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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lcfrlem20 | Structured version Visualization version GIF version |
Description: Lemma for lcfr 40542. (Contributed by NM, 11-Mar-2015.) |
Ref | Expression |
---|---|
lcfrlem17.h | β’ π» = (LHypβπΎ) |
lcfrlem17.o | β’ β₯ = ((ocHβπΎ)βπ) |
lcfrlem17.u | β’ π = ((DVecHβπΎ)βπ) |
lcfrlem17.v | β’ π = (Baseβπ) |
lcfrlem17.p | β’ + = (+gβπ) |
lcfrlem17.z | β’ 0 = (0gβπ) |
lcfrlem17.n | β’ π = (LSpanβπ) |
lcfrlem17.a | β’ π΄ = (LSAtomsβπ) |
lcfrlem17.k | β’ (π β (πΎ β HL β§ π β π»)) |
lcfrlem17.x | β’ (π β π β (π β { 0 })) |
lcfrlem17.y | β’ (π β π β (π β { 0 })) |
lcfrlem17.ne | β’ (π β (πβ{π}) β (πβ{π})) |
lcfrlem20.e | β’ (π β Β¬ π β ( β₯ β{(π + π)})) |
Ref | Expression |
---|---|
lcfrlem20 | β’ (π β ((πβ{π, π}) β© ( β₯ β{(π + π)})) β π΄) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lcfrlem17.v | . . . 4 β’ π = (Baseβπ) | |
2 | lcfrlem17.n | . . . 4 β’ π = (LSpanβπ) | |
3 | eqid 2732 | . . . 4 β’ (LSSumβπ) = (LSSumβπ) | |
4 | lcfrlem17.h | . . . . 5 β’ π» = (LHypβπΎ) | |
5 | lcfrlem17.u | . . . . 5 β’ π = ((DVecHβπΎ)βπ) | |
6 | lcfrlem17.k | . . . . 5 β’ (π β (πΎ β HL β§ π β π»)) | |
7 | 4, 5, 6 | dvhlmod 40067 | . . . 4 β’ (π β π β LMod) |
8 | lcfrlem17.x | . . . . 5 β’ (π β π β (π β { 0 })) | |
9 | 8 | eldifad 3960 | . . . 4 β’ (π β π β π) |
10 | lcfrlem17.y | . . . . 5 β’ (π β π β (π β { 0 })) | |
11 | 10 | eldifad 3960 | . . . 4 β’ (π β π β π) |
12 | 1, 2, 3, 7, 9, 11 | lsmpr 20705 | . . 3 β’ (π β (πβ{π, π}) = ((πβ{π})(LSSumβπ)(πβ{π}))) |
13 | 12 | ineq1d 4211 | . 2 β’ (π β ((πβ{π, π}) β© ( β₯ β{(π + π)})) = (((πβ{π})(LSSumβπ)(πβ{π})) β© ( β₯ β{(π + π)}))) |
14 | eqid 2732 | . . 3 β’ (LSubSpβπ) = (LSubSpβπ) | |
15 | eqid 2732 | . . 3 β’ (LSHypβπ) = (LSHypβπ) | |
16 | lcfrlem17.a | . . 3 β’ π΄ = (LSAtomsβπ) | |
17 | 4, 5, 6 | dvhlvec 40066 | . . 3 β’ (π β π β LVec) |
18 | lcfrlem17.o | . . . 4 β’ β₯ = ((ocHβπΎ)βπ) | |
19 | lcfrlem17.z | . . . 4 β’ 0 = (0gβπ) | |
20 | lcfrlem17.p | . . . . 5 β’ + = (+gβπ) | |
21 | lcfrlem17.ne | . . . . 5 β’ (π β (πβ{π}) β (πβ{π})) | |
22 | 4, 18, 5, 1, 20, 19, 2, 16, 6, 8, 10, 21 | lcfrlem17 40516 | . . . 4 β’ (π β (π + π) β (π β { 0 })) |
23 | 4, 18, 5, 1, 19, 15, 6, 22 | dochsnshp 40410 | . . 3 β’ (π β ( β₯ β{(π + π)}) β (LSHypβπ)) |
24 | 1, 2, 19, 16, 7, 8 | lsatlspsn 37949 | . . 3 β’ (π β (πβ{π}) β π΄) |
25 | 1, 2, 19, 16, 7, 10 | lsatlspsn 37949 | . . 3 β’ (π β (πβ{π}) β π΄) |
26 | lcfrlem20.e | . . . 4 β’ (π β Β¬ π β ( β₯ β{(π + π)})) | |
27 | 1, 20 | lmodvacl 20490 | . . . . . . . 8 β’ ((π β LMod β§ π β π β§ π β π) β (π + π) β π) |
28 | 7, 9, 11, 27 | syl3anc 1371 | . . . . . . 7 β’ (π β (π + π) β π) |
