Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lcd0v | Structured version Visualization version GIF version |
Description: The zero functional in the set of functionals with closed kernels. (Contributed by NM, 20-Mar-2015.) |
Ref | Expression |
---|---|
lcd0v.h | ⊢ 𝐻 = (LHyp‘𝐾) |
lcd0v.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
lcd0v.v | ⊢ 𝑉 = (Base‘𝑈) |
lcd0v.r | ⊢ 𝑅 = (Scalar‘𝑈) |
lcd0v.z | ⊢ 0 = (0g‘𝑅) |
lcd0v.c | ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) |
lcd0v.o | ⊢ 𝑂 = (0g‘𝐶) |
lcd0v.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
Ref | Expression |
---|---|
lcd0v | ⊢ (𝜑 → 𝑂 = (𝑉 × { 0 })) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lcd0v.o | . . 3 ⊢ 𝑂 = (0g‘𝐶) | |
2 | lcd0v.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
3 | eqid 2818 | . . . . 5 ⊢ ((ocH‘𝐾)‘𝑊) = ((ocH‘𝐾)‘𝑊) | |
4 | lcd0v.c | . . . . 5 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
5 | lcd0v.u | . . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
6 | eqid 2818 | . . . . 5 ⊢ (LFnl‘𝑈) = (LFnl‘𝑈) | |
7 | eqid 2818 | . . . . 5 ⊢ (LKer‘𝑈) = (LKer‘𝑈) | |
8 | eqid 2818 | . . . . 5 ⊢ (LDual‘𝑈) = (LDual‘𝑈) | |
9 | lcd0v.k | . . . . 5 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
10 | 2, 3, 4, 5, 6, 7, 8, 9 | lcdval 38605 | . . . 4 ⊢ (𝜑 → 𝐶 = ((LDual‘𝑈) ↾s {𝑓 ∈ (LFnl‘𝑈) ∣ (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓))) = ((LKer‘𝑈)‘𝑓)})) |
11 | 10 | fveq2d 6667 | . . 3 ⊢ (𝜑 → (0g‘𝐶) = (0g‘((LDual‘𝑈) ↾s {𝑓 ∈ (LFnl‘𝑈) ∣ (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓))) = ((LKer‘𝑈)‘𝑓)}))) |
12 | 1, 11 | syl5eq 2865 | . 2 ⊢ (𝜑 → 𝑂 = (0g‘((LDual‘𝑈) ↾s {𝑓 ∈ (LFnl‘𝑈) ∣ (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓))) = ((LKer‘𝑈)‘𝑓)}))) |
13 | 2, 5, 9 | dvhlmod 38126 | . . . 4 ⊢ (𝜑 → 𝑈 ∈ LMod) |
14 | 8, 13 | lduallmod 36169 | . . 3 ⊢ (𝜑 → (LDual‘𝑈) ∈ LMod) |
15 | eqid 2818 | . . . 4 ⊢ (LSubSp‘(LDual‘𝑈)) = (LSubSp‘(LDual‘𝑈)) | |
16 | eqid 2818 | . . . 4 ⊢ {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)} | |
17 | 2, 5, 3, 6, 7, 8, 15, 16, 9 | lclkr 38549 | . . 3 ⊢ (𝜑 → {𝑓 ∈ (LFnl‘𝑈) ∣ (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓))) = ((LKer‘𝑈)‘𝑓)} ∈ (LSubSp‘(LDual‘𝑈))) |
18 | eqid 2818 | . . . 4 ⊢ ((LDual‘𝑈) ↾s {𝑓 ∈ (LFnl‘𝑈) ∣ (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓))) = ((LKer‘𝑈)‘𝑓)}) = ((LDual‘𝑈) ↾s {𝑓 ∈ (LFnl‘𝑈) ∣ (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓))) = ((LKer‘𝑈)‘𝑓)}) | |
19 | eqid 2818 | . . . 4 ⊢ (0g‘(LDual‘𝑈)) = (0g‘(LDual‘𝑈)) | |
20 | eqid 2818 | . . . 4 ⊢ (0g‘((LDual‘𝑈) ↾s {𝑓 ∈ (LFnl‘𝑈) ∣ (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓))) = ((LKer‘𝑈)‘𝑓)})) = (0g‘((LDual‘𝑈) ↾s {𝑓 ∈ (LFnl‘𝑈) ∣ (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓))) = ((LKer‘𝑈)‘𝑓)})) | |
