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Mirrors > Home > MPE Home > Th. List > Mathboxes > lcfl6lem | Structured version Visualization version GIF version |
Description: Lemma for lcfl6 40874. A functional πΊ (whose kernel is closed by dochsnkr 40846) is comletely determined by a vector π in the orthocomplement in its kernel at which the functional value is 1. Note that the β { 0 } in the π hypothesis is redundant by the last hypothesis but allows easier use of other theorems. (Contributed by NM, 3-Jan-2015.) |
Ref | Expression |
---|---|
lcfl6lem.h | β’ π» = (LHypβπΎ) |
lcfl6lem.o | β’ β₯ = ((ocHβπΎ)βπ) |
lcfl6lem.u | β’ π = ((DVecHβπΎ)βπ) |
lcfl6lem.v | β’ π = (Baseβπ) |
lcfl6lem.a | β’ + = (+gβπ) |
lcfl6lem.t | β’ Β· = ( Β·π βπ) |
lcfl6lem.s | β’ π = (Scalarβπ) |
lcfl6lem.i | β’ 1 = (1rβπ) |
lcfl6lem.r | β’ π = (Baseβπ) |
lcfl6lem.z | β’ 0 = (0gβπ) |
lcfl6lem.f | β’ πΉ = (LFnlβπ) |
lcfl6lem.l | β’ πΏ = (LKerβπ) |
lcfl6lem.k | β’ (π β (πΎ β HL β§ π β π»)) |
lcfl6lem.g | β’ (π β πΊ β πΉ) |
lcfl6lem.x | β’ (π β π β (( β₯ β(πΏβπΊ)) β { 0 })) |
lcfl6lem.y | β’ (π β (πΊβπ) = 1 ) |
Ref | Expression |
---|---|
lcfl6lem | β’ (π β πΊ = (π£ β π β¦ (β©π β π βπ€ β ( β₯ β{π})π£ = (π€ + (π Β· π))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lcfl6lem.v | . 2 β’ π = (Baseβπ) | |
2 | lcfl6lem.s | . 2 β’ π = (Scalarβπ) | |
3 | lcfl6lem.r | . 2 β’ π = (Baseβπ) | |
4 | eqid 2724 | . 2 β’ (0gβπ) = (0gβπ) | |
5 | lcfl6lem.f | . 2 β’ πΉ = (LFnlβπ) | |
6 | lcfl6lem.l | . 2 β’ πΏ = (LKerβπ) | |
7 | lcfl6lem.h | . . 3 β’ π» = (LHypβπΎ) | |
8 | lcfl6lem.u | . . 3 β’ π = ((DVecHβπΎ)βπ) | |
9 | lcfl6lem.k | . . 3 β’ (π β (πΎ β HL β§ π β π»)) | |
10 | 7, 8, 9 | dvhlvec 40483 | . 2 β’ (π β π β LVec) |
11 | 7, 8, 9 | dvhlmod 40484 | . . . . 5 β’ (π β π β LMod) |
12 | lcfl6lem.g | . . . . 5 β’ (π β πΊ β πΉ) | |
13 | 1, 5, 6, 11, 12 | lkrssv 38469 | . . . 4 β’ (π β (πΏβπΊ) β π) |
14 | lcfl6lem.o | . . . . 5 β’ β₯ = ((ocHβπΎ)βπ) | |
15 | 7, 8, 1, 14 | dochssv 40729 | . . . 4 β’ (((πΎ β HL β§ π β π») β§ (πΏβπΊ) β π) β ( β₯ β(πΏβπΊ)) β π) |
16 | 9, 13, 15 | syl2anc 583 | . . 3 β’ (π β ( β₯ β(πΏβπΊ)) β π) |
17 | lcfl6lem.x | . . . 4 β’ (π β π β (( β₯ β(πΏβπΊ)) β { 0 })) | |
18 | 17 | eldifad 3953 | . . 3 β’ (π β π β ( β₯ β(πΏβπΊ))) |
19 | 16, 18 | sseldd 3976 | . 2 β’ (π β π β π) |
20 | lcfl6lem.z | . . 3 β’ 0 = (0gβπ) | |
21 | lcfl6lem.a | . . 3 β’ + = (+gβπ) | |
22 | lcfl6lem.t | . . 3 β’ Β· = ( Β·π βπ) | |
23 | eqid 2724 | . . 3 β’ (π£ β π β¦ (β©π β π βπ€ β ( β₯ β{π})π£ = (π€ + (π Β· π)))) = (π£ β π β¦ (β©π β π βπ€ β ( β₯ β{π})π£ = (π€ + (π Β· π)))) | |
24 | eldifsni 4786 | . . . . 5 β’ (π β (( β₯ β(πΏβπΊ)) β { 0 }) β π β 0 ) | |
25 | 17, 24 | syl 17 | . . . 4 β’ (π β π β 0 ) |
26 | eldifsn 4783 | . . . 4 β’ (π β (π β { 0 }) β (π β π β§ π β 0 )) | |
27 | 19, 25, 26 | sylanbrc 582 | . . 3 β’ (π β π β (π β { 0 })) |
28 | 7, 14, 8, 1, 20, 21, 22, 5, 2, 3, 23, 9, 27 | dochflcl 40849 | . 2 β’ (π β (π£ β π β¦ (β©π β π βπ€ β ( β₯ β{π})π£ = (π€ + (π Β· π)))) β πΉ) |
