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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lcfrlem6 | Structured version Visualization version Unicode version |
Description: Lemma for lcfr 35198. Closure of vector sum with colinear
vectors. TODO:
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Ref | Expression |
---|---|
lcfrlem6.h |
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lcfrlem6.o |
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lcfrlem6.u |
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lcfrlem6.p |
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lcfrlem6.n |
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lcfrlem6.l |
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lcfrlem6.d |
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lcfrlem6.q |
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lcfrlem6.k |
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lcfrlem6.g |
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lcfrlem6.e |
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lcfrlem6.x |
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lcfrlem6.y |
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lcfrlem6.en |
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Ref | Expression |
---|---|
lcfrlem6 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lcfrlem6.x |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
2 | lcfrlem6.e |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
3 | 1, 2 | syl6eleq 2550 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
4 | eliun 4297 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
5 | 3, 4 | sylib 201 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
6 | lcfrlem6.h |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
7 | lcfrlem6.u |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
8 | lcfrlem6.k |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
9 | 6, 7, 8 | dvhlmod 34723 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
10 | 9 | adantr 471 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
11 | 10 | adantr 471 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
12 | 8 | adantr 471 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
13 | eqid 2462 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
14 | eqid 2462 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
15 | lcfrlem6.l |
. . . . . . . . . 10
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16 | lcfrlem6.g |
. . . . . . . . . . . 12
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
17 | eqid 2462 |
. . . . . . . . . . . . 13
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
18 | lcfrlem6.q |
. . . . . . . . . . . . 13
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
19 | 17, 18 | lssel 18210 |
. . . . . . . . . . . 12
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
20 | 16, 19 | sylan 478 |
. . . . . . . . . . 11
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
21 | lcfrlem6.d |
. . . . . . . . . . . . 13
![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
22 | 14, 21, 17, 9 | ldualvbase 32737 |
. . . . . . . . . . . 12
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
23 | 22 | adantr 471 |
. . . . . . . . . . 11
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
24 | 20, 23 | eleqtrd 2542 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
25 | 13, 14, 15, 10, 24 | lkrssv 32707 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
26 | eqid 2462 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
27 | lcfrlem6.o |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
28 | 6, 7, 13, 26, 27 | dochlss 34967 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
29 | 12, 25, 28 | syl2anc 671 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
30 | 29 | adantr 471 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
31 | simpr 467 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
32 | lcfrlem6.en |
. . . . . . . . . . . . 13
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
33 | 32 | adantr 471 |
. . . . . . . . . . . 12
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
34 | 33 | adantr 471 |
. . . . . . . . . . 11
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
35 | simpr 467 |
. . . . . . . . . . 11
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
36 | 34, 35 | eqsstr3d 3479 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
37 | 36 | ex 440 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
38 | lcfrlem6.n |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
39 | 6, 27, 7, 13, 15, 21, 18, 2, 8, 16, 1 | lcfrlem4 35158 |
