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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lcfl4N | Structured version Visualization version GIF version |
Description: Property of a functional with a closed kernel. (Contributed by NM, 1-Jan-2015.) (New usage is discouraged.) |
Ref | Expression |
---|---|
lcfl4.h | ⊢ 𝐻 = (LHyp‘𝐾) |
lcfl4.o | ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) |
lcfl4.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
lcfl4.v | ⊢ 𝑉 = (Base‘𝑈) |
lcfl4.y | ⊢ 𝑌 = (LSHyp‘𝑈) |
lcfl4.f | ⊢ 𝐹 = (LFnl‘𝑈) |
lcfl4.l | ⊢ 𝐿 = (LKer‘𝑈) |
lcfl4.c | ⊢ 𝐶 = {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)} |
lcfl4.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
lcfl4.g | ⊢ (𝜑 → 𝐺 ∈ 𝐹) |
Ref | Expression |
---|---|
lcfl4N | ⊢ (𝜑 → (𝐺 ∈ 𝐶 ↔ (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌 ∨ (𝐿‘𝐺) = 𝑉))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lcfl4.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | lcfl4.o | . . 3 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) | |
3 | lcfl4.u | . . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
4 | lcfl4.v | . . 3 ⊢ 𝑉 = (Base‘𝑈) | |
5 | eqid 2752 | . . 3 ⊢ (LSAtoms‘𝑈) = (LSAtoms‘𝑈) | |
6 | lcfl4.f | . . 3 ⊢ 𝐹 = (LFnl‘𝑈) | |
7 | lcfl4.l | . . 3 ⊢ 𝐿 = (LKer‘𝑈) | |
8 | lcfl4.c | . . 3 ⊢ 𝐶 = {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)} | |
9 | lcfl4.k | . . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
10 | lcfl4.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ 𝐹) | |
11 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 | lcfl3 37277 | . 2 ⊢ (𝜑 → (𝐺 ∈ 𝐶 ↔ (( ⊥ ‘(𝐿‘𝐺)) ∈ (LSAtoms‘𝑈) ∨ (𝐿‘𝐺) = 𝑉))) |
12 | eqid 2752 | . . . 4 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
13 | lcfl4.y | . . . 4 ⊢ 𝑌 = (LSHyp‘𝑈) | |
14 | 1, 3, 9 | dvhlmod 36893 | . . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LMod) |
15 | 4, 6, 7, 14, 10 | lkrssv 34878 | . . . . 5 ⊢ (𝜑 → (𝐿‘𝐺) ⊆ 𝑉) |
16 | 1, 3, 4, 12, 2 | dochlss 37137 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐿‘𝐺) ⊆ 𝑉) → ( ⊥ ‘(𝐿‘𝐺)) ∈ (LSubSp‘𝑈)) |
17 | 9, 15, 16 | syl2anc 696 | . . . 4 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘𝐺)) ∈ (LSubSp‘𝑈)) |
18 | 1, 2, 3, 12, 5, 13, 9, 17 | dochsatshpb 37235 | . . 3 ⊢ (𝜑 → (( ⊥ ‘(𝐿‘𝐺)) ∈ (LSAtoms‘𝑈) ↔ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌)) |
19 | 18 | orbi1d 741 | . 2 ⊢ (𝜑 → ((( ⊥ ‘(𝐿‘𝐺)) ∈ (LSAtoms‘𝑈) ∨ (𝐿‘𝐺) = 𝑉) ↔ (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌 ∨ (𝐿‘𝐺) = 𝑉))) |
20 | 11, 19 | bitrd 268 | 1 ⊢ (𝜑 → (𝐺 ∈ 𝐶 ↔ (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌 ∨ (𝐿‘𝐺) = 𝑉))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∨ wo 382 ∧ wa 383 = wceq 1624 ∈ wcel 2131 {crab 3046 ⊆ wss 3707 ‘cfv 6041 Basecbs 16051 LSubSpclss 19126 LSAtomsclsa 34756 LSHypclsh 34757 LFnlclfn 34839 LKerclk 34867 HLchlt 35132 LHypclh 35765 DVecHcdvh 36861 ocHcoch 37130 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1863 ax-4 1878 ax-5 1980 ax-6 2046 ax-7 2082 ax-8 2133 ax-9 2140 ax-10 