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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 36916 | . . . 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 6193 | . 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 | fvex 6201 | . . . . 5 ⊢ (LFnl‘𝑈) ∈ V | |
14 | 5, 13 | eqeltri 2697 | . . . 4 ⊢ 𝐹 ∈ V |
15 | 14 | rabex 4813 | . . 3 ⊢ {𝑓 ∈ 𝐹 ∣ ((𝑂‘(𝑂‘(𝐿‘𝑓))) = (𝐿‘𝑓) ∧ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑇)} ∈ V |
16 | sseq2 3627 | . . . . . 6 ⊢ (𝑠 = 𝑇 → ((𝑂‘(𝐿‘𝑓)) ⊆ 𝑠 ↔ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑇)) | |
17 | 16 | anbi2d 740 | . . . . 5 ⊢ (𝑠 = 𝑇 → (((𝑂‘(𝑂‘(𝐿‘𝑓))) = (𝐿‘𝑓) ∧ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑠) ↔ ((𝑂‘(𝑂‘(𝐿‘𝑓))) = (𝐿‘𝑓) ∧ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑇))) |
18 | 17 | rabbidv 3189 | . . . 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)}) |
19 | eqid 2622 | . . . 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)}) | |
20 | 18, 19 | fvmptg 6280 | . . 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)}) |
21 | 12, 15, 20 | sylancl 694 | . 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)}) |
22 | 11, 21 | eqtrd 2656 | 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 384 = wceq 1483 ∈ wcel 1990 {crab 2916 Vcvv 3200 ⊆ wss 3574 ↦ cmpt 4729 ‘cfv 5888 LSubSpclss 18932 LFnlclfn 34344 LKerclk 34372 LHypclh 35270 DVecHcdvh 36367 ocHcoch 36636 mapdcmpd 36913 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-mapd 36914 |
This theorem is referenced by: mapdvalc 36918 mapddlssN 36929 mapdsn 36930 mapd1o 36937 mapd0 36954 |
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