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Mirrors > Home > MPE Home > Th. List > Mathboxes > lcfl8b | Structured version Visualization version GIF version |
Description: Property of a nonzero functional with a closed kernel. (Contributed by NM, 4-Feb-2015.) |
Ref | Expression |
---|---|
lcfl8b.h | ⊢ 𝐻 = (LHyp‘𝐾) |
lcfl8b.o | ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) |
lcfl8b.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
lcfl8b.v | ⊢ 𝑉 = (Base‘𝑈) |
lcfl8b.n | ⊢ 𝑁 = (LSpan‘𝑈) |
lcfl8b.z | ⊢ 0 = (0g‘𝑈) |
lcfl8b.f | ⊢ 𝐹 = (LFnl‘𝑈) |
lcfl8b.l | ⊢ 𝐿 = (LKer‘𝑈) |
lcfl8b.d | ⊢ 𝐷 = (LDual‘𝑈) |
lcfl8b.y | ⊢ 𝑌 = (0g‘𝐷) |
lcfl8b.c | ⊢ 𝐶 = {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)} |
lcfl8b.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
lcfl8b.g | ⊢ (𝜑 → 𝐺 ∈ (𝐶 ∖ {𝑌})) |
Ref | Expression |
---|---|
lcfl8b | ⊢ (𝜑 → ∃𝑥 ∈ (𝑉 ∖ { 0 })( ⊥ ‘(𝐿‘𝐺)) = (𝑁‘{𝑥})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lcfl8b.g | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ (𝐶 ∖ {𝑌})) | |
2 | 1 | eldifad 3810 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ 𝐶) |
3 | lcfl8b.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
4 | lcfl8b.o | . . . . 5 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) | |
5 | lcfl8b.u | . . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
6 | lcfl8b.v | . . . . 5 ⊢ 𝑉 = (Base‘𝑈) | |
7 | lcfl8b.f | . . . . 5 ⊢ 𝐹 = (LFnl‘𝑈) | |
8 | lcfl8b.l | . . . . 5 ⊢ 𝐿 = (LKer‘𝑈) | |
9 | lcfl8b.c | . . . . 5 ⊢ 𝐶 = {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)} | |
10 | lcfl8b.k | . . . . 5 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
11 | 9 | lcfl1lem 37566 | . . . . . . 7 ⊢ (𝐺 ∈ 𝐶 ↔ (𝐺 ∈ 𝐹 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺))) |
12 | 11 | simplbi 493 | . . . . . 6 ⊢ (𝐺 ∈ 𝐶 → 𝐺 ∈ 𝐹) |
13 | 2, 12 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ 𝐹) |
14 | 3, 4, 5, 6, 7, 8, 9, 10, 13 | lcfl8 37577 | . . . 4 ⊢ (𝜑 → (𝐺 ∈ 𝐶 ↔ ∃𝑥 ∈ 𝑉 (𝐿‘𝐺) = ( ⊥ ‘{𝑥}))) |
15 | 2, 14 | mpbid 224 | . . 3 ⊢ (𝜑 → ∃𝑥 ∈ 𝑉 (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) |
16 | fveq2 6433 | . . . . . . . . . 10 ⊢ ((𝐿‘𝐺) = ( ⊥ ‘{𝑥}) → ( ⊥ ‘(𝐿‘𝐺)) = ( ⊥ ‘( ⊥ ‘{𝑥}))) | |
17 | 16 | adantl 475 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) → ( ⊥ ‘(𝐿‘𝐺)) = ( ⊥ ‘( ⊥ ‘{𝑥}))) |
18 | lcfl8b.n | . . . . . . . . . 10 ⊢ 𝑁 = (LSpan‘𝑈) | |
19 | 10 | ad2antrr 719 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
20 | simplr 787 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) → 𝑥 ∈ 𝑉) | |
