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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 41761 | . . 3 ⊢ (𝜑 → 𝑀 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑗 ∈ 𝑅 ∃𝑡 ∈ (𝑂‘{𝑥})𝑣 = (𝑡 + (𝑗 · 𝑥)))))) |
| 13 | 12 | fveq1d 6908 | . 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 6920 | . . . 4 ⊢ 𝑉 ∈ V |
| 16 | 15 | mptex 7243 | . . 3 ⊢ (𝑣 ∈ 𝑉 ↦ (℩𝑗 ∈ 𝑅 ∃𝑡 ∈ (𝑂‘{𝑋})𝑣 = (𝑡 + (𝑗 · 𝑋)))) ∈ V |
| 17 | sneq 4636 | . . . . . . . 8 ⊢ (𝑥 = 𝑋 → {𝑥} = {𝑋}) | |
| 18 | 17 | fveq2d 6910 | . . . . . . 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 2748 | . . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑣 = (𝑡 + (𝑗 · 𝑥)) ↔ 𝑣 = (𝑡 + (𝑗 · 𝑋)))) |
| 22 | 18, 21 | rexeqbidv 3347 | . . . . . 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 5244 | . . . 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 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 2777 | 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 1540 ∈ wcel 2108 ∃wrex 3070 Vcvv 3480 ∖ cdif 3948 {csn 4626 ↦ cmpt 5225 ‘cfv 6561 ℩crio 7387 (class class class)co 7431 Basecbs 17247 +gcplusg 17297 Scalarcsca 17300 ·𝑠 cvsca 17301 0gc0g 17484 LHypclh 39986 DVecHcdvh 41080 ocHcoch 41349 HVMapchvm 41758 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-hvmap 41759 |
| This theorem is referenced by: hvmapvalvalN 41763 hvmapidN 41764 hdmapevec2 41838 |
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