cvlcvr1 - Mathbox for Norm Megill

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Theorem cvlcvr1 34626

Description: The covering property. Proposition 1(ii) in [Kalmbach] p. 140 (and its converse). (chcv1 29214 analog.) (Contributed by NM, 5-Nov-2012.)

**Hypotheses**
Ref	Expression
cvlcvr1.b
cvlcvr1.l
cvlcvr1.j	$.\/$
cvlcvr1.c
cvlcvr1.a

**Assertion**
Ref	Expression
cvlcvr1	$/\$ $/\$ $/\$ $/\$ $.\/$

Proof of Theorem cvlcvr1 Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp13 1093	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
2		cvllat 34613	. . . . . . 7
3	1, 2	syl 17	. . . . . 6 $/\$ $/\$ $/\$ $/\$
4		simp2 1062	. . . . . 6 $/\$ $/\$ $/\$ $/\$
5		cvlcvr1.b	. . . . . . . 8
6		cvlcvr1.a	. . . . . . . 8
7	5, 6	atbase 34576	. . . . . . 7
8	7	3ad2ant3 1084	. . . . . 6 $/\$ $/\$ $/\$ $/\$
9		cvlcvr1.l	. . . . . . 7
10		eqid 2622	. . . . . . 7
11		cvlcvr1.j	. . . . . . 7 $.\/$
12	5, 9, 10, 11	latnle 17085	. . . . . 6 $/\$ $/\$ $.\/$
13	3, 4, 8, 12	syl3anc 1326	. . . . 5 $/\$ $/\$ $/\$ $/\$ $.\/$
14	13	biimpd 219	. . . 4 $/\$ $/\$ $/\$ $/\$ $.\/$
15		simpl13 1138	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
16	15, 2	syl 17	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
17		simprll 802	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
18		simpl2 1065	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
19		simpl3 1066	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
20	19, 7	syl 17	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
21	5, 11	latjcl 17051	. . . . . . . . 9 $/\$ $/\$ $.\/$
22	16, 18, 20, 21	syl3anc 1326	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$
23		simprrr 805	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$
24		simprrl 804	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
25		simpl11 1136	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
26		simpl12 1137	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
27		cvlatl 34612	. . . . . . . . . . . 12
28	15, 27	syl 17	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
29	5, 9, 10, 6	atlrelat1 34608	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
30	25, 26, 28, 18, 17, 29	syl311anc 1340	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$
31	24, 30	mpd 15	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$
32	16	adantr 481	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$
33	5, 6	atbase 34576	. . . . . . . . . . . . . 14
34	33	ad2antrl 764	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$
35	17	adantr 481	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$
36	22	adantr 481	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $.\/$
37		simprrr 805	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$
38	23	adantr 481	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $.\/$
39	5, 9, 32, 34, 35, 36, 37, 38	lattrd 17058	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $.\/$
40	15	adantr 481	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$
41		simprl 794	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$
42		simpll3 1102	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$
43		simpll2 1101	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$
44		simprrl 804	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$
45	5, 9, 11, 6	cvlexch1 34615	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$
46	40, 41, 42, 43, 44, 45	syl131anc 1339	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $.\/$ $.\/$
47	39, 46	mpd 15	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $.\/$
48		simprlr 803	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
49	48	adantr 481	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$
50	5, 9, 11, 6	cvlexchb1 34617	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $.\/$
51	40, 42, 41, 43, 49, 50	syl131anc 1339	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $.\/$
52	47, 51	mpbid 222	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $.\/$ $.\/$
53	9, 10	pltle 16961	. . . . . . . . . . . . . 14 $/\$ $/\$
54	25, 18, 17, 53	syl3anc 1326	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
55	24, 54	mpd 15	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
56	55	adantr 481	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$
57	5, 9, 11	latjle12 17062	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $.\/$
58	32, 43, 34, 35, 57	syl13anc 1328	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $.\/$
59	56, 37, 58	mpbi2and 956	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $.\/$
60	52, 59	eqbrtrd 4675	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $.\/$
61	31, 60	rexlimddv 3035	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$
62	5, 9, 16, 17, 22, 23, 61	latasymd 17057	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$
63	62	exp44 641	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$
64	63	imp 445	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$
65	64	ralrimdva 2969	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$
66	14, 65	jcad 555	. . 3 $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$
67	3, 4, 8, 21	syl3anc 1326	. . . 4 $/\$ $/\$ $/\$ $/\$ $.\/$
68		cvlcvr1.c	. . . . 5
69	5, 9, 10, 68	cvrval2 34561	. . . 4 $/\$ $/\$ $.\/$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$
70	3, 4, 67, 69	syl3anc 1326	. . 3 $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$
71	66, 70	sylibrd 249	. 2 $/\$ $/\$ $/\$ $/\$ $.\/$
72	3	adantr 481	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
73		simpl2 1065	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
74	67	adantr 481	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$
75		simpr 477	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$
76	5, 10, 68	cvrlt 34557	. . . . 5 $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$
77	72, 73, 74, 75, 76	syl31anc 1329	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$
78	77	ex 450	. . 3 $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$
79	78, 13	sylibrd 249	. 2 $/\$ $/\$ $/\$ $/\$ $.\/$
80	71, 79	impbid 202	1 $/\$ $/\$ $/\$ $/\$ $.\/$

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4

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

wceq 1483

wcel 1990

wral 2912

wrex 2913 class class class wbr 4653

cfv 5888 (class class class)co 6650

cbs 15857

cple 15948

cplt 16941

cjn 16944

clat 17045

ccla 17107

coml 34462

ccvr 34549

catm 34550

cal 34551

clc 34552

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

This theorem is referenced by: cvlcvrp 34627 cvr1 34696

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