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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lsatcvat3 | Unicode version |
Description: A condition implying that a certain subspace is an atom. Part of Lemma 3.2.20 of [PtakPulmannova] p. 68. (atcvat3i 23856 analog.) (Contributed by NM, 11-Jan-2015.) |
Ref | Expression |
---|---|
lsatcvat3.s |
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lsatcvat3.p |
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lsatcvat3.a |
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lsatcvat3.w |
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lsatcvat3.u |
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lsatcvat3.q |
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lsatcvat3.r |
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lsatcvat3.n |
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lsatcvat3.m |
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lsatcvat3.l |
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Ref | Expression |
---|---|
lsatcvat3 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lsatcvat3.s |
. 2
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2 | lsatcvat3.p |
. 2
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3 | lsatcvat3.a |
. 2
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4 | eqid 2408 |
. 2
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5 | lsatcvat3.w |
. 2
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6 | lveclmod 16137 |
. . . 4
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7 | 5, 6 | syl 16 |
. . 3
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8 | lsatcvat3.u |
. . 3
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9 | lsatcvat3.q |
. . . . 5
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10 | 1, 3, 7, 9 | lsatlssel 29484 |
. . . 4
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11 | lsatcvat3.r |
. . . . 5
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12 | 1, 3, 7, 11 | lsatlssel 29484 |
. . . 4
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13 | 1, 2 | lsmcl 16114 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
14 | 7, 10, 12, 13 | syl3anc 1184 |
. . 3
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15 | 1 | lssincl 16000 |
. . 3
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16 | 7, 8, 14, 15 | syl3anc 1184 |
. 2
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17 | lsatcvat3.n |
. 2
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18 | lsatcvat3.m |
. . . . 5
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19 | 1, 2, 3, 4, 5, 8, 11 | lcv1 29528 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
20 | 18, 19 | mpbid 202 |
. . . 4
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21 | lmodabl 15950 |
. . . . . . . . . . 11
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22 | 7, 21 | syl 16 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
23 | 1 | lsssssubg 15993 |
. . . . . . . . . . . 12
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24 | 7, 23 | syl 16 |
. . . . . . . . . . 11
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
25 | 24, 10 | sseldd 3313 |
. . . . . . . . . 10
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26 | 24, 12 | sseldd 3313 |
. . . . . . . . . 10
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27 | 2 | lsmcom 15432 |
. . . . . . . . . 10
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28 | 22, 25, 26, 27 | syl3anc 1184 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
29 | 28 | oveq2d 6060 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
30 | 24, 8 | sseldd 3313 |
. . . . . . . . 9
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31 | 2 | lsmass 15261 |
. . . . . . . . 9
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32 | 30, 26, 25, 31 | syl3anc 1184 |
. . . . . . . 8
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33 | 29, 32 | eqtr4d 2443 |
. . . . . . 7
