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Mirrors > Home > MPE Home > Th. List > Mathboxes > lcfl6lem | Structured version Visualization version GIF version |
Description: Lemma for lcfl6 37638. A functional 𝐺 (whose kernel is closed by dochsnkr 37610) is comletely determined by a vector 𝑋 in the orthocomplement in its kernel at which the functional value is 1. Note that the ∖ { 0 } in the 𝑋 hypothesis is redundant by the last hypothesis but allows easier use of other theorems. (Contributed by NM, 3-Jan-2015.) |
Ref | Expression |
---|---|
lcfl6lem.h | ⊢ 𝐻 = (LHyp‘𝐾) |
lcfl6lem.o | ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) |
lcfl6lem.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
lcfl6lem.v | ⊢ 𝑉 = (Base‘𝑈) |
lcfl6lem.a | ⊢ + = (+g‘𝑈) |
lcfl6lem.t | ⊢ · = ( ·𝑠 ‘𝑈) |
lcfl6lem.s | ⊢ 𝑆 = (Scalar‘𝑈) |
lcfl6lem.i | ⊢ 1 = (1r‘𝑆) |
lcfl6lem.r | ⊢ 𝑅 = (Base‘𝑆) |
lcfl6lem.z | ⊢ 0 = (0g‘𝑈) |
lcfl6lem.f | ⊢ 𝐹 = (LFnl‘𝑈) |
lcfl6lem.l | ⊢ 𝐿 = (LKer‘𝑈) |
lcfl6lem.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
lcfl6lem.g | ⊢ (𝜑 → 𝐺 ∈ 𝐹) |
lcfl6lem.x | ⊢ (𝜑 → 𝑋 ∈ (( ⊥ ‘(𝐿‘𝐺)) ∖ { 0 })) |
lcfl6lem.y | ⊢ (𝜑 → (𝐺‘𝑋) = 1 ) |
Ref | Expression |
---|---|
lcfl6lem | ⊢ (𝜑 → 𝐺 = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑋})𝑣 = (𝑤 + (𝑘 · 𝑋))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lcfl6lem.v | . 2 ⊢ 𝑉 = (Base‘𝑈) | |
2 | lcfl6lem.s | . 2 ⊢ 𝑆 = (Scalar‘𝑈) | |
3 | lcfl6lem.r | . 2 ⊢ 𝑅 = (Base‘𝑆) | |
4 | eqid 2777 | . 2 ⊢ (0g‘𝑆) = (0g‘𝑆) | |
5 | lcfl6lem.f | . 2 ⊢ 𝐹 = (LFnl‘𝑈) | |
6 | lcfl6lem.l | . 2 ⊢ 𝐿 = (LKer‘𝑈) | |
7 | lcfl6lem.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
8 | lcfl6lem.u | . . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
9 | lcfl6lem.k | . . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
10 | 7, 8, 9 | dvhlvec 37247 | . 2 ⊢ (𝜑 → 𝑈 ∈ LVec) |
11 | 7, 8, 9 | dvhlmod 37248 | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod) |
12 | lcfl6lem.g | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ 𝐹) | |
13 | 1, 5, 6, 11, 12 | lkrssv 35234 | . . . 4 ⊢ (𝜑 → (𝐿‘𝐺) ⊆ 𝑉) |
14 | lcfl6lem.o | . . . . 5 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) | |
15 | 7, 8, 1, 14 | dochssv 37493 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐿‘𝐺) ⊆ 𝑉) → ( ⊥ ‘(𝐿‘𝐺)) ⊆ 𝑉) |
16 | 9, 13, 15 | syl2anc 579 | . . 3 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘𝐺)) ⊆ 𝑉) |
17 | lcfl6lem.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ (( ⊥ ‘(𝐿‘𝐺)) ∖ { 0 })) | |
18 | 17 | eldifad 3803 | . . 3 ⊢ (𝜑 → 𝑋 ∈ ( ⊥ ‘(𝐿‘𝐺))) |
19 | 16, 18 | sseldd 3821 | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
20 | lcfl6lem.z | . . 3 ⊢ 0 = (0g‘𝑈) | |
21 | lcfl6lem.a | . . 3 ⊢ + = (+g‘𝑈) | |
22 | lcfl6lem.t | . . 3 ⊢ · = ( ·𝑠 ‘𝑈) | |
23 | eqid 2777 | . . 3 ⊢ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑋})𝑣 = (𝑤 + (𝑘 · 𝑋)))) = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑋})𝑣 = (𝑤 + (𝑘 · 𝑋)))) | |
24 | eldifsni 4553 | . . . . 5 ⊢ (𝑋 ∈ (( ⊥ ‘(𝐿‘𝐺)) ∖ { 0 }) → 𝑋 ≠ 0 ) | |
25 | 17, 24 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑋 ≠ 0 ) |
26 | eldifsn 4549 | . . . 4 ⊢ (𝑋 ∈ (𝑉 ∖ { 0 }) ↔ (𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 )) | |
27 | 19, 25, 26 | sylanbrc 578 | . . 3 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) |
