paddasslem12 - Mathbox for Norm Megill

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Theorem paddasslem12 33442

Description: Lemma for paddass 33449. 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 1065	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$
2		simpl21 1092	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$
3		simpl22 1093	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$
4		paddasslem.a	. . . . . 6
5		paddasslem.p	. . . . . 6
6	4, 5	paddssat 33425	. . . . 5 $/\$ $/\$
7	1, 2, 3, 6	syl3anc 1276	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$
8		simpl23 1094	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$
9	1, 7, 8	3jca 1194	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$
10	4, 5	sspadd2 33427	. . . 4 $/\$ $/\$
11	1, 3, 2, 10	syl3anc 1276	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$
12	4, 5	paddss1 33428	. . 3 $/\$ $/\$
13	9, 11, 12	sylc 62	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$
14		hllat 32975	. . . 4
15	1, 14	syl 17	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$
16		simprll 777	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$
17		simprlr 778	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$
18		simpl3l 1069	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$
19		eqid 2462	. . . 4
20		paddasslem.l	. . . 4
21	19, 4	atbase 32901	. . . . 5
22	18, 21	syl 17	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$
23	3, 16	sseldd 3445	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$
24	19, 4	atbase 32901	. . . . . 6
25	23, 24	syl 17	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$
26		simpl3r 1070	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$
27	19, 4	atbase 32901	. . . . . 6
28	26, 27	syl 17	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$
29		paddasslem.j	. . . . . 6 $.\/$
30	19, 29	latjcl 16352	. . . . 5 $/\$ $/\$ $.\/$
31	15, 25, 28, 30	syl3anc 1276	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$
32	8, 17	sseldd 3445	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$
33	19, 4	atbase 32901	. . . . . 6
34	32, 33	syl 17	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$
35	19, 29	latjcl 16352	. . . . 5 $/\$ $/\$ $.\/$
36	15, 25, 34, 35	syl3anc 1276	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$
37		simpl1r 1066	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$
38		simprrl 779	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$
39		oveq1 6327	. . . . . . 7 $.\/$ $.\/$
40	39	breq2d 4430	. . . . . 6 $.\/$ $.\/$
41	40	biimpa 491	. . . . 5 $/\$ $.\/$ $.\/$
42	37, 38, 41	syl2anc 671	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$
43	19, 20, 29	latlej1 16361	. . . . . 6 $/\$ $/\$ $.\/$
44	15, 25, 34, 43	syl3anc 1276	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$
45		simprrr 780	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$
46	19, 20, 29	latjle12 16363	. . . . . 6 $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$
47	15, 25, 28, 36, 46	syl13anc 1278	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$
48	44, 45, 47	mpbi2and 937	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$
49	19, 20, 15, 22, 31, 36, 42, 48	lattrd 16359	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$
50	20, 29, 4, 5	elpaddri 33413	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
51	15, 3, 8, 16, 17, 18, 49, 50	syl322anc 1304	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$
52	13, 51	sseldd 3445	1 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$

Colors of variables: wff setvar class

Syntax hints:

wi 4

wb 189 $/\$ wa 375 $/\$ w3a 991

wceq 1455

wcel 1898

wss 3416 class class class wbr 4418

cfv 5605 (class class class)co 6320

cbs 15176

cple 15252

cjn 16244

clat 16346

catm 32875

chlt 32962

cpadd 33406

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1680 ax-4 1693 ax-5 1769 ax-6 1816 ax-7 1862 ax-8 1900 ax-9 1907 ax-10 1926 ax-11 1931 ax-12 1944 ax-13 2102 ax-ext 2442 ax-rep 4531 ax-sep 4541 ax-nul 4550 ax-pow 4598 ax-pr 4656 ax-un 6615

This theorem depends on definitions: df-bi 190 df-or 376 df-an 377 df-3an 993 df-tru 1458 df-ex 1675 df-nf 1679 df-sb 1809 df-eu 2314 df-mo 2315 df-clab 2449 df-cleq 2455 df-clel 2458 df-nfc 2592 df-ne 2635 df-ral 2754 df-rex 2755 df-reu 2756 df-rab 2758 df-v 3059 df-sbc 3280 df-csb 3376 df-dif 3419 df-un 3421 df-in 3423 df-ss 3430 df-nul 3744 df-if 3894 df-pw 3965 df-sn 3981 df-pr 3983 df-op 3987 df-uni 4213 df-iun 4294 df-br 4419 df-opab 4478 df-mpt 4479 df-id 4771 df-xp 4862 df-rel 4863 df-cnv 4864 df-co 4865 df-dm 4866 df-rn 4867 df-res 4868 df-ima 4869 df-iota 5569 df-fun 5607 df-fn 5608 df-f 5609 df-f1 5610 df-fo 5611 df-f1o 5612 df-fv 5613 df-riota 6282 df-ov 6323 df-oprab 6324 df-mpt2 6325 df-1st 6825 df-2nd 6826 df-poset 16246 df-lub 16275 df-glb 16276 df-join 16277 df-meet 16278 df-lat 16347 df-ats 32879 df-atl 32910 df-cvlat 32934 df-hlat 32963 df-padd 33407

This theorem is referenced by: paddasslem14 33444

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