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Mirrors > Home > MPE Home > Th. List > Mathboxes > lcfl6lem | Structured version Visualization version GIF version |
Description: Lemma for lcfl6 39200. A functional 𝐺 (whose kernel is closed by dochsnkr 39172) 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 2736 | . 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 38809 | . 2 ⊢ (𝜑 → 𝑈 ∈ LVec) |
11 | 7, 8, 9 | dvhlmod 38810 | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod) |
12 | lcfl6lem.g | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ 𝐹) | |
13 | 1, 5, 6, 11, 12 | lkrssv 36796 | . . . 4 ⊢ (𝜑 → (𝐿‘𝐺) ⊆ 𝑉) |
14 | lcfl6lem.o | . . . . 5 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) | |
15 | 7, 8, 1, 14 | dochssv 39055 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐿‘𝐺) ⊆ 𝑉) → ( ⊥ ‘(𝐿‘𝐺)) ⊆ 𝑉) |
16 | 9, 13, 15 | syl2anc 587 | . . 3 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘𝐺)) ⊆ 𝑉) |
17 | lcfl6lem.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ (( ⊥ ‘(𝐿‘𝐺)) ∖ { 0 })) | |
18 | 17 | eldifad 3865 | . . 3 ⊢ (𝜑 → 𝑋 ∈ ( ⊥ ‘(𝐿‘𝐺))) |
19 | 16, 18 | sseldd 3888 | . 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 2736 | . . 3 ⊢ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑋})𝑣 = (𝑤 + (𝑘 · 𝑋)))) = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑋})𝑣 = (𝑤 + (𝑘 · 𝑋)))) | |
24 | eldifsni 4689 | . . . . 5 ⊢ (𝑋 ∈ (( ⊥ ‘(𝐿‘𝐺)) ∖ { 0 }) → 𝑋 ≠ 0 ) | |
25 | 17, 24 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑋 ≠ 0 ) |
26 | eldifsn 4686 | . . . 4 ⊢ (𝑋 ∈ (𝑉 ∖ { 0 }) ↔ (𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 )) | |
27 | 19, 25, 26 | sylanbrc 586 | . . 3 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) |
28 | 7, 14, 8, 1, 20, 21, 22, 5, 2, 3, 23, 9, 27 | dochflcl 39175 | . 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 39172 | . . 3 ⊢ (𝜑 → (𝐿‘𝐺) = ( ⊥ ‘{𝑋})) |
30 | 7, 14, 8, 1, 20, 21, 22, 6, 2, 3, 23, 9, 27 | dochsnkr2 39173 | . . 3 ⊢ (𝜑 → (𝐿‘(𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑋})𝑣 = (𝑤 + (𝑘 · 𝑋))))) = ( ⊥ ‘{𝑋})) |
31 | 29, 30 | eqtr4d 2774 | . 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 39176 | . . 3 ⊢ (𝜑 → ((𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑋})𝑣 = (𝑤 + (𝑘 · 𝑋))))‘𝑋) = 1 ) |
35 | 32, 34 | eqtr4d 2774 | . 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 39177 | . 2 ⊢ (𝜑 → (𝐺‘𝑋) ≠ (0g‘𝑆)) |
37 | 1, 2, 3, 4, 5, 6, 10, 19, 12, 28, 31, 35, 36 | eqlkr3 36801 | 1 ⊢ (𝜑 → 𝐺 = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑋})𝑣 = (𝑤 + (𝑘 · 𝑋))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ≠ wne 2932 ∃wrex 3052 ∖ cdif 3850 ⊆ wss 3853 {csn 4527 ↦ cmpt 5120 ‘cfv 6358 ℩crio 7147 (class class class)co 7191 Basecbs 16666 +gcplusg 16749 Scalarcsca 16752 ·𝑠 cvsca 16753 0gc0g 16898 1rcur 19470 LFnlclfn 36757 LKerclk 36785 HLchlt 37050 LHypclh 37684 DVecHcdvh 38778 ocHcoch 39047 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 ax-riotaBAD 36653 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-int 4846 df-iun 4892 df-iin 4893 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-1st 7739 df-2nd 7740 df-tpos 7946 df-undef 7993 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-er 8369 df-map 8488 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-nn 11796 df-2 11858 df-3 11859 df-4 11860 df-5 11861 df-6 11862 df-n0 12056 df-z 12142 df-uz 12404 df-fz 13061 df-struct 16668 df-ndx 16669 df-slot 16670 df-base 16672 df-sets 16673 df-ress 16674 df-plusg 16762 df-mulr 16763 df-sca 16765 df-vsca 16766 df-0g 16900 df-proset 17756 df-poset 17774 df-plt 17790 df-lub 17806 df-glb 17807 df-join 17808 df-meet 17809 df-p0 17885 df-p1 17886 df-lat 17892 df-clat 17959 df-mgm 18068 df-sgrp 18117 df-mnd 18128 df-submnd 18173 df-grp 18322 df-minusg 18323 df-sbg 18324 df-subg 18494 df-cntz 18665 df-lsm 18979 df-cmn 19126 df-abl 19127 df-mgp 19459 df-ur 19471 df-ring 19518 df-oppr 19595 df-dvdsr 19613 df-unit 19614 df-invr 19644 df-dvr 19655 df-drng 19723 df-lmod 19855 df-lss 19923 df-lsp 19963 df-lvec 20094 df-lsatoms 36676 df-lshyp 36677 df-lfl 36758 df-lkr 36786 df-oposet 36876 df-ol 36878 df-oml 36879 df-covers 36966 df-ats 36967 df-atl 36998 df-cvlat 37022 df-hlat 37051 df-llines 37198 df-lplanes 37199 df-lvols 37200 df-lines 37201 df-psubsp 37203 df-pmap 37204 df-padd 37496 df-lhyp 37688 df-laut 37689 df-ldil 37804 df-ltrn 37805 df-trl 37859 df-tgrp 38443 df-tendo 38455 df-edring 38457 df-dveca 38703 df-disoa 38729 df-dvech 38779 df-dib 38839 df-dic 38873 df-dih 38929 df-doch 39048 df-djh 39095 |
This theorem is referenced by: lcfl6 39200 |
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