28 | 7, 14, 8, 1, 20, 21, 22, 5, 2, 3, 23, 9, 27 | dochflcl 37613 | . 2 ⊢ (𝜑 → (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑋})𝑣 = (𝑤 + (𝑘 · 𝑋)))) ∈ 𝐹) |
29 | 7, 14, 8, 1, 20, 5, 6, 9, 12, 17 | dochsnkr 37610 | . . 3 ⊢ (𝜑 → (𝐿‘𝐺) = ( ⊥ ‘{𝑋})) |
30 | 7, 14, 8, 1, 20, 21, 22, 6, 2, 3, 23, 9, 27 | dochsnkr2 37611 | . . 3 ⊢ (𝜑 → (𝐿‘(𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑋})𝑣 = (𝑤 + (𝑘 · 𝑋))))) = ( ⊥ ‘{𝑋})) |
31 | 29, 30 | eqtr4d 2816 | . 2 ⊢ (𝜑 → (𝐿‘𝐺) = (𝐿‘(𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑋})𝑣 = (𝑤 + (𝑘 · 𝑋)))))) |
32 | lcfl6lem.y | . . 3 ⊢ (𝜑 → (𝐺‘𝑋) = 1 ) | |
33 | lcfl6lem.i | . . . 4 ⊢ 1 = (1r‘𝑆) | |
34 | 7, 14, 8, 1, 21, 22, 20, 2, 3, 33, 9, 27, 23 | dochfl1 37614 | . . 3 ⊢ (𝜑 → ((𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑋})𝑣 = (𝑤 + (𝑘 · 𝑋))))‘𝑋) = 1 ) |
35 | 32, 34 | eqtr4d 2816 | . 2 ⊢ (𝜑 → (𝐺‘𝑋) = ((𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑋})𝑣 = (𝑤 + (𝑘 · 𝑋))))‘𝑋)) |
36 | 7, 14, 8, 1, 2, 4, 20, 5, 6, 9, 12, 17 | dochfln0 37615 | . 2 ⊢ (𝜑 → (𝐺‘𝑋) ≠ (0g‘𝑆)) |
37 | 1, 2, 3, 4, 5, 6, 10, 19, 12, 28, 31, 35, 36 | eqlkr3 35239 | 1 ⊢ (𝜑 → 𝐺 = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑋})𝑣 = (𝑤 + (𝑘 · 𝑋))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2106 ≠ wne 2968 ∃wrex 3090 ∖ cdif 3788 ⊆ wss 3791 {csn 4397 ↦ cmpt 4965 ‘cfv 6135 ℩crio 6882 (class class class)co 6922 Basecbs 16255 +gcplusg 16338 Scalarcsca 16341 ·𝑠 cvsca 16342 0gc0g 16486 1rcur 18888 LFnlclfn 35195 LKerclk 35223 HLchlt 35488 LHypclh 36122 DVecHcdvh 37216 ocHcoch 37485 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 ax-riotaBAD 35091 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-fal 1615 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4672 df-int 4711 df-iun 4755 df-iin 4756 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-tpos 7634 df-undef 7681 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-oadd 7847 df-er 8026 df-map 8142 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-n0 11643 df-z 11729 df-uz 11993 df-fz 12644 df-struct 16257 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-ress 16263 df-plusg 16351 df-mulr 16352 df-sca 16354 df-vsca 16355 df-0g 16488 df-proset 17314 df-poset 17332 df-plt 17344 df-lub 17360 df-glb 17361 df-join 17362 df-meet 17363 df-p0 17425 df-p1 17426 df-lat 17432 df-clat 17494 df-mgm 17628 df-sgrp 17670 df-mnd 17681 df-submnd 17722 df-grp 17812 df-minusg 17813 df-sbg 17814 df-subg 17975 df-cntz 18133 df-lsm 18435 df-cmn 18581 df-abl 18582 df-mgp 18877 df-ur 18889 df-ring 18936 df-oppr 19010 df-dvdsr 19028 df-unit 19029 df-invr 19059 df-dvr 19070 df-drng 19141 df-lmod 19257 df-lss 19325 df-lsp 19367 df-lvec 19498 df-lsatoms 35114 df-lshyp 35115 df-lfl 35196 df-lkr 35224 df-oposet 35314 df-ol 35316 df-oml 35317 df-covers 35404 df-ats 35405 df-atl 35436 df-cvlat 35460 df-hlat 35489 df-llines 35636 df-lplanes 35637 df-lvols 35638 df-lines 35639 df-psubsp 35641 df-pmap 35642 df-padd 35934 df-lhyp 36126 df-laut 36127 df-ldil 36242 df-ltrn 36243 df-trl 36297 df-tgrp 36881 df-tendo 36893 df-edring 36895 df-dveca 37141 df-disoa 37167 df-dvech 37217 df-dib 37277 df-dic 37311 df-dih 37367 df-doch 37486 df-djh 37533 |
This theorem is referenced by: lcfl6 37638 |
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