paddasslem12 - Mathbox for Norm Megill

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Theorem paddasslem12 35117

Description: Lemma for paddass 35124. 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 1112	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$
2		simpl21 1139	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$
3		simpl22 1140	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$
4		paddasslem.a	. . . . . 6
5		paddasslem.p	. . . . . 6
6	4, 5	paddssat 35100	. . . . 5 $/\$ $/\$
7	1, 2, 3, 6	syl3anc 1326	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$
8		simpl23 1141	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$
9	1, 7, 8	3jca 1242	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$
10	4, 5	sspadd2 35102	. . . 4 $/\$ $/\$
11	1, 3, 2, 10	syl3anc 1326	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$
12	4, 5	paddss1 35103	. . 3 $/\$ $/\$
13	9, 11, 12	sylc 65	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$
14		hllat 34650	. . . 4
15	1, 14	syl 17	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$
16		simprll 802	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$
17		simprlr 803	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$
18		simpl3l 1116	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$
19		eqid 2622	. . . 4
20		paddasslem.l	. . . 4
21	19, 4	atbase 34576	. . . . 5
22	18, 21	syl 17	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$
23	3, 16	sseldd 3604	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$
24	19, 4	atbase 34576	. . . . . 6
25	23, 24	syl 17	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$
26		simpl3r 1117	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$
27	19, 4	atbase 34576	. . . . . 6
28	26, 27	syl 17	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$
29		paddasslem.j	. . . . . 6 $.\/$
30	19, 29	latjcl 17051	. . . . 5 $/\$ $/\$ $.\/$
31	15, 25, 28, 30	syl3anc 1326	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$
32	8, 17	sseldd 3604	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$
33	19, 4	atbase 34576	. . . . . 6
34	32, 33	syl 17	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$
35	19, 29	latjcl 17051	. . . . 5 $/\$ $/\$ $.\/$
36	15, 25, 34, 35	syl3anc 1326	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$
37		simpl1r 1113	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$
38		simprrl 804	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$
39		oveq1 6657	. . . . . . 7 $.\/$ $.\/$
40	39	breq2d 4665	. . . . . 6 $.\/$ $.\/$
41	40	biimpa 501	. . . . 5 $/\$ $.\/$ $.\/$
42	37, 38, 41	syl2anc 693	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$
43	19, 20, 29	latlej1 17060	. . . . . 6 $/\$ $/\$ $.\/$
44	15, 25, 34, 43	syl3anc 1326	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$
45		simprrr 805	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$
46	19, 20, 29	latjle12 17062	. . . . . 6 $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$
47	15, 25, 28, 36, 46	syl13anc 1328	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$
48	44, 45, 47	mpbi2and 956	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$
49	19, 20, 15, 22, 31, 36, 42, 48	lattrd 17058	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$
50	20, 29, 4, 5	elpaddri 35088	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
51	15, 3, 8, 16, 17, 18, 49, 50	syl322anc 1354	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$
52	13, 51	sseldd 3604	1 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$

Colors of variables: wff setvar class

Syntax hints:

wi 4

wb 196 $/\$ wa 384 $/\$ w3a 1037

wceq 1483

wcel 1990

wss 3574 class class class wbr 4653

cfv 5888 (class class class)co 6650

cbs 15857

cple 15948

cjn 16944

clat 17045

catm 34550

chlt 34637

cpadd 35081

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-poset 16946 df-lub 16974 df-glb 16975 df-join 16976 df-meet 16977 df-lat 17046 df-ats 34554 df-atl 34585 df-cvlat 34609 df-hlat 34638 df-padd 35082

This theorem is referenced by: paddasslem14 35119

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