4atexlemc - Mathbox for Norm Megill

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Theorem 4atexlemc 30551

Description: Lemma for 4atexlem7 30557. (Contributed by NM, 24-Nov-2012.)

**Hypotheses**
Ref	Expression
4thatlem.ph	$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$
4thatlem0.l
4thatlem0.j	$.\/$
4thatlem0.m	$./\$
4thatlem0.a
4thatlem0.h
4thatlem0.u	$.\/$ $./\$
4thatlem0.v	$.\/$ $./\$
4thatlem0.c	$.\/$ $./\$ $.\/$

**Assertion**
Ref	Expression
4atexlemc

**Proof of Theorem 4atexlemc**
Step	Hyp	Ref	Expression
1		4thatlem0.c	. . 3 $.\/$ $./\$ $.\/$
2		4thatlem.ph	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$
3	2	4atexlemkl 30539	. . . 4
4		4thatlem0.j	. . . . 5 $.\/$
5		4thatlem0.a	. . . . 5
6	2, 4, 5	4atexlemqtb 30543	. . . 4 $.\/$
7	2, 4, 5	4atexlempsb 30542	. . . 4 $.\/$
8		eqid 2404	. . . . 5
9		4thatlem0.m	. . . . 5 $./\$
10	8, 9	latmcom 14459	. . . 4 $/\$ $.\/$ $/\$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$
11	3, 6, 7, 10	syl3anc 1184	. . 3 $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$
12	1, 11	syl5eq 2448	. 2 $.\/$ $./\$ $.\/$
13	2	4atexlemk 30529	. . 3
14	2	4atexlemp 30532	. . 3
15	2	4atexlems 30534	. . 3
16	2	4atexlemq 30533	. . 3
17	2	4atexlemt 30535	. . 3
18		4thatlem0.l	. . . 4
19	2, 18, 4, 5	4atexlempns 30544	. . 3
20		4thatlem0.h	. . . . 5
21		4thatlem0.u	. . . . 5 $.\/$ $./\$
22		4thatlem0.v	. . . . 5 $.\/$ $./\$
23	2, 18, 4, 9, 5, 20, 21, 22	4atexlemntlpq 30550	. . . 4 $.\/$
24	18, 4, 5	atnlej2 29862	. . . . 5 $/\$ $/\$ $/\$ $/\$ $.\/$
25	24	necomd 2650	. . . 4 $/\$ $/\$ $/\$ $/\$ $.\/$
26	13, 17, 14, 16, 23, 25	syl131anc 1197	. . 3
27	2	4atexlempnq 30537	. . . 4
28	2	4atexlemnslpq 30538	. . . 4 $.\/$
29	18, 4, 5	4atlem0ae 30076	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$
30	13, 14, 16, 15, 27, 28, 29	syl132anc 1202	. . 3 $.\/$
31	8, 5	atbase 29772	. . . . 5
32	17, 31	syl 16	. . . 4
33	2, 18, 4, 9, 5, 20, 21	4atexlemu 30546	. . . . 5
34	2, 18, 4, 9, 5, 20, 21, 22	4atexlemv 30547	. . . . 5
35	8, 4, 5	hlatjcl 29849	. . . . 5 $/\$ $/\$ $.\/$
36	13, 33, 34, 35	syl3anc 1184	. . . 4 $.\/$
37	8, 5	atbase 29772	. . . . . 6
38	16, 37	syl 16	. . . . 5
39	8, 4	latjcl 14434	. . . . 5 $/\$ $.\/$ $/\$ $.\/$ $.\/$
40	3, 7, 38, 39	syl3anc 1184	. . . 4 $.\/$ $.\/$
41	2	4atexlemkc 30540	. . . . 5
42	2, 18, 4, 9, 5, 20, 21, 22	4atexlemunv 30548	. . . . 5
43	2	4atexlemutvt 30536	. . . . 5 $.\/$ $.\/$
44	5, 18, 4	cvlsupr4 29828	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $.\/$
45	41, 33, 34, 17, 42, 43, 44	syl132anc 1202	. . . 4 $.\/$
46	8, 4, 5	hlatjcl 29849	. . . . . . . . 9 $/\$ $/\$ $.\/$
47	13, 14, 16, 46	syl3anc 1184	. . . . . . . 8 $.\/$
48	2, 20	4atexlemwb 30541	. . . . . . . 8
49	8, 18, 9	latmle1 14460	. . . . . . . 8 $/\$ $.\/$ $/\$ $.\/$ $./\$ $.\/$
50	3, 47, 48, 49	syl3anc 1184	. . . . . . 7 $.\/$ $./\$ $.\/$
51	21, 50	syl5eqbr 4205	. . . . . 6 $.\/$
52	8, 18, 9	latmle1 14460	. . . . . . . 8 $/\$ $.\/$ $/\$ $.\/$ $./\$ $.\/$
53	3, 7, 48, 52	syl3anc 1184	. . . . . . 7 $.\/$ $./\$ $.\/$
54	22, 53	syl5eqbr 4205	. . . . . 6 $.\/$
55	8, 5	atbase 29772	. . . . . . . 8
56	33, 55	syl 16	. . . . . . 7
57	8, 5	atbase 29772	. . . . . . . 8
58	34, 57	syl 16	. . . . . . 7
59	8, 18, 4	latjlej12 14451	. . . . . . 7 $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$
60	3, 56, 47, 58, 7, 59	syl122anc 1193	. . . . . 6 $.\/$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$
61	51, 54, 60	mp2and 661	. . . . 5 $.\/$ $.\/$ $.\/$ $.\/$
62	4, 5	hlatjass 29852	. . . . . . 7 $/\$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$
63	13, 14, 16, 15, 62	syl13anc 1186	. . . . . 6 $.\/$ $.\/$ $.\/$ $.\/$
64	8, 5	atbase 29772	. . . . . . . 8
65	14, 64	syl 16	. . . . . . 7
66	8, 5	atbase 29772	. . . . . . . 8
67	15, 66	syl 16	. . . . . . 7
68	8, 4	latj32 14481	. . . . . . 7 $/\$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$
69	3, 65, 38, 67, 68	syl13anc 1186	. . . . . 6 $.\/$ $.\/$ $.\/$ $.\/$
70	8, 4	latjjdi 14487	. . . . . . 7 $/\$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$
71	3, 65, 38, 67, 70	syl13anc 1186	. . . . . 6 $.\/$ $.\/$ $.\/$ $.\/$ $.\/$
72	63, 69, 71	3eqtr3rd 2445	. . . . 5 $.\/$ $.\/$ $.\/$ $.\/$ $.\/$
73	61, 72	breqtrd 4196	. . . 4 $.\/$ $.\/$ $.\/$
74	8, 18, 3, 32, 36, 40, 45, 73	lattrd 14442	. . 3 $.\/$ $.\/$
75	18, 4, 9, 5	2atmat 30043	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$ $./\$ $.\/$
76	13, 14, 15, 16, 17, 19, 26, 30, 74, 75	syl333anc 1216	. 2 $.\/$ $./\$ $.\/$
77	12, 76	eqeltrd 2478	1