2160 ax-11 2175 ax-12 2188 ax-13 2383 ax-ext 2732 ax-rep 4915 ax-sep 4925 ax-nul 4933 ax-pow 4984 ax-pr 5047 ax-un 7106 ax-cnex 10176 ax-resscn 10177 ax-1cn 10178 ax-icn 10179 ax-addcl 10180 ax-addrcl 10181 ax-mulcl 10182 ax-mulrcl 10183 ax-mulcom 10184 ax-addass 10185 ax-mulass 10186 ax-distr 10187 ax-i2m1 10188 ax-1ne0 10189 ax-1rid 10190 ax-rnegex 10191 ax-rrecex 10192 ax-cnre 10193 ax-pre-lttri 10194 ax-pre-lttrn 10195 ax-pre-ltadd 10196 ax-pre-mulgt0 10197 ax-riotaBAD 34734 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1627 df-fal 1630 df-ex 1846 df-nf 1851 df-sb 2039 df-eu 2603 df-mo 2604 df-clab 2739 df-cleq 2745 df-clel 2748 df-nfc 2883 df-ne 2925 df-nel 3028 df-ral 3047 df-rex 3048 df-reu 3049 df-rmo 3050 df-rab 3051 df-v 3334 df-sbc 3569 df-csb 3667 df-dif 3710 df-un 3712 df-in 3714 df-ss 3721 df-pss 3723 df-nul 4051 df-if 4223 df-pw 4296 df-sn 4314 df-pr 4316 df-tp 4318 df-op 4320 df-uni 4581 df-int 4620 df-iun 4666 df-iin 4667 df-br 4797 df-opab 4857 df-mpt 4874 df-tr 4897 df-id 5166 df-eprel 5171 df-po 5179 df-so 5180 df-fr 5217 df-we 5219 df-xp 5264 df-rel 5265 df-cnv 5266 df-co 5267 df-dm 5268 df-rn 5269 df-res 5270 df-ima 5271 df-pred 5833 df-ord 5879 df-on 5880 df-lim 5881 df-suc 5882 df-iota 6004 df-fun 6043 df-fn 6044 df-f 6045 df-f1 6046 df-fo 6047 df-f1o 6048 df-fv 6049 df-riota 6766 df-ov 6808 df-oprab 6809 df-mpt2 6810 df-om 7223 df-1st 7325 df-2nd 7326 df-tpos 7513 df-undef 7560 df-wrecs 7568 df-recs 7629 df-rdg 7667 df-1o 7721 df-oadd 7725 df-er 7903 df-map 8017 df-en 8114 df-dom 8115 df-sdom 8116 df-fin 8117 df-pnf 10260 df-mnf 10261 df-xr 10262 df-ltxr 10263 df-le 10264 df-sub 10452 df-neg 10453 df-nn 11205 df-2 11263 df-3 11264 df-4 11265 df-5 11266 df-6 11267 df-n0 11477 df-z 11562 df-uz 11872 df-fz 12512 df-struct 16053 df-ndx 16054 df-slot 16055 df-base 16057 df-sets 16058 df-ress 16059 df-plusg 16148 df-mulr 16149 df-sca 16151 df-vsca 16152 df-0g 16296 df-preset 17121 df-poset 17139 df-plt 17151 df-lub 17167 df-glb 17168 df-join 17169 df-meet 17170 df-p0 17232 df-p1 17233 df-lat 17239 df-clat 17301 df-mgm 17435 df-sgrp 17477 df-mnd 17488 df-submnd 17529 df-grp 17618 df-minusg 17619 df-sbg 17620 df-subg 17784 df-cntz 17942 df-lsm 18243 df-cmn 18387 df-abl 18388 df-mgp 18682 df-ur 18694 df-ring 18741 df-oppr 18815 df-dvdsr 18833 df-unit 18834 df-invr 18864 df-dvr 18875 df-drng 18943 df-lmod 19059 df-lss 19127 df-lsp 19166 df-lvec 19297 df-lsatoms 34758 df-lshyp 34759 df-lfl 34840 df-lkr 34868 df-oposet 34958 df-ol 34960 df-oml 34961 df-covers 35048 df-ats 35049 df-atl 35080 df-cvlat 35104 df-hlat 35133 df-llines 35279 df-lplanes 35280 df-lvols 35281 df-lines 35282 df-psubsp 35284 df-pmap 35285 df-padd 35577 df-lhyp 35769 df-laut 35770 df-ldil 35885 df-ltrn 35886 df-trl 35941 df-tgrp 36525 df-tendo 36537 df-edring 36539 df-dveca 36785 df-disoa 36812 df-dvech 36862 df-dib 36922 df-dic 36956 df-dih 37012 df-doch 37131 df-djh 37178 |
This theorem is referenced by: (None) |
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