Theorem dochss 36654

Description: Subset law for orthocomplement. (Contributed by NM, 16-Apr-2014.)

Hypotheses

Ref	Expression
dochss.h	|- H = ( LHyp ` K )
dochss.u	|- U = ( ( DVecH ` K ) ` W )
dochss.v	|- V = ( Base ` U )
dochss.o	|- ._|_ = ( ( ocH ` K ) ` W )

Assertion

Ref	Expression
dochss	|- ( ( ( K e. HL /\ W e. H ) /\ Y C_ V /\ X C_ Y ) -> ( ._|_ ` Y ) C_ ( ._|_ ` X ) )

Proof of Theorem dochss

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1l 1085	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ Y C_ V /\ X C_ Y ) -> K e. HL )
2		hlclat 34645	. . . . . 6 |- ( K e. HL -> K e. CLat )
3	1, 2	syl 17	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ Y C_ V /\ X C_ Y ) -> K e. CLat )
4		ssrab2 3687	. . . . . 6 |- { z e. ( Base ` K ) | X C_ ( ( ( DIsoH ` K ) ` W ) ` z ) } C_ ( Base ` K )
5	4	a1i 11	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ Y C_ V /\ X C_ Y ) -> { z e. ( Base ` K ) | X C_ ( ( ( DIsoH ` K ) ` W ) ` z ) } C_ ( Base ` K ) )
6		simpll3 1102	. . . . . . . 8 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ Y C_ V /\ X C_ Y ) /\ z e. ( Base ` K ) ) /\ Y C_ ( ( ( DIsoH ` K ) ` W ) ` z ) ) -> X C_ Y )
7		simpr 477	. . . . . . . 8 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ Y C_ V /\ X C_ Y ) /\ z e. ( Base ` K ) ) /\ Y C_ ( ( ( DIsoH ` K ) ` W ) ` z ) ) -> Y C_ ( ( ( DIsoH ` K ) ` W ) ` z ) )
8	6, 7	sstrd 3613	. . . . . . 7 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ Y C_ V /\ X C_ Y ) /\ z e. ( Base ` K ) ) /\ Y C_ ( ( ( DIsoH ` K ) ` W ) ` z ) ) -> X C_ ( ( ( DIsoH ` K ) ` W ) ` z ) )
9	8	ex 450	. . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ Y C_ V /\ X C_ Y ) /\ z e. ( Base ` K ) ) -> ( Y C_ ( ( ( DIsoH ` K ) ` W ) ` z ) -> X C_ ( ( ( DIsoH ` K ) ` W ) ` z ) ) )
10	9	ss2rabdv 3683	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ Y C_ V /\ X C_ Y ) -> { z e. ( Base ` K ) | Y C_ ( ( ( DIsoH ` K ) ` W ) ` z ) } C_ { z e. ( Base ` K ) | X C_ ( ( ( DIsoH ` K ) ` W ) ` z ) } )
11		eqid 2622	. . . . . 6 |- ( Base ` K ) = ( Base ` K )
12		eqid 2622	. . . . . 6 |- ( le ` K ) = ( le ` K )
13		eqid 2622	. . . . . 6 |- ( glb ` K ) = ( glb ` K )
14	11, 12, 13	clatglbss 17127	. . . . 5 |- ( ( K e. CLat /\ { z e. ( Base ` K ) | X C_ ( ( ( DIsoH ` K ) ` W ) ` z ) } C_ ( Base ` K ) /\ { z e. ( Base ` K ) | Y C_ ( ( ( DIsoH ` K ) ` W ) ` z ) } C_ { z e. ( Base ` K ) | X C_ ( ( ( DIsoH ` K ) ` W ) ` z ) } ) -> ( ( glb ` K ) ` { z e. ( Base ` K ) | X C_ ( ( ( DIsoH ` K ) ` W ) ` z ) } ) ( le ` K ) ( ( glb ` K ) ` { z e. ( Base ` K ) | Y C_ ( ( ( DIsoH ` K ) ` W ) ` z ) } ) )
15	3, 5, 10, 14	syl3anc 1326	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ Y C_ V /\ X C_ Y ) -> ( ( glb ` K ) ` { z e. ( Base ` K ) | X C_ ( ( ( DIsoH ` K ) ` W ) ` z ) } ) ( le ` K ) ( ( glb ` K ) ` { z e. ( Base ` K ) | Y C_ ( ( ( DIsoH ` K ) ` W ) ` z ) } ) )
16		hlop 34649	. . . . . 6 |- ( K e. HL -> K e. OP )
17	1, 16	syl 17	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ Y C_ V /\ X C_ Y ) -> K e. OP )
18	11, 13	clatglbcl 17114	. . . . . 6 |- ( ( K e. CLat /\ { z e. ( Base ` K ) | X C_ ( ( ( DIsoH ` K ) ` W ) ` z ) } C_ ( Base ` K ) ) -> ( ( glb ` K ) ` { z e. ( Base ` K ) | X C_ ( ( ( DIsoH ` K ) ` W ) ` z ) } ) e. ( Base ` K ) )
19	3, 4, 18	sylancl 694	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ Y C_ V /\ X C_ Y ) -> ( ( glb ` K ) ` { z e. ( Base ` K ) | X C_ ( ( ( DIsoH ` K ) ` W ) ` z ) } ) e. ( Base ` K ) )
20		ssrab2 3687	. . . . . 6 |- { z e. ( Base ` K ) | Y C_ ( ( ( DIsoH ` K ) ` W ) ` z ) } C_ ( Base ` K )
21	11, 13	clatglbcl 17114	. . . . . 6 |- ( ( K e. CLat /\ { z e. ( Base ` K ) | Y C_ ( ( ( DIsoH ` K ) ` W ) ` z ) } C_ ( Base ` K ) ) -> ( ( glb ` K ) ` { z e. ( Base ` K ) | Y C_ ( ( ( DIsoH ` K ) ` W ) ` z ) } ) e. ( Base ` K ) )
22	3, 20, 21	sylancl 694	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ Y C_ V /\ X C_ Y ) -> ( ( glb ` K ) ` { z e. ( Base ` K ) | Y C_ ( ( ( DIsoH ` K ) ` W ) ` z ) } ) e. ( Base ` K ) )
23		eqid 2622	. . . . . 6 |- ( oc ` K ) = ( oc ` K )
24	11, 12, 23	oplecon3b 34487	. . . . 5 |- ( ( K e. OP /\ ( ( glb ` K ) ` { z e. ( Base ` K ) | X C_ ( ( ( DIsoH ` K ) ` W ) ` z ) } ) e. ( Base ` K ) /\ ( ( glb ` K ) ` { z e. ( Base ` K ) | Y C_ ( ( ( DIsoH ` K ) ` W ) ` z ) } ) e. ( Base ` K ) ) -> ( ( ( glb ` K ) ` { z e. ( Base ` K ) | X C_ ( ( ( DIsoH ` K ) ` W ) ` z ) } ) ( le ` K ) ( ( glb ` K ) ` { z e. ( Base ` K ) | Y C_ ( ( ( DIsoH ` K ) ` W ) ` z ) } ) <-> ( ( oc ` K ) ` ( ( glb ` K ) ` { z e. ( Base ` K ) | Y C_ ( ( ( DIsoH ` K ) ` W ) ` z ) } ) ) ( le ` K ) ( ( oc ` K ) ` ( ( glb ` K ) ` { z e. ( Base ` K ) | X C_ ( ( ( DIsoH ` K ) ` W ) ` z ) } ) ) ) )
25	17, 19, 22, 24	syl3anc 1326	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ Y C_ V /\ X C_ Y ) -> ( ( ( glb ` K ) ` { z e. ( Base ` K ) | X C_ ( ( ( DIsoH ` K ) ` W ) ` z ) } ) ( le ` K ) ( ( glb ` K ) ` { z e. ( Base ` K ) | Y C_ ( ( ( DIsoH ` K ) ` W ) ` z ) } ) <-> ( ( oc ` K ) ` ( ( glb ` K ) ` { z e. ( Base ` K ) | Y C_ ( ( ( DIsoH ` K ) ` W ) ` z ) } ) ) ( le ` K ) ( ( oc ` K ) ` ( ( glb ` K ) ` { z e. ( Base ` K ) | X C_ ( ( ( DIsoH ` K ) ` W ) ` z ) } ) ) ) )
26	15, 25	mpbid 222	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ Y C_ V /\ X C_ Y ) -> ( ( oc ` K ) ` ( ( glb ` K ) ` { z e. ( Base ` K ) | Y C_ ( ( ( DIsoH ` K ) ` W ) ` z ) } ) ) ( le ` K ) ( ( oc ` K ) ` ( ( glb ` K ) ` { z e. ( Base ` K ) | X C_ ( ( ( DIsoH ` K ) ` W ) ` z ) } ) ) )
27		simp1 1061	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ Y C_ V /\ X C_ Y ) -> ( K e. HL /\ W e. H ) )
28	11, 23	opoccl 34481	. . . . 5 |- ( ( K e. OP /\ ( ( glb ` K ) ` { z e. ( Base ` K ) | Y C_ ( ( ( DIsoH ` K ) ` W ) ` z ) } ) e. ( Base ` K ) ) -> ( ( oc ` K ) ` ( ( glb ` K ) ` { z e. ( Base ` K ) | Y C_ ( ( ( DIsoH ` K ) ` W ) ` z ) } ) ) e. ( Base ` K ) )
29	17, 22, 28	syl2anc 693	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ Y C_ V /\ X C_ Y ) -> ( ( oc ` K ) ` ( ( glb ` K ) ` { z e. ( Base ` K ) | Y C_ ( ( ( DIsoH ` K ) ` W ) ` z ) } ) ) e. ( Base ` K ) )
