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| Mirrors > Home > MPE Home > Th. List > Mathboxes > lcfrlem19 | Structured version Visualization version GIF version | ||
| Description: Lemma for lcfr 41114. (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 41088 | . . . 4 β’ (π β (π + π) β (π β { 0 })) |
| 14 | 1, 2, 3, 4, 5, 6, 13 | dochnel 40922 | . . 3 β’ (π β Β¬ (π + π) β ( β₯ β{(π + π)})) |
| 15 | 1, 3, 6 | dvhlmod 40639 | . . . . 5 β’ (π β π β LMod) |
| 16 | 15 | adantr 479 | . . . 4 β’ ((π β§ (π β ( β₯ β{(π + π)}) β§ π β ( β₯ β{(π + π)}))) β π β LMod) |
| 17 | 10 | eldifad 3951 | . . . . . . . 8 β’ (π β π β π) |
| 18 | 11 | eldifad 3951 | . . . . . . . 8 β’ (π β π β π) |
| 19 | 4, 7 | lmodvacl 20762 | . . . . . . . 8 β’ ((π β LMod β§ π β π β§ π β π) β (π + π) β π) |
| 20 | 15, 17, 18, 19 | syl3anc 1368 | . . . . . . 7 β’ (π β (π + π) β π) |
| 21 | 20 | snssd 4808 | . . . . . 6 β’ (π β {(π + π)} β π) |
| 22 | eqid 2725 | . . . . . . 7 β’ (LSubSpβπ) = (LSubSpβπ) | |
| 23 | 1, 3, 4, 22, 2 | dochlss 40883 | . . . . . 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 582 | . . . . 5 β’ (π β ( β₯ β{(π + π)}) β (LSubSpβπ)) |
| 25 | 24 | adantr 479 | . . . 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 483 | . . . 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 20831 | . . . 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 836 | . . 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 814 | . 2 β’ (π β Β¬ (π β ( β₯ β{(π + π)}) β§ π β ( β₯ β{(π + π)}))) |
| 30 | ianor 979 | . 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 394 β¨ wo 845 = wceq 1533 β wcel 2098 β wne 2930 β cdif 3936 β wss 3939 {csn 4624 βcfv 6543 (class class class)co 7416 Basecbs 17179 +gcplusg 17232 0gc0g 17420 LModclmod 20747 LSubSpclss 20819 LSpanclspn 20859 LSAtomsclsa 38502 HLchlt 38878 LHypclh 39513 DVecHcdvh 40607 ocHcoch 40876 |
| 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 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-riotaBAD 38481 |
| 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 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 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 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7991 df-2nd 7992 df-tpos 8230 df-undef 8277 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8723 df-map 8845 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-n0 12503 df-z 12589 df-uz 12853 df-fz 13517 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-ress 17209 df-plusg 17245 df-mulr 17246 df-sca 17248 df-vsca 17249 df-0g 17422 df-proset 18286 df-poset 18304 df-plt 18321 df-lub 18337 df-glb 18338 df-join 18339 df-meet 18340 df-p0 18416 df-p1 18417 df-lat 18423 df-clat 18490 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-submnd 18740 df-grp 18897 df-minusg 18898 df-sbg 18899 df-subg 19082 df-cntz 19272 df-lsm 19595 df-cmn 19741 df-abl 19742 df-mgp 20079 df-rng 20097 df-ur 20126 df-ring 20179 df-oppr 20277 df-dvdsr 20300 df-unit 20301 df-invr 20331 df-dvr 20344 df-drng 20630 df-lmod 20749 df-lss 20820 df-lsp 20860 df-lvec 20992 df-lsatoms 38504 df-oposet 38704 df-ol 38706 df-oml 38707 df-covers 38794 df-ats 38795 df-atl 38826 df-cvlat 38850 df-hlat 38879 df-llines 39027 df-lplanes 39028 df-lvols 39029 df-lines 39030 df-psubsp 39032 df-pmap 39033 df-padd 39325 df-lhyp 39517 df-laut 39518 df-ldil 39633 df-ltrn 39634 df-trl 39688 df-tendo 40284 df-edring 40286 df-disoa 40558 df-dvech 40608 df-dib 40668 df-dic 40702 df-dih 40758 df-doch 40877 |
| This theorem is referenced by: lcfrlem21 41092 |
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