40079

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 40078	. . 3 ⊢ (𝜑 → 𝑀 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑗 ∈ 𝑅 ∃𝑡 ∈ (𝑂‘{𝑥})𝑣 = (𝑡 + (𝑗 · 𝑥))))))
13	12	fveq1d 6836	. 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 6848	. . . 4 ⊢ 𝑉 ∈ V
16	15	mptex 7164	. . 3 ⊢ (𝑣 ∈ 𝑉 ↦ (℩𝑗 ∈ 𝑅 ∃𝑡 ∈ (𝑂‘{𝑋})𝑣 = (𝑡 + (𝑗 · 𝑋)))) ∈ V
17		sneq 4591	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → {𝑥} = {𝑋})
18	17	fveq2d 6838	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑂‘{𝑥}) = (𝑂‘{𝑋}))
19		oveq2 7354	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (𝑗 · 𝑥) = (𝑗 · 𝑋))
20	19	oveq2d 7362	. . . . . . . 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 3318	. . . . . 6 ⊢ (𝑥 = 𝑋 → (∃𝑡 ∈ (𝑂‘{𝑥})𝑣 = (𝑡 + (𝑗 · 𝑥)) ↔ ∃𝑡 ∈ (𝑂‘{𝑋})𝑣 = (𝑡 + (𝑗 · 𝑋))))
23	22	riotabidv 7304	. . . . 5 ⊢ (𝑥 = 𝑋 → (℩𝑗 ∈ 𝑅 ∃𝑡 ∈ (𝑂‘{𝑥})𝑣 = (𝑡 + (𝑗 · 𝑥))) = (℩𝑗 ∈ 𝑅 ∃𝑡 ∈ (𝑂‘{𝑋})𝑣 = (𝑡 + (𝑗 · 𝑋))))
24	23	mpteq2dv 5202	. . . 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 6938	. . 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 2777	1 ⊢ (𝜑 → (𝑀‘𝑋) = (𝑣 ∈ 𝑉 ↦ (℩𝑗 ∈ 𝑅 ∃𝑡 ∈ (𝑂‘{𝑋})𝑣 = (𝑡 + (𝑗 · 𝑋)))))

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1541   ∈ wcel 2106  ∃wrex 3071  Vcvv 3443   ∖ cdif 3902  {csn 4581   ↦ cmpt 5183  ‘cfv 6488  ℩crio 7301  (class class class)co 7346  Basecbs 17014  +_gcplusg 17064  Scalarcsca 17067   ·_𝑠 cvsca 17068  0_gc0g 17252  LHypclh 38303  DVecHcdvh 39397  ocHcoch 39666  HVMapchvm 40075

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-rep 5237  ax-sep 5251  ax-nul 5258  ax-pr 5379

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3735  df-csb 3851  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4278  df-if 4482  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4861  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5184  df-id 5525  df-xp 5633  df-rel 5634  df-cnv 5635  df-co 5636  df-dm 5637  df-rn 5638  df-res 5639  df-ima 5640  df-iota 6440  df-fun 6490  df-fn 6491  df-f 6492  df-f1 6493  df-fo 6494  df-f1o 6495  df-fv 6496  df-riota 7302  df-ov 7349  df-hvmap 40076

This theorem is referenced by:  hvmapvalvalN  40080  hvmapidN  40081  hdmapevec2  40155