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4

wb 177 $/\$ wa 359 $/\$ w3a 936

wceq 1649

wcel 1721

wne 2567 class class class wbr 4172

cfv 5413 (class class class)co 6040

cbs 13424

cple 13491

cjn 14356

cmee 14357

clat 14429

catm 29746

clc 29748

chlt 29833

clh 30466

This theorem is referenced by: 4atexlemnclw 30552 4atexlemex2 30553 4atexlemcnd 30554

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1552 ax-5 1563 ax-17 1623 ax-9 1662 ax-8 1683 ax-13 1723 ax-14 1725 ax-6 1740 ax-7 1745 ax-11 1757 ax-12 1946 ax-ext 2385 ax-rep 4280 ax-sep 4290 ax-nul 4298 ax-pow 4337 ax-pr 4363 ax-un 4660

This theorem depends on definitions: df-bi 178 df-or 360 df-an 361 df-3an 938 df-tru 1325 df-ex 1548 df-nf 1551 df-sb 1656 df-eu 2258 df-mo 2259 df-clab 2391 df-cleq 2397 df-clel 2400 df-nfc 2529 df-ne 2569 df-nel 2570 df-ral 2671 df-rex 2672 df-reu 2673 df-rab 2675 df-v 2918 df-sbc 3122 df-csb 3212 df-dif 3283 df-un 3285 df-in 3287 df-ss 3294 df-nul 3589 df-if 3700 df-pw 3761 df-sn 3780 df-pr 3781 df-op 3783 df-uni 3976 df-iun 4055 df-br 4173 df-opab 4227 df-mpt 4228 df-id 4458 df-xp 4843 df-rel 4844 df-cnv 4845 df-co 4846 df-dm 4847 df-rn 4848 df-res 4849 df-ima 4850 df-iota 5377 df-fun 5415 df-fn 5416 df-f 5417 df-f1 5418 df-fo 5419 df-f1o 5420 df-fv 5421 df-ov 6043 df-oprab 6044 df-mpt2 6045 df-1st 6308 df-2nd 6309 df-undef 6502 df-riota 6508 df-poset 14358 df-plt 14370 df-lub 14386 df-glb 14387 df-join 14388 df-meet 14389 df-p0 14423 df-p1 14424 df-lat 14430 df-clat 14492 df-oposet 29659 df-ol 29661 df-oml 29662 df-covers 29749 df-ats 29750 df-atl 29781 df-cvlat 29805 df-hlat 29834 df-llines 29980 df-lplanes 29981 df-lhyp 30470

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