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| Mirrors > Home > MPE Home > Th. List > Mathboxes > lcfrlem19 | Structured version Visualization version GIF version | ||
| Description: Lemma for lcfr 40456. (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})) |
| Ref | Expression |
|---|---|
| lcfrlem19 | β’ (π β (Β¬ π β ( β₯ β{(π + π)}) β¨ Β¬ π β ( β₯ β{(π + π)}))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | lcfrlem17.h | . . . 4 β’ π» = (LHypβπΎ) | |
| 2 | lcfrlem17.o | . . . 4 β’ β₯ = ((ocHβπΎ)βπ) | |
| 3 | lcfrlem17.u | . . . 4 β’ π = ((DVecHβπΎ)βπ) | |
| 4 | lcfrlem17.v | . . . 4 β’ π = (Baseβπ) | |
| 5 | lcfrlem17.z | . . . 4 β’ 0 = (0gβπ) | |
| 6 | lcfrlem17.k | . . . 4 β’ (π β (πΎ β HL β§ π β π»)) | |
| 7 | lcfrlem17.p | . . . . 5 β’ + = (+gβπ) | |
| 8 | lcfrlem17.n | . . . . 5 β’ π = (LSpanβπ) | |
| 9 | lcfrlem17.a | . . . . 5 β’ π΄ = (LSAtomsβπ) | |
| 10 | lcfrlem17.x | . . . . 5 β’ (π β π β (π β { 0 })) | |
| 11 | lcfrlem17.y | . . . . 5 β’ (π β π β (π β { 0 })) | |
| 12 | lcfrlem17.ne | . . . . 5 β’ (π β (πβ{π}) β (πβ{π})) | |
| 13 | 1, 2, 3, 4, 7, 5, 8, 9, 6, 10, 11, 12 | lcfrlem17 40430 | . . . 4 β’ (π β (π + π) β (π β { 0 })) |
| 14 | 1, 2, 3, 4, 5, 6, 13 | dochnel 40264 | . . 3 β’ (π β Β¬ (π + π) β ( β₯ β{(π + π)})) |
| 15 | 1, 3, 6 | dvhlmod 39981 | . . . . 5 β’ (π β π β LMod) |
| 16 | 15 | adantr 482 | . . . 4 β’ ((π β§ (π β ( β₯ β{(π + π)}) β§ π β ( β₯ β{(π + π)}))) β π β LMod) |
| 17 | 10 | eldifad 3961 | . . . . . . . 8 β’ (π β π β π) |
| 18 | 11 | eldifad 3961 | . . . . . . . 8 β’ (π β π β π) |
| 19 | 4, 7 | lmodvacl 20486 | . . . . . . . 8 β’ ((π β LMod β§ π β π β§ π β π) β (π + π) β π) |
| 20 | 15, 17, 18, 19 | syl3anc 1372 | . . . . . . 7 β’ (π β (π + π) β π) |
| 21 | 20 | snssd 4813 | . . . . . 6 β’ (π β {(π + π)} β π) |
| 22 | eqid 2733 | . . . . . . 7 β’ (LSubSpβπ) = (LSubSpβπ) | |
| 23 | 1, 3, 4, 22, 2 | dochlss 40225 | . . . . . 6 β’ (((πΎ β HL β§ π β π») β§ {(π + π)} β π) β ( β₯ β{(π + π)}) β (LSubSpβπ)) |
| 24 | 6, 21, 23 | syl2anc 585 | . . . . 5 β’ (π β ( β₯ β{(π + π)}) β (LSubSpβπ)) |
| 25 | 24 | adantr 482 | . . . 4 β’ ((π β§ (π β ( β₯ β{(π + π)}) β§ π β ( β₯ β{(π + π)}))) β ( β₯ β{(π + π)}) β (LSubSpβπ)) |
| 26 | simpr 486 | . . . 4 β’ ((π β§ (π β ( β₯ β{(π + π)}) β§ π β ( β₯ β{(π + π)}))) β (π β ( β₯ β{(π + π)}) β§ π β ( β₯ β{(π + π)}))) | |
| 27 | 7, 22 | lssvacl 20565 | . . . 4 β’ (((π β LMod β§ ( β₯ β{(π + π)}) β (LSubSpβπ)) β§ (π β ( β₯ β{(π + π)}) β§ π β ( β₯ β{(π + π)}))) β (π + π) β ( β₯ β{(π + π)})) |
