cdleme11g - Mathbox for Norm Megill

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Theorem cdleme11g 33631

Description: Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme11 33636. (Contributed by NM, 14-Jun-2012.)

**Hypotheses**
Ref	Expression
cdleme11.l
cdleme11.j	$.\/$
cdleme11.m	$./\$
cdleme11.a
cdleme11.h
cdleme11.u	$.\/$ $./\$
cdleme11.c	$.\/$ $./\$
cdleme11.d	$.\/$ $./\$
cdleme11.f	$.\/$ $./\$ $.\/$ $.\/$ $./\$

**Assertion**
Ref	Expression
cdleme11g	$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$

**Proof of Theorem cdleme11g**
Step	Hyp	Ref	Expression
1		cdleme11.f	. . . 4 $.\/$ $./\$ $.\/$ $.\/$ $./\$
2	1	oveq2i 6101	. . 3 $.\/$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $./\$
3		simp1l 1007	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
4		simp22l 1102	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
5		hllat 32730	. . . . . 6
6	3, 5	syl 16	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
7		simp23 1018	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
8		eqid 2441	. . . . . . 7
9		cdleme11.a	. . . . . . 7
10	8, 9	atbase 32656	. . . . . 6
11	7, 10	syl 16	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
12		simp1 983	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
13		simp21 1016	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
14		cdleme11.l	. . . . . . 7
15		cdleme11.j	. . . . . . 7 $.\/$
16		cdleme11.m	. . . . . . 7 $./\$
17		cdleme11.h	. . . . . . 7
18		cdleme11.u	. . . . . . 7 $.\/$ $./\$
19	14, 15, 16, 9, 17, 18, 8	cdleme0aa 33576	. . . . . 6 $/\$ $/\$ $/\$
20	12, 13, 4, 19	syl3anc 1213	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
21	8, 15	latjcl 15217	. . . . 5 $/\$ $/\$ $.\/$
22	6, 11, 20, 21	syl3anc 1213	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
23	8, 9	atbase 32656	. . . . . 6
24	4, 23	syl 16	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
25	8, 9	atbase 32656	. . . . . . . 8
26	13, 25	syl 16	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
27	8, 15	latjcl 15217	. . . . . . 7 $/\$ $/\$ $.\/$
28	6, 26, 11, 27	syl3anc 1213	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
29		simp1r 1008	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
30	8, 17	lhpbase 33364	. . . . . . 7
31	29, 30	syl 16	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
32	8, 16	latmcl 15218	. . . . . 6 $/\$ $.\/$ $/\$ $.\/$ $./\$
33	6, 28, 31, 32	syl3anc 1213	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$
34	8, 15	latjcl 15217	. . . . 5 $/\$ $/\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$
35	6, 24, 33, 34	syl3anc 1213	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $./\$
36	8, 14, 15	latlej1 15226	. . . . 5 $/\$ $/\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$
37	6, 24, 33, 36	syl3anc 1213	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $./\$
38	8, 14, 15, 16, 9	atmod1i1 33223	. . . 4 $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $./\$ $/\$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $./\$
39	3, 4, 22, 35, 37, 38	syl131anc 1226	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $./\$
40	2, 39	syl5eq 2485	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $./\$
41		simp22 1017	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
42	14, 15, 16, 9, 17, 18	cdleme0cq 33581	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$
43	12, 13, 41, 42	syl12anc 1211	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$
44	43	oveq2d 6106	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$
45	8, 15	latj12 15262	. . . . 5 $/\$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$
46	6, 24, 11, 20, 45	syl13anc 1215	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$
47	8, 15	latj13 15264	. . . . 5 $/\$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$
48	6, 24, 26, 11, 47	syl13anc 1215	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$
49	44, 46, 48	3eqtr4d 2483	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$
50	49	oveq1d 6105	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $./\$
51	8, 14, 16	latmle1 15242	. . . . . 6 $/\$ $.\/$ $/\$ $.\/$ $./\$ $.\/$
52	6, 28, 31, 51	syl3anc 1213	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$
53	8, 14, 15	latjlej2 15232	. . . . . 6 $/\$ $.\/$ $./\$ $/\$ $.\/$ $/\$ $.\/$ $./\$ $.\/$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$
54	6, 33, 28, 24, 53	syl13anc 1215	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$
55	52, 54	mpd 15	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$
56	8, 15	latjcl 15217	. . . . . 6 $/\$ $/\$ $.\/$ $.\/$ $.\/$
57	6, 24, 28, 56	syl3anc 1213	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$
58	8, 14, 16	latleeqm2 15246	. . . . 5 $/\$ $.\/$ $.\/$ $./\$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $./\$
59	6, 35, 57, 58	syl3anc 1213	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $./\$
60	55, 59	mpbid 210	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $./\$
61		cdleme11.c	. . . 4 $.\/$ $./\$
62	61	oveq2i 6101	. . 3 $.\/$ $.\/$ $.\/$ $./\$
63	60, 62	syl6eqr 2491	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$
64	40, 50, 63	3eqtrd 2477	1 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4

wb 184 $/\$ wa 369 $/\$ w3a 960

wceq 1364

wcel 1761

wne 2604 class class class wbr 4289

cfv 5415 (class class class)co 6090

cbs 14170

cple 14241

cjn 15110

cmee 15111

clat 15211

catm 32630

chlt 32717

clh 33350

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1596 ax-4 1607 ax-5 1675 ax-6 1713 ax-7 1733 ax-8 1763 ax-9 1765 ax-10 1780 ax-11 1785 ax-12 1797 ax-13 1948 ax-ext 2422 ax-rep 4400 ax-sep 4410 ax-nul 4418 ax-pow 4467 ax-pr 4528 ax-un 6371

This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3an 962 df-tru 1367 df-ex 1592 df-nf 1595 df-sb 1706 df-eu 2261 df-mo 2262 df-clab 2428 df-cleq 2434 df-clel 2437 df-nfc 2566 df-ne 2606 df-ral 2718 df-rex 2719 df-reu 2720 df-rab 2722 df-v 2972 df-sbc 3184 df-csb 3286 df-dif 3328 df-un 3330 df-in 3332 df-ss 3339 df-nul 3635 df-if 3789 df-pw 3859 df-sn 3875 df-pr 3877 df-op 3881 df-uni 4089 df-iun 4170 df-iin 4171 df-br 4290 df-opab 4348 df-mpt 4349 df-id 4632 df-xp 4842 df-rel 4843 df-cnv 4844 df-co 4845 df-dm 4846 df-rn 4847 df-res 4848 df-ima 4849 df-iota 5378 df-fun 5417 df-fn 5418 df-f 5419 df-f1 5420 df-fo 5421 df-f1o 5422 df-fv 5423 df-riota 6049 df-ov 6093 df-oprab 6094 df-mpt2 6095 df-1st 6576 df-2nd 6577 df-poset 15112 df-plt 15124 df-lub 15140 df-glb 15141 df-join 15142 df-meet 15143 df-p0 15205 df-p1 15206 df-lat 15212 df-clat 15274 df-oposet 32543 df-ol 32545 df-oml 32546 df-covers 32633 df-ats 32634 df-atl 32665 df-cvlat 32689 df-hlat 32718 df-psubsp 32869 df-pmap 32870 df-padd 33162 df-lhyp 33354

This theorem is referenced by: cdleme11h 33632 cdleme11j 33633 cdleme15a 33640

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