cdleme7ga - Mathbox for Norm Megill

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Theorem cdleme7ga 35535

Description: Part of proof of Lemma E in [Crawley] p. 113. See cdleme7 35536. (Contributed by NM, 8-Jun-2012.)

**Hypotheses**
Ref	Expression
cdleme4.l
cdleme4.j	$.\/$
cdleme4.m	$./\$
cdleme4.a
cdleme4.h
cdleme4.u	$.\/$ $./\$
cdleme4.f	$.\/$ $./\$ $.\/$ $.\/$ $./\$
cdleme4.g	$.\/$ $./\$ $.\/$ $.\/$ $./\$

**Assertion**
Ref	Expression
cdleme7ga	$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$

**Proof of Theorem cdleme7ga**
Step	Hyp	Ref	Expression
1		cdleme4.g	. 2 $.\/$ $./\$ $.\/$ $.\/$ $./\$
2		simp11l 1172	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$
3		simp12l 1174	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$
4		simp13l 1176	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$
5		eqid 2622	. . . . 5
6		cdleme4.j	. . . . 5 $.\/$
7		cdleme4.a	. . . . 5
8	5, 6, 7	hlatjcl 34653	. . . 4 $/\$ $/\$ $.\/$
9	2, 3, 4, 8	syl3anc 1326	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$
10		simp11 1091	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$
11		simp12 1092	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$
12		simp13 1093	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$
13		simp2r 1088	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$
14		simp31 1097	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$
15		simp33 1099	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$
16		cdleme4.l	. . . . . 6
17		cdleme4.m	. . . . . 6 $./\$
18		cdleme4.h	. . . . . 6
19		cdleme4.u	. . . . . 6 $.\/$ $./\$
20		cdleme4.f	. . . . . 6 $.\/$ $./\$ $.\/$ $.\/$ $./\$
21	16, 6, 17, 7, 18, 19, 20	cdleme3fa 35523	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
22	10, 11, 12, 13, 14, 15, 21	syl132anc 1344	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$
23		simp2l 1087	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$
24		simp2rl 1130	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$
25		simp32 1098	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$
26		eqid 2622	. . . . . 6 $.\/$ $./\$ $.\/$ $./\$
27	16, 6, 17, 7, 18, 19, 20, 1, 26	cdleme7b 35531	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $./\$
28	10, 23, 24, 15, 25, 27	syl113anc 1338	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $./\$
29	5, 6, 7	hlatjcl 34653	. . . 4 $/\$ $/\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$
30	2, 22, 28, 29	syl3anc 1326	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$ $./\$
31		hllat 34650	. . . . 5
32	2, 31	syl 17	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$
33		eqid 2622	. . . . 5
34		eqid 2622	. . . . 5
35	6, 7, 33, 34	linepmap 35061	. . . 4 $/\$ $/\$ $/\$ $.\/$
36	32, 3, 4, 14, 35	syl31anc 1329	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$
37		simp2ll 1128	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$
38	5, 6, 7	hlatjcl 34653	. . . . . . 7 $/\$ $/\$ $.\/$
39	2, 37, 24, 38	syl3anc 1326	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$
40		simp11r 1173	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$
41	5, 18	lhpbase 35284	. . . . . . 7
42	40, 41	syl 17	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$
43	5, 16, 17	latmle2 17077	. . . . . 6 $/\$ $.\/$ $/\$ $.\/$ $./\$
44	32, 39, 42, 43	syl3anc 1326	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $./\$
45	16, 6, 17, 7, 18, 19, 20	cdleme3 35524	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
46	10, 11, 12, 13, 14, 15, 45	syl132anc 1344	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$
47		nbrne2 4673	. . . . . 6 $.\/$ $./\$ $/\$ $.\/$ $./\$
48	47	necomd 2849	. . . . 5 $.\/$ $./\$ $/\$ $.\/$ $./\$
49	44, 46, 48	syl2anc 693	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $./\$
50	6, 7, 33, 34	linepmap 35061	. . . 4 $/\$ $/\$ $.\/$ $./\$ $/\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$
51	32, 22, 28, 49, 50	syl31anc 1329	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$ $./\$
52	5, 7	atbase 34576	. . . . . 6
53	22, 52	syl 17	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$
54	5, 17	latmcl 17052	. . . . . 6 $/\$ $.\/$ $/\$ $.\/$ $./\$
55	32, 39, 42, 54	syl3anc 1326	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $./\$
56	5, 16, 6	latlej2 17061	. . . . 5 $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$
57	32, 53, 55, 56	syl3anc 1326	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $./\$
58	16, 6, 17, 7, 18, 19, 20, 1, 26	cdleme7c 35532	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $./\$
59	10, 11, 4, 23, 13, 14, 25, 15, 58	syl323anc 1356	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $./\$
60	59	necomd 2849	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $./\$
61		hlatl 34647	. . . . . . . 8
62	2, 61	syl 17	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$
63	16, 6, 17, 7, 18, 19	lhpat2 35331	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
64	10, 11, 4, 14, 63	syl112anc 1330	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$
65	16, 7	atncmp 34599	. . . . . . 7 $/\$ $.\/$ $./\$ $/\$ $.\/$ $./\$ $.\/$ $./\$
66	62, 28, 64, 65	syl3anc 1326	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$
67	60, 66	mpbird 247	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $./\$
68	5, 16, 17	latlem12 17078	. . . . . . . . 9 $/\$ $.\/$ $./\$ $/\$ $.\/$ $/\$ $.\/$ $./\$ $.\/$ $/\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$
69	32, 55, 9, 42, 68	syl13anc 1328	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$
70	69	biimpd 219	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$
71	44, 70	mpan2d 710	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$
72	19	breq2i 4661	. . . . . 6 $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$
73	71, 72	syl6ibr 242	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $./\$
74	67, 73	mtod 189	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $./\$ $.\/$
75		nbrne1 4672	. . . . 5 $.\/$ $./\$ $.\/$ $.\/$ $./\$ $/\$ $.\/$ $./\$ $.\/$ $.\/$ $.\/$ $./\$ $.\/$
76	75	necomd 2849	. . . 4 $.\/$ $./\$ $.\/$ $.\/$ $./\$ $/\$ $.\/$ $./\$ $.\/$ $.\/$ $.\/$ $.\/$ $./\$
77	57, 74, 76	syl2anc 693	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$ $./\$
78	16, 6, 17, 7, 18, 19, 20, 1, 26	cdleme7e 35534	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$
79	1, 78	syl5eqner 2869	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $./\$
80		eqid 2622	. . . 4
81	5, 17, 80, 7, 33, 34	2lnat 35070	. . 3 $/\$ $.\/$ $/\$ $.\/$ $.\/$ $./\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $./\$ $/\$ $.\/$ $.\/$ $.\/$ $./\$ $/\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$
82	2, 9, 30, 36, 51, 77, 79, 81	syl322anc 1354	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $./\$
83	1, 82	syl5eqel 2705	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 class class class wbr 4653

cfv 5888 (class class class)co 6650

cbs 15857

cple 15948

cjn 16944

cmee 16945

cp0 17037

clat 17045

catm 34550

cal 34551

chlt 34637

clines 34780

cpmap 34783

clh 35270

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-iin 4523 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-mpt2 6655 df-1st 7168 df-2nd 7169 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-oposet 34463 df-ol 34465 df-oml 34466 df-covers 34553 df-ats 34554 df-atl 34585 df-cvlat 34609 df-hlat 34638 df-lines 34787 df-psubsp 34789 df-pmap 34790 df-padd 35082 df-lhyp 35274

This theorem is referenced by: cdleme7 35536 cdleme18c 35580 cdleme22f2 35635 cdlemefs32sn1aw 35702

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