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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 37776 | . . . 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 6448 | . 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 6460 | . . . 4 ⊢ 𝐹 ∈ V |
14 | 13 | rabex 5049 | . . 3 ⊢ {𝑓 ∈ 𝐹 ∣ ((𝑂‘(𝑂‘(𝐿‘𝑓))) = (𝐿‘𝑓) ∧ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑇)} ∈ V |
15 | sseq2 3845 | . . . . . 6 ⊢ (𝑠 = 𝑇 → ((𝑂‘(𝐿‘𝑓)) ⊆ 𝑠 ↔ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑇)) | |
16 | 15 | anbi2d 622 | . . . . 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 3385 | . . . 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 2777 | . . . 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 6540 | . . 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 580 | . 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 2813 | 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 386 = wceq 1601 ∈ wcel 2106 {crab 3093 Vcvv 3397 ⊆ wss 3791 ↦ cmpt 4965 ‘cfv 6135 LSubSpclss 19324 LFnlclfn 35206 LKerclk 35234 LHypclh 36133 DVecHcdvh 37227 ocHcoch 37496 mapdcmpd 37773 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pr 5138 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-nul 4141 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-mapd 37774 |
This theorem is referenced by: mapdvalc 37778 mapddlssN 37789 mapdsn 37790 mapd1o 37797 mapd0 37814 |
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