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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lclkrlem2g | Structured version Visualization version GIF version |
Description: Lemma for lclkr 40046. Comparable hyperplanes are equal, so the kernel of the sum is closed. (Contributed by NM, 16-Jan-2015.) |
Ref | Expression |
---|---|
lclkrlem2f.h | β’ π» = (LHypβπΎ) |
lclkrlem2f.o | β’ β₯ = ((ocHβπΎ)βπ) |
lclkrlem2f.u | β’ π = ((DVecHβπΎ)βπ) |
lclkrlem2f.v | β’ π = (Baseβπ) |
lclkrlem2f.s | β’ π = (Scalarβπ) |
lclkrlem2f.q | β’ π = (0gβπ) |
lclkrlem2f.z | β’ 0 = (0gβπ) |
lclkrlem2f.a | β’ β = (LSSumβπ) |
lclkrlem2f.n | β’ π = (LSpanβπ) |
lclkrlem2f.f | β’ πΉ = (LFnlβπ) |
lclkrlem2f.j | β’ π½ = (LSHypβπ) |
lclkrlem2f.l | β’ πΏ = (LKerβπ) |
lclkrlem2f.d | β’ π· = (LDualβπ) |
lclkrlem2f.p | β’ + = (+gβπ·) |
lclkrlem2f.k | β’ (π β (πΎ β HL β§ π β π»)) |
lclkrlem2f.b | β’ (π β π΅ β (π β { 0 })) |
lclkrlem2f.e | β’ (π β πΈ β πΉ) |
lclkrlem2f.g | β’ (π β πΊ β πΉ) |
lclkrlem2f.le | β’ (π β (πΏβπΈ) = ( β₯ β{π})) |
lclkrlem2f.lg | β’ (π β (πΏβπΊ) = ( β₯ β{π})) |
lclkrlem2f.kb | β’ (π β ((πΈ + πΊ)βπ΅) = π) |
lclkrlem2f.nx | β’ (π β (Β¬ π β ( β₯ β{π΅}) β¨ Β¬ π β ( β₯ β{π΅}))) |
lclkrlem2f.x | β’ (π β π β (π β { 0 })) |
lclkrlem2f.y | β’ (π β π β (π β { 0 })) |
lclkrlem2f.ne | β’ (π β (πΏβπΈ) β (πΏβπΊ)) |
lclkrlem2f.lp | β’ (π β (πΏβ(πΈ + πΊ)) β π½) |
Ref | Expression |
---|---|
lclkrlem2g | β’ (π β ( β₯ β( β₯ β(πΏβ(πΈ + πΊ)))) = (πΏβ(πΈ + πΊ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lclkrlem2f.h | . . . . 5 β’ π» = (LHypβπΎ) | |
2 | lclkrlem2f.o | . . . . 5 β’ β₯ = ((ocHβπΎ)βπ) | |
3 | lclkrlem2f.u | . . . . 5 β’ π = ((DVecHβπΎ)βπ) | |
4 | lclkrlem2f.v | . . . . 5 β’ π = (Baseβπ) | |
5 | lclkrlem2f.s | . . . . 5 β’ π = (Scalarβπ) | |
6 | lclkrlem2f.q | . . . . 5 β’ π = (0gβπ) | |
7 | lclkrlem2f.z | . . . . 5 β’ 0 = (0gβπ) | |
8 | lclkrlem2f.a | . . . . 5 β’ β = (LSSumβπ) | |
9 | lclkrlem2f.n | . . . . 5 β’ π = (LSpanβπ) | |
10 | lclkrlem2f.f | . . . . 5 β’ πΉ = (LFnlβπ) | |
11 | lclkrlem2f.j | . . . . 5 β’ π½ = (LSHypβπ) | |
12 | lclkrlem2f.l | . . . . 5 β’ πΏ = (LKerβπ) | |
13 | lclkrlem2f.d | . . . . 5 β’ π· = (LDualβπ) | |
14 | lclkrlem2f.p | . . . . 5 β’ + = (+gβπ·) | |
15 | lclkrlem2f.k | . . . . 5 β’ (π β (πΎ β HL β§ π β π»)) | |
16 | lclkrlem2f.b | . . . . 5 β’ (π β π΅ β (π β { 0 })) | |
17 | lclkrlem2f.e | . . . . 5 β’ (π β πΈ β πΉ) | |
