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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 2731 | . . . . 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 2731 | . . . . 5 ⊢ (LFnl‘𝑈) = (LFnl‘𝑈) | |
7 | eqid 2731 | . . . . 5 ⊢ (LKer‘𝑈) = (LKer‘𝑈) | |
8 | eqid 2731 | . . . . 5 ⊢ (LDual‘𝑈) = (LDual‘𝑈) | |
9 | lcd0v.k | . . . . 5 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
10 | 2, 3, 4, 5, 6, 7, 8, 9 | lcdval 40165 | . . . 4 ⊢ (𝜑 → 𝐶 = ((LDual‘𝑈) ↾s {𝑓 ∈ (LFnl‘𝑈) ∣ (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓))) = ((LKer‘𝑈)‘𝑓)})) |
11 | 10 | fveq2d 6873 | . . 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 | eqtrid 2783 | . 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 39686 | . . . 4 ⊢ (𝜑 → 𝑈 ∈ LMod) |
14 | 8, 13 | lduallmod 37728 | . . 3 ⊢ (𝜑 → (LDual‘𝑈) ∈ LMod) |
15 | eqid 2731 | . . . 4 ⊢ (LSubSp‘(LDual‘𝑈)) = (LSubSp‘(LDual‘𝑈)) | |
16 | eqid 2731 | . . . 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 40109 | . . 3 ⊢ (𝜑 → {𝑓 ∈ (LFnl‘𝑈) ∣ (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓))) = ((LKer‘𝑈)‘𝑓)} ∈ (LSubSp‘(LDual‘𝑈))) |
18 | eqid 2731 | . . . 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 2731 | . . . 4 ⊢ (0g‘(LDual‘𝑈)) = (0g‘(LDual‘𝑈)) | |
20 | eqid 2731 | . . . 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 20556 | . . 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 37725 | . 2 ⊢ (𝜑 → (0g‘(LDual‘𝑈)) = (𝑉 × { 0 })) |
27 | 12, 22, 26 | 3eqtrd 2775 | 1 ⊢ (𝜑 → 𝑂 = (𝑉 × { 0 })) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 {crab 3425 {csn 4613 × cxp 5658 ‘cfv 6523 (class class class)co 7384 Basecbs 17116 ↾s cress 17145 Scalarcsca 17172 0gc0g 17357 LModclmod 20400 LSubSpclss 20471 LFnlclfn 37632 LKerclk 37660 LDualcld 37698 HLchlt 37925 LHypclh 38560 DVecHcdvh 39654 ocHcoch 39923 LCDualclcd 40162 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5269 ax-sep 5283 ax-nul 5290 ax-pow 5347 ax-pr 5411 ax-un 7699 ax-cnex 11138 ax-resscn 11139 ax-1cn 11140 ax-icn 11141 ax-addcl 11142 ax-addrcl 11143 ax-mulcl 11144 ax-mulrcl 11145 ax-mulcom 11146 ax-addass 11147 ax-mulass 11148 ax-distr 11149 ax-i2m1 11150 ax-1ne0 11151 ax-1rid 11152 ax-rnegex 11153 ax-rrecex 11154 ax-cnre 11155 ax-pre-lttri 11156 ax-pre-lttrn 11157 ax-pre-ltadd 11158 ax-pre-mulgt0 11159 ax-riotaBAD 37528 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3371 df-reu 3372 df-rab 3426 df-v 3468 df-sbc 3765 df-csb 3881 df-dif 3938 df-un 3940 df-in 3942 df-ss 3952 df-pss 3954 df-nul 4310 df-if 4514 df-pw 4589 df-sn 4614 df-pr 4616 df-tp 4618 df-op 4620 df-uni 4893 df-int 4935 df-iun 4983 df-iin 4984 df-br 5133 df-opab 5195 df-mpt 5216 df-tr 5250 df-id 5558 df-eprel 5564 df-po 5572 df-so 5573 df-fr 5615 df-we 5617 df-xp 5666 df-rel 5667 df-cnv 5668 df-co 5669 df-dm 5670 df-rn 5671 df-res 5672 df-ima 5673 df-pred 6280 df-ord 6347 df-on 6348 df-lim 6349 df-suc 6350 df-iota 6475 df-fun 6525 df-fn 6526 df-f 6527 df-f1 6528 df-fo 6529 df-f1o 6530 df-fv 6531 df-riota 7340 df-ov 7387 df-oprab 7388 df-mpo 7389 df-of 7644 df-om 7830 df-1st 7948 df-2nd 7949 df-tpos 8184 df-undef 8231 df-frecs 8239 df-wrecs 8270 df-recs 8344 df-rdg 8383 df-1o 8439 df-er 8677 df-map 8796 df-en 8913 df-dom 8914 df-sdom 8915 df-fin 8916 df-pnf 11222 df-mnf 11223 df-xr 11224 df-ltxr 11225 df-le 11226 df-sub 11418 df-neg 11419 df-nn 12185 df-2 12247 df-3 12248 df-4 12249 df-5 12250 df-6 12251 df-n0 12445 df-z 12531 df-uz 12795 df-fz 13457 df-struct 17052 df-sets 17069 df-slot 17087 df-ndx 17099 df-base 17117 df-ress 17146 df-plusg 17182 df-mulr 17183 df-sca 17185 df-vsca 17186 df-0g 17359 df-mre 17502 df-mrc 17503 df-acs 17505 df-proset 18220 df-poset 18238 df-plt 18255 df-lub 18271 df-glb 18272 df-join 18273 df-meet 18274 df-p0 18350 df-p1 18351 df-lat 18357 df-clat 18424 df-mgm 18533 df-sgrp 18582 df-mnd 18593 df-submnd 18638 df-grp 18787 df-minusg 18788 df-sbg 18789 df-subg 18961 df-cntz 19133 df-oppg 19160 df-lsm 19454 df-cmn 19600 df-abl 19601 df-mgp 19933 df-ur 19950 df-ring 20002 df-oppr 20085 df-dvdsr 20106 df-unit 20107 df-invr 20137 df-dvr 20148 df-drng 20249 df-lmod 20402 df-lss 20472 df-lsp 20512 df-lvec 20643 df-lsatoms 37551 df-lshyp 37552 df-lcv 37594 df-lfl 37633 df-lkr 37661 df-ldual 37699 df-oposet 37751 df-ol 37753 df-oml 37754 df-covers 37841 df-ats 37842 df-atl 37873 df-cvlat 37897 df-hlat 37926 df-llines 38074 df-lplanes 38075 df-lvols 38076 df-lines 38077 df-psubsp 38079 df-pmap 38080 df-padd 38372 df-lhyp 38564 df-laut 38565 df-ldil 38680 df-ltrn 38681 df-trl 38735 df-tgrp 39319 df-tendo 39331 df-edring 39333 df-dveca 39579 df-disoa 39605 df-dvech 39655 df-dib 39715 df-dic 39749 df-dih 39805 df-doch 39924 df-djh 39971 df-lcdual 40163 |
This theorem is referenced by: lcd0v2 40188 lcd0vvalN 40189 mapd0 40241 hdmaplkr 40489 |
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