cdleme32e - Mathbox for Norm Megill

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Theorem cdleme32e 34083

Description: Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 20-Feb-2013.)

**Hypotheses**
Ref	Expression
cdleme32.b
cdleme32.l
cdleme32.j	$.\/$
cdleme32.m	$./\$
cdleme32.a
cdleme32.h
cdleme32.u	$.\/$ $./\$
cdleme32.c	$.\/$ $./\$ $.\/$ $.\/$ $./\$
cdleme32.d	$.\/$ $./\$ $.\/$ $.\/$ $./\$
cdleme32.e	$.\/$ $./\$ $.\/$ $.\/$ $./\$
cdleme32.i	$/\$ $.\/$
cdleme32.n	$.\/$
cdleme32.o	$/\$ $.\/$ $./\$ $.\/$ $./\$
cdleme32.f	$/\$

**Assertion**
Ref	Expression
cdleme32e	$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$

Distinct variable groups:

$.\/$ ,

$./\$ ,

Allowed substitution hints:

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**Proof of Theorem cdleme32e**
Step	Hyp	Ref	Expression
1		simp23l 1151	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$
2	1	pm2.24d 139	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $.\/$ $./\$
3		simp11l 1141	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$
4		hllat 33000	. . . . . 6
5	3, 4	syl 17	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$
6		simp21l 1147	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$
7		simp11r 1142	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$
8		cdleme32.b	. . . . . . 7
9		cdleme32.h	. . . . . . 7
10	8, 9	lhpbase 33634	. . . . . 6
11	7, 10	syl 17	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$
12		cdleme32.l	. . . . . 6
13		cdleme32.m	. . . . . 6 $./\$
14	8, 12, 13	latleeqm1 16403	. . . . 5 $/\$ $/\$ $./\$
15	5, 6, 11, 14	syl3anc 1292	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $./\$
16	8, 13	latmcl 16376	. . . . . . 7 $/\$ $/\$ $./\$
17	5, 6, 11, 16	syl3anc 1292	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $./\$
18		simp21r 1148	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$
19	8, 13	latmcl 16376	. . . . . . 7 $/\$ $/\$ $./\$
20	5, 18, 11, 19	syl3anc 1292	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $./\$
21		simp11 1060	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $/\$
22		simp12 1061	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $/\$
23		simp13 1062	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $/\$
24		simp31 1066	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $/\$
25		cdleme32.j	. . . . . . . . 9 $.\/$
26		cdleme32.a	. . . . . . . . 9
27		cdleme32.u	. . . . . . . . 9 $.\/$ $./\$
28		cdleme32.c	. . . . . . . . 9 $.\/$ $./\$ $.\/$ $.\/$ $./\$
29		cdleme32.d	. . . . . . . . 9 $.\/$ $./\$ $.\/$ $.\/$ $./\$
30		cdleme32.e	. . . . . . . . 9 $.\/$ $./\$ $.\/$ $.\/$ $./\$
31		cdleme32.i	. . . . . . . . 9 $/\$ $.\/$
32		cdleme32.n	. . . . . . . . 9 $.\/$
33	8, 12, 25, 13, 26, 9, 27, 28, 29, 30, 31, 32	cdleme27cl 34004	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
34	21, 22, 23, 24, 1, 33	syl122anc 1301	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$
35	8, 25	latjcl 16375	. . . . . . 7 $/\$ $/\$ $./\$ $.\/$ $./\$
36	5, 34, 20, 35	syl3anc 1292	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $.\/$ $./\$
37		simp33 1068	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$
38	8, 12, 13	latmlem1 16405	. . . . . . . 8 $/\$ $/\$ $/\$ $./\$ $./\$
39	5, 6, 18, 11, 38	syl13anc 1294	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $./\$ $./\$
40	37, 39	mpd 15	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $./\$ $./\$
41	8, 12, 25	latlej2 16385	. . . . . . 7 $/\$ $/\$ $./\$ $./\$ $.\/$ $./\$
42	5, 34, 20, 41	syl3anc 1292	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $./\$ $.\/$ $./\$
43	8, 12, 5, 17, 20, 36, 40, 42	lattrd 16382	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $./\$ $.\/$ $./\$
44		breq1 4398	. . . . 5 $./\$ $./\$ $.\/$ $./\$ $.\/$ $./\$
45	43, 44	syl5ibcom 228	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $./\$ $.\/$ $./\$
46	15, 45	sylbid 223	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $.\/$ $./\$
47		simp22 1064	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $/\$
48		pm4.53 500	. . . 4 $/\$ $\/$
49	47, 48	sylib 201	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $\/$
50	2, 46, 49	mpjaod 388	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $.\/$ $./\$
51		cdleme32.f	. . . 4 $/\$
52	51	cdleme31fv2 34031	. . 3 $/\$ $/\$
53	6, 47, 52	syl2anc 673	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$
54		simp1 1030	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
55		simp23 1065	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $/\$
56		simp32 1067	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $.\/$ $./\$
57		cdleme32.o	. . . 4 $/\$ $.\/$ $./\$ $.\/$ $./\$
58	8, 12, 25, 13, 26, 9, 27, 28, 29, 30, 31, 32, 57, 51	cdleme32a 34079	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$
59	54, 18, 55, 24, 56, 58	syl122anc 1301	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $.\/$ $./\$
60	50, 53, 59	3brtr4d 4426	1 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4

