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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > 2lplnmj | Structured version Visualization version Unicode version |
Description: The meet of two lattice planes is a lattice line iff their join is a lattice volume. (Contributed by NM, 13-Jul-2012.) |
Ref | Expression |
---|---|
2lplnmj.j |
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2lplnmj.m |
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2lplnmj.n |
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2lplnmj.p |
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2lplnmj.v |
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Ref | Expression |
---|---|
2lplnmj |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1009 |
. . 3
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2 | eqid 2453 |
. . . . 5
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3 | 2lplnmj.p |
. . . . 5
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4 | 2, 3 | lplnbase 33111 |
. . . 4
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5 | 4 | 3ad2ant2 1031 |
. . 3
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6 | 2, 3 | lplnbase 33111 |
. . . 4
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7 | 6 | 3ad2ant3 1032 |
. . 3
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8 | 2lplnmj.j |
. . . 4
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9 | 2lplnmj.m |
. . . 4
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10 | eqid 2453 |
. . . 4
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11 | 2, 8, 9, 10 | cvrexch 32997 |
. . 3
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12 | 1, 5, 7, 11 | syl3anc 1269 |
. 2
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13 | simpl1 1012 |
. . . 4
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14 | simpr 463 |
. . . 4
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15 | simpl3 1014 |
. . . 4
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16 | hllat 32941 |
. . . . . 6
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17 | eqid 2453 |
. . . . . . 7
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18 | 2, 17, 9 | latmle2 16335 |
. . . . . 6
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19 | 16, 4, 6, 18 | syl3an 1311 |
. . . . 5
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20 | 19 | adantr 467 |
. . . 4
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21 | 2lplnmj.n |
. . . . 5
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22 | 17, 10, 21, 3 | llncvrlpln2 33134 |
. . . 4
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23 | 13, 14, 15, 20, 22 | syl31anc 1272 |
. . 3
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24 | simpl3 1014 |
. . . 4
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25 | 2, 9 | latmcl 16310 |
. . . . . . 7
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26 | 16, 4, 6, 25 | syl3an 1311 |
. . . . . 6
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27 | 1, 26, 7 | 3jca 1189 |
. . . . 5
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28 | 2, 10, 21, 3 | llncvrlpln 33135 |
. . . . 5
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29 | 27, 28 | sylan 474 |
. . . 4
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30 | 24, 29 | mpbird 236 |
. . 3
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31 | 23, 30 | impbida 844 |
. 2
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32 | simpl1 1012 |
. . . 4
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33 | simpl2 1013 |
. . . 4
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34 | simpr 463 |
. . . 4
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35 | 2, 17, 8 | latlej1 16318 |
. . . . . 6
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36 | 16, 4, 6, 35 | syl3an 1311 |
. . . . 5
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37 | 36 | adantr 467 |
. . . 4
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38 | 2lplnmj.v |
. . . . 5
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39 | 17, 10, 3, 38 | lplncvrlvol2 33192 |
. . . 4
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40 | 32, 33, 34, 37, 39 | syl31anc 1272 |
. . 3
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41 | simpl2 1013 |
. . . 4
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42 | 2, 8 | latjcl 16309 |
. . . . . . 7
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43 | 16, 4, 6, 42 | syl3an 1311 |
. . . . . 6
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44 | 1, 5, 43 | 3jca 1189 |
. . . . 5
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45 | 2, 10, 3, 38 | lplncvrlvol 33193 |
. . . . 5
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46 | 44, 45 | sylan 474 |
. . . 4
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47 | 41, 46 | mpbid 214 |
. . 3
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48 | 40, 47 | impbida 844 |
. 2
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49 | 12, 31, 48 | 3bitr4d 289 |
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 1671 ax-4 1684 ax-5 1760 ax-6 1807 ax-7 1853 ax-8 1891 ax-9 1898 ax-10 1917 ax-11 1922 ax-12 1935 ax-13 2093 ax-ext 2433 ax-rep 4518 ax-sep 4528 ax-nul 4537 ax-pow 4584 ax-pr 4642 ax-un 6588 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3or 987 df-3an 988 df-tru 1449 df-ex 1666 df-nf 1670 df-sb 1800 df-eu 2305 df-mo 2306 df-clab 2440 df-cleq 2446 df-clel 2449 df-nfc 2583 df-ne 2626 df-ral 2744 df-rex 2745 df-reu 2746 df-rab 2748 df-v 3049 df-sbc 3270 df-csb 3366 df-dif 3409 df-un 3411 df-in 3413 df-ss 3420 df-nul 3734 df-if 3884 df-pw 3955 df-sn 3971 df-pr 3973 df-op 3977 df-uni 4202 df-iun 4283 df-br 4406 df-opab 4465 df-mpt 4466 df-id 4752 df-xp 4843 df-rel 4844 df-cnv 4845 df-co 4846 df-dm 4847 df-rn 4848 df-res 4849 df-ima 4850 df-iota 5549 df-fun 5587 df-fn 5588 df-f 5589 df-f1 5590 df-fo 5591 df-f1o 5592 df-fv 5593 df-riota 6257 df-ov 6298 df-oprab 6299 df-preset 16185 df-poset 16203 df-plt 16216 df-lub 16232 df-glb 16233 df-join 16234 df-meet 16235 df-p0 16297 df-lat 16304 df-clat 16366 df-oposet 32754 df-ol 32756 df-oml 32757 df-covers 32844 df-ats 32845 df-atl 32876 df-cvlat 32900 df-hlat 32929 df-llines 33075 df-lplanes 33076 df-lvols 33077 |
This theorem is referenced by: dalem15 33255 |
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