Theorem odudlatb 17196

Description: The dual of a distributive lattice is a distributive lattice and conversely. (Contributed by Stefan O'Rear, 30-Jan-2015.)

Hypothesis

Ref	Expression
odudlat.d	|- D = (ODual ` K )

Assertion

Ref	Expression
odudlatb	|- ( K e. V -> ( K e. DLat <-> D e. DLat ) )

Proof of Theorem odudlatb

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. . . . . 6 |- ( Base ` K ) = ( Base ` K )
2		eqid 2622	. . . . . 6 |- ( join ` K ) = ( join ` K )
3		eqid 2622	. . . . . 6 |- ( meet ` K ) = ( meet ` K )
4	1, 2, 3	latdisd 17190	. . . . 5 |- ( K e. Lat -> ( A. x e. ( Base ` K ) A. y e. ( Base ` K ) A. z e. ( Base ` K ) ( x ( join ` K ) ( y ( meet ` K ) z ) ) = ( ( x ( join ` K ) y ) ( meet ` K ) ( x ( join ` K ) z ) ) <-> A. x e. ( Base ` K ) A. y e. ( Base ` K ) A. z e. ( Base ` K ) ( x ( meet ` K ) ( y ( join ` K ) z ) ) = ( ( x ( meet ` K ) y ) ( join ` K ) ( x ( meet ` K ) z ) ) ) )
5	4	bicomd 213	. . . 4 |- ( K e. Lat -> ( A. x e. ( Base ` K ) A. y e. ( Base ` K ) A. z e. ( Base ` K ) ( x ( meet ` K ) ( y ( join ` K ) z ) ) = ( ( x ( meet ` K ) y ) ( join ` K ) ( x ( meet ` K ) z ) ) <-> A. x e. ( Base ` K ) A. y e. ( Base ` K ) A. z e. ( Base ` K ) ( x ( join ` K ) ( y ( meet ` K ) z ) ) = ( ( x ( join ` K ) y ) ( meet ` K ) ( x ( join ` K ) z ) ) ) )
6	5	pm5.32i 669	. . 3 |- ( ( K e. Lat /\ A. x e. ( Base ` K ) A. y e. ( Base ` K ) A. z e. ( Base ` K ) ( x ( meet ` K ) ( y ( join ` K ) z ) ) = ( ( x ( meet ` K ) y ) ( join ` K ) ( x ( meet ` K ) z ) ) ) <-> ( K e. Lat /\ A. x e. ( Base ` K ) A. y e. ( Base ` K ) A. z e. ( Base ` K ) ( x ( join ` K ) ( y ( meet ` K ) z ) ) = ( ( x ( join ` K ) y ) ( meet ` K ) ( x ( join ` K ) z ) ) ) )
7		odudlat.d	. . . . 5 |- D = (ODual ` K )
8	7	odulatb 17143	. . . 4 |- ( K e. V -> ( K e. Lat <-> D e. Lat ) )
9	8	anbi1d 741	. . 3 |- ( K e. V -> ( ( K e. Lat /\ A. x e. ( Base ` K ) A. y e. ( Base ` K ) A. z e. ( Base ` K ) ( x ( join ` K ) ( y ( meet ` K ) z ) ) = ( ( x ( join ` K ) y ) ( meet ` K ) ( x ( join ` K ) z ) ) ) <-> ( D e. Lat /\ A. x e. ( Base ` K ) A. y e. ( Base ` K ) A. z e. ( Base ` K ) ( x ( join ` K ) ( y ( meet ` K ) z ) ) = ( ( x ( join ` K ) y ) ( meet ` K ) ( x ( join ` K ) z ) ) ) ) )
10	6, 9	syl5bb 272	. 2 |- ( K e. V -> ( ( K e. Lat /\ A. x e. ( Base ` K ) A. y e. ( Base ` K ) A. z e. ( Base ` K ) ( x ( meet ` K ) ( y ( join ` K ) z ) ) = ( ( x ( meet ` K ) y ) ( join ` K ) ( x ( meet ` K ) z ) ) ) <-> ( D e. Lat /\ A. x e. ( Base ` K ) A. y e. ( Base ` K ) A. z e. ( Base ` K ) ( x ( join ` K ) ( y ( meet ` K ) z ) ) = ( ( x ( join ` K ) y ) ( meet ` K ) ( x ( join ` K ) z ) ) ) ) )
11	1, 2, 3	isdlat 17193	. 2 |- ( K e. DLat <-> ( K e. Lat /\ A. x e. ( Base ` K ) A. y e. ( Base ` K ) A. z e. ( Base ` K ) ( x ( meet ` K ) ( y ( join ` K ) z ) ) = ( ( x ( meet ` K ) y ) ( join ` K ) ( x ( meet ` K ) z ) ) ) )
12	7, 1	odubas 17133	. . 3 |- ( Base ` K ) = ( Base ` D )
13	7, 3	odujoin 17142	. . 3 |- ( meet ` K ) = ( join ` D )
14	7, 2	odumeet 17140	. . 3 |- ( join ` K ) = ( meet ` D )
15	12, 13, 14	isdlat 17193	. 2 |- ( D e. DLat <-> ( D e. Lat /\ A. x e. ( Base ` K ) A. y e. ( Base ` K ) A. z e. ( Base ` K ) ( x ( join ` K ) ( y ( meet ` K ) z ) ) = ( ( x ( join ` K ) y ) ( meet ` K ) ( x ( join ` K ) z ) ) ) )
16	10, 11, 15	3bitr4g 303	1 |- ( K e. V -> ( K e. DLat <-> D e. DLat ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   ` cfv 5888  (class class class)co 6650   Basecbs 15857   joincjn 16944   meetcmee 16945   Latclat 17045  ODualcodu 17128  DLatcdlat 17191

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  df-dlat 17192

This theorem is referenced by:  dlatjmdi  17197