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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 3908 | . . . 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 39731 | . . . . . . 7 ⊢ (𝐺 ∈ 𝐶 ↔ (𝐺 ∈ 𝐹 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺))) |
12 | 11 | simplbi 498 | . . . . . 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 39742 | . . . 4 ⊢ (𝜑 → (𝐺 ∈ 𝐶 ↔ ∃𝑥 ∈ 𝑉 (𝐿‘𝐺) = ( ⊥ ‘{𝑥}))) |
15 | 2, 14 | mpbid 231 | . . 3 ⊢ (𝜑 → ∃𝑥 ∈ 𝑉 (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) |
16 | fveq2 6811 | . . . . . . . . . 10 ⊢ ((𝐿‘𝐺) = ( ⊥ ‘{𝑥}) → ( ⊥ ‘(𝐿‘𝐺)) = ( ⊥ ‘( ⊥ ‘{𝑥}))) | |
17 | 16 | adantl 482 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) → ( ⊥ ‘(𝐿‘𝐺)) = ( ⊥ ‘( ⊥ ‘{𝑥}))) |
18 | lcfl8b.n | . . . . . . . . . 10 ⊢ 𝑁 = (LSpan‘𝑈) | |
19 | 10 | ad2antrr 723 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
20 | simplr 766 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) → 𝑥 ∈ 𝑉) | |
21 | 3, 5, 4, 6, 18, 19, 20 | dochocsn 39621 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) → ( ⊥ ‘( ⊥ ‘{𝑥})) = (𝑁‘{𝑥})) |
22 | 17, 21 | eqtrd 2776 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) → ( ⊥ ‘(𝐿‘𝐺)) = (𝑁‘{𝑥})) |
23 | 2, 11 | sylib 217 | . . . . . . . . . . . 12 ⊢ (𝜑 → (𝐺 ∈ 𝐹 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺))) |
24 | 23 | simprd 496 | . . . . . . . . . . 11 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺)) |
25 | eldifsni 4734 | . . . . . . . . . . . . 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 39350 | . . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑈 ∈ LMod) |
30 | 6, 7, 8, 27, 28, 29, 13 | lkr0f2 37400 | . . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐿‘𝐺) = 𝑉 ↔ 𝐺 = 𝑌)) |
31 | 30 | necon3bid 2985 | . . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐿‘𝐺) ≠ 𝑉 ↔ 𝐺 ≠ 𝑌)) |
32 | 26, 31 | mpbird 256 | . . . . . . . . . . 11 ⊢ (𝜑 → (𝐿‘𝐺) ≠ 𝑉) |
33 | 24, 32 | eqnetrd 3008 | . . . . . . . . . 10 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ≠ 𝑉) |
34 | 33 | ad2antrr 723 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ≠ 𝑉) |
35 | eqid 2736 | . . . . . . . . . 10 ⊢ (LSAtoms‘𝑈) = (LSAtoms‘𝑈) | |
36 | 13 | ad2antrr 723 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) → 𝐺 ∈ 𝐹) |
37 | 3, 4, 5, 6, 35, 7, 8, 19, 36 | dochkrsat2 39696 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ≠ 𝑉 ↔ ( ⊥ ‘(𝐿‘𝐺)) ∈ (LSAtoms‘𝑈))) |
38 | 34, 37 | mpbid 231 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) → ( ⊥ ‘(𝐿‘𝐺)) ∈ (LSAtoms‘𝑈)) |
39 | 22, 38 | eqeltrrd 2838 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) → (𝑁‘{𝑥}) ∈ (LSAtoms‘𝑈)) |
40 | lcfl8b.z | . . . . . . . 8 ⊢ 0 = (0g‘𝑈) | |
41 | 29 | ad2antrr 723 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) → 𝑈 ∈ LMod) |
42 | 6, 18, 40, 35, 41, 20 | lsatspn0 37239 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) → ((𝑁‘{𝑥}) ∈ (LSAtoms‘𝑈) ↔ 𝑥 ≠ 0 )) |
43 | 39, 42 | mpbid 231 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) → 𝑥 ≠ 0 ) |
44 | 43, 22 | jca 512 | . . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) → (𝑥 ≠ 0 ∧ ( ⊥ ‘(𝐿‘𝐺)) = (𝑁‘{𝑥}))) |
45 | 44 | ex 413 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ((𝐿‘𝐺) = ( ⊥ ‘{𝑥}) → (𝑥 ≠ 0 ∧ ( ⊥ ‘(𝐿‘𝐺)) = (𝑁‘{𝑥})))) |
46 | 45 | reximdva 3161 | . . 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 4738 | . 2 ⊢ (∃𝑥 ∈ (𝑉 ∖ { 0 })( ⊥ ‘(𝐿‘𝐺)) = (𝑁‘{𝑥}) ↔ ∃𝑥 ∈ 𝑉 (𝑥 ≠ 0 ∧ ( ⊥ ‘(𝐿‘𝐺)) = (𝑁‘{𝑥}))) | |