18 | lclkrlem2f.g | . . . . 5 β’ (π β πΊ β πΉ) | |
19 | lclkrlem2f.le | . . . . 5 β’ (π β (πΏβπΈ) = ( β₯ β{π})) | |
20 | lclkrlem2f.lg | . . . . 5 β’ (π β (πΏβπΊ) = ( β₯ β{π})) | |
21 | lclkrlem2f.kb | . . . . 5 β’ (π β ((πΈ + πΊ)βπ΅) = π) | |
22 | lclkrlem2f.nx | . . . . 5 β’ (π β (Β¬ π β ( β₯ β{π΅}) β¨ Β¬ π β ( β₯ β{π΅}))) | |
23 | lclkrlem2f.x | . . . . 5 β’ (π β π β (π β { 0 })) | |
24 | lclkrlem2f.y | . . . . 5 β’ (π β π β (π β { 0 })) | |
25 | lclkrlem2f.ne | . . . . 5 β’ (π β (πΏβπΈ) β (πΏβπΊ)) | |
26 | lclkrlem2f.lp | . . . . 5 β’ (π β (πΏβ(πΈ + πΊ)) β π½) | |
27 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26 | lclkrlem2f 40025 | . . . 4 β’ (π β (((πΏβπΈ) β© (πΏβπΊ)) β (πβ{π΅})) β (πΏβ(πΈ + πΊ))) |
28 | 1, 3, 15 | dvhlvec 39622 | . . . . 5 β’ (π β π β LVec) |
29 | 19, 20 | ineq12d 4177 | . . . . . . 7 β’ (π β ((πΏβπΈ) β© (πΏβπΊ)) = (( β₯ β{π}) β© ( β₯ β{π}))) |
30 | 29 | oveq1d 7376 | . . . . . 6 β’ (π β (((πΏβπΈ) β© (πΏβπΊ)) β (πβ{π΅})) = ((( β₯ β{π}) β© ( β₯ β{π})) β (πβ{π΅}))) |
31 | eqid 2733 | . . . . . . 7 β’ (LSAtomsβπ) = (LSAtomsβπ) | |
32 | 25, 19, 20 | 3netr3d 3017 | . . . . . . 7 β’ (π β ( β₯ β{π}) β ( β₯ β{π})) |
33 | 1, 2, 3, 4, 7, 8, 9, 31, 15, 16, 23, 24, 32, 22, 11 | lclkrlem2c 40022 | . . . . . 6 β’ (π β ((( β₯ β{π}) β© ( β₯ β{π})) β (πβ{π΅})) β π½) |
34 | 30, 33 | eqeltrd 2834 | . . . . 5 β’ (π β (((πΏβπΈ) β© (πΏβπΊ)) β (πβ{π΅})) β π½) |
35 | 11, 28, 34, 26 | lshpcmp 37500 | . . . 4 β’ (π β ((((πΏβπΈ) β© (πΏβπΊ)) β (πβ{π΅})) β (πΏβ(πΈ + πΊ)) β (((πΏβπΈ) β© (πΏβπΊ)) β (πβ{π΅})) = (πΏβ(πΈ + πΊ)))) |
36 | 27, 35 | mpbid 231 | . . 3 β’ (π β (((πΏβπΈ) β© (πΏβπΊ)) β (πβ{π΅})) = (πΏβ(πΈ + πΊ))) |
37 | eqid 2733 | . . . . 5 β’ ((DIsoHβπΎ)βπ) = ((DIsoHβπΎ)βπ) | |
38 | 1, 2, 3, 4, 7, 8, 9, 31, 15, 16, 23, 24, 32, 22, 37 | lclkrlem2d 40023 | . . . 4 β’ (π β ((( β₯ β{π}) β© ( β₯ β{π})) β (πβ{π΅})) β ran ((DIsoHβπΎ)βπ)) |
39 | 30, 38 | eqeltrd 2834 | . . 3 β’ (π β (((πΏβπΈ) β© (πΏβπΊ)) β (πβ{π΅})) β ran ((DIsoHβπΎ)βπ)) |
40 | 36, 39 | eqeltrrd 2835 | . 2 β’ (π β (πΏβ(πΈ + πΊ)) β ran ((DIsoHβπΎ)βπ)) |
41 | 1, 3, 37, 4 | dihrnss 39791 | . . . 4 β’ (((πΎ β HL β§ π β π») β§ (πΏβ(πΈ + πΊ)) β ran ((DIsoHβπΎ)βπ)) β (πΏβ(πΈ + πΊ)) β π) |
42 | 15, 40, 41 | syl2anc 585 | . . 3 β’ (π β (πΏβ(πΈ + πΊ)) β π) |
43 | 1, 37, 3, 4, 2, 15, 42 | dochoccl 39882 | . 2 β’ (π β ((πΏβ(πΈ + πΊ)) β ran ((DIsoHβπΎ)βπ) β ( β₯ β( β₯ β(πΏβ(πΈ + πΊ)))) = (πΏβ(πΈ + πΊ)))) |
