![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > lcdlss2N | Structured version Visualization version GIF version |
Description: Subspaces of a dual vector space of functionals with closed kernels. (Contributed by NM, 13-Mar-2015.) (New usage is discouraged.) |
Ref | Expression |
---|---|
lcdlss2.h | ⊢ 𝐻 = (LHyp‘𝐾) |
lcdlss2.c | ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) |
lcdlss2.s | ⊢ 𝑆 = (LSubSp‘𝐶) |
lcdlss2.v | ⊢ 𝑉 = (Base‘𝐶) |
lcdlss2.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
lcdlss2.d | ⊢ 𝐷 = (LDual‘𝑈) |
lcdlss2.t | ⊢ 𝑇 = (LSubSp‘𝐷) |
lcdlss2.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
Ref | Expression |
---|---|
lcdlss2N | ⊢ (𝜑 → 𝑆 = (𝑇 ∩ 𝒫 𝑉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lcdlss2.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | eqid 2726 | . . 3 ⊢ ((ocH‘𝐾)‘𝑊) = ((ocH‘𝐾)‘𝑊) | |
3 | lcdlss2.c | . . 3 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
4 | lcdlss2.s | . . 3 ⊢ 𝑆 = (LSubSp‘𝐶) | |
5 | lcdlss2.u | . . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
6 | eqid 2726 | . . 3 ⊢ (LFnl‘𝑈) = (LFnl‘𝑈) | |
7 | eqid 2726 | . . 3 ⊢ (LKer‘𝑈) = (LKer‘𝑈) | |
8 | lcdlss2.d | . . 3 ⊢ 𝐷 = (LDual‘𝑈) | |
9 | lcdlss2.t | . . 3 ⊢ 𝑇 = (LSubSp‘𝐷) | |
10 | eqid 2726 | . . 3 ⊢ {𝑓 ∈ (LFnl‘𝑈) ∣ (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓))) = ((LKer‘𝑈)‘𝑓)} = {𝑓 ∈ (LFnl‘𝑈) ∣ (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓))) = ((LKer‘𝑈)‘𝑓)} | |
11 | lcdlss2.k | . . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
12 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 | lcdlss 41328 | . 2 ⊢ (𝜑 → 𝑆 = (𝑇 ∩ 𝒫 {𝑓 ∈ (LFnl‘𝑈) ∣ (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓))) = ((LKer‘𝑈)‘𝑓)})) |
13 | lcdlss2.v | . . . . 5 ⊢ 𝑉 = (Base‘𝐶) | |
14 | 1, 2, 3, 13, 5, 6, 7, 10, 11 | lcdvbase 41302 | . . . 4 ⊢ (𝜑 → 𝑉 = {𝑓 ∈ (LFnl‘𝑈) ∣ (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓))) = ((LKer‘𝑈)‘𝑓)}) |
15 | 14 | pweqd 4614 | . . 3 ⊢ (𝜑 → 𝒫 𝑉 = 𝒫 {𝑓 ∈ (LFnl‘𝑈) ∣ (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓))) = ((LKer‘𝑈)‘𝑓)}) |
16 | 15 | ineq2d 4210 | . 2 ⊢ (𝜑 → (𝑇 ∩ 𝒫 𝑉) = (𝑇 ∩ 𝒫 {𝑓 ∈ (LFnl‘𝑈) ∣ (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓))) = ((LKer‘𝑈)‘𝑓)})) |
17 | 12, 16 | eqtr4d 2769 | 1 ⊢ (𝜑 → 𝑆 = (𝑇 ∩ 𝒫 𝑉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 ∈ wcel 2099 {crab 3419 ∩ cin 3945 𝒫 cpw 4597 ‘cfv 6543 Basecbs 17205 LSubSpclss 20901 LFnlclfn 38765 LKerclk 38793 LDualcld 38831 HLchlt 39058 LHypclh 39693 DVecHcdvh 40787 ocHcoch 41056 LCDualclcd 41295 |
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 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7735 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-riotaBAD 38661 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 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-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4906 df-int 4947 df-iun 4995 df-iin 4996 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6302 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-of 7679 df-om 7866 df-1st 7992 df-2nd 7993 df-tpos 8230 df-undef 8277 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-2o 8486 df-er 8723 df-map 8846 df-en 8964 df-dom 8965 df-sdom 8966 df-fin 8967 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12256 df-2 12318 df-3 12319 df-4 12320 df-5 12321 df-6 12322 df-n0 12516 df-z 12602 df-uz 12866 df-fz 13530 df-struct 17141 df-sets 17158 df-slot 17176 df-ndx 17188 df-base 17206 df-ress 17235 df-plusg 17271 df-mulr 17272 df-sca 17274 df-vsca 17275 df-0g 17448 df-mre 17591 df-mrc 17592 df-acs 17594 df-proset 18312 df-poset 18330 df-plt 18347 df-lub 18363 df-glb 18364 df-join 18365 df-meet 18366 df-p0 18442 df-p1 18443 df-lat 18449 df-clat 18516 df-mgm 18625 df-sgrp 18704 df-mnd 18720 df-submnd 18766 df-grp 18923 df-minusg 18924 df-sbg 18925 df-subg 19110 df-cntz 19304 df-oppg 19333 df-lsm 19627 df-cmn 19773 df-abl 19774 df-mgp 20111 df-rng 20129 df-ur 20158 df-ring 20211 df-oppr 20309 df-dvdsr 20332 df-unit 20333 df-invr 20363 df-dvr 20376 df-nzr 20488 df-rlreg 20665 df-domn 20666 df-drng 20702 df-lmod 20831 df-lss 20902 df-lsp 20942 df-lvec 21074 df-lsatoms 38684 df-lshyp 38685 df-lcv 38727 df-lfl 38766 df-lkr 38794 df-ldual 38832 df-oposet 38884 df-ol 38886 df-oml 38887 df-covers 38974 df-ats 38975 df-atl 39006 df-cvlat 39030 df-hlat 39059 df-llines 39207 df-lplanes 39208 df-lvols 39209 df-lines 39210 df-psubsp 39212 df-pmap 39213 df-padd 39505 df-lhyp 39697 df-laut 39698 df-ldil 39813 df-ltrn 39814 df-trl 39868 df-tgrp 40452 df-tendo 40464 df-edring 40466 df-dveca 40712 df-disoa 40738 df-dvech 40788 df-dib 40848 df-dic 40882 df-dih 40938 df-doch 41057 df-djh 41104 df-lcdual 41296 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |