Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lcfrlem21 | Structured version Visualization version GIF version |
Description: Lemma for lcfr 39943. (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 |
---|---|
lcfrlem21 | β’ (π β ((πβ{π, π}) β© ( β₯ β{(π + π)})) β π΄) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lcfrlem17.h | . . 3 β’ π» = (LHypβπΎ) | |
2 | lcfrlem17.o | . . 3 β’ β₯ = ((ocHβπΎ)βπ) | |
3 | lcfrlem17.u | . . 3 β’ π = ((DVecHβπΎ)βπ) | |
4 | lcfrlem17.v | . . 3 β’ π = (Baseβπ) | |
5 | lcfrlem17.p | . . 3 β’ + = (+gβπ) | |
6 | lcfrlem17.z | . . 3 β’ 0 = (0gβπ) | |
7 | lcfrlem17.n | . . 3 β’ π = (LSpanβπ) | |
8 | lcfrlem17.a | . . 3 β’ π΄ = (LSAtomsβπ) | |
9 | lcfrlem17.k | . . . 4 β’ (π β (πΎ β HL β§ π β π»)) | |
10 | 9 | adantr 481 | . . 3 β’ ((π β§ Β¬ π β ( β₯ β{(π + π)})) β (πΎ β HL β§ π β π»)) |
11 | lcfrlem17.x | . . . 4 β’ (π β π β (π β { 0 })) | |
12 | 11 | adantr 481 | . . 3 β’ ((π β§ Β¬ π β ( β₯ β{(π + π)})) β π β (π β { 0 })) |
13 | lcfrlem17.y | . . . 4 β’ (π β π β (π β { 0 })) | |
14 | 13 | adantr 481 | . . 3 β’ ((π β§ Β¬ π β ( β₯ β{(π + π)})) β π β (π β { 0 })) |
15 | lcfrlem17.ne | . . . 4 β’ (π β (πβ{π}) β (πβ{π})) | |
16 | 15 | adantr 481 | . . 3 β’ ((π β§ Β¬ π β ( β₯ β{(π + π)})) β (πβ{π}) β (πβ{π})) |
17 | simpr 485 | . . 3 β’ ((π β§ Β¬ π β ( β₯ β{(π + π)})) β Β¬ π β ( β₯ β{(π + π)})) | |
18 | 1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 16, 17 | lcfrlem20 39920 | . 2 β’ ((π β§ Β¬ π β ( β₯ β{(π + π)})) β ((πβ{π, π}) β© ( β₯ β{(π + π)})) β π΄) |
19 | 1, 3, 9 | dvhlmod 39468 | . . . . . . . . 9 β’ (π β π β LMod) |
20 | 11 | eldifad 3920 | . . . . . . . . 9 β’ (π β π β π) |
21 | 13 | eldifad 3920 | . . . . . . . . 9 β’ (π β π β π) |
22 | 4, 5 | lmodcom 20291 | . . . . . . . . 9 β’ ((π β LMod β§ π β π β§ π β π) β (π + π) = (π + π)) |
23 | 19, 20, 21, 22 | syl3anc 1371 | . . . . . . . 8 β’ (π β (π + π) = (π + π)) |
24 | 23 | sneqd 4596 | . . . . . . 7 β’ (π β {(π + π)} = {(π + π)}) |
25 | 24 | fveq2d 6841 | . . . . . 6 β’ (π β ( β₯ β{(π + π)}) = ( β₯ β{(π + π)})) |
26 | 25 | eleq2d 2823 | . . . . 5 β’ (π β (π β ( β₯ β{(π + π)}) β π β ( β₯ β{(π + π)}))) |
27 | 26 | biimprd 247 | . . . 4 β’ (π β (π β ( β₯ β{(π + π)}) β π β ( β₯ β{(π + π)}))) |
28 | 27 | con3dimp 409 | . . 3 β’ ((π β§ Β¬ π β ( β₯ β{(π + π)})) β Β¬ π β ( β₯ β{(π + π)})) |
29 | prcom 4691 | . . . . . . . 8 β’ {π, π} = {π, π} | |
30 | 29 | fveq2i 6840 | . . . . . . 7 β’ (πβ{π, π}) = (πβ{π, π}) |
31 | 30 | a1i 11 | . . . . . 6 β’ (π β (πβ{π, π}) = (πβ{π, π})) |
32 | 31, 25 | ineq12d 4171 | . . . . 5 β’ (π β ((πβ{π, π}) β© ( β₯ β{(π + π)})) = ((πβ{π, π}) β© ( β₯ β{(π + π)}))) |
33 | 32 | adantr 481 | . . . 4 β’ ((π β§ Β¬ π β ( β₯ β{(π + π)})) β ((πβ{π, π}) β© ( β₯ β{(π + π)})) = ((πβ{π, π}) β© ( β₯ β{(π + π)}))) |
34 | 9 | adantr 481 | . . . . 5 β’ ((π β§ Β¬ π β ( β₯ β{(π + π)})) β (πΎ β HL β§ π β π»)) |
35 | 13 | adantr 481 | . . . . 5 β’ ((π β§ Β¬ π β ( β₯ β{(π + π)})) β π β (π β { 0 })) |
36 | 11 | adantr 481 | . . . . 5 β’ ((π β§ Β¬ π β ( β₯ β{(π + π)})) β π β (π β { 0 })) |
37 | 15 | necomd 2997 | . . . . . 6 β’ (π β (πβ{π}) β (πβ{π})) |
38 | 37 | adantr 481 | . . . . 5 β’ ((π β§ Β¬ π β ( β₯ β{(π + π)})) β (πβ{π}) β (πβ{π})) |
39 | simpr 485 | . . . . 5 β’ ((π β§ Β¬ π β ( β₯ β{(π + π)})) β Β¬ π β ( β₯ β{(π + π)})) | |
40 | 1, 2, 3, 4, 5, 6, 7, 8, 34, 35, 36, 38, 39 | lcfrlem20 39920 | . . . 4 β’ ((π β§ Β¬ π β ( β₯ β{(π + π)})) β ((πβ{π, π}) β© ( β₯ β{(π + π)})) β π΄) |
