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Mirrors > Home > MPE Home > Th. List > Mathboxes > hvmapvalvalN | Structured version Visualization version GIF version |
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.) |
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) |
Ref | Expression |
---|---|
hvmapvalvalN | ⊢ (𝜑 → ((𝑀‘𝑋)‘𝑌) = (℩𝑗 ∈ 𝑅 ∃𝑡 ∈ (𝑂‘{𝑋})𝑌 = (𝑡 + (𝑗 · 𝑋)))) |
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 39468 | . . 3 ⊢ (𝜑 → (𝑀‘𝑋) = (𝑦 ∈ 𝑉 ↦ (℩𝑗 ∈ 𝑅 ∃𝑡 ∈ (𝑂‘{𝑋})𝑦 = (𝑡 + (𝑗 · 𝑋))))) |
14 | 13 | fveq1d 6708 | . 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 7163 | . . 3 ⊢ (℩𝑗 ∈ 𝑅 ∃𝑡 ∈ (𝑂‘{𝑋})𝑌 = (𝑡 + (𝑗 · 𝑋))) ∈ V | |
17 | eqeq1 2738 | . . . . . 6 ⊢ (𝑦 = 𝑌 → (𝑦 = (𝑡 + (𝑗 · 𝑋)) ↔ 𝑌 = (𝑡 + (𝑗 · 𝑋)))) | |
18 | 17 | rexbidv 3209 | . . . . 5 ⊢ (𝑦 = 𝑌 → (∃𝑡 ∈ (𝑂‘{𝑋})𝑦 = (𝑡 + (𝑗 · 𝑋)) ↔ ∃𝑡 ∈ (𝑂‘{𝑋})𝑌 = (𝑡 + (𝑗 · 𝑋)))) |
19 | 18 | riotabidv 7161 | . . . 4 ⊢ (𝑦 = 𝑌 → (℩𝑗 ∈ 𝑅 ∃𝑡 ∈ (𝑂‘{𝑋})𝑦 = (𝑡 + (𝑗 · 𝑋))) = (℩𝑗 ∈ 𝑅 ∃𝑡 ∈ (𝑂‘{𝑋})𝑌 = (𝑡 + (𝑗 · 𝑋)))) |
20 | eqid 2734 | . . . 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 6805 | . . 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 589 | . 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 2774 | 1 ⊢ (𝜑 → ((𝑀‘𝑋)‘𝑌) = (℩𝑗 ∈ 𝑅 ∃𝑡 ∈ (𝑂‘{𝑋})𝑌 = (𝑡 + (𝑗 · 𝑋)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ∃wrex 3055 Vcvv 3401 ∖ cdif 3854 {csn 4531 ↦ cmpt 5124 ‘cfv 6369 ℩crio 7158 (class class class)co 7202 Basecbs 16684 +gcplusg 16767 Scalarcsca 16770 ·𝑠 cvsca 16771 0gc0g 16916 LHypclh 37692 DVecHcdvh 38786 ocHcoch 39055 HVMapchvm 39464 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pr 5311 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-nul 4228 df-if 4430 df-sn 4532 df-pr 4534 df-op 4538 df-uni 4810 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-id 5444 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-riota 7159 df-ov 7205 df-hvmap 39465 |
This theorem is referenced by: (None) |
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