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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lcdvbase | Structured version Visualization version GIF version |
Description: Vector base set of a dual vector space of functionals with closed kernels. (Contributed by NM, 13-Mar-2015.) |
Ref | Expression |
---|---|
lcdvbase.h | ⊢ 𝐻 = (LHyp‘𝐾) |
lcdvbase.o | ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) |
lcdvbase.c | ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) |
lcdvbase.v | ⊢ 𝑉 = (Base‘𝐶) |
lcdvbase.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
lcdvbase.f | ⊢ 𝐹 = (LFnl‘𝑈) |
lcdvbase.l | ⊢ 𝐿 = (LKer‘𝑈) |
lcdvbase.b | ⊢ 𝐵 = {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)} |
lcdvbase.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
Ref | Expression |
---|---|
lcdvbase | ⊢ (𝜑 → 𝑉 = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lcdvbase.v | . . 3 ⊢ 𝑉 = (Base‘𝐶) | |
2 | lcdvbase.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
3 | lcdvbase.o | . . . . 5 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) | |
4 | lcdvbase.c | . . . . 5 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
5 | lcdvbase.u | . . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
6 | lcdvbase.f | . . . . 5 ⊢ 𝐹 = (LFnl‘𝑈) | |
7 | lcdvbase.l | . . . . 5 ⊢ 𝐿 = (LKer‘𝑈) | |
8 | eqid 2826 | . . . . 5 ⊢ (LDual‘𝑈) = (LDual‘𝑈) | |
9 | lcdvbase.k | . . . . 5 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
10 | lcdvbase.b | . . . . 5 ⊢ 𝐵 = {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)} | |
11 | 2, 3, 4, 5, 6, 7, 8, 9, 10 | lcdval2 37666 | . . . 4 ⊢ (𝜑 → 𝐶 = ((LDual‘𝑈) ↾s 𝐵)) |
12 | 11 | fveq2d 6438 | . . 3 ⊢ (𝜑 → (Base‘𝐶) = (Base‘((LDual‘𝑈) ↾s 𝐵))) |
13 | 1, 12 | syl5eq 2874 | . 2 ⊢ (𝜑 → 𝑉 = (Base‘((LDual‘𝑈) ↾s 𝐵))) |
14 | ssrab2 3913 | . . . . 5 ⊢ {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)} ⊆ 𝐹 | |
15 | 10, 14 | eqsstri 3861 | . . . 4 ⊢ 𝐵 ⊆ 𝐹 |
16 | eqid 2826 | . . . . 5 ⊢ (Base‘(LDual‘𝑈)) = (Base‘(LDual‘𝑈)) | |
17 | 2, 5, 9 | dvhlmod 37186 | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod) |
18 | 6, 8, 16, 17 | ldualvbase 35202 | . . . 4 ⊢ (𝜑 → (Base‘(LDual‘𝑈)) = 𝐹) |
19 | 15, 18 | syl5sseqr 3880 | . . 3 ⊢ (𝜑 → 𝐵 ⊆ (Base‘(LDual‘𝑈))) |
20 | eqid 2826 | . . . 4 ⊢ ((LDual‘𝑈) ↾s 𝐵) = ((LDual‘𝑈) ↾s 𝐵) | |
21 | 20, 16 | ressbas2 16295 | . . 3 ⊢ (𝐵 ⊆ (Base‘(LDual‘𝑈)) → 𝐵 = (Base‘((LDual‘𝑈) ↾s 𝐵))) |
22 | 19, 21 | syl 17 | . 2 ⊢ (𝜑 → 𝐵 = (Base‘((LDual‘𝑈) ↾s 𝐵))) |