wb 189 $\/$ wo 375 $/\$ wa 376 $/\$ w3a 1007

wceq 1452

wcel 1904

wne 2641

wral 2756

cif 3872 class class class wbr 4395

cmpt 4454

cfv 5589

crio 6269 (class class class)co 6308

cbs 15199

cple 15275

cjn 16267

cmee 16268

clat 16369

catm 32900

chlt 32987

clh 33620

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1677 ax-4 1690 ax-5 1766 ax-6 1813 ax-7 1859 ax-8 1906 ax-9 1913 ax-10 1932 ax-11 1937 ax-12 1950 ax-13 2104 ax-ext 2451 ax-rep 4508 ax-sep 4518 ax-nul 4527 ax-pow 4579 ax-pr 4639 ax-un 6602 ax-riotaBAD 32589

This theorem depends on definitions: df-bi 190 df-or 377 df-an 378 df-3or 1008 df-3an 1009 df-tru 1455 df-ex 1672 df-nf 1676 df-sb 1806 df-eu 2323 df-mo 2324 df-clab 2458 df-cleq 2464 df-clel 2467 df-nfc 2601 df-ne 2643 df-nel 2644 df-ral 2761 df-rex 2762 df-reu 2763 df-rmo 2764 df-rab 2765 df-v 3033 df-sbc 3256 df-csb 3350 df-dif 3393 df-un 3395 df-in 3397 df-ss 3404 df-nul 3723 df-if 3873 df-pw 3944 df-sn 3960 df-pr 3962 df-op 3966 df-uni 4191 df-iun 4271 df-iin 4272 df-br 4396 df-opab 4455 df-mpt 4456 df-id 4754 df-xp 4845 df-rel 4846 df-cnv 4847 df-co 4848 df-dm 4849 df-rn 4850 df-res 4851 df-ima 4852 df-iota 5553 df-fun 5591 df-fn 5592 df-f 5593 df-f1 5594 df-fo 5595 df-f1o 5596 df-fv 5597 df-riota 6270 df-ov 6311 df-oprab 6312 df-mpt2 6313 df-1st 6812 df-2nd 6813 df-undef 7038 df-preset 16251 df-poset 16269 df-plt 16282 df-lub 16298 df-glb 16299 df-join 16300 df-meet 16301 df-p0 16363 df-p1 16364 df-lat 16370 df-clat 16432 df-oposet 32813 df-ol 32815 df-oml 32816 df-covers 32903 df-ats 32904 df-atl 32935 df-cvlat 32959 df-hlat 32988 df-llines 33134 df-lplanes 33135 df-lvols 33136 df-lines 33137 df-psubsp 33139 df-pmap 33140 df-padd 33432 df-lhyp 33624

This theorem is referenced by: cdleme32f 34084

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