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Theorem mod1ile 17105

Description: The weak direction of the modular law (e.g., pmod1i 35134, atmod1i1 35143) that holds in any lattice. (Contributed by NM, 11-May-2012.)

**Hypotheses**
Ref	Expression
modle.b
modle.l
modle.j	$.\/$
modle.m	$./\$

**Assertion**
Ref	Expression
mod1ile	$/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$

**Proof of Theorem mod1ile**
Step	Hyp	Ref	Expression
1		simpll 790	. . . . 5 $/\$ $/\$ $/\$ $/\$
2		simplr1 1103	. . . . 5 $/\$ $/\$ $/\$ $/\$
3		simplr2 1104	. . . . 5 $/\$ $/\$ $/\$ $/\$
4		modle.b	. . . . . 6
5		modle.l	. . . . . 6
6		modle.j	. . . . . 6 $.\/$
7	4, 5, 6	latlej1 17060	. . . . 5 $/\$ $/\$ $.\/$
8	1, 2, 3, 7	syl3anc 1326	. . . 4 $/\$ $/\$ $/\$ $/\$ $.\/$
9		simpr 477	. . . 4 $/\$ $/\$ $/\$ $/\$
10	4, 6	latjcl 17051	. . . . . 6 $/\$ $/\$ $.\/$
11	1, 2, 3, 10	syl3anc 1326	. . . . 5 $/\$ $/\$ $/\$ $/\$ $.\/$
12		simplr3 1105	. . . . 5 $/\$ $/\$ $/\$ $/\$
13		modle.m	. . . . . 6 $./\$
14	4, 5, 13	latlem12 17078	. . . . 5 $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $./\$
15	1, 2, 11, 12, 14	syl13anc 1328	. . . 4 $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $./\$
16	8, 9, 15	mpbi2and 956	. . 3 $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$
17	4, 5, 6, 13	latmlej12 17091	. . . . 5 $/\$ $/\$ $/\$ $./\$ $.\/$
18	1, 3, 12, 2, 17	syl13anc 1328	. . . 4 $/\$ $/\$ $/\$ $/\$ $./\$ $.\/$
19	4, 5, 13	latmle2 17077	. . . . 5 $/\$ $/\$ $./\$
20	1, 3, 12, 19	syl3anc 1326	. . . 4 $/\$ $/\$ $/\$ $/\$ $./\$
21	4, 13	latmcl 17052	. . . . . 6 $/\$ $/\$ $./\$
22	1, 3, 12, 21	syl3anc 1326	. . . . 5 $/\$ $/\$ $/\$ $/\$ $./\$
23	4, 5, 13	latlem12 17078	. . . . 5 $/\$ $./\$ $/\$ $.\/$ $/\$ $./\$ $.\/$ $/\$ $./\$ $./\$ $.\/$ $./\$
24	1, 22, 11, 12, 23	syl13anc 1328	. . . 4 $/\$ $/\$ $/\$ $/\$ $./\$ $.\/$ $/\$ $./\$ $./\$ $.\/$ $./\$
25	18, 20, 24	mpbi2and 956	. . 3 $/\$ $/\$ $/\$ $/\$ $./\$ $.\/$ $./\$
26	4, 13	latmcl 17052	. . . . 5 $/\$ $.\/$ $/\$ $.\/$ $./\$
27	1, 11, 12, 26	syl3anc 1326	. . . 4 $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$
28	4, 5, 6	latjle12 17062	. . . 4 $/\$ $/\$ $./\$ $/\$ $.\/$ $./\$ $.\/$ $./\$ $/\$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$
29	1, 2, 22, 27, 28	syl13anc 1328	. . 3 $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$
30	16, 25, 29	mpbi2and 956	. 2 $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$
31	30	ex 450	1 $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$

Colors of variables: wff setvar class

Syntax hints:

wi 4

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

wceq 1483

wcel 1990 class class class wbr 4653

cfv 5888 (class class class)co 6650

cbs 15857

cple 15948

cjn 16944

cmee 16945

clat 17045

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-poset 16946 df-lub 16974 df-glb 16975 df-join 16976 df-meet 16977 df-lat 17046

This theorem is referenced by: mod2ile 17106 hlmod1i 35142

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