44 | 40, 43 | mpbid 231 | 1 β’ (π β ( β₯ β( β₯ β(πΏβ(πΈ + πΊ)))) = (πΏβ(πΈ + πΊ))) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 397 β¨ wo 846 = wceq 1542 β wcel 2107 β wne 2940 β cdif 3911 β© cin 3913 β wss 3914 {csn 4590 ran crn 5638 βcfv 6500 (class class class)co 7361 Basecbs 17091 +gcplusg 17141 Scalarcsca 17144 0gc0g 17329 LSSumclsm 19424 LSpanclspn 20476 LSAtomsclsa 37486 LSHypclsh 37487 LFnlclfn 37569 LKerclk 37597 LDualcld 37635 HLchlt 37862 LHypclh 38497 DVecHcdvh 39591 DIsoHcdih 39741 ocHcoch 39860 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 ax-riotaBAD 37465 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-tp 4595 df-op 4597 df-uni 4870 df-int 4912 df-iun 4960 df-iin 4961 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-of 7621 df-om 7807 df-1st 7925 df-2nd 7926 df-tpos 8161 df-undef 8208 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-er 8654 df-map 8773 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-nn 12162 df-2 12224 df-3 12225 df-4 12226 df-5 12227 df-6 12228 df-n0 12422 df-z 12508 df-uz 12772 df-fz 13434 df-struct 17027 df-sets 17044 df-slot 17062 df-ndx 17074 df-base 17092 df-ress 17121 df-plusg 17154 df-mulr 17155 df-sca 17157 df-vsca 17158 df-0g 17331 df-mre 17474 df-mrc 17475 df-acs 17477 df-proset 18192 df-poset 18210 df-plt 18227 df-lub 18243 df-glb 18244 df-join 18245 df-meet 18246 df-p0 18322 df-p1 18323 df-lat 18329 df-clat 18396 df-mgm 18505 df-sgrp 18554 df-mnd 18565 df-submnd 18610 df-grp 18759 df-minusg 18760 df-sbg 18761 df-subg 18933 df-cntz 19105 df-oppg 19132 df-lsm 19426 df-cmn 19572 df-abl 19573 df-mgp 19905 df-ur 19922 df-ring 19974 df-oppr 20057 df-dvdsr 20078 df-unit 20079 df-invr 20109 df-dvr 20120 df-drng 20221 df-lmod 20367 df-lss 20437 df-lsp 20477 df-lvec 20608 df-lsatoms 37488 df-lshyp 37489 df-lcv 37531 df-lfl 37570 df-lkr 37598 df-ldual 37636 df-oposet 37688 df-ol 37690 df-oml 37691 df-covers 37778 df-ats 37779 df-atl 37810 df-cvlat 37834 df-hlat 37863 df-llines 38011 df-lplanes 38012 df-lvols 38013 df-lines 38014 df-psubsp 38016 df-pmap 38017 df-padd 38309 df-lhyp 38501 df-laut 38502 df-ldil 38617 df-ltrn 38618 df-trl 38672 df-tgrp 39256 df-tendo 39268 df-edring 39270 df-dveca 39516 df-disoa 39542 df-dvech 39592 df-dib 39652 df-dic 39686 df-dih 39742 df-doch 39861 df-djh 39908 |
This theorem is referenced by: lclkrlem2h 40027 |
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