21 | 3, 5, 4, 6, 18, 19, 20 | dochocsn 37456 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) → ( ⊥ ‘( ⊥ ‘{𝑥})) = (𝑁‘{𝑥})) |
22 | 17, 21 | eqtrd 2861 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) → ( ⊥ ‘(𝐿‘𝐺)) = (𝑁‘{𝑥})) |
23 | 2, 11 | sylib 210 | . . . . . . . . . . . 12 ⊢ (𝜑 → (𝐺 ∈ 𝐹 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺))) |
24 | 23 | simprd 491 | . . . . . . . . . . 11 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺)) |
25 | eldifsni 4540 | . . . . . . . . . . . . 13 ⊢ (𝐺 ∈ (𝐶 ∖ {𝑌}) → 𝐺 ≠ 𝑌) | |
26 | 1, 25 | syl 17 | . . . . . . . . . . . 12 ⊢ (𝜑 → 𝐺 ≠ 𝑌) |
27 | lcfl8b.d | . . . . . . . . . . . . . 14 ⊢ 𝐷 = (LDual‘𝑈) | |
28 | lcfl8b.y | . . . . . . . . . . . . . 14 ⊢ 𝑌 = (0g‘𝐷) | |
29 | 3, 5, 10 | dvhlmod 37185 | . . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑈 ∈ LMod) |
30 | 6, 7, 8, 27, 28, 29, 13 | lkr0f2 35236 | . . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐿‘𝐺) = 𝑉 ↔ 𝐺 = 𝑌)) |
31 | 30 | necon3bid 3043 | . . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐿‘𝐺) ≠ 𝑉 ↔ 𝐺 ≠ 𝑌)) |
32 | 26, 31 | mpbird 249 | . . . . . . . . . . 11 ⊢ (𝜑 → (𝐿‘𝐺) ≠ 𝑉) |
33 | 24, 32 | eqnetrd 3066 | . . . . . . . . . 10 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ≠ 𝑉) |
34 | 33 | ad2antrr 719 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ≠ 𝑉) |
35 | eqid 2825 | . . . . . . . . . 10 ⊢ (LSAtoms‘𝑈) = (LSAtoms‘𝑈) | |
36 | 13 | ad2antrr 719 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) → 𝐺 ∈ 𝐹) |
37 | 3, 4, 5, 6, 35, 7, 8, 19, 36 | dochkrsat2 37531 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ≠ 𝑉 ↔ ( ⊥ ‘(𝐿‘𝐺)) ∈ (LSAtoms‘𝑈))) |
38 | 34, 37 | mpbid 224 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) → ( ⊥ ‘(𝐿‘𝐺)) ∈ (LSAtoms‘𝑈)) |
39 | 22, 38 | eqeltrrd 2907 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) → (𝑁‘{𝑥}) ∈ (LSAtoms‘𝑈)) |
40 | lcfl8b.z | . . . . . . . 8 ⊢ 0 = (0g‘𝑈) | |
41 | 29 | ad2antrr 719 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) → 𝑈 ∈ LMod) |
42 | 6, 18, 40, 35, 41, 20 | lsatspn0 35075 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) → ((𝑁‘{𝑥}) ∈ (LSAtoms‘𝑈) ↔ 𝑥 ≠ 0 )) |
43 | 39, 42 | mpbid 224 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) → 𝑥 ≠ 0 ) |
44 | 43, 22 | jca 509 | . . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) → (𝑥 ≠ 0 ∧ ( ⊥ ‘(𝐿‘𝐺)) = (𝑁‘{𝑥}))) |
45 | 44 | ex 403 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ((𝐿‘𝐺) = ( ⊥ ‘{𝑥}) → (𝑥 ≠ 0 ∧ ( ⊥ ‘(𝐿‘𝐺)) = (𝑁‘{𝑥})))) |
46 | 45 | reximdva 3225 | . . 3 ⊢ (𝜑 → (∃𝑥 ∈ 𝑉 (𝐿‘𝐺) = ( ⊥ ‘{𝑥}) → ∃𝑥 ∈ 𝑉 (𝑥 ≠ 0 ∧ ( ⊥ ‘(𝐿‘𝐺)) = (𝑁‘{𝑥})))) |
