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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 41326 | . . . 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 6903 | . 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 6915 | . . . 4 ⊢ 𝐹 ∈ V |
14 | 13 | rabex 5339 | . . 3 ⊢ {𝑓 ∈ 𝐹 ∣ ((𝑂‘(𝑂‘(𝐿‘𝑓))) = (𝐿‘𝑓) ∧ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑇)} ∈ V |
15 | sseq2 4006 | . . . . . 6 ⊢ (𝑠 = 𝑇 → ((𝑂‘(𝐿‘𝑓)) ⊆ 𝑠 ↔ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑇)) | |
16 | 15 | anbi2d 628 | . . . . 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 3427 | . . . 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 2726 | . . . 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 7007 | . . 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 584 | . 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 2766 | 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 394 = wceq 1534 ∈ wcel 2099 {crab 3419 Vcvv 3462 ⊆ wss 3947 ↦ cmpt 5236 ‘cfv 6554 LSubSpclss 20908 LFnlclfn 38755 LKerclk 38783 LHypclh 39683 DVecHcdvh 40777 ocHcoch 41046 mapdcmpd 41323 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-mapd 41324 |
This theorem is referenced by: mapdvalc 41328 mapddlssN 41339 mapdsn 41340 mapd1o 41347 mapd0 41364 |
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