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Mirrors > Home > MPE Home > Th. List > Mathboxes > dochexmidlem5 | Structured version Visualization version GIF version |
Description: Lemma for dochexmid 40327. (Contributed by NM, 15-Jan-2015.) |
Ref | Expression |
---|---|
dochexmidlem1.h | β’ π» = (LHypβπΎ) |
dochexmidlem1.o | β’ β₯ = ((ocHβπΎ)βπ) |
dochexmidlem1.u | β’ π = ((DVecHβπΎ)βπ) |
dochexmidlem1.v | β’ π = (Baseβπ) |
dochexmidlem1.s | β’ π = (LSubSpβπ) |
dochexmidlem1.n | β’ π = (LSpanβπ) |
dochexmidlem1.p | β’ β = (LSSumβπ) |
dochexmidlem1.a | β’ π΄ = (LSAtomsβπ) |
dochexmidlem1.k | β’ (π β (πΎ β HL β§ π β π»)) |
dochexmidlem1.x | β’ (π β π β π) |
dochexmidlem5.pp | β’ (π β π β π΄) |
dochexmidlem5.z | β’ 0 = (0gβπ) |
dochexmidlem5.m | β’ π = (π β π) |
dochexmidlem5.xn | β’ (π β π β { 0 }) |
dochexmidlem5.pl | β’ (π β Β¬ π β (π β ( β₯ βπ))) |
Ref | Expression |
---|---|
dochexmidlem5 | β’ (π β (( β₯ βπ) β© π) = { 0 }) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dochexmidlem5.pl | . 2 β’ (π β Β¬ π β (π β ( β₯ βπ))) | |
2 | dochexmidlem1.s | . . . . . 6 β’ π = (LSubSpβπ) | |
3 | dochexmidlem5.z | . . . . . 6 β’ 0 = (0gβπ) | |
4 | dochexmidlem1.a | . . . . . 6 β’ π΄ = (LSAtomsβπ) | |
5 | dochexmidlem1.h | . . . . . . . 8 β’ π» = (LHypβπΎ) | |
6 | dochexmidlem1.u | . . . . . . . 8 β’ π = ((DVecHβπΎ)βπ) | |
7 | dochexmidlem1.k | . . . . . . . 8 β’ (π β (πΎ β HL β§ π β π»)) | |
8 | 5, 6, 7 | dvhlmod 39969 | . . . . . . 7 β’ (π β π β LMod) |
9 | 8 | adantr 481 | . . . . . 6 β’ ((π β§ (( β₯ βπ) β© π) β { 0 }) β π β LMod) |
10 | dochexmidlem1.x | . . . . . . . . . 10 β’ (π β π β π) | |
11 | dochexmidlem1.v | . . . . . . . . . . 11 β’ π = (Baseβπ) | |
12 | 11, 2 | lssss 20539 | . . . . . . . . . 10 β’ (π β π β π β π) |
13 | 10, 12 | syl 17 | . . . . . . . . 9 β’ (π β π β π) |
14 | dochexmidlem1.o | . . . . . . . . . 10 β’ β₯ = ((ocHβπΎ)βπ) | |
15 | 5, 6, 11, 2, 14 | dochlss 40213 | . . . . . . . . 9 β’ (((πΎ β HL β§ π β π») β§ π β π) β ( β₯ βπ) β π) |
16 | 7, 13, 15 | syl2anc 584 | . . . . . . . 8 β’ (π β ( β₯ βπ) β π) |
17 | dochexmidlem5.m | . . . . . . . . 9 β’ π = (π β π) | |
18 | dochexmidlem5.pp | . . . . . . . . . . 11 β’ (π β π β π΄) | |
19 | 2, 4, 8, 18 | lsatlssel 37855 | . . . . . . . . . 10 β’ (π β π β π) |
20 | dochexmidlem1.p | . . . . . . . . . . 11 β’ β = (LSSumβπ) | |
21 | 2, 20 | lsmcl 20686 | . . . . . . . . . 10 β’ ((π β LMod β§ π β π β§ π β π) β (π β π) β π) |
22 | 8, 10, 19, 21 | syl3anc 1371 | . . . . . . . . 9 β’ (π β (π β π) β π) |
23 | 17, 22 | eqeltrid 2837 | . . . . . . . 8 β’ (π β π β π) |
24 | 2 | lssincl 20568 | . . . . . . . 8 β’ ((π β LMod β§ ( β₯ βπ) β π β§ π β π) β (( β₯ βπ) β© π) β π) |
25 | 8, 16, 23, 24 | syl3anc 1371 | . . . . . . 7 β’ (π β (( β₯ βπ) β© π) β π) |
26 | 25 | adantr 481 | . . . . . 6 β’ ((π β§ (( β₯ βπ) β© π) β { 0 }) β (( β₯ βπ) β© π) β π) |
27 | simpr 485 | . . . . . 6 β’ ((π β§ (( β₯ βπ) β© π) β { 0 }) β (( β₯ βπ) β© π) β { 0 }) | |
28 | 2, 3, 4, 9, 26, 27 | lssatomic 37869 | . . . . 5 β’ ((π β§ (( β₯ βπ) β© π) β { 0 }) β βπ β π΄ π β (( β₯ βπ) β© π)) |
29 | 28 | ex 413 | . . . 4 β’ (π β ((( β₯ βπ) β© π) β { 0 } β βπ β π΄ π β (( β₯ βπ) β© π))) |
