Theorem cmt2N 34537

Description: Commutation with orthocomplement. Theorem 2.3(i) of [Beran] p. 39. (cmcm2i 28452 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
cmt2.b	|- B = ( Base ` K )
cmt2.o	|- ._|_ = ( oc ` K )
cmt2.c	|- C = ( cm ` K )

Assertion

Ref	Expression
cmt2N	|- ( ( K e. OML /\ X e. B /\ Y e. B ) -> ( X C Y <-> X C ( ._|_ ` Y ) ) )

Proof of Theorem cmt2N

Step	Hyp	Ref	Expression
1		omllat 34529	. . . . . 6 |- ( K e. OML -> K e. Lat )
2	1	3ad2ant1 1082	. . . . 5 |- ( ( K e. OML /\ X e. B /\ Y e. B ) -> K e. Lat )
3		cmt2.b	. . . . . . 7 |- B = ( Base ` K )
4		eqid 2622	. . . . . . 7 |- ( meet ` K ) = ( meet ` K )
5	3, 4	latmcl 17052	. . . . . 6 |- ( ( K e. Lat /\ X e. B /\ Y e. B ) -> ( X ( meet ` K ) Y ) e. B )
6	1, 5	syl3an1 1359	. . . . 5 |- ( ( K e. OML /\ X e. B /\ Y e. B ) -> ( X ( meet ` K ) Y ) e. B )
7		simp2 1062	. . . . . 6 |- ( ( K e. OML /\ X e. B /\ Y e. B ) -> X e. B )
8		omlop 34528	. . . . . . . 8 |- ( K e. OML -> K e. OP )
9	8	3ad2ant1 1082	. . . . . . 7 |- ( ( K e. OML /\ X e. B /\ Y e. B ) -> K e. OP )
10		simp3 1063	. . . . . . 7 |- ( ( K e. OML /\ X e. B /\ Y e. B ) -> Y e. B )
11		cmt2.o	. . . . . . . 8 |- ._|_ = ( oc ` K )
12	3, 11	opoccl 34481	. . . . . . 7 |- ( ( K e. OP /\ Y e. B ) -> ( ._|_ ` Y ) e. B )
13	9, 10, 12	syl2anc 693	. . . . . 6 |- ( ( K e. OML /\ X e. B /\ Y e. B ) -> ( ._|_ ` Y ) e. B )
14	3, 4	latmcl 17052	. . . . . 6 |- ( ( K e. Lat /\ X e. B /\ ( ._|_ ` Y ) e. B ) -> ( X ( meet ` K ) ( ._|_ ` Y ) ) e. B )
15	2, 7, 13, 14	syl3anc 1326	. . . . 5 |- ( ( K e. OML /\ X e. B /\ Y e. B ) -> ( X ( meet ` K ) ( ._|_ ` Y ) ) e. B )
16		eqid 2622	. . . . . 6 |- ( join ` K ) = ( join ` K )
17	3, 16	latjcom 17059	. . . . 5 |- ( ( K e. Lat /\ ( X ( meet ` K ) Y ) e. B /\ ( X ( meet ` K ) ( ._|_ ` Y ) ) e. B ) -> ( ( X ( meet ` K ) Y ) ( join ` K ) ( X ( meet ` K ) ( ._|_ ` Y ) ) ) = ( ( X ( meet ` K ) ( ._|_ ` Y ) ) ( join ` K ) ( X ( meet ` K ) Y ) ) )
18	2, 6, 15, 17	syl3anc 1326	. . . 4 |- ( ( K e. OML /\ X e. B /\ Y e. B ) -> ( ( X ( meet ` K ) Y ) ( join ` K ) ( X ( meet ` K ) ( ._|_ ` Y ) ) ) = ( ( X ( meet ` K ) ( ._|_ ` Y ) ) ( join ` K ) ( X ( meet ` K ) Y ) ) )
19	3, 11	opococ 34482	. . . . . . 7 |- ( ( K e. OP /\ Y e. B ) -> ( ._|_ ` ( ._|_ ` Y ) ) = Y )
20	9, 10, 19	syl2anc 693	. . . . . 6 |- ( ( K e. OML /\ X e. B /\ Y e. B ) -> ( ._|_ ` ( ._|_ ` Y ) ) = Y )
21	20	oveq2d 6666	. . . . 5 |- ( ( K e. OML /\ X e. B /\ Y e. B ) -> ( X ( meet ` K ) ( ._|_ ` ( ._|_ ` Y ) ) ) = ( X ( meet ` K ) Y ) )
22	21	oveq2d 6666	. . . 4 |- ( ( K e. OML /\ X e. B /\ Y e. B ) -> ( ( X ( meet ` K ) ( ._|_ ` Y ) ) ( join ` K ) ( X ( meet ` K ) ( ._|_ ` ( ._|_ ` Y ) ) ) ) = ( ( X ( meet ` K ) ( ._|_ ` Y ) ) ( join ` K ) ( X ( meet ` K ) Y ) ) )
23	18, 22	eqtr4d 2659	. . 3 |- ( ( K e. OML /\ X e. B /\ Y e. B ) -> ( ( X ( meet ` K ) Y ) ( join ` K ) ( X ( meet ` K ) ( ._|_ ` Y ) ) ) = ( ( X ( meet ` K ) ( ._|_ ` Y ) ) ( join ` K ) ( X ( meet ` K ) ( ._|_ ` ( ._|_ ` Y ) ) ) ) )
24	23	eqeq2d 2632	. 2 |- ( ( K e. OML /\ X e. B /\ Y e. B ) -> ( X = ( ( X ( meet ` K ) Y ) ( join ` K ) ( X ( meet ` K ) ( ._|_ ` Y ) ) ) <-> X = ( ( X ( meet ` K ) ( ._|_ ` Y ) ) ( join ` K ) ( X ( meet ` K ) ( ._|_ ` ( ._|_ ` Y ) ) ) ) ) )
25		cmt2.c	. . 3 |- C = ( cm ` K )
26	3, 16, 4, 11, 25	cmtvalN 34498	. 2 |- ( ( K e. OML /\ X e. B /\ Y e. B ) -> ( X C Y <-> X = ( ( X ( meet ` K ) Y ) ( join ` K ) ( X ( meet ` K ) ( ._|_ ` Y ) ) ) ) )
27	3, 16, 4, 11, 25	cmtvalN 34498	. . 3 |- ( ( K e. OML /\ X e. B /\ ( ._|_ ` Y ) e. B ) -> ( X C ( ._|_ ` Y ) <-> X = ( ( X ( meet ` K ) ( ._|_ ` Y ) ) ( join ` K ) ( X ( meet ` K ) ( ._|_ ` ( ._|_ ` Y ) ) ) ) ) )
28	13, 27	syld3an3 1371	. 2 |- ( ( K e. OML /\ X e. B /\ Y e. B ) -> ( X C ( ._|_ ` Y ) <-> X = ( ( X ( meet ` K ) ( ._|_ ` Y ) ) ( join ` K ) ( X ( meet ` K ) ( ._|_ ` ( ._|_ ` Y ) ) ) ) ) )
29	24, 26, 28	3bitr4d 300	1 |- ( ( K e. OML /\ X e. B /\ Y e. B ) -> ( X C Y <-> X C ( ._|_ ` Y ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ w3a 1037   = wceq 1483   e. wcel 1990   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Basecbs 15857   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-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:  cmt3N  34538  cmt4N  34539  omlfh1N  34545