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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dochkr1OLDN | Structured version Unicode version |
Description: A non-zero functional has a value of 1 at some argument belonging to the orthocomplement of its kernel (when its kernel is a closed hyperplane). Tighter version of lfl1 33024. (Contributed by NM, 2-Jan-2015.) (New usage is discouraged.) |
Ref | Expression |
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dochkr1OLD.h |
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dochkr1OLD.o |
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dochkr1OLD.u |
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dochkr1OLD.v |
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dochkr1OLD.r |
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dochkr1OLD.z |
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dochkr1OLD.i |
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dochkr1OLD.f |
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dochkr1OLD.l |
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dochkr1OLD.k |
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dochkr1OLD.g |
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dochkr1OLD.n |
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Ref | Expression |
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dochkr1OLDN |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2451 |
. . . 4
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2 | eqid 2451 |
. . . 4
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3 | dochkr1OLD.h |
. . . . 5
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4 | dochkr1OLD.u |
. . . . 5
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5 | dochkr1OLD.k |
. . . . 5
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6 | 3, 4, 5 | dvhlmod 35064 |
. . . 4
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7 | dochkr1OLD.n |
. . . . 5
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8 | dochkr1OLD.o |
. . . . . 6
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9 | dochkr1OLD.v |
. . . . . 6
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10 | dochkr1OLD.f |
. . . . . 6
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11 | dochkr1OLD.l |
. . . . . 6
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12 | dochkr1OLD.g |
. . . . . 6
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13 | 3, 8, 4, 9, 2, 10, 11, 5, 12 | dochkrsat2 35410 |
. . . . 5
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14 | 7, 13 | mpbid 210 |
. . . 4
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15 | 1, 2, 6, 14 | lsateln0 32949 |
. . 3
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16 | dochkr1OLD.r |
. . . . . 6
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17 | dochkr1OLD.z |
. . . . . 6
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18 | 5 | ad2antrr 725 |
. . . . . 6
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19 | 12 | ad2antrr 725 |
. . . . . 6
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20 | eldifsn 4101 |
. . . . . . . 8
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21 | 20 | biimpri 206 |
. . . . . . 7
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22 | 21 | adantll 713 |
. . . . . 6
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23 | 3, 8, 4, 9, 16, 17, 1, 10, 11, 18, 19, 22 | dochfln0 35431 |
. . . . 5
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24 | 23 | ex 434 |
. . . 4
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25 | 24 | reximdva 2927 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
26 | 15, 25 | mpd 15 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
27 | 9, 10, 11, 6, 12 | lkrssv 33050 |
. . . . . . . 8
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28 | eqid 2451 |
. . . . . . . . 9
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29 | 3, 4, 9, 28, 8 | dochlss 35308 |
. . . . . . . 8
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30 | 5, 27, 29 | syl2anc 661 |
. . . . . . 7
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31 | 6, 30 | jca 532 |
. . . . . 6
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32 | 31 | 3ad2ant1 1009 |
. . . . 5
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33 | 3, 4, 5 | dvhlvec 35063 |
. . . . . . . 8
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34 | 33 | 3ad2ant1 1009 |
. . . . . . 7
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35 | 16 | lvecdrng 17301 |
. . . . . . 7
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36 | 34, 35 | syl 16 |
. . . . . 6
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37 | 6 | 3ad2ant1 1009 |
. . . . . . 7
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38 | 12 | 3ad2ant1 1009 |
. . . . . . 7
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39 | 3, 4, 9, 8 | dochssv 35309 |
. . . . . . . . . 10
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40 | 5, 27, 39 | syl2anc 661 |
. . . . . . . . 9
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41 | 40 | sselda 3457 |
. . . . . . . 8
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42 | 41 | 3adant3 1008 |
