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Mathbox for Norm Megill |
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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 3956 | . . . 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 41094 | . . . . . . 7 ⊢ (𝐺 ∈ 𝐶 ↔ (𝐺 ∈ 𝐹 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺))) |
12 | 11 | simplbi 496 | . . . . . 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 41105 | . . . 4 ⊢ (𝜑 → (𝐺 ∈ 𝐶 ↔ ∃𝑥 ∈ 𝑉 (𝐿‘𝐺) = ( ⊥ ‘{𝑥}))) |
15 | 2, 14 | mpbid 231 | . . 3 ⊢ (𝜑 → ∃𝑥 ∈ 𝑉 (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) |
16 | fveq2 6896 | . . . . . . . . . 10 ⊢ ((𝐿‘𝐺) = ( ⊥ ‘{𝑥}) → ( ⊥ ‘(𝐿‘𝐺)) = ( ⊥ ‘( ⊥ ‘{𝑥}))) | |
17 | 16 | adantl 480 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) → ( ⊥ ‘(𝐿‘𝐺)) = ( ⊥ ‘( ⊥ ‘{𝑥}))) |
18 | lcfl8b.n | . . . . . . . . . 10 ⊢ 𝑁 = (LSpan‘𝑈) | |
19 | 10 | ad2antrr 724 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
20 | simplr 767 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) → 𝑥 ∈ 𝑉) | |
21 | 3, 5, 4, 6, 18, 19, 20 | dochocsn 40984 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) → ( ⊥ ‘( ⊥ ‘{𝑥})) = (𝑁‘{𝑥})) |
22 | 17, 21 | eqtrd 2765 | . . . . . . . 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 494 | . . . . . . . . . . 11 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺)) |
25 | eldifsni 4795 | . . . . . . . . . . . . 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 40713 | . . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑈 ∈ LMod) |
30 | 6, 7, 8, 27, 28, 29, 13 | lkr0f2 38763 | . . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐿‘𝐺) = 𝑉 ↔ 𝐺 = 𝑌)) |
31 | 30 | necon3bid 2974 | . . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐿‘𝐺) ≠ 𝑉 ↔ 𝐺 ≠ 𝑌)) |
32 | 26, 31 | mpbird 256 | . . . . . . . . . . 11 ⊢ (𝜑 → (𝐿‘𝐺) ≠ 𝑉) |
33 | 24, 32 | eqnetrd 2997 | . . . . . . . . . 10 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ≠ 𝑉) |
34 | 33 | ad2antrr 724 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ≠ 𝑉) |
35 | eqid 2725 | . . . . . . . . . 10 ⊢ (LSAtoms‘𝑈) = (LSAtoms‘𝑈) | |
36 | 13 | ad2antrr 724 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) → 𝐺 ∈ 𝐹) |
37 | 3, 4, 5, 6, 35, 7, 8, 19, 36 | dochkrsat2 41059 | . . . . . . . . 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 2826 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) → (𝑁‘{𝑥}) ∈ (LSAtoms‘𝑈)) |
40 | lcfl8b.z | . . . . . . . 8 ⊢ 0 = (0g‘𝑈) | |
41 | 29 | ad2antrr 724 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) → 𝑈 ∈ LMod) |
42 | 6, 18, 40, 35, 41, 20 | lsatspn0 38602 | . . . . . . 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 510 | . . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) → (𝑥 ≠ 0 ∧ ( ⊥ ‘(𝐿‘𝐺)) = (𝑁‘{𝑥}))) |
45 | 44 | ex 411 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ((𝐿‘𝐺) = ( ⊥ ‘{𝑥}) → (𝑥 ≠ 0 ∧ ( ⊥ ‘(𝐿‘𝐺)) = (𝑁‘{𝑥})))) |
46 | 45 | reximdva 3157 | . . 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 4799 | . 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 394 = wceq 1533 ∈ wcel 2098 ≠ wne 2929 ∃wrex 3059 {crab 3418 ∖ cdif 3941 {csn 4630 ‘cfv 6549 Basecbs 17183 0gc0g 17424 LModclmod 20755 LSpanclspn 20867 LSAtomsclsa 38576 LFnlclfn 38659 LKerclk 38687 LDualcld 38725 HLchlt 38952 LHypclh 39587 DVecHcdvh 40681 ocHcoch 40950 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 ax-riotaBAD 38555 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-iin 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7685 df-om 7872 df-1st 7994 df-2nd 7995 df-tpos 8232 df-undef 8279 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-n0 12506 df-z 12592 df-uz 12856 df-fz 13520 df-struct 17119 df-sets 17136 df-slot 17154 df-ndx 17166 df-base 17184 df-ress 17213 df-plusg 17249 df-mulr 17250 df-sca 17252 df-vsca 17253 df-0g 17426 df-proset 18290 df-poset 18308 df-plt 18325 df-lub 18341 df-glb 18342 df-join 18343 df-meet 18344 df-p0 18420 df-p1 18421 df-lat 18427 df-clat 18494 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-submnd 18744 df-grp 18901 df-minusg 18902 df-sbg 18903 df-subg 19086 df-cntz 19280 df-lsm 19603 df-cmn 19749 df-abl 19750 df-mgp 20087 df-rng 20105 df-ur 20134 df-ring 20187 df-oppr 20285 df-dvdsr 20308 df-unit 20309 df-invr 20339 df-dvr 20352 df-drng 20638 df-lmod 20757 df-lss 20828 df-lsp 20868 df-lvec 21000 df-lsatoms 38578 df-lshyp 38579 df-lfl 38660 df-lkr 38688 df-ldual 38726 df-oposet 38778 df-ol 38780 df-oml 38781 df-covers 38868 df-ats 38869 df-atl 38900 df-cvlat 38924 df-hlat 38953 df-llines 39101 df-lplanes 39102 df-lvols 39103 df-lines 39104 df-psubsp 39106 df-pmap 39107 df-padd 39399 df-lhyp 39591 df-laut 39592 df-ldil 39707 df-ltrn 39708 df-trl 39762 df-tgrp 40346 df-tendo 40358 df-edring 40360 df-dveca 40606 df-disoa 40632 df-dvech 40682 df-dib 40742 df-dic 40776 df-dih 40832 df-doch 40951 df-djh 40998 |
This theorem is referenced by: mapdrvallem2 41248 |
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