47 | 15, 46 | mpd 15 | . 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝑉 (𝑥 ≠ 0 ∧ ( ⊥ ‘(𝐿‘𝐺)) = (𝑁‘{𝑥}))) |
48 | rexdifsn 4543 | . 2 ⊢ (∃𝑥 ∈ (𝑉 ∖ { 0 })( ⊥ ‘(𝐿‘𝐺)) = (𝑁‘{𝑥}) ↔ ∃𝑥 ∈ 𝑉 (𝑥 ≠ 0 ∧ ( ⊥ ‘(𝐿‘𝐺)) = (𝑁‘{𝑥}))) | |
49 | 47, 48 | sylibr 226 | 1 ⊢ (𝜑 → ∃𝑥 ∈ (𝑉 ∖ { 0 })( ⊥ ‘(𝐿‘𝐺)) = (𝑁‘{𝑥})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1658 ∈ wcel 2166 ≠ wne 2999 ∃wrex 3118 {crab 3121 ∖ cdif 3795 {csn 4397 ‘cfv 6123 Basecbs 16222 0gc0g 16453 LModclmod 19219 LSpanclspn 19330 LSAtomsclsa 35049 LFnlclfn 35132 LKerclk 35160 LDualcld 35198 HLchlt 35425 LHypclh 36059 DVecHcdvh 37153 ocHcoch 37422 |
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 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 ax-riotaBAD 35028 |
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 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-int 4698 df-iun 4742 df-iin 4743 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-of 7157 df-om 7327 df-1st 7428 df-2nd 7429 df-tpos 7617 df-undef 7664 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-1o 7826 df-oadd 7830 df-er 8009 df-map 8124 df-en 8223 df-dom 8224 df-sdom 8225 df-fin 8226 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-nn 11351 df-2 11414 df-3 11415 df-4 11416 df-5 11417 df-6 11418 df-n0 11619 df-z 11705 df-uz 11969 df-fz 12620 df-struct 16224 df-ndx 16225 df-slot 16226 df-base 16228 df-sets 16229 df-ress 16230 df-plusg 16318 df-mulr 16319 df-sca 16321 df-vsca 16322 df-0g 16455 df-proset 17281 df-poset 17299 df-plt 17311 df-lub 17327 df-glb 17328 df-join 17329 df-meet 17330 df-p0 17392 df-p1 17393 df-lat 17399 df-clat 17461 df-mgm 17595 df-sgrp 17637 df-mnd 17648 df-submnd 17689 df-grp 17779 df-minusg 17780 df-sbg 17781 df-subg 17942 df-cntz 18100 df-lsm 18402 df-cmn 18548 df-abl 18549 df-mgp 18844 df-ur 18856 df-ring 18903 df-oppr 18977 df-dvdsr 18995 df-unit 18996 df-invr 19026 df-dvr 19037 df-drng 19105 df-lmod 19221 df-lss 19289 df-lsp 19331 df-lvec 19462 df-lsatoms 35051 df-lshyp 35052 df-lfl 35133 df-lkr 35161 df-ldual 35199 df-oposet 35251 df-ol 35253 df-oml 35254 df-covers 35341 df-ats 35342 df-atl 35373 df-cvlat 35397 df-hlat 35426 df-llines 35573 df-lplanes 35574 df-lvols 35575 df-lines 35576 df-psubsp 35578 df-pmap 35579 df-padd 35871 df-lhyp 36063 df-laut 36064 df-ldil 36179 df-ltrn 36180 df-trl 36234 df-tgrp 36818 df-tendo 36830 df-edring 36832 df-dveca 37078 df-disoa 37104 df-dvech 37154 df-dib 37214 df-dic 37248 df-dih 37304 df-doch 37423 df-djh 37470 |
This theorem is referenced by: mapdrvallem2 37720 |
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