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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 36067 | . . 3 ⊢ (𝜑 → (𝑀‘𝑋) = (𝑦 ∈ 𝑉 ↦ (℩𝑗 ∈ 𝑅 ∃𝑡 ∈ (𝑂‘{𝑋})𝑦 = (𝑡 + (𝑗 · 𝑋))))) |
14 | 13 | fveq1d 6105 | . 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 6515 | . . 3 ⊢ (℩𝑗 ∈ 𝑅 ∃𝑡 ∈ (𝑂‘{𝑋})𝑌 = (𝑡 + (𝑗 · 𝑋))) ∈ V | |
17 | eqeq1 2614 | . . . . . 6 ⊢ (𝑦 = 𝑌 → (𝑦 = (𝑡 + (𝑗 · 𝑋)) ↔ 𝑌 = (𝑡 + (𝑗 · 𝑋)))) | |
18 | 17 | rexbidv 3034 | . . . . 5 ⊢ (𝑦 = 𝑌 → (∃𝑡 ∈ (𝑂‘{𝑋})𝑦 = (𝑡 + (𝑗 · 𝑋)) ↔ ∃𝑡 ∈ (𝑂‘{𝑋})𝑌 = (𝑡 + (𝑗 · 𝑋)))) |
19 | 18 | riotabidv 6513 | . . . 4 ⊢ (𝑦 = 𝑌 → (℩𝑗 ∈ 𝑅 ∃𝑡 ∈ (𝑂‘{𝑋})𝑦 = (𝑡 + (𝑗 · 𝑋))) = (℩𝑗 ∈ 𝑅 ∃𝑡 ∈ (𝑂‘{𝑋})𝑌 = (𝑡 + (𝑗 · 𝑋)))) |
20 | eqid 2610 | . . . 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 6189 | . . 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 693 | . 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 2644 | 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 383 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 Vcvv 3173 ∖ cdif 3537 {csn 4125 ↦ cmpt 4643 ‘cfv 5804 ℩crio 6510 (class class class)co 6549 Basecbs 15695 +gcplusg 15768 Scalarcsca 15771 ·𝑠 cvsca 15772 0gc0g 15923 LHypclh 34288 DVecHcdvh 35385 ocHcoch 35654 HVMapchvm 36063 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-hvmap 36064 |
This theorem is referenced by: (None) |
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