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| Mirrors > Home > MPE Home > Th. List > Mathboxes > lcfrlem19 | Structured version Visualization version GIF version | ||
| Description: Lemma for lcfr 40969. (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 40943 | . . . 4 β’ (π β (π + π) β (π β { 0 })) |
| 14 | 1, 2, 3, 4, 5, 6, 13 | dochnel 40777 | . . 3 β’ (π β Β¬ (π + π) β ( β₯ β{(π + π)})) |
| 15 | 1, 3, 6 | dvhlmod 40494 | . . . . 5 β’ (π β π β LMod) |
| 16 | 15 | adantr 480 | . . . 4 β’ ((π β§ (π β ( β₯ β{(π + π)}) β§ π β ( β₯ β{(π + π)}))) β π β LMod) |
| 17 | 10 | eldifad 3955 | . . . . . . . 8 β’ (π β π β π) |
| 18 | 11 | eldifad 3955 | . . . . . . . 8 β’ (π β π β π) |
| 19 | 4, 7 | lmodvacl 20721 | . . . . . . . 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 4807 | . . . . . 6 β’ (π β {(π + π)} β π) |
| 22 | eqid 2726 | . . . . . . 7 β’ (LSubSpβπ) = (LSubSpβπ) | |
| 23 | 1, 3, 4, 22, 2 | dochlss 40738 | . . . . . 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 583 | . . . . 5 β’ (π β ( β₯ β{(π + π)}) β (LSubSpβπ)) |
| 25 | 24 | adantr 480 | . . . 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 484 | . . . 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 20790 | . . . 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 835 | . . 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 813 | . 2 β’ (π β Β¬ (π β ( β₯ β{(π + π)}) β§ π β ( β₯ β{(π + π)}))) |
| 30 | ianor 978 | . 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 395 β¨ wo 844 = wceq 1533 β wcel 2098 β wne 2934 β cdif 3940 β wss 3943 {csn 4623 βcfv 6537 (class class class)co 7405 Basecbs 17153 +gcplusg 17206 0gc0g 17394 LModclmod 20706 LSubSpclss 20778 LSpanclspn 20818 LSAtomsclsa 38357 HLchlt 38733 LHypclh 39368 DVecHcdvh 40462 ocHcoch 40731 |
| 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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 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 38336 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-tpos 8212 df-undef 8259 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-n0 12477 df-z 12563 df-uz 12827 df-fz 13491 df-struct 17089 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-mulr 17220 df-sca 17222 df-vsca 17223 df-0g 17396 df-proset 18260 df-poset 18278 df-plt 18295 df-lub 18311 df-glb 18312 df-join 18313 df-meet 18314 df-p0 18390 df-p1 18391 df-lat 18397 df-clat 18464 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-submnd 18714 df-grp 18866 df-minusg 18867 df-sbg 18868 df-subg 19050 df-cntz 19233 df-lsm 19556 df-cmn 19702 df-abl 19703 df-mgp 20040 df-rng 20058 df-ur 20087 df-ring 20140 df-oppr 20236 df-dvdsr 20259 df-unit 20260 df-invr 20290 df-dvr 20303 df-drng 20589 df-lmod 20708 df-lss 20779 df-lsp 20819 df-lvec 20951 df-lsatoms 38359 df-oposet 38559 df-ol 38561 df-oml 38562 df-covers 38649 df-ats 38650 df-atl 38681 df-cvlat 38705 df-hlat 38734 df-llines 38882 df-lplanes 38883 df-lvols 38884 df-lines 38885 df-psubsp 38887 df-pmap 38888 df-padd 39180 df-lhyp 39372 df-laut 39373 df-ldil 39488 df-ltrn 39489 df-trl 39543 df-tendo 40139 df-edring 40141 df-disoa 40413 df-dvech 40463 df-dib 40523 df-dic 40557 df-dih 40613 df-doch 40732 |
| This theorem is referenced by: lcfrlem21 40947 |
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