29 | 28 | snssd 4812 | . . . . . 6 β’ (π β {(π + π)} β π) |
30 | 4, 5, 1, 14, 18 | dochlss 40311 | . . . . . 6 β’ (((πΎ β HL β§ π β π») β§ {(π + π)} β π) β ( β₯ β{(π + π)}) β (LSubSpβπ)) |
31 | 6, 29, 30 | syl2anc 584 | . . . . 5 β’ (π β ( β₯ β{(π + π)}) β (LSubSpβπ)) |
32 | 1, 14, 2, 7, 31, 9 | lspsnel5 20611 | . . . 4 β’ (π β (π β ( β₯ β{(π + π)}) β (πβ{π}) β ( β₯ β{(π + π)}))) |
33 | 26, 32 | mtbid 323 | . . 3 β’ (π β Β¬ (πβ{π}) β ( β₯ β{(π + π)})) |
34 | 14, 3, 15, 16, 17, 23, 24, 25, 21, 33 | lshpat 38012 | . 2 β’ (π β (((πβ{π})(LSSumβπ)(πβ{π})) β© ( β₯ β{(π + π)})) β π΄) |
35 | 13, 34 | eqeltrd 2833 | 1 β’ (π β ((πβ{π, π}) β© ( β₯ β{(π + π)})) β π΄) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 396 = wceq 1541 β wcel 2106 β wne 2940 β cdif 3945 β© cin 3947 β wss 3948 {csn 4628 {cpr 4630 βcfv 6543 (class class class)co 7411 Basecbs 17146 +gcplusg 17199 0gc0g 17387 LSSumclsm 19504 LModclmod 20475 LSubSpclss 20547 LSpanclspn 20587 LSAtomsclsa 37930 LSHypclsh 37931 HLchlt 38306 LHypclh 38941 DVecHcdvh 40035 ocHcoch 40304 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-riotaBAD 37909 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-tpos 8213 df-undef 8260 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-pnf 11252 df-mnf 11253 df-xr 11254 df-ltxr 11255 df-le 11256 df-sub 11448 df-neg 11449 df-nn 12215 df-2 12277 df-3 12278 df-4 12279 df-5 12280 df-6 12281 df-n0 12475 df-z 12561 df-uz 12825 df-fz 13487 df-struct 17082 df-sets 17099 df-slot 17117 df-ndx 17129 df-base 17147 df-ress 17176 df-plusg 17212 df-mulr 17213 df-sca 17215 df-vsca 17216 df-0g 17389 df-mre 17532 df-mrc 17533 df-acs 17535 df-proset 18250 df-poset 18268 df-plt 18285 df-lub 18301 df-glb 18302 df-join 18303 df-meet 18304 df-p0 18380 df-p1 18381 df-lat 18387 df-clat 18454 df-mgm 18563 df-sgrp 18612 df-mnd 18628 df-submnd 18674 df-grp 18824 df-minusg 18825 df-sbg 18826 df-subg 19005 df-cntz 19183 df-oppg 19212 df-lsm 19506 df-cmn 19652 df-abl 19653 df-mgp 19990 df-ur 20007 df-ring 20060 df-oppr 20154 df-dvdsr 20175 df-unit 20176 df-invr 20206 df-dvr 20219 df-drng 20363 df-lmod 20477 df-lss 20548 df-lsp 20588 df-lvec 20719 df-lsatoms 37932 df-lshyp 37933 df-lcv 37975 df-oposet 38132 df-ol 38134 df-oml 38135 df-covers 38222 df-ats 38223 df-atl 38254 df-cvlat 38278 df-hlat 38307 df-llines 38455 df-lplanes 38456 df-lvols 38457 df-lines 38458 df-psubsp 38460 df-pmap 38461 df-padd 38753 df-lhyp 38945 df-laut 38946 df-ldil 39061 df-ltrn 39062 df-trl 39116 df-tgrp 39700 df-tendo 39712 df-edring 39714 df-dveca 39960 df-disoa 39986 df-dvech 40036 df-dib 40096 df-dic 40130 df-dih 40186 df-doch 40305 df-djh 40352 |
This theorem is referenced by: lcfrlem21 40520 |
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