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34 | 1, 2 | lsmcl 16114 |
. . . . . . . . . 10
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35 | 7, 8, 12, 34 | syl3anc 1184 |
. . . . . . . . 9
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36 | 24, 35 | sseldd 3313 |
. . . . . . . 8
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37 | lsatcvat3.l |
. . . . . . . 8
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38 | 2 | lsmless2 15253 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
39 | 36, 36, 37, 38 | syl3anc 1184 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
40 | 33, 39 | eqsstrd 3346 |
. . . . . 6
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41 | 2 | lsmidm 15255 |
. . . . . . 7
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42 | 36, 41 | syl 16 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
43 | 40, 42 | sseqtrd 3348 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
44 | 24, 14 | sseldd 3313 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
45 | 2 | lsmub2 15250 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
46 | 25, 26, 45 | syl2anc 643 |
. . . . . 6
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47 | 2 | lsmless2 15253 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
48 | 30, 44, 46, 47 | syl3anc 1184 |
. . . . 5
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49 | 43, 48 | eqssd 3329 |
. . . 4
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50 | 20, 49 | breqtrrd 4202 |
. . 3
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51 | 1, 2, 4, 7, 8, 14, 50 | lcvexchlem4 29524 |
. 2
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52 | 1, 2, 3, 4, 5, 16, 9, 11, 17, 51 | lsatcvat2 29538 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem is referenced by: l1cvat 29542 |
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 2389 ax-rep 4284 ax-sep 4294 ax-nul 4302 ax-pow 4341 ax-pr 4367 ax-un 4664 ax-cnex 9006 ax-resscn 9007 ax-1cn 9008 ax-icn 9009 ax-addcl 9010 ax-addrcl 9011 ax-mulcl 9012 ax-mulrcl 9013 ax-mulcom 9014 ax-addass 9015 ax-mulass 9016 ax-distr 9017 ax-i2m1 9018 ax-1ne0 9019 ax-1rid 9020 ax-rnegex 9021 ax-rrecex 9022 ax-cnre 9023 ax-pre-lttri 9024 ax-pre-lttrn 9025 ax-pre-ltadd 9026 ax-pre-mulgt0 9027 |
This theorem depends on definitions: df-bi 178 df-or 360 df-an 361 df-3or 937 df-3an 938 df-tru 1325 df-ex 1548 df-nf 1551 df-sb 1656 df-eu 2262 df-mo 2263 df-clab 2395 df-cleq 2401 df-clel 2404 df-nfc 2533 df-ne 2573 df-nel 2574 df-ral 2675 df-rex 2676 df-reu 2677 df-rmo 2678 df-rab 2679 df-v 2922 df-sbc 3126 df-csb 3216 df-dif 3287 df-un 3289 df-in 3291 df-ss 3298 df-pss 3300 df-nul 3593 df-if 3704 df-pw 3765 df-sn 3784 df-pr 3785 df-tp 3786 df-op 3787 df-uni 3980 df-int 4015 df-iun 4059 df-iin 4060 df-br 4177 df-opab 4231 df-mpt 4232 df-tr 4267 df-eprel 4458 df-id 4462 df-po 4467 df-so 4468 df-fr 4505 df-we 4507 df-ord 4548 df-on 4549 df-lim 4550 df-suc 4551 df-om 4809 df-xp 4847 df-rel 4848 df-cnv 4849 df-co 4850 df-dm 4851 df-rn 4852 df-res 4853 df-ima 4854 df-iota 5381 df-fun 5419 df-fn 5420 df-f 5421 df-f1 5422 df-fo 5423 df-f1o 5424 df-fv 5425 df-ov 6047 df-oprab 6048 df-mpt2 6049 df-1st 6312 df-2nd 6313 df-tpos 6442 df-riota 6512 df-recs 6596 df-rdg 6631 df-1o 6687 df-oadd 6691 df-er 6868 df-en 7073 df-dom 7074 df-sdom 7075 df-fin 7076 df-pnf 9082 df-mnf 9083 df-xr 9084 df-ltxr 9085 df-le 9086 df-sub 9253 df-neg 9254 df-nn 9961 df-2 10018 df-3 10019 df-ndx 13431 df-slot 13432 df-base 13433 df-sets 13434 df-ress 13435 df-plusg 13501 df-mulr 13502 df-0g 13686 df-mre 13770 df-mrc 13771 df-acs 13773 df-mnd 14649 df-submnd 14698 df-grp 14771 df-minusg 14772 df-sbg 14773 df-subg 14900 df-cntz 15075 df-oppg 15101 df-lsm 15229 df-cmn 15373 df-abl 15374 df-mgp 15608 df-rng 15622 df-ur 15624 df-oppr 15687 df-dvdsr 15705 df-unit 15706 df-invr 15736 df-drng 15796 df-lmod 15911 df-lss 15968 df-lsp 16007 df-lvec 16134 df-lsatoms 29463 df-lcv 29506 |
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