paddasslem12 - Mathbox for Norm Megill

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Theorem paddasslem12 33838

Description: Lemma for paddass 33845. The case when

. (Contributed by NM, 11-Jan-2012.)

**Hypotheses**
Ref	Expression
paddasslem.l
paddasslem.j	$.\/$
paddasslem.a
paddasslem.p

**Assertion**
Ref	Expression
paddasslem12	$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$

**Proof of Theorem paddasslem12**
Step	Hyp	Ref	Expression
1		simpl1l 1039	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$
2		simpl21 1066	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$
3		simpl22 1067	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$
4		paddasslem.a	. . . . . 6
5		paddasslem.p	. . . . . 6
6	4, 5	paddssat 33821	. . . . 5 $/\$ $/\$
7	1, 2, 3, 6	syl3anc 1219	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$
8		simpl23 1068	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$
9	1, 7, 8	3jca 1168	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$
10	4, 5	sspadd2 33823	. . . 4 $/\$ $/\$
11	1, 3, 2, 10	syl3anc 1219	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$
12	4, 5	paddss1 33824	. . 3 $/\$ $/\$
13	9, 11, 12	sylc 60	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$
14		hllat 33371	. . . 4
15	1, 14	syl 16	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$
16		simprll 761	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$
17		simprlr 762	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$
18		simpl3l 1043	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$
19		eqid 2454	. . . 4
20		paddasslem.l	. . . 4
21	19, 4	atbase 33297	. . . . 5
22	18, 21	syl 16	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$
23	3, 16	sseldd 3468	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$
24	19, 4	atbase 33297	. . . . . 6
25	23, 24	syl 16	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$
26		simpl3r 1044	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$
27	19, 4	atbase 33297	. . . . . 6
28	26, 27	syl 16	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$
29		paddasslem.j	. . . . . 6 $.\/$
30	19, 29	latjcl 15344	. . . . 5 $/\$ $/\$ $.\/$
31	15, 25, 28, 30	syl3anc 1219	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$
32	8, 17	sseldd 3468	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$
33	19, 4	atbase 33297	. . . . . 6
34	32, 33	syl 16	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$
35	19, 29	latjcl 15344	. . . . 5 $/\$ $/\$ $.\/$
36	15, 25, 34, 35	syl3anc 1219	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$
37		simpl1r 1040	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$
38		simprrl 763	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$
39		oveq1 6210	. . . . . . 7 $.\/$ $.\/$
40	39	breq2d 4415	. . . . . 6 $.\/$ $.\/$
41	40	biimpa 484	. . . . 5 $/\$ $.\/$ $.\/$
42	37, 38, 41	syl2anc 661	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$
43	19, 20, 29	latlej1 15353	. . . . . 6 $/\$ $/\$ $.\/$
44	15, 25, 34, 43	syl3anc 1219	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$
45		simprrr 764	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$
46	19, 20, 29	latjle12 15355	. . . . . 6 $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$
47	15, 25, 28, 36, 46	syl13anc 1221	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$
48	44, 45, 47	mpbi2and 912	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$
49	19, 20, 15, 22, 31, 36, 42, 48	lattrd 15351	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$
50	20, 29, 4, 5	elpaddri 33809	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
51	15, 3, 8, 16, 17, 18, 49, 50	syl322anc 1247	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$
52	13, 51	sseldd 3468	1 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$

Colors of variables: wff setvar class

Syntax hints:

wi 4

wb 184 $/\$ wa 369 $/\$ w3a 965

wceq 1370

wcel 1758

wss 3439 class class class wbr 4403

cfv 5529 (class class class)co 6203

cbs 14296

cple 14368

cjn 15237

clat 15338

catm 33271

chlt 33358

cpadd 33802

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1592 ax-4 1603 ax-5 1671 ax-6 1710 ax-7 1730 ax-8 1760 ax-9 1762 ax-10 1777 ax-11 1782 ax-12 1794 ax-13 1955 ax-ext 2432 ax-rep 4514 ax-sep 4524 ax-nul 4532 ax-pow 4581 ax-pr 4642 ax-un 6485

This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3an 967 df-tru 1373 df-ex 1588 df-nf 1591 df-sb 1703 df-eu 2266 df-mo 2267 df-clab 2440 df-cleq 2446 df-clel 2449 df-nfc 2604 df-ne 2650 df-ral 2804 df-rex 2805 df-reu 2806 df-rab 2808 df-v 3080 df-sbc 3295 df-csb 3399 df-dif 3442 df-un 3444 df-in 3446 df-ss 3453 df-nul 3749 df-if 3903 df-pw 3973 df-sn 3989 df-pr 3991 df-op 3995 df-uni 4203 df-iun 4284 df-br 4404 df-opab 4462 df-mpt 4463 df-id 4747 df-xp 4957 df-rel 4958 df-cnv 4959 df-co 4960 df-dm 4961 df-rn 4962 df-res 4963 df-ima 4964 df-iota 5492 df-fun 5531 df-fn 5532 df-f 5533 df-f1 5534 df-fo 5535 df-f1o 5536 df-fv 5537 df-riota 6164 df-ov 6206 df-oprab 6207 df-mpt2 6208 df-1st 6690 df-2nd 6691 df-poset 15239 df-lub 15267 df-glb 15268 df-join 15269 df-meet 15270 df-lat 15339 df-ats 33275 df-atl 33306 df-cvlat 33330 df-hlat 33359 df-padd 33803

This theorem is referenced by: paddasslem14 33840

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