Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > 2lplnm2N | Structured version Visualization version Unicode version |
Description: The meet of two different lattice planes in a lattice volume is a lattice line. (Contributed by NM, 12-Jul-2012.) (New usage is discouraged.) |
Ref | Expression |
---|---|
2lplnm2.l | |
2lplnm2.m | |
2lplnm2.a | |
2lplnm2.p | |
2lplnm2.v |
Ref | Expression |
---|---|
2lplnm2N |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp22 1095 | . 2 | |
2 | simp1 1061 | . . 3 | |
3 | hllat 34650 | . . . . 5 | |
4 | 3 | 3ad2ant1 1082 | . . . 4 |
5 | simp21 1094 | . . . . 5 | |
6 | eqid 2622 | . . . . . 6 | |
7 | 2lplnm2.p | . . . . . 6 | |
8 | 6, 7 | lplnbase 34820 | . . . . 5 |
9 | 5, 8 | syl 17 | . . . 4 |
10 | 6, 7 | lplnbase 34820 | . . . . 5 |
11 | 1, 10 | syl 17 | . . . 4 |
12 | 2lplnm2.m | . . . . 5 | |
13 | 6, 12 | latmcl 17052 | . . . 4 |
14 | 4, 9, 11, 13 | syl3anc 1326 | . . 3 |
15 | 2lplnm2.l | . . . . . . 7 | |
16 | eqid 2622 | . . . . . . 7 | |
17 | 2lplnm2.v | . . . . . . 7 | |
18 | 15, 16, 7, 17 | 2lplnj 34906 | . . . . . 6 |
19 | simp23 1096 | . . . . . 6 | |
20 | 18, 19 | eqeltrd 2701 | . . . . 5 |
21 | 6, 15, 16 | latlej1 17060 | . . . . . 6 |
22 | 4, 9, 11, 21 | syl3anc 1326 | . . . . 5 |
23 | eqid 2622 | . . . . . 6 | |
24 | 15, 23, 7, 17 | lplncvrlvol2 34901 | . . . . 5 |
25 | 2, 5, 20, 22, 24 | syl31anc 1329 | . . . 4 |
26 | 6, 16, 12, 23 | cvrexch 34706 | . . . . 5 |
27 | 2, 9, 11, 26 | syl3anc 1326 | . . . 4 |
28 | 25, 27 | mpbird 247 | . . 3 |
29 | 2lplnm2.a | . . . 4 | |
30 | 6, 23, 29, 7 | llncvrlpln 34844 | . . 3 |
31 | 2, 14, 11, 28, 30 | syl31anc 1329 | . 2 |
32 | 1, 31 | mpbird 247 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 w3a 1037 wceq 1483 wcel 1990 wne 2794 class class class wbr 4653 cfv 5888 (class class class)co 6650 cbs 15857 cple 15948 cjn 16944 cmee 16945 clat 17045 ccvr 34549 chlt 34637 clln 34777 clpl 34778 clvol 34779 |
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 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 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-ral 2917 df-rex 2918 df-reu 2919 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-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 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-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-preset 16928 df-poset 16946 df-plt 16958 df-lub 16974 df-glb 16975 df-join 16976 df-meet 16977 df-p0 17039 df-lat 17046 df-clat 17108 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 |
This theorem is referenced by: (None) |
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