. . . . . . . . . . 11
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
40 | 39 | adantr 471 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
41 | 13, 26, 38, 10, 29, 40 | lspsnel5 18267 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
42 | lcfrlem6.y |
. . . . . . . . . . . 12
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
43 | 6, 27, 7, 13, 15, 21, 18, 2, 8, 16, 42 | lcfrlem4 35158 |
. . . . . . . . . . 11
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
44 | 43 | adantr 471 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
45 | 13, 26, 38, 10, 29, 44 | lspsnel5 18267 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
46 | 37, 41, 45 | 3imtr4d 276 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
47 | 46 | imp 435 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
48 | lcfrlem6.p |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
49 | 48, 26 | lssvacl 18226 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
50 | 11, 30, 31, 47, 49 | syl22anc 1277 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
51 | 50 | ex 440 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
52 | 51 | reximdva 2874 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
53 | 5, 52 | mpd 15 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
54 | eliun 4297 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
55 | 53, 54 | sylibr 217 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
56 | 55, 2 | syl6eleqr 2551 |
1
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1680 ax-4 1693 ax-5 1769 ax-6 1816 ax-7 1862 ax-8 1900 ax-9 1907 ax-10 1926 ax-11 1931 ax-12 1944 ax-13 2102 ax-ext 2442 ax-rep 4529 ax-sep 4539 ax-nul 4548 ax-pow 4595 ax-pr 4653 ax-un 6610 ax-cnex 9621 ax-resscn 9622 ax-1cn 9623 ax-icn 9624 ax-addcl 9625 ax-addrcl 9626 ax-mulcl 9627 ax-mulrcl 9628 ax-mulcom 9629 ax-addass 9630 ax-mulass 9631 ax-distr 9632 ax-i2m1 9633 ax-1ne0 9634 ax-1rid 9635 ax-rnegex 9636 ax-rrecex 9637 ax-cnre 9638 ax-pre-lttri 9639 ax-pre-lttrn 9640 ax-pre-ltadd 9641 ax-pre-mulgt0 9642 ax-riotaBAD 32570 |
This theorem depends on definitions: df-bi 190 df-or 376 df-an 377 df-3or 992 df-3an 993 df-tru 1458 df-fal 1461 df-ex 1675 df-nf 1679 df-sb 1809 df-eu 2314 df-mo 2315 df-clab 2449 df-cleq 2455 df-clel 2458 df-nfc 2592 df-ne 2635 df-nel 2636 df-ral 2754 df-rex 2755 df-reu 2756 df-rmo 2757 df-rab 2758 df-v 3059 df-sbc 3280 df-csb 3376 df-dif 3419 df-un 3421 df-in 3423 df-ss 3430 df-pss 3432 df-nul 3744 df-if 3894 df-pw 3965 df-sn 3981 df-pr 3983 df-tp 3985 df-op 3987 df-uni 4213 df-int 4249 df-iun 4294 df-iin 4295 df-br 4417 df-opab 4476 df-mpt 4477 df-tr 4512 df-eprel 4764 df-id 4768 df-po 4774 df-so 4775 df-fr 4812 df-we 4814 df-xp 4859 df-rel 4860 df-cnv 4861 df-co 4862 df-dm 4863 df-rn 4864 df-res 4865 df-ima 4866 df-pred 5399 df-ord 5445 df-on 5446 df-lim 5447 df-suc 5448 df-iota 5565 df-fun 5603 df-fn 5604 df-f 5605 df-f1 5606 df-fo 5607 df-f1o 5608 df-fv 5609 df-riota 6277 df-ov 6318 df-oprab 6319 df-mpt2 6320 df-of 6558 df-om 6720 df-1st 6820 df-2nd 6821 df-tpos 6999 df-undef 7046 df-wrecs 7054 df-recs 7116 df-rdg 7154 df-1o 7208 df-oadd 7212 df-er 7389 df-map 7500 df-en 7596 df-dom 7597 df-sdom 7598 df-fin 7599 df-pnf 9703 df-mnf 9704 df-xr 9705 df-ltxr 9706 df-le 9707 df-sub 9888 df-neg 9889 df-nn 10638 df-2 10696 df-3 10697 df-4 10698 df-5 10699 df-6 10700 df-n0 10899 df-z 10967 df-uz 11189 df-fz 11814 df-struct 15172 df-ndx 15173 df-slot 15174 df-base 15175 df-sets 15176 df-ress 15177 df-plusg 15252 df-mulr 15253 df-sca 15255 df-vsca 15256 df-0g 15389 df-preset 16222 df-poset 16240 df-plt 16253 df-lub 16269 df-glb 16270 df-join 16271 df-meet 16272 df-p0 16334 df-p1 16335 df-lat 16341 df-clat 16403 df-mgm 16537 df-sgrp 16576 df-mnd 16586 df-submnd 16632 df-grp 16722 df-minusg 16723 df-sbg 16724 df-subg 16863 df-cntz 17020 df-lsm 17337 df-cmn 17481 df-abl 17482 df-mgp 17773 df-ur 17785 df-ring 17831 df-oppr 17900 df-dvdsr 17918 df-unit 17919 df-invr 17949 df-dvr 17960 df-drng 18026 df-lmod 18142 df-lss 18205 df-lsp 18244 df-lvec 18375 df-lfl 32669 df-lkr 32697 df-ldual 32735 df-oposet 32787 df-ol 32789 df-oml 32790 df-covers 32877 df-ats 32878 df-atl 32909 df-cvlat 32933 df-hlat 32962 df-llines 33108 df-lplanes 33109 df-lvols 33110 df-lines 33111 df-psubsp 33113 df-pmap 33114 df-padd 33406 df-lhyp 33598 df-laut 33599 df-ldil 33714 df-ltrn 33715 df-trl 33770 df-tendo 34367 df-edring 34369 df-disoa 34642 df-dvech 34692 df-dib 34752 df-dic 34786 df-dih 34842 df-doch 34961 |
This theorem is referenced by: lcfrlem41 35196 |
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