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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 41038. 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 41017 | . . . 4 β’ (π β (((πΏβπΈ) β© (πΏβπΊ)) β (πβ{π΅})) β (πΏβ(πΈ + πΊ))) |
28 | 1, 3, 15 | dvhlvec 40614 | . . . . 5 β’ (π β π β LVec) |
29 | 19, 20 | ineq12d 4215 | . . . . . . 7 β’ (π β ((πΏβπΈ) β© (πΏβπΊ)) = (( β₯ β{π}) β© ( β₯ β{π}))) |
30 | 29 | oveq1d 7441 | . . . . . 6 β’ (π β (((πΏβπΈ) β© (πΏβπΊ)) β (πβ{π΅})) = ((( β₯ β{π}) β© ( β₯ β{π})) β (πβ{π΅}))) |
31 | eqid 2728 | . . . . . . 7 β’ (LSAtomsβπ) = (LSAtomsβπ) | |
32 | 25, 19, 20 | 3netr3d 3014 | . . . . . . 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 41014 | . . . . . 6 β’ (π β ((( β₯ β{π}) β© ( β₯ β{π})) β (πβ{π΅})) β π½) |
34 | 30, 33 | eqeltrd 2829 | . . . . 5 β’ (π β (((πΏβπΈ) β© (πΏβπΊ)) β (πβ{π΅})) β π½) |
35 | 11, 28, 34, 26 | lshpcmp 38492 | . . . 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 2728 | . . . . 5 β’ ((DIsoHβπΎ)βπ) = ((DIsoHβπΎ)βπ) | |
38 | 1, 2, 3, 4, 7, 8, 9, 31, 15, 16, 23, 24, 32, 22, 37 | lclkrlem2d 41015 | . . . 4 β’ (π β ((( β₯ β{π}) β© ( β₯ β{π})) β (πβ{π΅})) β ran ((DIsoHβπΎ)βπ)) |
39 | 30, 38 | eqeltrd 2829 | . . 3 β’ (π β (((πΏβπΈ) β© (πΏβπΊ)) β (πβ{π΅})) β ran ((DIsoHβπΎ)βπ)) |
40 | 36, 39 | eqeltrrd 2830 | . 2 β’ (π β (πΏβ(πΈ + πΊ)) β ran ((DIsoHβπΎ)βπ)) |
41 | 1, 3, 37, 4 | dihrnss 40783 | . . . 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 582 | . . 3 β’ (π β (πΏβ(πΈ + πΊ)) β π) |
43 | 1, 37, 3, 4, 2, 15, 42 | dochoccl 40874 | . 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 394 β¨ wo 845 = wceq 1533 β wcel 2098 β wne 2937 β cdif 3946 β© cin 3948 β wss 3949 {csn 4632 ran crn 5683 βcfv 6553 (class class class)co 7426 Basecbs 17187 +gcplusg 17240 Scalarcsca 17243 0gc0g 17428 LSSumclsm 19596 LSpanclspn 20862 LSAtomsclsa 38478 LSHypclsh 38479 LFnlclfn 38561 LKerclk 38589 LDualcld 38627 HLchlt 38854 LHypclh 39489 DVecHcdvh 40583 DIsoHcdih 40733 ocHcoch 40852 |
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 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-riotaBAD 38457 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-iin 5003 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-of 7691 df-om 7877 df-1st 7999 df-2nd 8000 df-tpos 8238 df-undef 8285 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-er 8731 df-map 8853 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-n0 12511 df-z 12597 df-uz 12861 df-fz 13525 df-struct 17123 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-ress 17217 df-plusg 17253 df-mulr 17254 df-sca 17256 df-vsca 17257 df-0g 17430 df-mre 17573 df-mrc 17574 df-acs 17576 df-proset 18294 df-poset 18312 df-plt 18329 df-lub 18345 df-glb 18346 df-join 18347 df-meet 18348 df-p0 18424 df-p1 18425 df-lat 18431 df-clat 18498 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-submnd 18748 df-grp 18900 df-minusg 18901 df-sbg 18902 df-subg 19085 df-cntz 19275 df-oppg 19304 df-lsm 19598 df-cmn 19744 df-abl 19745 df-mgp 20082 df-rng 20100 df-ur 20129 df-ring 20182 df-oppr 20280 df-dvdsr 20303 df-unit 20304 df-invr 20334 df-dvr 20347 df-drng 20633 df-lmod 20752 df-lss 20823 df-lsp 20863 df-lvec 20995 df-lsatoms 38480 df-lshyp 38481 df-lcv 38523 df-lfl 38562 df-lkr 38590 df-ldual 38628 df-oposet 38680 df-ol 38682 df-oml 38683 df-covers 38770 df-ats 38771 df-atl 38802 df-cvlat 38826 df-hlat 38855 df-llines 39003 df-lplanes 39004 df-lvols 39005 df-lines 39006 df-psubsp 39008 df-pmap 39009 df-padd 39301 df-lhyp 39493 df-laut 39494 df-ldil 39609 df-ltrn 39610 df-trl 39664 df-tgrp 40248 df-tendo 40260 df-edring 40262 df-dveca 40508 df-disoa 40534 df-dvech 40584 df-dib 40644 df-dic 40678 df-dih 40734 df-doch 40853 df-djh 40900 |
This theorem is referenced by: lclkrlem2h 41019 |
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