Theorem lecmtN 34543

Description: Ordered elements commute. (lecmi 28461 analog.) (Contributed by NM, 10-Nov-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
lecmt.b	|- B = ( Base ` K )
lecmt.l	|- .<_ = ( le ` K )
lecmt.c	|- C = ( cm ` K )

Assertion

Ref	Expression
lecmtN	|- ( ( K e. OML /\ X e. B /\ Y e. B ) -> ( X .<_ Y -> X C Y ) )

Proof of Theorem lecmtN

Step	Hyp	Ref	Expression
1		omllat 34529	. . . . 5 |- ( K e. OML -> K e. Lat )
2	1	3ad2ant1 1082	. . . 4 |- ( ( K e. OML /\ X e. B /\ Y e. B ) -> K e. Lat )
3		simp2 1062	. . . 4 |- ( ( K e. OML /\ X e. B /\ Y e. B ) -> X e. B )
4		omlop 34528	. . . . . . 7 |- ( K e. OML -> K e. OP )
5	4	3ad2ant1 1082	. . . . . 6 |- ( ( K e. OML /\ X e. B /\ Y e. B ) -> K e. OP )
6		lecmt.b	. . . . . . 7 |- B = ( Base ` K )
7		eqid 2622	. . . . . . 7 |- ( oc ` K ) = ( oc ` K )
8	6, 7	opoccl 34481	. . . . . 6 |- ( ( K e. OP /\ X e. B ) -> ( ( oc ` K ) ` X ) e. B )
9	5, 3, 8	syl2anc 693	. . . . 5 |- ( ( K e. OML /\ X e. B /\ Y e. B ) -> ( ( oc ` K ) ` X ) e. B )
10		simp3 1063	. . . . 5 |- ( ( K e. OML /\ X e. B /\ Y e. B ) -> Y e. B )
11		eqid 2622	. . . . . 6 |- ( join ` K ) = ( join ` K )
12	6, 11	latjcl 17051	. . . . 5 |- ( ( K e. Lat /\ ( ( oc ` K ) ` X ) e. B /\ Y e. B ) -> ( ( ( oc ` K ) ` X ) ( join ` K ) Y ) e. B )
13	2, 9, 10, 12	syl3anc 1326	. . . 4 |- ( ( K e. OML /\ X e. B /\ Y e. B ) -> ( ( ( oc ` K ) ` X ) ( join ` K ) Y ) e. B )
14		lecmt.l	. . . . 5 |- .<_ = ( le ` K )
15		eqid 2622	. . . . 5 |- ( meet ` K ) = ( meet ` K )
16	6, 14, 15	latmle1 17076	. . . 4 |- ( ( K e. Lat /\ X e. B /\ ( ( ( oc ` K ) ` X ) ( join ` K ) Y ) e. B ) -> ( X ( meet ` K ) ( ( ( oc ` K ) ` X ) ( join ` K ) Y ) ) .<_ X )
17	2, 3, 13, 16	syl3anc 1326	. . 3 |- ( ( K e. OML /\ X e. B /\ Y e. B ) -> ( X ( meet ` K ) ( ( ( oc ` K ) ` X ) ( join ` K ) Y ) ) .<_ X )
18	6, 15	latmcl 17052	. . . . 5 |- ( ( K e. Lat /\ X e. B /\ ( ( ( oc ` K ) ` X ) ( join ` K ) Y ) e. B ) -> ( X ( meet ` K ) ( ( ( oc ` K ) ` X ) ( join ` K ) Y ) ) e. B )
19	2, 3, 13, 18	syl3anc 1326	. . . 4 |- ( ( K e. OML /\ X e. B /\ Y e. B ) -> ( X ( meet ` K ) ( ( ( oc ` K ) ` X ) ( join ` K ) Y ) ) e. B )
20	6, 14	lattr 17056	. . . 4 |- ( ( K e. Lat /\ ( ( X ( meet ` K ) ( ( ( oc ` K ) ` X ) ( join ` K ) Y ) ) e. B /\ X e. B /\ Y e. B ) ) -> ( ( ( X ( meet ` K ) ( ( ( oc ` K ) ` X ) ( join ` K ) Y ) ) .<_ X /\ X .<_ Y ) -> ( X ( meet ` K ) ( ( ( oc ` K ) ` X ) ( join ` K ) Y ) ) .<_ Y ) )
21	2, 19, 3, 10, 20	syl13anc 1328	. . 3 |- ( ( K e. OML /\ X e. B /\ Y e. B ) -> ( ( ( X ( meet ` K ) ( ( ( oc ` K ) ` X ) ( join ` K ) Y ) ) .<_ X /\ X .<_ Y ) -> ( X ( meet ` K ) ( ( ( oc ` K ) ` X ) ( join ` K ) Y ) ) .<_ Y ) )
22	17, 21	mpand 711	. 2 |- ( ( K e. OML /\ X e. B /\ Y e. B ) -> ( X .<_ Y -> ( X ( meet ` K ) ( ( ( oc ` K ) ` X ) ( join ` K ) Y ) ) .<_ Y ) )
23		lecmt.c	. . 3 |- C = ( cm ` K )
24	6, 14, 11, 15, 7, 23	cmtbr4N 34542	. 2 |- ( ( K e. OML /\ X e. B /\ Y e. B ) -> ( X C Y <-> ( X ( meet ` K ) ( ( ( oc ` K ) ` X ) ( join ` K ) Y ) ) .<_ Y ) )
25	22, 24	sylibrd 249	1 |- ( ( K e. OML /\ X e. B /\ Y e. B ) -> ( X .<_ Y -> X C Y ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Basecbs 15857   lecple 15948   occoc 15949   joincjn 16944   meetcmee 16945   Latclat 17045   OPcops 34459   cmccmtN 34460   OMLcoml 34462

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  df-oposet 34463  df-cmtN 34464  df-ol 34465  df-oml 34466

This theorem is referenced by:  cmtidN  34544  omlmod1i2N  34547