21 | 18, 19, 20, 15 | lss0v 19717 | . . 3 ⊢ (((LDual‘𝑈) ∈ LMod ∧ {𝑓 ∈ (LFnl‘𝑈) ∣ (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓))) = ((LKer‘𝑈)‘𝑓)} ∈ (LSubSp‘(LDual‘𝑈))) → (0g‘((LDual‘𝑈) ↾s {𝑓 ∈ (LFnl‘𝑈) ∣ (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓))) = ((LKer‘𝑈)‘𝑓)})) = (0g‘(LDual‘𝑈))) |
22 | 14, 17, 21 | syl2anc 584 | . 2 ⊢ (𝜑 → (0g‘((LDual‘𝑈) ↾s {𝑓 ∈ (LFnl‘𝑈) ∣ (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓))) = ((LKer‘𝑈)‘𝑓)})) = (0g‘(LDual‘𝑈))) |
23 | lcd0v.v | . . 3 ⊢ 𝑉 = (Base‘𝑈) | |
24 | lcd0v.r | . . 3 ⊢ 𝑅 = (Scalar‘𝑈) | |
25 | lcd0v.z | . . 3 ⊢ 0 = (0g‘𝑅) | |
26 | 23, 24, 25, 8, 19, 13 | ldual0v 36166 | . 2 ⊢ (𝜑 → (0g‘(LDual‘𝑈)) = (𝑉 × { 0 })) |
27 | 12, 22, 26 | 3eqtrd 2857 | 1 ⊢ (𝜑 → 𝑂 = (𝑉 × { 0 })) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 {crab 3139 {csn 4557 × cxp 5546 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 ↾s cress 16472 Scalarcsca 16556 0gc0g 16701 LModclmod 19563 LSubSpclss 19632 LFnlclfn 36073 LKerclk 36101 LDualcld 36139 HLchlt 36366 LHypclh 37000 DVecHcdvh 38094 ocHcoch 38363 LCDualclcd 38602 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-riotaBAD 35969 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 df-om 7570 df-1st 7678 df-2nd 7679 df-tpos 7881 df-undef 7928 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12881 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-sca 16569 df-vsca 16570 df-0g 16703 df-mre 16845 df-mrc 16846 df-acs 16848 df-proset 17526 df-poset 17544 df-plt 17556 df-lub 17572 df-glb 17573 df-join 17574 df-meet 17575 df-p0 17637 df-p1 17638 df-lat 17644 df-clat 17706 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-submnd 17945 df-grp 18044 df-minusg 18045 df-sbg 18046 df-subg 18214 df-cntz 18385 df-oppg 18412 df-lsm 18690 df-cmn 18837 df-abl 18838 df-mgp 19169 df-ur 19181 df-ring 19228 df-oppr 19302 df-dvdsr 19320 df-unit 19321 df-invr 19351 df-dvr 19362 df-drng 19433 df-lmod 19565 df-lss 19633 df-lsp 19673 df-lvec 19804 df-lsatoms 35992 df-lshyp 35993 df-lcv 36035 df-lfl 36074 df-lkr 36102 df-ldual 36140 df-oposet 36192 df-ol 36194 df-oml 36195 df-covers 36282 df-ats 36283 df-atl 36314 df-cvlat 36338 df-hlat 36367 df-llines 36514 df-lplanes 36515 df-lvols 36516 df-lines 36517 df-psubsp 36519 df-pmap 36520 df-padd 36812 df-lhyp 37004 df-laut 37005 df-ldil 37120 df-ltrn 37121 df-trl 37175 df-tgrp 37759 df-tendo 37771 df-edring 37773 df-dveca 38019 df-disoa 38045 df-dvech 38095 df-dib 38155 df-dic 38189 df-dih 38245 df-doch 38364 df-djh 38411 df-lcdual 38603 |
This theorem is referenced by: lcd0v2 38628 lcd0vvalN 38629 mapd0 38681 hdmaplkr 38929 |
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