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Mathbox for Norm Megill |
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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 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)))) |
Colors of variables: wff setvar class |
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 0gc0g 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) |
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