Theorem latmidm 17086

Description: Lattice join is idempotent. (inidm 3822 analog.) (Contributed by NM, 8-Nov-2011.)

Hypotheses

Ref	Expression
latmidm.b	|- B = ( Base ` K )
latmidm.m	|- ./\ = ( meet ` K )

Assertion

Ref	Expression
latmidm	|- ( ( K e. Lat /\ X e. B ) -> ( X ./\ X ) = X )

Proof of Theorem latmidm

Step	Hyp	Ref	Expression
1		latmidm.b	. 2 |- B = ( Base ` K )
2		eqid 2622	. 2 |- ( le ` K ) = ( le ` K )
3		simpl 473	. 2 |- ( ( K e. Lat /\ X e. B ) -> K e. Lat )
4		latmidm.m	. . . 4 |- ./\ = ( meet ` K )
5	1, 4	latmcl 17052	. . 3 |- ( ( K e. Lat /\ X e. B /\ X e. B ) -> ( X ./\ X ) e. B )
6	5	3anidm23 1385	. 2 |- ( ( K e. Lat /\ X e. B ) -> ( X ./\ X ) e. B )
7		simpr 477	. 2 |- ( ( K e. Lat /\ X e. B ) -> X e. B )
8	1, 2, 4	latmle1 17076	. . 3 |- ( ( K e. Lat /\ X e. B /\ X e. B ) -> ( X ./\ X ) ( le ` K ) X )
9	8	3anidm23 1385	. 2 |- ( ( K e. Lat /\ X e. B ) -> ( X ./\ X ) ( le ` K ) X )
10	1, 2	latref 17053	. . 3 |- ( ( K e. Lat /\ X e. B ) -> X ( le ` K ) X )
11	1, 2, 4	latlem12 17078	. . . 4 |- ( ( K e. Lat /\ ( X e. B /\ X e. B /\ X e. B ) ) -> ( ( X ( le ` K ) X /\ X ( le ` K ) X ) <-> X ( le ` K ) ( X ./\ X ) ) )
12	3, 7, 7, 7, 11	syl13anc 1328	. . 3 |- ( ( K e. Lat /\ X e. B ) -> ( ( X ( le ` K ) X /\ X ( le ` K ) X ) <-> X ( le ` K ) ( X ./\ X ) ) )
13	10, 10, 12	mpbi2and 956	. 2 |- ( ( K e. Lat /\ X e. B ) -> X ( le ` K ) ( X ./\ X ) )
14	1, 2, 3, 6, 7, 9, 13	latasymd 17057	1 |- ( ( K e. Lat /\ X e. B ) -> ( X ./\ X ) = X )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Basecbs 15857   lecple 15948   meetcmee 16945   Latclat 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-preset 16928  df-poset 16946  df-lub 16974  df-glb 16975  df-join 16976  df-meet 16977  df-lat 17046

This theorem is referenced by:  latmmdiN  34521  latmmdir  34522  2llnm3N  34855