llncvrlpln2 - Mathbox for Norm Megill

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Theorem llncvrlpln2 34843

Description: A lattice line under a lattice plane is covered by it. (Contributed by NM, 24-Jun-2012.)

**Hypotheses**
Ref	Expression
llncvrlpln2.l
llncvrlpln2.c
llncvrlpln2.n
llncvrlpln2.p

**Assertion**
Ref	Expression
llncvrlpln2	$/\$ $/\$ $/\$

Proof of Theorem llncvrlpln2 Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 477	. . 3 $/\$ $/\$ $/\$
2		simpl1 1064	. . . . 5 $/\$ $/\$ $/\$
3		simpl3 1066	. . . . 5 $/\$ $/\$ $/\$
4		llncvrlpln2.n	. . . . . 6
5		llncvrlpln2.p	. . . . . 6
6	4, 5	lplnnelln 34832	. . . . 5 $/\$
7	2, 3, 6	syl2anc 693	. . . 4 $/\$ $/\$ $/\$
8		simpl2 1065	. . . . . 6 $/\$ $/\$ $/\$
9		eleq1 2689	. . . . . 6
10	8, 9	syl5ibcom 235	. . . . 5 $/\$ $/\$ $/\$
11	10	necon3bd 2808	. . . 4 $/\$ $/\$ $/\$
12	7, 11	mpd 15	. . 3 $/\$ $/\$ $/\$
13		llncvrlpln2.l	. . . . 5
14		eqid 2622	. . . . 5
15	13, 14	pltval 16960	. . . 4 $/\$ $/\$ $/\$
16	15	adantr 481	. . 3 $/\$ $/\$ $/\$ $/\$
17	1, 12, 16	mpbir2and 957	. 2 $/\$ $/\$ $/\$
18		simpl1 1064	. . . 4 $/\$ $/\$ $/\$
19		simpl2 1065	. . . . 5 $/\$ $/\$ $/\$
20		eqid 2622	. . . . . 6
21	20, 4	llnbase 34795	. . . . 5
22	19, 21	syl 17	. . . 4 $/\$ $/\$ $/\$
23		simpl3 1066	. . . . 5 $/\$ $/\$ $/\$
24	20, 5	lplnbase 34820	. . . . 5
25	23, 24	syl 17	. . . 4 $/\$ $/\$ $/\$
26		simpr 477	. . . 4 $/\$ $/\$ $/\$
27		eqid 2622	. . . . 5
28		llncvrlpln2.c	. . . . 5
29		eqid 2622	. . . . 5
30	20, 13, 14, 27, 28, 29	hlrelat3 34698	. . . 4 $/\$ $/\$ $/\$ $/\$
31	18, 22, 25, 26, 30	syl31anc 1329	. . 3 $/\$ $/\$ $/\$ $/\$
32	20, 13, 27, 29, 5	islpln2 34822	. . . . . . . 8 $/\$ $/\$ $/\$
33	32	adantr 481	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
34		simp3 1063	. . . . . . . . . . 11 $/\$ $/\$
35	20, 27, 29, 4	islln2 34797	. . . . . . . . . . . . 13 $/\$ $/\$
36		simp3l 1089	. . . . . . . . . . . . . . . . . . . 20 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
37		simp3r 1090	. . . . . . . . . . . . . . . . . . . . . . 23 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
38		simp12r 1175	. . . . . . . . . . . . . . . . . . . . . . . 24 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
39	38	oveq1d 6665	. . . . . . . . . . . . . . . . . . . . . . 23 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
40		simp22 1095	. . . . . . . . . . . . . . . . . . . . . . 23 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
41	37, 39, 40	3brtr3d 4684	. . . . . . . . . . . . . . . . . . . . . 22 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
42		simp111 1190	. . . . . . . . . . . . . . . . . . . . . . 23 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
43		simp112 1191	. . . . . . . . . . . . . . . . . . . . . . . 24 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
44		simp113 1192	. . . . . . . . . . . . . . . . . . . . . . . 24 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
45		simp23 1096	. . . . . . . . . . . . . . . . . . . . . . . 24 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
46	43, 44, 45	3jca 1242	. . . . . . . . . . . . . . . . . . . . . . 23 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
47		simp13l 1176	. . . . . . . . . . . . . . . . . . . . . . . 24 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
48		simp13r 1177	. . . . . . . . . . . . . . . . . . . . . . . 24 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
49		simp21 1094	. . . . . . . . . . . . . . . . . . . . . . . 24 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
50	47, 48, 49	3jca 1242	. . . . . . . . . . . . . . . . . . . . . . 23 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
51	36, 38, 39	3brtr3d 4684	. . . . . . . . . . . . . . . . . . . . . . . 24 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
52	20, 27, 29	hlatjcl 34653	. . . . . . . . . . . . . . . . . . . . . . . . . 26 $/\$ $/\$
53	42, 43, 44, 52	syl3anc 1326	. . . . . . . . . . . . . . . . . . . . . . . . 25 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
54	20, 13, 27, 28, 29	cvr1 34696	. . . . . . . . . . . . . . . . . . . . . . . . 25 $/\$ $/\$
55	42, 53, 45, 54	syl3anc 1326	. . . . . . . . . . . . . . . . . . . . . . . 24 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
56	51, 55	mpbird 247	. . . . . . . . . . . . . . . . . . . . . . 23 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
57		simp12l 1174	. . . . . . . . . . . . . . . . . . . . . . 23 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
58	13, 27, 29	3at 34776	. . . . . . . . . . . . . . . . . . . . . . 23 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
59	42, 46, 50, 56, 57, 58	syl32anc 1334	. . . . . . . . . . . . . . . . . . . . . 22 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
60	41, 59	mpbid 222	. . . . . . . . . . . . . . . . . . . . 21 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
61	60, 39, 40	3eqtr4d 2666	. . . . . . . . . . . . . . . . . . . 20 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
62	36, 61	breqtrd 4679	. . . . . . . . . . . . . . . . . . 19 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
63	62	3exp 1264	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
64	63	3expd 1284	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
65	64	3exp 1264	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$
66	65	3expib 1268	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$
67	66	rexlimdvv 3037	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
68	67	adantld 483	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
69	35, 68	sylbid 230	. . . . . . . . . . . 12 $/\$ $/\$
70	69	imp31 448	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
71	34, 70	syl7 74	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
72	71	rexlimdv 3030	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
73	72	rexlimdvva 3038	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
74	73	adantld 483	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
75	33, 74	sylbid 230	. . . . . 6 $/\$ $/\$
76	75	3impia 1261	. . . . 5 $/\$ $/\$ $/\$
77	76	rexlimdv 3030	. . . 4 $/\$ $/\$ $/\$
78	77	imp 445	. . 3 $/\$ $/\$ $/\$ $/\$
79	31, 78	syldan 487	. 2 $/\$ $/\$ $/\$
80	17, 79	syldan 487	1 $/\$ $/\$ $/\$

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4

wb 196 $/\$ wa 384 $/\$ w3a 1037

wceq 1483

wcel 1990

wne 2794

wrex 2913 class class class wbr 4653

cfv 5888 (class class class)co 6650

cbs 15857

cple 15948

cplt 16941

cjn 16944

ccvr 34549

catm 34550

chlt 34637

clln 34777

clpl 34778

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-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

This theorem is referenced by: llncvrlpln 34844 2llnmj 34846 lplncmp 34848 lplnexatN 34849 2llnm2N 34854 2lplnmj 34908

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