| 28 | 16, 25, 26, 27 | syl21anc 837 | . . 3 β’ ((π β§ (π β ( β₯ β{(π + π)}) β§ π β ( β₯ β{(π + π)}))) β (π + π) β ( β₯ β{(π + π)})) |
| 29 | 14, 28 | mtand 815 | . 2 β’ (π β Β¬ (π β ( β₯ β{(π + π)}) β§ π β ( β₯ β{(π + π)}))) |
| 30 | ianor 981 | . 2 β’ (Β¬ (π β ( β₯ β{(π + π)}) β§ π β ( β₯ β{(π + π)})) β (Β¬ π β ( β₯ β{(π + π)}) β¨ Β¬ π β ( β₯ β{(π + π)}))) | |
| 31 | 29, 30 | sylib 217 | 1 β’ (π β (Β¬ π β ( β₯ β{(π + π)}) β¨ Β¬ π β ( β₯ β{(π + π)}))) |
| 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 2941 β cdif 3946 β wss 3949 {csn 4629 βcfv 6544 (class class class)co 7409 Basecbs 17144 +gcplusg 17197 0gc0g 17385 LModclmod 20471 LSubSpclss 20542 LSpanclspn 20582 LSAtomsclsa 37844 HLchlt 38220 LHypclh 38855 DVecHcdvh 39949 ocHcoch 40218 |
| 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 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-riotaBAD 37823 |
| 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 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-iin 5001 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-tpos 8211 df-undef 8258 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-er 8703 df-map 8822 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-nn 12213 df-2 12275 df-3 12276 df-4 12277 df-5 12278 df-6 12279 df-n0 12473 df-z 12559 df-uz 12823 df-fz 13485 df-struct 17080 df-sets 17097 df-slot 17115 df-ndx 17127 df-base 17145 df-ress 17174 df-plusg 17210 df-mulr 17211 df-sca 17213 df-vsca 17214 df-0g 17387 df-proset 18248 df-poset 18266 df-plt 18283 df-lub 18299 df-glb 18300 df-join 18301 df-meet 18302 df-p0 18378 df-p1 18379 df-lat 18385 df-clat 18452 df-mgm 18561 df-sgrp 18610 df-mnd 18626 df-submnd 18672 df-grp 18822 df-minusg 18823 df-sbg 18824 df-subg 19003 df-cntz 19181 df-lsm 19504 df-cmn 19650 df-abl 19651 df-mgp 19988 df-ur 20005 df-ring 20058 df-oppr 20150 df-dvdsr 20171 df-unit 20172 df-invr 20202 df-dvr 20215 df-drng 20359 df-lmod 20473 df-lss 20543 df-lsp 20583 df-lvec 20714 df-lsatoms 37846 df-oposet 38046 df-ol 38048 df-oml 38049 df-covers 38136 df-ats 38137 df-atl 38168 df-cvlat 38192 df-hlat 38221 df-llines 38369 df-lplanes 38370 df-lvols 38371 df-lines 38372 df-psubsp 38374 df-pmap 38375 df-padd 38667 df-lhyp 38859 df-laut 38860 df-ldil 38975 df-ltrn 38976 df-trl 39030 df-tendo 39626 df-edring 39628 df-disoa 39900 df-dvech 39950 df-dib 40010 df-dic 40044 df-dih 40100 df-doch 40219 |
| This theorem is referenced by: lcfrlem21 40434 |
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