41 | 33, 40 | eqeltrd 2838 | . . 3 β’ ((π β§ Β¬ π β ( β₯ β{(π + π)})) β ((πβ{π, π}) β© ( β₯ β{(π + π)})) β π΄) |
42 | 28, 41 | syldan 591 | . 2 β’ ((π β§ Β¬ π β ( β₯ β{(π + π)})) β ((πβ{π, π}) β© ( β₯ β{(π + π)})) β π΄) |
43 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 15 | lcfrlem19 39919 | . 2 β’ (π β (Β¬ π β ( β₯ β{(π + π)}) β¨ Β¬ π β ( β₯ β{(π + π)}))) |
44 | 18, 42, 43 | mpjaodan 957 | 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 2941 β cdif 3905 β© cin 3907 {csn 4584 {cpr 4586 βcfv 6491 (class class class)co 7349 Basecbs 17017 +gcplusg 17067 0gc0g 17255 LModclmod 20245 LSpanclspn 20355 LSAtomsclsa 37331 HLchlt 37707 LHypclh 38342 DVecHcdvh 39436 ocHcoch 39705 |
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 2708 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7662 ax-cnex 11040 ax-resscn 11041 ax-1cn 11042 ax-icn 11043 ax-addcl 11044 ax-addrcl 11045 ax-mulcl 11046 ax-mulrcl 11047 ax-mulcom 11048 ax-addass 11049 ax-mulass 11050 ax-distr 11051 ax-i2m1 11052 ax-1ne0 11053 ax-1rid 11054 ax-rnegex 11055 ax-rrecex 11056 ax-cnre 11057 ax-pre-lttri 11058 ax-pre-lttrn 11059 ax-pre-ltadd 11060 ax-pre-mulgt0 11061 ax-riotaBAD 37310 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4864 df-int 4906 df-iun 4954 df-iin 4955 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5528 df-eprel 5534 df-po 5542 df-so 5543 df-fr 5585 df-we 5587 df-xp 5636 df-rel 5637 df-cnv 5638 df-co 5639 df-dm 5640 df-rn 5641 df-res 5642 df-ima 5643 df-pred 6249 df-ord 6316 df-on 6317 df-lim 6318 df-suc 6319 df-iota 6443 df-fun 6493 df-fn 6494 df-f 6495 df-f1 6496 df-fo 6497 df-f1o 6498 df-fv 6499 df-riota 7305 df-ov 7352 df-oprab 7353 df-mpo 7354 df-om 7793 df-1st 7911 df-2nd 7912 df-tpos 8124 df-undef 8171 df-frecs 8179 df-wrecs 8210 df-recs 8284 df-rdg 8323 df-1o 8379 df-er 8581 df-map 8700 df-en 8817 df-dom 8818 df-sdom 8819 df-fin 8820 df-pnf 11124 df-mnf 11125 df-xr 11126 df-ltxr 11127 df-le 11128 df-sub 11320 df-neg 11321 df-nn 12087 df-2 12149 df-3 12150 df-4 12151 df-5 12152 df-6 12153 df-n0 12347 df-z 12433 df-uz 12696 df-fz 13353 df-struct 16953 df-sets 16970 df-slot 16988 df-ndx 17000 df-base 17018 df-ress 17047 df-plusg 17080 df-mulr 17081 df-sca 17083 df-vsca 17084 df-0g 17257 df-mre 17400 df-mrc 17401 df-acs 17403 df-proset 18118 df-poset 18136 df-plt 18153 df-lub 18169 df-glb 18170 df-join 18171 df-meet 18172 df-p0 18248 df-p1 18249 df-lat 18255 df-clat 18322 df-mgm 18431 df-sgrp 18480 df-mnd 18491 df-submnd 18536 df-grp 18685 df-minusg 18686 df-sbg 18687 df-subg 18857 df-cntz 19029 df-oppg 19056 df-lsm 19347 df-cmn 19493 df-abl 19494 df-mgp 19826 df-ur 19843 df-ring 19890 df-oppr 19972 df-dvdsr 19993 df-unit 19994 df-invr 20024 df-dvr 20035 df-drng 20110 df-lmod 20247 df-lss 20316 df-lsp 20356 df-lvec 20487 df-lsatoms 37333 df-lshyp 37334 df-lcv 37376 df-oposet 37533 df-ol 37535 df-oml 37536 df-covers 37623 df-ats 37624 df-atl 37655 df-cvlat 37679 df-hlat 37708 df-llines 37856 df-lplanes 37857 df-lvols 37858 df-lines 37859 df-psubsp 37861 df-pmap 37862 df-padd 38154 df-lhyp 38346 df-laut 38347 df-ldil 38462 df-ltrn 38463 df-trl 38517 df-tgrp 39101 df-tendo 39113 df-edring 39115 df-dveca 39361 df-disoa 39387 df-dvech 39437 df-dib 39497 df-dic 39531 df-dih 39587 df-doch 39706 df-djh 39753 |
This theorem is referenced by: lcfrlem22 39922 lcfrlem40 39940 |
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