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Theorem mod1ile 15374

Description: The weak direction of the modular law (e.g. pmod1i 33795, atmod1i1 33804) 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 753	. . . . 5 $/\$ $/\$ $/\$ $/\$
2		simplr1 1030	. . . . 5 $/\$ $/\$ $/\$ $/\$
3		simplr2 1031	. . . . 5 $/\$ $/\$ $/\$ $/\$
4		modle.b	. . . . . 6
5		modle.l	. . . . . 6
6		modle.j	. . . . . 6 $.\/$
7	4, 5, 6	latlej1 15329	. . . . 5 $/\$ $/\$ $.\/$
8	1, 2, 3, 7	syl3anc 1219	. . . 4 $/\$ $/\$ $/\$ $/\$ $.\/$
9		simpr 461	. . . 4 $/\$ $/\$ $/\$ $/\$
10	4, 6	latjcl 15320	. . . . . 6 $/\$ $/\$ $.\/$
11	1, 2, 3, 10	syl3anc 1219	. . . . 5 $/\$ $/\$ $/\$ $/\$ $.\/$
12		simplr3 1032	. . . . 5 $/\$ $/\$ $/\$ $/\$
13		modle.m	. . . . . 6 $./\$
14	4, 5, 13	latlem12 15347	. . . . 5 $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $./\$
15	1, 2, 11, 12, 14	syl13anc 1221	. . . 4 $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $./\$
16	8, 9, 15	mpbi2and 912	. . 3 $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$
17	4, 5, 6, 13	latmlej12 15360	. . . . 5 $/\$ $/\$ $/\$ $./\$ $.\/$
18	1, 3, 12, 2, 17	syl13anc 1221	. . . 4 $/\$ $/\$ $/\$ $/\$ $./\$ $.\/$
19	4, 5, 13	latmle2 15346	. . . . 5 $/\$ $/\$ $./\$
20	1, 3, 12, 19	syl3anc 1219	. . . 4 $/\$ $/\$ $/\$ $/\$ $./\$
21	4, 13	latmcl 15321	. . . . . 6 $/\$ $/\$ $./\$
22	1, 3, 12, 21	syl3anc 1219	. . . . 5 $/\$ $/\$ $/\$ $/\$ $./\$
23	4, 5, 13	latlem12 15347	. . . . 5 $/\$ $./\$ $/\$ $.\/$ $/\$ $./\$ $.\/$ $/\$ $./\$ $./\$ $.\/$ $./\$
24	1, 22, 11, 12, 23	syl13anc 1221	. . . 4 $/\$ $/\$ $/\$ $/\$ $./\$ $.\/$ $/\$ $./\$ $./\$ $.\/$ $./\$
25	18, 20, 24	mpbi2and 912	. . 3 $/\$ $/\$ $/\$ $/\$ $./\$ $.\/$ $./\$
26	4, 13	latmcl 15321	. . . . 5 $/\$ $.\/$ $/\$ $.\/$ $./\$
27	1, 11, 12, 26	syl3anc 1219	. . . 4 $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$
28	4, 5, 6	latjle12 15331	. . . 4 $/\$ $/\$ $./\$ $/\$ $.\/$ $./\$ $.\/$ $./\$ $/\$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$
29	1, 2, 22, 27, 28	syl13anc 1221	. . 3 $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$
30	16, 25, 29	mpbi2and 912	. 2 $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$
31	30	ex 434	1 $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$

Colors of variables: wff setvar class

Syntax hints:

wi 4

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

wceq 1370

wcel 1758 class class class wbr 4387

cfv 5513 (class class class)co 6187

cbs 14273

cple 14344

cjn 15213

cmee 15214

clat 15314

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 1952 ax-ext 2430 ax-rep 4498 ax-sep 4508 ax-nul 4516 ax-pow 4565 ax-pr 4626 ax-un 6469

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 2264 df-mo 2265 df-clab 2437 df-cleq 2443 df-clel 2446 df-nfc 2599 df-ne 2644 df-ral 2798 df-rex 2799 df-reu 2800 df-rab 2802 df-v 3067 df-sbc 3282 df-csb 3384 df-dif 3426 df-un 3428 df-in 3430 df-ss 3437 df-nul 3733 df-if 3887 df-pw 3957 df-sn 3973 df-pr 3975 df-op 3979 df-uni 4187 df-iun 4268 df-br 4388 df-opab 4446 df-mpt 4447 df-id 4731 df-xp 4941 df-rel 4942 df-cnv 4943 df-co 4944 df-dm 4945 df-rn 4946 df-res 4947 df-ima 4948 df-iota 5476 df-fun 5515 df-fn 5516 df-f 5517 df-f1 5518 df-fo 5519 df-f1o 5520 df-fv 5521 df-riota 6148 df-ov 6190 df-oprab 6191 df-poset 15215 df-lub 15243 df-glb 15244 df-join 15245 df-meet 15246 df-lat 15315

This theorem is referenced by: mod2ile 15375 hlmod1i 33803

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