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Mirrors > Home > MPE Home > Th. List > Mathboxes > dia2dim | Unicode version |
Description: A two-dimensional subspace of partial vector space A is closed, or equivalently, the isomorphism of a join of two atoms is a subset of the subspace sum of the isomorphisms of each atom (and thus they are equal, as shown later for the full vector space H). (Contributed by NM, 9-Sep-2014.) |
Ref | Expression |
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dia2dim.l |
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dia2dim.j |
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dia2dim.a |
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dia2dim.h |
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dia2dim.y |
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dia2dim.pl |
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dia2dim.i |
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dia2dim.k |
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dia2dim.u |
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dia2dim.v |
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Ref | Expression |
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dia2dim |
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Step | Hyp | Ref | Expression |
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1 | dia2dim.l |
. 2
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2 | dia2dim.j |
. 2
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3 | eqid 2412 |
. 2
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4 | dia2dim.a |
. 2
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5 | dia2dim.h |
. 2
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6 | eqid 2412 |
. 2
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7 | eqid 2412 |
. 2
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8 | dia2dim.y |
. 2
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9 | eqid 2412 |
. 2
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10 | dia2dim.pl |
. 2
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11 | eqid 2412 |
. 2
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12 | dia2dim.i |
. 2
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13 | dia2dim.k |
. 2
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14 | dia2dim.u |
. 2
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15 | dia2dim.v |
. 2
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16 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 | dia2dimlem13 31571 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem is referenced by: dib2dim 31738 dih2dimbALTN 31740 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1552 ax-5 1563 ax-17 1623 ax-9 1662 ax-8 1683 ax-13 1723 ax-14 1725 ax-6 1740 ax-7 1745 ax-11 1757 ax-12 1946 ax-ext 2393 ax-rep 4288 ax-sep 4298 ax-nul 4306 ax-pow 4345 ax-pr 4371 ax-un 4668 ax-cnex 9010 ax-resscn 9011 ax-1cn 9012 ax-icn 9013 ax-addcl 9014 ax-addrcl 9015 ax-mulcl 9016 ax-mulrcl 9017 ax-mulcom 9018 ax-addass 9019 ax-mulass 9020 ax-distr 9021 ax-i2m1 9022 ax-1ne0 9023 ax-1rid 9024 ax-rnegex 9025 ax-rrecex 9026 ax-cnre 9027 ax-pre-lttri 9028 ax-pre-lttrn 9029 ax-pre-ltadd 9030 ax-pre-mulgt0 9031 |
This theorem depends on definitions: df-bi 178 df-or 360 df-an 361 df-3or 937 df-3an 938 df-tru 1325 df-fal 1326 df-ex 1548 df-nf 1551 df-sb 1656 df-eu 2266 df-mo 2267 df-clab 2399 df-cleq 2405 df-clel 2408 df-nfc 2537 df-ne 2577 df-nel 2578 df-ral 2679 df-rex 2680 df-reu 2681 df-rmo 2682 df-rab 2683 df-v 2926 df-sbc 3130 df-csb 3220 df-dif 3291 df-un 3293 df-in 3295 df-ss 3302 df-pss 3304 df-nul 3597 df-if 3708 df-pw 3769 df-sn 3788 df-pr 3789 df-tp 3790 df-op 3791 df-uni 3984 df-int 4019 df-iun 4063 df-iin 4064 df-br 4181 df-opab 4235 df-mpt 4236 df-tr 4271 df-eprel 4462 df-id 4466 df-po 4471 df-so 4472 df-fr 4509 df-we 4511 df-ord 4552 df-on 4553 df-lim 4554 df-suc 4555 df-om 4813 df-xp 4851 df-rel 4852 df-cnv 4853 df-co 4854 df-dm 4855 df-rn 4856 df-res 4857 df-ima 4858 df-iota 5385 df-fun 5423 df-fn 5424 df-f 5425 df-f1 5426 df-fo 5427 df-f1o 5428 df-fv 5429 df-ov 6051 df-oprab 6052 df-mpt2 6053 df-1st 6316 df-2nd 6317 df-tpos 6446 df-undef 6510 df-riota 6516 df-recs 6600 df-rdg 6635 df-1o 6691 df-oadd 6695 df-er 6872 df-map 6987 df-en 7077 df-dom 7078 df-sdom 7079 df-fin 7080 df-pnf 9086 df-mnf 9087 df-xr 9088 df-ltxr 9089 df-le 9090 df-sub 9257 df-neg 9258 df-nn 9965 df-2 10022 df-3 10023 df-4 10024 df-5 10025 df-6 10026 df-n0 10186 df-z 10247 df-uz 10453 df-fz 11008 df-struct 13434 df-ndx 13435 df-slot 13436 df-base 13437 df-sets 13438 df-ress 13439 df-plusg 13505 df-mulr 13506 df-sca 13508 df-vsca 13509 df-0g 13690 df-poset 14366 df-plt 14378 df-lub 14394 df-glb 14395 df-join 14396 df-meet 14397 df-p0 14431 df-p1 14432 df-lat 14438 df-clat 14500 df-mnd 14653 df-submnd 14702 df-grp 14775 df-minusg 14776 df-sbg 14777 df-subg 14904 df-cntz 15079 df-lsm 15233 df-cmn 15377 df-abl 15378 df-mgp 15612 df-rng 15626 df-ur 15628 df-oppr 15691 df-dvdsr 15709 df-unit 15710 df-invr 15740 df-dvr 15751 df-drng 15800 df-lmod 15915 df-lss 15972 df-lsp 16011 df-lvec 16138 df-oposet 29671 df-ol 29673 df-oml 29674 df-covers 29761 df-ats 29762 df-atl 29793 df-cvlat 29817 df-hlat 29846 df-llines 29992 df-lplanes 29993 df-lvols 29994 df-lines 29995 df-psubsp 29997 df-pmap 29998 df-padd 30290 df-lhyp 30482 df-laut 30483 df-ldil 30598 df-ltrn 30599 df-trl 30653 df-tgrp 31237 df-tendo 31249 df-edring 31251 df-dveca 31497 df-disoa 31524 |
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