30 | dochexmidlem1.n | . . . . . 6 β’ π = (LSpanβπ) | |
31 | 7 | 3ad2ant1 1133 | . . . . . 6 β’ ((π β§ π β π΄ β§ π β (( β₯ βπ) β© π)) β (πΎ β HL β§ π β π»)) |
32 | 10 | 3ad2ant1 1133 | . . . . . 6 β’ ((π β§ π β π΄ β§ π β (( β₯ βπ) β© π)) β π β π) |
33 | 18 | 3ad2ant1 1133 | . . . . . 6 β’ ((π β§ π β π΄ β§ π β (( β₯ βπ) β© π)) β π β π΄) |
34 | simp2 1137 | . . . . . 6 β’ ((π β§ π β π΄ β§ π β (( β₯ βπ) β© π)) β π β π΄) | |
35 | dochexmidlem5.xn | . . . . . . 7 β’ (π β π β { 0 }) | |
36 | 35 | 3ad2ant1 1133 | . . . . . 6 β’ ((π β§ π β π΄ β§ π β (( β₯ βπ) β© π)) β π β { 0 }) |
37 | simp3 1138 | . . . . . 6 β’ ((π β§ π β π΄ β§ π β (( β₯ βπ) β© π)) β π β (( β₯ βπ) β© π)) | |
38 | 5, 14, 6, 11, 2, 30, 20, 4, 31, 32, 33, 34, 3, 17, 36, 37 | dochexmidlem4 40322 | . . . . 5 β’ ((π β§ π β π΄ β§ π β (( β₯ βπ) β© π)) β π β (π β ( β₯ βπ))) |
39 | 38 | rexlimdv3a 3159 | . . . 4 β’ (π β (βπ β π΄ π β (( β₯ βπ) β© π) β π β (π β ( β₯ βπ)))) |
40 | 29, 39 | syld 47 | . . 3 β’ (π β ((( β₯ βπ) β© π) β { 0 } β π β (π β ( β₯ βπ)))) |
41 | 40 | necon1bd 2958 | . 2 β’ (π β (Β¬ π β (π β ( β₯ βπ)) β (( β₯ βπ) β© π) = { 0 })) |
42 | 1, 41 | mpd 15 | 1 β’ (π β (( β₯ βπ) β© π) = { 0 }) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 396 β§ w3a 1087 = wceq 1541 β wcel 2106 β wne 2940 βwrex 3070 β© cin 3946 β wss 3947 {csn 4627 βcfv 6540 (class class class)co 7405 Basecbs 17140 0gc0g 17381 LSSumclsm 19496 LModclmod 20463 LSubSpclss 20534 LSpanclspn 20574 LSAtomsclsa 37832 HLchlt 38208 LHypclh 38843 DVecHcdvh 39937 ocHcoch 40206 |
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 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-riotaBAD 37811 |
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 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-tpos 8207 df-undef 8254 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8699 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-n0 12469 df-z 12555 df-uz 12819 df-fz 13481 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-sca 17209 df-vsca 17210 df-0g 17383 df-mre 17526 df-mrc 17527 df-acs 17529 df-proset 18244 df-poset 18262 df-plt 18279 df-lub 18295 df-glb 18296 df-join 18297 df-meet 18298 df-p0 18374 df-p1 18375 df-lat 18381 df-clat 18448 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-submnd 18668 df-grp 18818 df-minusg 18819 df-sbg 18820 df-subg 18997 df-cntz 19175 df-oppg 19204 df-lsm 19498 df-cmn 19644 df-abl 19645 df-mgp 19982 df-ur 19999 df-ring 20051 df-oppr 20142 df-dvdsr 20163 df-unit 20164 df-invr 20194 df-dvr 20207 df-drng 20309 df-lmod 20465 df-lss 20535 df-lsp 20575 df-lvec 20706 df-lsatoms 37834 df-lcv 37877 df-oposet 38034 df-ol 38036 df-oml 38037 df-covers 38124 df-ats 38125 df-atl 38156 df-cvlat 38180 df-hlat 38209 df-llines 38357 df-lplanes 38358 df-lvols 38359 df-lines 38360 df-psubsp 38362 df-pmap 38363 df-padd 38655 df-lhyp 38847 df-laut 38848 df-ldil 38963 df-ltrn 38964 df-trl 39018 df-tendo 39614 df-edring 39616 df-disoa 39888 df-dvech 39938 df-dib 39998 df-dic 40032 df-dih 40088 df-doch 40207 |
This theorem is referenced by: dochexmidlem6 40324 |
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