. . . . . . 7
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43 | eqid 2451 |
. . . . . . . 8
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44 | 16, 43, 9, 10 | lflcl 33018 |
. . . . . . 7
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45 | 37, 38, 42, 44 | syl3anc 1219 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
46 | simp3 990 |
. . . . . 6
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47 | eqid 2451 |
. . . . . . 7
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48 | 43, 17, 47 | drnginvrcl 16964 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
49 | 36, 45, 46, 48 | syl3anc 1219 |
. . . . 5
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50 | simp2 989 |
. . . . 5
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51 | eqid 2451 |
. . . . . 6
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52 | 16, 51, 43, 28 | lssvscl 17151 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
53 | 32, 49, 50, 52 | syl12anc 1217 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
54 | eqid 2451 |
. . . . . . 7
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55 | 16, 43, 54, 9, 51, 10 | lflmul 33022 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
56 | 37, 38, 49, 42, 55 | syl112anc 1223 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
57 | dochkr1OLD.i |
. . . . . . 7
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58 | 43, 17, 54, 57, 47 | drnginvrl 16966 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
59 | 36, 45, 46, 58 | syl3anc 1219 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
60 | 56, 59 | eqtrd 2492 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
61 | fveq2 5792 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
62 | 61 | eqeq1d 2453 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
63 | 62 | rspcev 3172 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
64 | 53, 60, 63 | syl2anc 661 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
65 | 64 | rexlimdv3a 2942 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
66 | 26, 65 | mpd 15 |
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 1592 ax-4 1603 ax-5 1671 ax-6 1710 ax-7 1730 ax-8 1760 ax-9 1762 ax-10 1777 ax-11 1782 ax-12 1794 ax-13 1952 ax-ext 2430 ax-rep 4504 ax-sep 4514 ax-nul 4522 ax-pow 4571 ax-pr 4632 ax-un 6475 ax-cnex 9442 ax-resscn 9443 ax-1cn 9444 ax-icn 9445 ax-addcl 9446 ax-addrcl 9447 ax-mulcl 9448 ax-mulrcl 9449 ax-mulcom 9450 ax-addass 9451 ax-mulass 9452 ax-distr 9453 ax-i2m1 9454 ax-1ne0 9455 ax-1rid 9456 ax-rnegex 9457 ax-rrecex 9458 ax-cnre 9459 ax-pre-lttri 9460 ax-pre-lttrn 9461 ax-pre-ltadd 9462 ax-pre-mulgt0 9463 ax-riotaBAD 32913 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3or 966 df-3an 967 df-tru 1373 df-fal 1376 df-ex 1588 df-nf 1591 df-sb 1703 df-eu 2264 df-mo 2265 df-clab 2437 df-cleq 2443 df-clel 2446 df-nfc 2601 df-ne 2646 df-nel 2647 df-ral 2800 df-rex 2801 df-reu 2802 df-rmo 2803 df-rab 2804 df-v 3073 df-sbc 3288 df-csb 3390 df-dif 3432 df-un 3434 df-in 3436 df-ss 3443 df-pss 3445 df-nul 3739 df-if 3893 df-pw 3963 df-sn 3979 df-pr 3981 df-tp 3983 df-op 3985 df-uni 4193 df-int 4230 df-iun 4274 df-iin 4275 df-br 4394 df-opab 4452 df-mpt 4453 df-tr 4487 df-eprel 4733 df-id 4737 df-po 4742 df-so 4743 df-fr 4780 df-we 4782 df-ord 4823 df-on 4824 df-lim 4825 df-suc 4826 df-xp 4947 df-rel 4948 df-cnv 4949 df-co 4950 df-dm 4951 df-rn 4952 df-res 4953 df-ima 4954 df-iota 5482 df-fun 5521 df-fn 5522 df-f 5523 df-f1 5524 df-fo 5525 df-f1o 5526 df-fv 5527 df-riota 6154 df-ov 6196 df-oprab 6197 df-mpt2 6198 df-om 6580 df-1st 6680 df-2nd 6681 df-tpos 6848 df-undef 6895 df-recs 6935 df-rdg 6969 df-1o 7023 df-oadd 7027 df-er 7204 df-map 7319 df-en 7414 df-dom 7415 df-sdom 7416 df-fin 7417 df-pnf 9524 df-mnf 9525 df-xr 9526 df-ltxr 9527 df-le 9528 df-sub 9701 df-neg 9702 df-nn 10427 df-2 10484 df-3 10485 df-4 10486 df-5 10487 df-6 10488 df-n0 10684 df-z 10751 df-uz 10966 df-fz 11548 df-struct 14287 df-ndx 14288 df-slot 14289 df-base 14290 df-sets 14291 df-ress 14292 df-plusg 14362 df-mulr 14363 df-sca 14365 df-vsca 14366 df-0g 14491 df-poset 15227 df-plt 15239 df-lub 15255 df-glb 15256 df-join 15257 df-meet 15258 df-p0 15320 df-p1 15321 df-lat 15327 df-clat 15389 df-mnd 15526 df-submnd 15576 df-grp 15656 df-minusg 15657 df-sbg 15658 df-subg 15789 df-cntz 15946 df-lsm 16248 df-cmn 16392 df-abl 16393 df-mgp 16706 df-ur 16718 df-rng 16762 df-oppr 16830 df-dvdsr 16848 df-unit 16849 df-invr 16879 df-dvr 16890 df-drng 16949 df-lmod 17065 df-lss 17129 df-lsp 17168 df-lvec 17299 df-lsatoms 32930 df-lshyp 32931 df-lfl 33012 df-lkr 33040 df-oposet 33130 df-ol 33132 df-oml 33133 df-covers 33220 df-ats 33221 df-atl 33252 df-cvlat 33276 df-hlat 33305 df-llines 33451 df-lplanes 33452 df-lvols 33453 df-lines 33454 df-psubsp 33456 df-pmap 33457 df-padd 33749 df-lhyp 33941 df-laut 33942 df-ldil 34057 df-ltrn 34058 df-trl 34112 df-tgrp 34696 df-tendo 34708 df-edring 34710 df-dveca 34956 df-disoa 34983 df-dvech 35033 df-dib 35093 df-dic 35127 df-dih 35183 df-doch 35302 df-djh 35349 |
This theorem is referenced by: (None) |
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