latdisd - Metamath Proof Explorer

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Theorem latdisd 17190

Description: In a lattice, joins distribute over meets if and only if meets distribute over joins; the distributive property is self-dual. (Contributed by Stefan O'Rear, 29-Jan-2015.)

**Hypotheses**
Ref	Expression
latdisd.b
latdisd.j	$.\/$
latdisd.m	$./\$

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

Distinct variable groups:

, $.\/$ ,

, $./\$ ,

Proof of Theorem latdisd Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		latdisd.b	. . . 4
2		latdisd.j	. . . 4 $.\/$
3		latdisd.m	. . . 4 $./\$
4	1, 2, 3	latdisdlem 17189	. . 3 $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$
5		eqid 2622	. . . . 5 ODual ODual
6	5	odulat 17145	. . . 4 ODual
7	5, 1	odubas 17133	. . . . 5 ODual
8	5, 3	odujoin 17142	. . . . 5 $./\$ ODual
9	5, 2	odumeet 17140	. . . . 5 $.\/$ ODual
10	7, 8, 9	latdisdlem 17189	. . . 4 ODual $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$
11	6, 10	syl 17	. . 3 $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$
12	4, 11	impbid 202	. 2 $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$
13		oveq1 6657	. . . 4 $./\$ $.\/$ $./\$ $.\/$
14		oveq1 6657	. . . . 5 $./\$ $./\$
15		oveq1 6657	. . . . 5 $./\$ $./\$
16	14, 15	oveq12d 6668	. . . 4 $./\$ $.\/$ $./\$ $./\$ $.\/$ $./\$
17	13, 16	eqeq12d 2637	. . 3 $./\$ $.\/$ $./\$ $.\/$ $./\$ $./\$ $.\/$ $./\$ $.\/$ $./\$
18		oveq1 6657	. . . . 5 $.\/$ $.\/$
19	18	oveq2d 6666	. . . 4 $./\$ $.\/$ $./\$ $.\/$
20		oveq2 6658	. . . . 5 $./\$ $./\$
21	20	oveq1d 6665	. . . 4 $./\$ $.\/$ $./\$ $./\$ $.\/$ $./\$
22	19, 21	eqeq12d 2637	. . 3 $./\$ $.\/$ $./\$ $.\/$ $./\$ $./\$ $.\/$ $./\$ $.\/$ $./\$
23		oveq2 6658	. . . . 5 $.\/$ $.\/$
24	23	oveq2d 6666	. . . 4 $./\$ $.\/$ $./\$ $.\/$
25		oveq2 6658	. . . . 5 $./\$ $./\$
26	25	oveq2d 6666	. . . 4 $./\$ $.\/$ $./\$ $./\$ $.\/$ $./\$
27	24, 26	eqeq12d 2637	. . 3 $./\$ $.\/$ $./\$ $.\/$ $./\$ $./\$ $.\/$ $./\$ $.\/$ $./\$
28	17, 22, 27	cbvral3v 3181	. 2 $./\$ $.\/$ $./\$ $.\/$ $./\$ $./\$ $.\/$ $./\$ $.\/$ $./\$
29	12, 28	syl6bb 276	1 $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$

Colors of variables: wff setvar class

Syntax hints:

wi 4

wb 196

wceq 1483

wcel 1990

wral 2912

cfv 5888 (class class class)co 6650

cbs 15857

cjn 16944

cmee 16945

clat 17045 ODualcodu 17128

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 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 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-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 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-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 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-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 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-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-dec 11494 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ple 15961 df-preset 16928 df-poset 16946 df-lub 16974 df-glb 16975 df-join 16976 df-meet 16977 df-lat 17046 df-odu 17129

This theorem is referenced by: odudlatb 17196

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