23 | 13, 22 | eqtr4d 2865 | 1 ⊢ (𝜑 → 𝑉 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1658 ∈ wcel 2166 {crab 3122 ⊆ wss 3799 ‘cfv 6124 (class class class)co 6906 Basecbs 16223 ↾s cress 16224 LModclmod 19220 LFnlclfn 35133 LKerclk 35161 LDualcld 35199 HLchlt 35426 LHypclh 36060 DVecHcdvh 37154 ocHcoch 37423 LCDualclcd 37662 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-rep 4995 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 ax-un 7210 ax-cnex 10309 ax-resscn 10310 ax-1cn 10311 ax-icn 10312 ax-addcl 10313 ax-addrcl 10314 ax-mulcl 10315 ax-mulrcl 10316 ax-mulcom 10317 ax-addass 10318 ax-mulass 10319 ax-distr 10320 ax-i2m1 10321 ax-1ne0 10322 ax-1rid 10323 ax-rnegex 10324 ax-rrecex 10325 ax-cnre 10326 ax-pre-lttri 10327 ax-pre-lttrn 10328 ax-pre-ltadd 10329 ax-pre-mulgt0 10330 ax-riotaBAD 35029 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-fal 1672 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-nel 3104 df-ral 3123 df-rex 3124 df-reu 3125 df-rmo 3126 df-rab 3127 df-v 3417 df-sbc 3664 df-csb 3759 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-pss 3815 df-nul 4146 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4660 df-int 4699 df-iun 4743 df-iin 4744 df-br 4875 df-opab 4937 df-mpt 4954 df-tr 4977 df-id 5251 df-eprel 5256 df-po 5264 df-so 5265 df-fr 5302 df-we 5304 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 df-pred 5921 df-ord 5967 df-on 5968 df-lim 5969 df-suc 5970 df-iota 6087 df-fun 6126 df-fn 6127 df-f 6128 df-f1 6129 df-fo 6130 df-f1o 6131 df-fv 6132 df-riota 6867 df-ov 6909 df-oprab 6910 df-mpt2 6911 df-of 7158 df-om 7328 df-1st 7429 df-2nd 7430 df-tpos 7618 df-undef 7665 df-wrecs 7673 df-recs 7735 df-rdg 7773 df-1o 7827 df-oadd 7831 df-er 8010 df-map 8125 df-en 8224 df-dom 8225 df-sdom 8226 df-fin 8227 df-pnf 10394 df-mnf 10395 df-xr 10396 df-ltxr 10397 df-le 10398 df-sub 10588 df-neg 10589 df-nn 11352 df-2 11415 df-3 11416 df-4 11417 df-5 11418 df-6 11419 df-n0 11620 df-z 11706 df-uz 11970 df-fz 12621 df-struct 16225 df-ndx 16226 df-slot 16227 df-base 16229 df-sets 16230 df-ress 16231 df-plusg 16319 df-mulr 16320 df-sca 16322 df-vsca 16323 df-0g 16456 df-proset 17282 df-poset 17300 df-plt 17312 df-lub 17328 df-glb 17329 df-join 17330 df-meet 17331 df-p0 17393 df-p1 17394 df-lat 17400 df-clat 17462 df-mgm 17596 df-sgrp 17638 df-mnd 17649 df-grp 17780 df-minusg 17781 df-mgp 18845 df-ur 18857 df-ring 18904 df-oppr 18978 df-dvdsr 18996 df-unit 18997 df-invr 19027 df-dvr 19038 df-drng 19106 df-lmod 19222 df-lvec 19463 df-ldual 35200 df-oposet 35252 df-ol 35254 df-oml 35255 df-covers 35342 df-ats 35343 df-atl 35374 df-cvlat 35398 df-hlat 35427 df-llines 35574 df-lplanes 35575 df-lvols 35576 df-lines 35577 df-psubsp 35579 df-pmap 35580 df-padd 35872 df-lhyp 36064 df-laut 36065 df-ldil 36180 df-ltrn 36181 df-trl 36235 df-tendo 36831 df-edring 36833 df-dvech 37155 df-lcdual 37663 |
This theorem is referenced by: lcdvbasess 37670 lcdlss2N 37696 lcdlsp 37697 hvmap1o2 37841 |
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