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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > hvmapval | Structured version Visualization version GIF version |
Description: Value of map from nonzero vectors to nonzero functionals in the closed kernel dual space. (Contributed by NM, 23-Mar-2015.) |
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 })) |
Ref | Expression |
---|---|
hvmapval | ⊢ (𝜑 → (𝑀‘𝑋) = (𝑣 ∈ 𝑉 ↦ (℩𝑗 ∈ 𝑅 ∃𝑡 ∈ (𝑂‘{𝑋})𝑣 = (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 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 | hvmapfval 41742 | . . 3 ⊢ (𝜑 → 𝑀 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑗 ∈ 𝑅 ∃𝑡 ∈ (𝑂‘{𝑥})𝑣 = (𝑡 + (𝑗 · 𝑥)))))) |
13 | 12 | fveq1d 6909 | . 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 6921 | . . . 4 ⊢ 𝑉 ∈ V |
16 | 15 | mptex 7243 | . . 3 ⊢ (𝑣 ∈ 𝑉 ↦ (℩𝑗 ∈ 𝑅 ∃𝑡 ∈ (𝑂‘{𝑋})𝑣 = (𝑡 + (𝑗 · 𝑋)))) ∈ V |
17 | sneq 4641 | . . . . . . . 8 ⊢ (𝑥 = 𝑋 → {𝑥} = {𝑋}) | |
18 | 17 | fveq2d 6911 | . . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑂‘{𝑥}) = (𝑂‘{𝑋})) |
19 | oveq2 7439 | . . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (𝑗 · 𝑥) = (𝑗 · 𝑋)) | |
20 | 19 | oveq2d 7447 | . . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝑡 + (𝑗 · 𝑥)) = (𝑡 + (𝑗 · 𝑋))) |
21 | 20 | eqeq2d 2746 | . . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑣 = (𝑡 + (𝑗 · 𝑥)) ↔ 𝑣 = (𝑡 + (𝑗 · 𝑋)))) |
22 | 18, 21 | rexeqbidv 3345 | . . . . . 6 ⊢ (𝑥 = 𝑋 → (∃𝑡 ∈ (𝑂‘{𝑥})𝑣 = (𝑡 + (𝑗 · 𝑥)) ↔ ∃𝑡 ∈ (𝑂‘{𝑋})𝑣 = (𝑡 + (𝑗 · 𝑋)))) |
23 | 22 | riotabidv 7390 | . . . . 5 ⊢ (𝑥 = 𝑋 → (℩𝑗 ∈ 𝑅 ∃𝑡 ∈ (𝑂‘{𝑥})𝑣 = (𝑡 + (𝑗 · 𝑥))) = (℩𝑗 ∈ 𝑅 ∃𝑡 ∈ (𝑂‘{𝑋})𝑣 = (𝑡 + (𝑗 · 𝑋)))) |
24 | 23 | mpteq2dv 5250 | . . . 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 2735 | . . . 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 7014 | . . 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 586 | . 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 2775 | 1 ⊢ (𝜑 → (𝑀‘𝑋) = (𝑣 ∈ 𝑉 ↦ (℩𝑗 ∈ 𝑅 ∃𝑡 ∈ (𝑂‘{𝑋})𝑣 = (𝑡 + (𝑗 · 𝑋))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∃wrex 3068 Vcvv 3478 ∖ cdif 3960 {csn 4631 ↦ cmpt 5231 ‘cfv 6563 ℩crio 7387 (class class class)co 7431 Basecbs 17245 +gcplusg 17298 Scalarcsca 17301 ·𝑠 cvsca 17302 0gc0g 17486 LHypclh 39967 DVecHcdvh 41061 ocHcoch 41330 HVMapchvm 41739 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-hvmap 41740 |
This theorem is referenced by: hvmapvalvalN 41744 hvmapidN 41745 hdmapevec2 41819 |
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