49 | 47, 48 | sylibr 233 | 1 ⊢ (𝜑 → ∃𝑥 ∈ (𝑉 ∖ { 0 })( ⊥ ‘(𝐿‘𝐺)) = (𝑁‘{𝑥})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1540 ∈ wcel 2105 ≠ wne 2940 ∃wrex 3070 {crab 3403 ∖ cdif 3893 {csn 4570 ‘cfv 6465 Basecbs 16986 0gc0g 17224 LModclmod 20203 LSpanclspn 20313 LSAtomsclsa 37213 LFnlclfn 37296 LKerclk 37324 LDualcld 37362 HLchlt 37589 LHypclh 38224 DVecHcdvh 39318 ocHcoch 39587 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-rep 5223 ax-sep 5237 ax-nul 5244 ax-pow 5302 ax-pr 5366 ax-un 7629 ax-cnex 11006 ax-resscn 11007 ax-1cn 11008 ax-icn 11009 ax-addcl 11010 ax-addrcl 11011 ax-mulcl 11012 ax-mulrcl 11013 ax-mulcom 11014 ax-addass 11015 ax-mulass 11016 ax-distr 11017 ax-i2m1 11018 ax-1ne0 11019 ax-1rid 11020 ax-rnegex 11021 ax-rrecex 11022 ax-cnre 11023 ax-pre-lttri 11024 ax-pre-lttrn 11025 ax-pre-ltadd 11026 ax-pre-mulgt0 11027 ax-riotaBAD 37192 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3349 df-reu 3350 df-rab 3404 df-v 3442 df-sbc 3726 df-csb 3842 df-dif 3899 df-un 3901 df-in 3903 df-ss 3913 df-pss 3915 df-nul 4267 df-if 4471 df-pw 4546 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4850 df-int 4892 df-iun 4938 df-iin 4939 df-br 5087 df-opab 5149 df-mpt 5170 df-tr 5204 df-id 5506 df-eprel 5512 df-po 5520 df-so 5521 df-fr 5562 df-we 5564 df-xp 5613 df-rel 5614 df-cnv 5615 df-co 5616 df-dm 5617 df-rn 5618 df-res 5619 df-ima 5620 df-pred 6224 df-ord 6291 df-on 6292 df-lim 6293 df-suc 6294 df-iota 6417 df-fun 6467 df-fn 6468 df-f 6469 df-f1 6470 df-fo 6471 df-f1o 6472 df-fv 6473 df-riota 7273 df-ov 7319 df-oprab 7320 df-mpo 7321 df-of 7574 df-om 7759 df-1st 7877 df-2nd 7878 df-tpos 8090 df-undef 8137 df-frecs 8145 df-wrecs 8176 df-recs 8250 df-rdg 8289 df-1o 8345 df-er 8547 df-map 8666 df-en 8783 df-dom 8784 df-sdom 8785 df-fin 8786 df-pnf 11090 df-mnf 11091 df-xr 11092 df-ltxr 11093 df-le 11094 df-sub 11286 df-neg 11287 df-nn 12053 df-2 12115 df-3 12116 df-4 12117 df-5 12118 df-6 12119 df-n0 12313 df-z 12399 df-uz 12662 df-fz 13319 df-struct 16922 df-sets 16939 df-slot 16957 df-ndx 16969 df-base 16987 df-ress 17016 df-plusg 17049 df-mulr 17050 df-sca 17052 df-vsca 17053 df-0g 17226 df-proset 18087 df-poset 18105 df-plt 18122 df-lub 18138 df-glb 18139 df-join 18140 df-meet 18141 df-p0 18217 df-p1 18218 df-lat 18224 df-clat 18291 df-mgm 18400 df-sgrp 18449 df-mnd 18460 df-submnd 18505 df-grp 18653 df-minusg 18654 df-sbg 18655 df-subg 18825 df-cntz 18996 df-lsm 19314 df-cmn 19460 df-abl 19461 df-mgp 19793 df-ur 19810 df-ring 19857 df-oppr 19934 df-dvdsr 19955 df-unit 19956 df-invr 19986 df-dvr 19997 df-drng 20069 df-lmod 20205 df-lss 20274 df-lsp 20314 df-lvec 20445 df-lsatoms 37215 df-lshyp 37216 df-lfl 37297 df-lkr 37325 df-ldual 37363 df-oposet 37415 df-ol 37417 df-oml 37418 df-covers 37505 df-ats 37506 df-atl 37537 df-cvlat 37561 df-hlat 37590 df-llines 37738 df-lplanes 37739 df-lvols 37740 df-lines 37741 df-psubsp 37743 df-pmap 37744 df-padd 38036 df-lhyp 38228 df-laut 38229 df-ldil 38344 df-ltrn 38345 df-trl 38399 df-tgrp 38983 df-tendo 38995 df-edring 38997 df-dveca 39243 df-disoa 39269 df-dvech 39319 df-dib 39379 df-dic 39413 df-dih 39469 df-doch 39588 df-djh 39635 |
This theorem is referenced by: mapdrvallem2 39885 |
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