29 | 7, 14, 8, 1, 20, 5, 6, 9, 12, 17 | dochsnkr 40846 | . . 3 β’ (π β (πΏβπΊ) = ( β₯ β{π})) |
30 | 7, 14, 8, 1, 20, 21, 22, 6, 2, 3, 23, 9, 27 | dochsnkr2 40847 | . . 3 β’ (π β (πΏβ(π£ β π β¦ (β©π β π βπ€ β ( β₯ β{π})π£ = (π€ + (π Β· π))))) = ( β₯ β{π})) |
31 | 29, 30 | eqtr4d 2767 | . 2 β’ (π β (πΏβπΊ) = (πΏβ(π£ β π β¦ (β©π β π βπ€ β ( β₯ β{π})π£ = (π€ + (π Β· π)))))) |
32 | lcfl6lem.y | . . 3 β’ (π β (πΊβπ) = 1 ) | |
33 | lcfl6lem.i | . . . 4 β’ 1 = (1rβπ) | |
34 | 7, 14, 8, 1, 21, 22, 20, 2, 3, 33, 9, 27, 23 | dochfl1 40850 | . . 3 β’ (π β ((π£ β π β¦ (β©π β π βπ€ β ( β₯ β{π})π£ = (π€ + (π Β· π))))βπ) = 1 ) |
35 | 32, 34 | eqtr4d 2767 | . 2 β’ (π β (πΊβπ) = ((π£ β π β¦ (β©π β π βπ€ β ( β₯ β{π})π£ = (π€ + (π Β· π))))βπ)) |
36 | 7, 14, 8, 1, 2, 4, 20, 5, 6, 9, 12, 17 | dochfln0 40851 | . 2 β’ (π β (πΊβπ) β (0gβπ)) |
37 | 1, 2, 3, 4, 5, 6, 10, 19, 12, 28, 31, 35, 36 | eqlkr3 38474 | 1 β’ (π β πΊ = (π£ β π β¦ (β©π β π βπ€ β ( β₯ β{π})π£ = (π€ + (π Β· π))))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 β wne 2932 βwrex 3062 β cdif 3938 β wss 3941 {csn 4621 β¦ cmpt 5222 βcfv 6534 β©crio 7357 (class class class)co 7402 Basecbs 17149 +gcplusg 17202 Scalarcsca 17205 Β·π cvsca 17206 0gc0g 17390 1rcur 20082 LFnlclfn 38430 LKerclk 38458 HLchlt 38723 LHypclh 39358 DVecHcdvh 40452 ocHcoch 40721 |
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 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-riotaBAD 38326 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-tp 4626 df-op 4628 df-uni 4901 df-int 4942 df-iun 4990 df-iin 4991 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-1st 7969 df-2nd 7970 df-tpos 8207 df-undef 8254 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8700 df-map 8819 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-n0 12472 df-z 12558 df-uz 12822 df-fz 13486 df-struct 17085 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-mulr 17216 df-sca 17218 df-vsca 17219 df-0g 17392 df-proset 18256 df-poset 18274 df-plt 18291 df-lub 18307 df-glb 18308 df-join 18309 df-meet 18310 df-p0 18386 df-p1 18387 df-lat 18393 df-clat 18460 df-mgm 18569 df-sgrp 18648 df-mnd 18664 df-submnd 18710 df-grp 18862 df-minusg 18863 df-sbg 18864 df-subg 19046 df-cntz 19229 df-lsm 19552 df-cmn 19698 df-abl 19699 df-mgp 20036 df-rng 20054 df-ur 20083 df-ring 20136 df-oppr 20232 df-dvdsr 20255 df-unit 20256 df-invr 20286 df-dvr 20299 df-drng 20585 df-lmod 20704 df-lss 20775 df-lsp 20815 df-lvec 20947 df-lsatoms 38349 df-lshyp 38350 df-lfl 38431 df-lkr 38459 df-oposet 38549 df-ol 38551 df-oml 38552 df-covers 38639 df-ats 38640 df-atl 38671 df-cvlat 38695 df-hlat 38724 df-llines 38872 df-lplanes 38873 df-lvols 38874 df-lines 38875 df-psubsp 38877 df-pmap 38878 df-padd 39170 df-lhyp 39362 df-laut 39363 df-ldil 39478 df-ltrn 39479 df-trl 39533 df-tgrp 40117 df-tendo 40129 df-edring 40131 df-dveca 40377 df-disoa 40403 df-dvech 40453 df-dib 40513 df-dic 40547 df-dih 40603 df-doch 40722 df-djh 40769 |
This theorem is referenced by: lcfl6 40874 |
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