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Mirrors > Home > MPE Home > Th. List > Mathboxes > lcfrlem21 | Structured version Visualization version GIF version |
Description: Lemma for lcfr 39934. (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 482 | . . 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 482 | . . 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 482 | . . 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 482 | . . 3 β’ ((π β§ Β¬ π β ( β₯ β{(π + π)})) β (πβ{π}) β (πβ{π})) |
17 | simpr 486 | . . 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 39911 | . 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 39459 | . . . . . . . . 9 β’ (π β π β LMod) |
20 | 11 | eldifad 3921 | . . . . . . . . 9 β’ (π β π β π) |
21 | 13 | eldifad 3921 | . . . . . . . . 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 1372 | . . . . . . . 8 β’ (π β (π + π) = (π + π)) |
24 | 23 | sneqd 4597 | . . . . . . 7 β’ (π β {(π + π)} = {(π + π)}) |
25 | 24 | fveq2d 6842 | . . . . . 6 β’ (π β ( β₯ β{(π + π)}) = ( β₯ β{(π + π)})) |
26 | 25 | eleq2d 2824 | . . . . 5 β’ (π β (π β ( β₯ β{(π + π)}) β π β ( β₯ β{(π + π)}))) |
27 | 26 | biimprd 248 | . . . 4 β’ (π β (π β ( β₯ β{(π + π)}) β π β ( β₯ β{(π + π)}))) |
28 | 27 | con3dimp 410 | . . 3 β’ ((π β§ Β¬ π β ( β₯ β{(π + π)})) β Β¬ π β ( β₯ β{(π + π)})) |
29 | prcom 4692 | . . . . . . . 8 β’ {π, π} = {π, π} | |
30 | 29 | fveq2i 6841 | . . . . . . 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 4172 | . . . . 5 β’ (π β ((πβ{π, π}) β© ( β₯ β{(π + π)})) = ((πβ{π, π}) β© ( β₯ β{(π + π)}))) |
33 | 32 | adantr 482 | . . . 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 482 | . . . . 5 β’ ((π β§ Β¬ π β ( β₯ β{(π + π)})) β (πΎ β HL β§ π β π»)) |
35 | 13 | adantr 482 | . . . . 5 β’ ((π β§ Β¬ π β ( β₯ β{(π + π)})) β π β (π β { 0 })) |
36 | 11 | adantr 482 | . . . . 5 β’ ((π β§ Β¬ π β ( β₯ β{(π + π)})) β π β (π β { 0 })) |
37 | 15 | necomd 2998 | . . . . . 6 β’ (π β (πβ{π}) β (πβ{π})) |
38 | 37 | adantr 482 | . . . . 5 β’ ((π β§ Β¬ π β ( β₯ β{(π + π)})) β (πβ{π}) β (πβ{π})) |
39 | simpr 486 | . . . . 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 39911 | . . . 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 2839 | . . 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 592 | . 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 39910 | . 2 β’ (π β (Β¬ π β ( β₯ β{(π + π)}) β¨ Β¬ π β ( β₯ β{(π + π)}))) |
44 | 18, 42, 43 | mpjaodan 958 | 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 397 = wceq 1542 β wcel 2107 β wne 2942 β cdif 3906 β© cin 3908 {csn 4585 {cpr 4587 βcfv 6492 (class class class)co 7350 Basecbs 17018 +gcplusg 17068 0gc0g 17256 LModclmod 20245 LSpanclspn 20355 LSAtomsclsa 37322 HLchlt 37698 LHypclh 38333 DVecHcdvh 39427 ocHcoch 39696 |
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 2709 ax-rep 5241 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7663 ax-cnex 11041 ax-resscn 11042 ax-1cn 11043 ax-icn 11044 ax-addcl 11045 ax-addrcl 11046 ax-mulcl 11047 ax-mulrcl 11048 ax-mulcom 11049 ax-addass 11050 ax-mulass 11051 ax-distr 11052 ax-i2m1 11053 ax-1ne0 11054 ax-1rid 11055 ax-rnegex 11056 ax-rrecex 11057 ax-cnre 11058 ax-pre-lttri 11059 ax-pre-lttrn 11060 ax-pre-ltadd 11061 ax-pre-mulgt0 11062 ax-riotaBAD 37301 |
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 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4865 df-int 4907 df-iun 4955 df-iin 4956 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6250 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6444 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7306 df-ov 7353 df-oprab 7354 df-mpo 7355 df-om 7794 df-1st 7912 df-2nd 7913 df-tpos 8125 df-undef 8172 df-frecs 8180 df-wrecs 8211 df-recs 8285 df-rdg 8324 df-1o 8380 df-er 8582 df-map 8701 df-en 8818 df-dom 8819 df-sdom 8820 df-fin 8821 df-pnf 11125 df-mnf 11126 df-xr 11127 df-ltxr 11128 df-le 11129 df-sub 11321 df-neg 11322 df-nn 12088 df-2 12150 df-3 12151 df-4 12152 df-5 12153 df-6 12154 df-n0 12348 df-z 12434 df-uz 12697 df-fz 13354 df-struct 16954 df-sets 16971 df-slot 16989 df-ndx 17001 df-base 17019 df-ress 17048 df-plusg 17081 df-mulr 17082 df-sca 17084 df-vsca 17085 df-0g 17258 df-mre 17401 df-mrc 17402 df-acs 17404 df-proset 18119 df-poset 18137 df-plt 18154 df-lub 18170 df-glb 18171 df-join 18172 df-meet 18173 df-p0 18249 df-p1 18250 df-lat 18256 df-clat 18323 df-mgm 18432 df-sgrp 18481 df-mnd 18492 df-submnd 18537 df-grp 18686 df-minusg 18687 df-sbg 18688 df-subg 18858 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 37324 df-lshyp 37325 df-lcv 37367 df-oposet 37524 df-ol 37526 df-oml 37527 df-covers 37614 df-ats 37615 df-atl 37646 df-cvlat 37670 df-hlat 37699 df-llines 37847 df-lplanes 37848 df-lvols 37849 df-lines 37850 df-psubsp 37852 df-pmap 37853 df-padd 38145 df-lhyp 38337 df-laut 38338 df-ldil 38453 df-ltrn 38454 df-trl 38508 df-tgrp 39092 df-tendo 39104 df-edring 39106 df-dveca 39352 df-disoa 39378 df-dvech 39428 df-dib 39488 df-dic 39522 df-dih 39578 df-doch 39697 df-djh 39744 |
This theorem is referenced by: lcfrlem22 39913 lcfrlem40 39931 |
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