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Mirrors > Home > MPE Home > Th. List > Mathboxes > lcfl6lem | Structured version Visualization version GIF version |
Description: Lemma for lcfl6 38516. A functional 𝐺 (whose kernel is closed by dochsnkr 38488) 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 2818 | . 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 38125 | . 2 ⊢ (𝜑 → 𝑈 ∈ LVec) |
11 | 7, 8, 9 | dvhlmod 38126 | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod) |
12 | lcfl6lem.g | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ 𝐹) | |
13 | 1, 5, 6, 11, 12 | lkrssv 36112 | . . . 4 ⊢ (𝜑 → (𝐿‘𝐺) ⊆ 𝑉) |
14 | lcfl6lem.o | . . . . 5 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) | |
15 | 7, 8, 1, 14 | dochssv 38371 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐿‘𝐺) ⊆ 𝑉) → ( ⊥ ‘(𝐿‘𝐺)) ⊆ 𝑉) |
16 | 9, 13, 15 | syl2anc 584 | . . 3 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘𝐺)) ⊆ 𝑉) |
17 | lcfl6lem.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ (( ⊥ ‘(𝐿‘𝐺)) ∖ { 0 })) | |
18 | 17 | eldifad 3945 | . . 3 ⊢ (𝜑 → 𝑋 ∈ ( ⊥ ‘(𝐿‘𝐺))) |
19 | 16, 18 | sseldd 3965 | . 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 2818 | . . 3 ⊢ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑋})𝑣 = (𝑤 + (𝑘 · 𝑋)))) = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑋})𝑣 = (𝑤 + (𝑘 · 𝑋)))) | |
24 | eldifsni 4714 | . . . . 5 ⊢ (𝑋 ∈ (( ⊥ ‘(𝐿‘𝐺)) ∖ { 0 }) → 𝑋 ≠ 0 ) | |
25 | 17, 24 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑋 ≠ 0 ) |
26 | eldifsn 4711 | . . . 4 ⊢ (𝑋 ∈ (𝑉 ∖ { 0 }) ↔ (𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 )) | |
27 | 19, 25, 26 | sylanbrc 583 | . . 3 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) |
28 | 7, 14, 8, 1, 20, 21, 22, 5, 2, 3, 23, 9, 27 | dochflcl 38491 | . 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 38488 | . . 3 ⊢ (𝜑 → (𝐿‘𝐺) = ( ⊥ ‘{𝑋})) |
30 | 7, 14, 8, 1, 20, 21, 22, 6, 2, 3, 23, 9, 27 | dochsnkr2 38489 | . . 3 ⊢ (𝜑 → (𝐿‘(𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑋})𝑣 = (𝑤 + (𝑘 · 𝑋))))) = ( ⊥ ‘{𝑋})) |
31 | 29, 30 | eqtr4d 2856 | . 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 38492 | . . 3 ⊢ (𝜑 → ((𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑋})𝑣 = (𝑤 + (𝑘 · 𝑋))))‘𝑋) = 1 ) |
35 | 32, 34 | eqtr4d 2856 | . 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 38493 | . 2 ⊢ (𝜑 → (𝐺‘𝑋) ≠ (0g‘𝑆)) |
37 | 1, 2, 3, 4, 5, 6, 10, 19, 12, 28, 31, 35, 36 | eqlkr3 36117 | 1 ⊢ (𝜑 → 𝐺 = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑋})𝑣 = (𝑤 + (𝑘 · 𝑋))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 ∃wrex 3136 ∖ cdif 3930 ⊆ wss 3933 {csn 4557 ↦ cmpt 5137 ‘cfv 6348 ℩crio 7102 (class class class)co 7145 Basecbs 16471 +gcplusg 16553 Scalarcsca 16556 ·𝑠 cvsca 16557 0gc0g 16701 1rcur 19180 LFnlclfn 36073 LKerclk 36101 HLchlt 36366 LHypclh 37000 DVecHcdvh 38094 ocHcoch 38363 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-riotaBAD 35969 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-tpos 7881 df-undef 7928 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12881 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-sca 16569 df-vsca 16570 df-0g 16703 df-proset 17526 df-poset 17544 df-plt 17556 df-lub 17572 df-glb 17573 df-join 17574 df-meet 17575 df-p0 17637 df-p1 17638 df-lat 17644 df-clat 17706 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-submnd 17945 df-grp 18044 df-minusg 18045 df-sbg 18046 df-subg 18214 df-cntz 18385 df-lsm 18690 df-cmn 18837 df-abl 18838 df-mgp 19169 df-ur 19181 df-ring 19228 df-oppr 19302 df-dvdsr 19320 df-unit 19321 df-invr 19351 df-dvr 19362 df-drng 19433 df-lmod 19565 df-lss 19633 df-lsp 19673 df-lvec 19804 df-lsatoms 35992 df-lshyp 35993 df-lfl 36074 df-lkr 36102 df-oposet 36192 df-ol 36194 df-oml 36195 df-covers 36282 df-ats 36283 df-atl 36314 df-cvlat 36338 df-hlat 36367 df-llines 36514 df-lplanes 36515 df-lvols 36516 df-lines 36517 df-psubsp 36519 df-pmap 36520 df-padd 36812 df-lhyp 37004 df-laut 37005 df-ldil 37120 df-ltrn 37121 df-trl 37175 df-tgrp 37759 df-tendo 37771 df-edring 37773 df-dveca 38019 df-disoa 38045 df-dvech 38095 df-dib 38155 df-dic 38189 df-dih 38245 df-doch 38364 df-djh 38411 |
This theorem is referenced by: lcfl6 38516 |
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