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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dvh1dim | Structured version Visualization version Unicode version |
Description: There exists a nonzero vector. (Contributed by NM, 26-Apr-2015.) |
Ref | Expression |
---|---|
dvh3dim.h |
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dvh3dim.u |
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dvh3dim.v |
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dvh1dim.o |
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dvh1dim.k |
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Ref | Expression |
---|---|
dvh1dim |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvh3dim.h |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
2 | dvh3dim.u |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
3 | eqid 2451 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
4 | dvh1dim.k |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
5 | 1, 2, 3, 4 | dvh1dimat 35009 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
6 | dvh1dim.o |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
7 | 1, 2, 4 | dvhlmod 34678 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
8 | 7 | adantr 467 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
9 | simpr 463 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
10 | 6, 3, 8, 9 | lsateln0 32561 |
. . 3
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11 | dvh3dim.v |
. . . . . . 7
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12 | 11, 3, 8, 9 | lsatssv 32564 |
. . . . . 6
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13 | 12 | sseld 3431 |
. . . . 5
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14 | 13 | anim1d 568 |
. . . 4
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15 | 14 | reximdv2 2858 |
. . 3
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16 | 10, 15 | mpd 15 |
. 2
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17 | 5, 16 | exlimddv 1781 |
1
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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 1669 ax-4 1682 ax-5 1758 ax-6 1805 ax-7 1851 ax-8 1889 ax-9 1896 ax-10 1915 ax-11 1920 ax-12 1933 ax-13 2091 ax-ext 2431 ax-rep 4515 ax-sep 4525 ax-nul 4534 ax-pow 4581 ax-pr 4639 ax-un 6583 ax-cnex 9595 ax-resscn 9596 ax-1cn 9597 ax-icn 9598 ax-addcl 9599 ax-addrcl 9600 ax-mulcl 9601 ax-mulrcl 9602 ax-mulcom 9603 ax-addass 9604 ax-mulass 9605 ax-distr 9606 ax-i2m1 9607 ax-1ne0 9608 ax-1rid 9609 ax-rnegex 9610 ax-rrecex 9611 ax-cnre 9612 ax-pre-lttri 9613 ax-pre-lttrn 9614 ax-pre-ltadd 9615 ax-pre-mulgt0 9616 ax-riotaBAD 32525 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3or 986 df-3an 987 df-tru 1447 df-fal 1450 df-ex 1664 df-nf 1668 df-sb 1798 df-eu 2303 df-mo 2304 df-clab 2438 df-cleq 2444 df-clel 2447 df-nfc 2581 df-ne 2624 df-nel 2625 df-ral 2742 df-rex 2743 df-reu 2744 df-rmo 2745 df-rab 2746 df-v 3047 df-sbc 3268 df-csb 3364 df-dif 3407 df-un 3409 df-in 3411 df-ss 3418 df-pss 3420 df-nul 3732 df-if 3882 df-pw 3953 df-sn 3969 df-pr 3971 df-tp 3973 df-op 3975 df-uni 4199 df-int 4235 df-iun 4280 df-iin 4281 df-br 4403 df-opab 4462 df-mpt 4463 df-tr 4498 df-eprel 4745 df-id 4749 df-po 4755 df-so 4756 df-fr 4793 df-we 4795 df-xp 4840 df-rel 4841 df-cnv 4842 df-co 4843 df-dm 4844 df-rn 4845 df-res 4846 df-ima 4847 df-pred 5380 df-ord 5426 df-on 5427 df-lim 5428 df-suc 5429 df-iota 5546 df-fun 5584 df-fn 5585 df-f 5586 df-f1 5587 df-fo 5588 df-f1o 5589 df-fv 5590 df-riota 6252 df-ov 6293 df-oprab 6294 df-mpt2 6295 df-om 6693 df-1st 6793 df-2nd 6794 df-tpos 6973 df-undef 7020 df-wrecs 7028 df-recs 7090 df-rdg 7128 df-1o 7182 df-oadd 7186 df-er 7363 df-map 7474 df-en 7570 df-dom 7571 df-sdom 7572 df-fin 7573 df-pnf 9677 df-mnf 9678 df-xr 9679 df-ltxr 9680 df-le 9681 df-sub 9862 df-neg 9863 df-nn 10610 df-2 10668 df-3 10669 df-4 10670 df-5 10671 df-6 10672 df-n0 10870 df-z 10938 df-uz 11160 df-fz 11785 df-struct 15123 df-ndx 15124 df-slot 15125 df-base 15126 df-sets 15127 df-ress 15128 df-plusg 15203 df-mulr 15204 df-sca 15206 df-vsca 15207 df-0g 15340 df-preset 16173 df-poset 16191 df-plt 16204 df-lub 16220 df-glb 16221 df-join 16222 df-meet 16223 df-p0 16285 df-p1 16286 df-lat 16292 df-clat 16354 df-mgm 16488 df-sgrp 16527 df-mnd 16537 df-submnd 16583 df-grp 16673 df-minusg 16674 df-sbg 16675 df-subg 16814 df-cntz 16971 df-lsm 17288 df-cmn 17432 df-abl 17433 df-mgp 17724 df-ur 17736 df-ring 17782 df-oppr 17851 df-dvdsr 17869 df-unit 17870 df-invr 17900 df-dvr 17911 df-drng 17977 df-lmod 18093 df-lss 18156 df-lsp 18195 df-lvec 18326 df-lsatoms 32542 df-oposet 32742 df-ol 32744 df-oml 32745 df-covers 32832 df-ats 32833 df-atl 32864 df-cvlat 32888 df-hlat 32917 df-llines 33063 df-lplanes 33064 df-lvols 33065 df-lines 33066 df-psubsp 33068 df-pmap 33069 df-padd 33361 df-lhyp 33553 df-laut 33554 df-ldil 33669 df-ltrn 33670 df-trl 33725 df-tendo 34322 df-edring 34324 df-disoa 34597 df-dvech 34647 df-dib 34707 df-dic 34741 df-dih 34797 |
This theorem is referenced by: dvh2dim 35013 hdmap14lem14 35452 hgmapval0 35463 hgmapval1 35464 hgmapadd 35465 hgmapmul 35466 hgmaprnlem5N 35471 hgmap11 35473 |
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