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Mirrors > Home > MPE Home > Th. List > Mathboxes > dochdmj1 | Structured version Visualization version Unicode version |
Description: De Morgan-like law for subspace orthocomplement. (Contributed by NM, 5-Aug-2014.) |
Ref | Expression |
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dochdmj1.h | |
dochdmj1.u | |
dochdmj1.v | |
dochdmj1.o |
Ref | Expression |
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dochdmj1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1061 | . . . 4 | |
2 | simp2 1062 | . . . . 5 | |
3 | simp3 1063 | . . . . 5 | |
4 | 2, 3 | unssd 3789 | . . . 4 |
5 | ssun1 3776 | . . . . 5 | |
6 | 5 | a1i 11 | . . . 4 |
7 | dochdmj1.h | . . . . 5 | |
8 | dochdmj1.u | . . . . 5 | |
9 | dochdmj1.v | . . . . 5 | |
10 | dochdmj1.o | . . . . 5 | |
11 | 7, 8, 9, 10 | dochss 36654 | . . . 4 |
12 | 1, 4, 6, 11 | syl3anc 1326 | . . 3 |
13 | ssun2 3777 | . . . . 5 | |
14 | 13 | a1i 11 | . . . 4 |
15 | 7, 8, 9, 10 | dochss 36654 | . . . 4 |
16 | 1, 4, 14, 15 | syl3anc 1326 | . . 3 |
17 | 12, 16 | ssind 3837 | . 2 |
18 | eqid 2622 | . . . . . . 7 | |
19 | 7, 18, 8, 9, 10 | dochcl 36642 | . . . . . 6 |
20 | 19 | 3adant3 1081 | . . . . 5 |
21 | 7, 18, 8, 9, 10 | dochcl 36642 | . . . . . 6 |
22 | 21 | 3adant2 1080 | . . . . 5 |
23 | 7, 18 | dihmeetcl 36634 | . . . . 5 |
24 | 1, 20, 22, 23 | syl12anc 1324 | . . . 4 |
25 | 7, 18, 10 | dochoc 36656 | . . . 4 |
26 | 1, 24, 25 | syl2anc 693 | . . 3 |
27 | 7, 8, 9, 10 | dochssv 36644 | . . . . . . 7 |
28 | 27 | 3adant3 1081 | . . . . . 6 |
29 | ssinss1 3841 | . . . . . 6 | |
30 | 28, 29 | syl 17 | . . . . 5 |
31 | 7, 8, 9, 10 | dochssv 36644 | . . . . 5 |
32 | 1, 30, 31 | syl2anc 693 | . . . 4 |
33 | 7, 8, 9, 10 | dochocss 36655 | . . . . . . 7 |
34 | 33 | 3adant3 1081 | . . . . . 6 |
35 | 7, 8, 9, 10 | dochocss 36655 | . . . . . . 7 |
36 | 35 | 3adant2 1080 | . . . . . 6 |
37 | unss12 3785 | . . . . . 6 | |
38 | 34, 36, 37 | syl2anc 693 | . . . . 5 |
39 | inss1 3833 | . . . . . . . 8 | |
40 | 39 | a1i 11 | . . . . . . 7 |
41 | 7, 8, 9, 10 | dochss 36654 | . . . . . . 7 |
42 | 1, 28, 40, 41 | syl3anc 1326 | . . . . . 6 |
43 | 7, 8, 9, 10 | dochssv 36644 | . . . . . . . 8 |
44 | 43 | 3adant2 1080 | . . . . . . 7 |
45 | inss2 3834 | . . . . . . . 8 | |
46 | 45 | a1i 11 | . . . . . . 7 |
47 | 7, 8, 9, 10 | dochss 36654 | . . . . . . 7 |
48 | 1, 44, 46, 47 | syl3anc 1326 | . . . . . 6 |
49 | 42, 48 | unssd 3789 | . . . . 5 |
50 | 38, 49 | sstrd 3613 | . . . 4 |
51 | 7, 8, 9, 10 | dochss 36654 | . . . 4 |
52 | 1, 32, 50, 51 | syl3anc 1326 | . . 3 |
53 | 26, 52 | eqsstr3d 3640 | . 2 |
54 | 17, 53 | eqssd 3620 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 w3a 1037 wceq 1483 wcel 1990 cun 3572 cin 3573 wss 3574 crn 5115 cfv 5888 cbs 15857 chlt 34637 clh 35270 cdvh 36367 cdih 36517 coch 36636 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-riotaBAD 34239 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-tpos 7352 df-undef 7399 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-sca 15957 df-vsca 15958 df-0g 16102 df-preset 16928 df-poset 16946 df-plt 16958 df-lub 16974 df-glb 16975 df-join 16976 df-meet 16977 df-p0 17039 df-p1 17040 df-lat 17046 df-clat 17108 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-grp 17425 df-minusg 17426 df-sbg 17427 df-subg 17591 df-cntz 17750 df-lsm 18051 df-cmn 18195 df-abl 18196 df-mgp 18490 df-ur 18502 df-ring 18549 df-oppr 18623 df-dvdsr 18641 df-unit 18642 df-invr 18672 df-dvr 18683 df-drng 18749 df-lmod 18865 df-lss 18933 df-lsp 18972 df-lvec 19103 df-lsatoms 34263 df-oposet 34463 df-ol 34465 df-oml 34466 df-covers 34553 df-ats 34554 df-atl 34585 df-cvlat 34609 df-hlat 34638 df-llines 34784 df-lplanes 34785 df-lvols 34786 df-lines 34787 df-psubsp 34789 df-pmap 34790 df-padd 35082 df-lhyp 35274 df-laut 35275 df-ldil 35390 df-ltrn 35391 df-trl 35446 df-tendo 36043 df-edring 36045 df-disoa 36318 df-dvech 36368 df-dib 36428 df-dic 36462 df-dih 36518 df-doch 36637 |
This theorem is referenced by: djhval2 36688 dochdmm1 36699 lclkrlem2c 36798 lclkrlem2v 36817 lcfrlem18 36849 |
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