| Mathbox for Norm Megill |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdval | Structured version Visualization version GIF version | ||
| Description: Value of projectivity from vector space H to dual space. (Contributed by NM, 27-Jan-2015.) |
| Ref | Expression |
|---|---|
| mapdval.h | ⊢ 𝐻 = (LHyp‘𝐾) |
| mapdval.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
| mapdval.s | ⊢ 𝑆 = (LSubSp‘𝑈) |
| mapdval.f | ⊢ 𝐹 = (LFnl‘𝑈) |
| mapdval.l | ⊢ 𝐿 = (LKer‘𝑈) |
| mapdval.o | ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊) |
| mapdval.m | ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) |
| mapdval.k | ⊢ (𝜑 → (𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻)) |
| mapdval.t | ⊢ (𝜑 → 𝑇 ∈ 𝑆) |
| Ref | Expression |
|---|---|
| mapdval | ⊢ (𝜑 → (𝑀‘𝑇) = {𝑓 ∈ 𝐹 ∣ ((𝑂‘(𝑂‘(𝐿‘𝑓))) = (𝐿‘𝑓) ∧ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑇)}) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | mapdval.k | . . . 4 ⊢ (𝜑 → (𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻)) | |
| 2 | mapdval.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
| 3 | mapdval.u | . . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
| 4 | mapdval.s | . . . . 5 ⊢ 𝑆 = (LSubSp‘𝑈) | |
| 5 | mapdval.f | . . . . 5 ⊢ 𝐹 = (LFnl‘𝑈) | |
| 6 | mapdval.l | . . . . 5 ⊢ 𝐿 = (LKer‘𝑈) | |
| 7 | mapdval.o | . . . . 5 ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊) | |
| 8 | mapdval.m | . . . . 5 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) | |
| 9 | 2, 3, 4, 5, 6, 7, 8 | mapdfval 41629 | . . . 4 ⊢ ((𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻) → 𝑀 = (𝑠 ∈ 𝑆 ↦ {𝑓 ∈ 𝐹 ∣ ((𝑂‘(𝑂‘(𝐿‘𝑓))) = (𝐿‘𝑓) ∧ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑠)})) |
| 10 | 1, 9 | syl 17 | . . 3 ⊢ (𝜑 → 𝑀 = (𝑠 ∈ 𝑆 ↦ {𝑓 ∈ 𝐹 ∣ ((𝑂‘(𝑂‘(𝐿‘𝑓))) = (𝐿‘𝑓) ∧ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑠)})) |
| 11 | 10 | fveq1d 6908 | . 2 ⊢ (𝜑 → (𝑀‘𝑇) = ((𝑠 ∈ 𝑆 ↦ {𝑓 ∈ 𝐹 ∣ ((𝑂‘(𝑂‘(𝐿‘𝑓))) = (𝐿‘𝑓) ∧ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑠)})‘𝑇)) |
| 12 | mapdval.t | . . 3 ⊢ (𝜑 → 𝑇 ∈ 𝑆) | |
| 13 | 5 | fvexi 6920 | . . . 4 ⊢ 𝐹 ∈ V |
| 14 | 13 | rabex 5339 | . . 3 ⊢ {𝑓 ∈ 𝐹 ∣ ((𝑂‘(𝑂‘(𝐿‘𝑓))) = (𝐿‘𝑓) ∧ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑇)} ∈ V |
| 15 | sseq2 4010 | . . . . . 6 ⊢ (𝑠 = 𝑇 → ((𝑂‘(𝐿‘𝑓)) ⊆ 𝑠 ↔ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑇)) | |
| 16 | 15 | anbi2d 630 | . . . . 5 ⊢ (𝑠 = 𝑇 → (((𝑂‘(𝑂‘(𝐿‘𝑓))) = (𝐿‘𝑓) ∧ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑠) ↔ ((𝑂‘(𝑂‘(𝐿‘𝑓))) = (𝐿‘𝑓) ∧ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑇))) |
| 17 | 16 | rabbidv 3444 | . . . 4 ⊢ (𝑠 = 𝑇 → {𝑓 ∈ 𝐹 ∣ ((𝑂‘(𝑂‘(𝐿‘𝑓))) = (𝐿‘𝑓) ∧ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑠)} = {𝑓 ∈ 𝐹 ∣ ((𝑂‘(𝑂‘(𝐿‘𝑓))) = (𝐿‘𝑓) ∧ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑇)}) |
| 18 | eqid 2737 | . . . 4 ⊢ (𝑠 ∈ 𝑆 ↦ {𝑓 ∈ 𝐹 ∣ ((𝑂‘(𝑂‘(𝐿‘𝑓))) = (𝐿‘𝑓) ∧ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑠)}) = (𝑠 ∈ 𝑆 ↦ {𝑓 ∈ 𝐹 ∣ ((𝑂‘(𝑂‘(𝐿‘𝑓))) = (𝐿‘𝑓) ∧ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑠)}) | |
| 19 | 17, 18 | fvmptg 7014 | . . 3 ⊢ ((𝑇 ∈ 𝑆 ∧ {𝑓 ∈ 𝐹 ∣ ((𝑂‘(𝑂‘(𝐿‘𝑓))) = (𝐿‘𝑓) ∧ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑇)} ∈ V) → ((𝑠 ∈ 𝑆 ↦ {𝑓 ∈ 𝐹 ∣ ((𝑂‘(𝑂‘(𝐿‘𝑓))) = (𝐿‘𝑓) ∧ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑠)})‘𝑇) = {𝑓 ∈ 𝐹 ∣ ((𝑂‘(𝑂‘(𝐿‘𝑓))) = (𝐿‘𝑓) ∧ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑇)}) |
| 20 | 12, 14, 19 | sylancl 586 | . 2 ⊢ (𝜑 → ((𝑠 ∈ 𝑆 ↦ {𝑓 ∈ 𝐹 ∣ ((𝑂‘(𝑂‘(𝐿‘𝑓))) = (𝐿‘𝑓) ∧ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑠)})‘𝑇) = {𝑓 ∈ 𝐹 ∣ ((𝑂‘(𝑂‘(𝐿‘𝑓))) = (𝐿‘𝑓) ∧ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑇)}) |
| 21 | 11, 20 | eqtrd 2777 | 1 ⊢ (𝜑 → (𝑀‘𝑇) = {𝑓 ∈ 𝐹 ∣ ((𝑂‘(𝑂‘(𝐿‘𝑓))) = (𝐿‘𝑓) ∧ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑇)}) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 {crab 3436 Vcvv 3480 ⊆ wss 3951 ↦ cmpt 5225 ‘cfv 6561 LSubSpclss 20929 LFnlclfn 39058 LKerclk 39086 LHypclh 39986 DVecHcdvh 41080 ocHcoch 41349 mapdcmpd 41626 |
| 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-pw 4602 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-mapd 41627 |
| This theorem is referenced by: mapdvalc 41631 mapddlssN 41642 mapdsn 41643 mapd1o 41650 mapd0 41667 |
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