30	11, 23	opoccl 34481	. . . . 5 |- ( ( K e. OP /\ ( ( glb ` K ) ` { z e. ( Base ` K ) | X C_ ( ( ( DIsoH ` K ) ` W ) ` z ) } ) e. ( Base ` K ) ) -> ( ( oc ` K ) ` ( ( glb ` K ) ` { z e. ( Base ` K ) | X C_ ( ( ( DIsoH ` K ) ` W ) ` z ) } ) ) e. ( Base ` K ) )
31	17, 19, 30	syl2anc 693	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ Y C_ V /\ X C_ Y ) -> ( ( oc ` K ) ` ( ( glb ` K ) ` { z e. ( Base ` K ) | X C_ ( ( ( DIsoH ` K ) ` W ) ` z ) } ) ) e. ( Base ` K ) )
32		dochss.h	. . . . 5 |- H = ( LHyp ` K )
33		eqid 2622	. . . . 5 |- ( ( DIsoH ` K ) ` W ) = ( ( DIsoH ` K ) ` W )
34	11, 12, 32, 33	dihord 36553	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( oc ` K ) ` ( ( glb ` K ) ` { z e. ( Base ` K ) | Y C_ ( ( ( DIsoH ` K ) ` W ) ` z ) } ) ) e. ( Base ` K ) /\ ( ( oc ` K ) ` ( ( glb ` K ) ` { z e. ( Base ` K ) | X C_ ( ( ( DIsoH ` K ) ` W ) ` z ) } ) ) e. ( Base ` K ) ) -> ( ( ( ( DIsoH ` K ) ` W ) ` ( ( oc ` K ) ` ( ( glb ` K ) ` { z e. ( Base ` K ) | Y C_ ( ( ( DIsoH ` K ) ` W ) ` z ) } ) ) ) C_ ( ( ( DIsoH ` K ) ` W ) ` ( ( oc ` K ) ` ( ( glb ` K ) ` { z e. ( Base ` K ) | X C_ ( ( ( DIsoH ` K ) ` W ) ` z ) } ) ) ) <-> ( ( oc ` K ) ` ( ( glb ` K ) ` { z e. ( Base ` K ) | Y C_ ( ( ( DIsoH ` K ) ` W ) ` z ) } ) ) ( le ` K ) ( ( oc ` K ) ` ( ( glb ` K ) ` { z e. ( Base ` K ) | X C_ ( ( ( DIsoH ` K ) ` W ) ` z ) } ) ) ) )
35	27, 29, 31, 34	syl3anc 1326	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ Y C_ V /\ X C_ Y ) -> ( ( ( ( DIsoH ` K ) ` W ) ` ( ( oc ` K ) ` ( ( glb ` K ) ` { z e. ( Base ` K ) | Y C_ ( ( ( DIsoH ` K ) ` W ) ` z ) } ) ) ) C_ ( ( ( DIsoH ` K ) ` W ) ` ( ( oc ` K ) ` ( ( glb ` K ) ` { z e. ( Base ` K ) | X C_ ( ( ( DIsoH ` K ) ` W ) ` z ) } ) ) ) <-> ( ( oc ` K ) ` ( ( glb ` K ) ` { z e. ( Base ` K ) | Y C_ ( ( ( DIsoH ` K ) ` W ) ` z ) } ) ) ( le ` K ) ( ( oc ` K ) ` ( ( glb ` K ) ` { z e. ( Base ` K ) | X C_ ( ( ( DIsoH ` K ) ` W ) ` z ) } ) ) ) )
36	26, 35	mpbird 247	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ Y C_ V /\ X C_ Y ) -> ( ( ( DIsoH ` K ) ` W ) ` ( ( oc ` K ) ` ( ( glb ` K ) ` { z e. ( Base ` K ) | Y C_ ( ( ( DIsoH ` K ) ` W ) ` z ) } ) ) ) C_ ( ( ( DIsoH ` K ) ` W ) ` ( ( oc ` K ) ` ( ( glb ` K ) ` { z e. ( Base ` K ) | X C_ ( ( ( DIsoH ` K ) ` W ) ` z ) } ) ) ) )
37		dochss.u	. . . 4 |- U = ( ( DVecH ` K ) ` W )
38		dochss.v	. . . 4 |- V = ( Base ` U )
39		dochss.o	. . . 4 |- ._|_ = ( ( ocH ` K ) ` W )
40	11, 13, 23, 32, 33, 37, 38, 39	dochval 36640	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ Y C_ V ) -> ( ._|_ ` Y ) = ( ( ( DIsoH ` K ) ` W ) ` ( ( oc ` K ) ` ( ( glb ` K ) ` { z e. ( Base ` K ) | Y C_ ( ( ( DIsoH ` K ) ` W ) ` z ) } ) ) ) )
41	40	3adant3 1081	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ Y C_ V /\ X C_ Y ) -> ( ._|_ ` Y ) = ( ( ( DIsoH ` K ) ` W ) ` ( ( oc ` K ) ` ( ( glb ` K ) ` { z e. ( Base ` K ) | Y C_ ( ( ( DIsoH ` K ) ` W ) ` z ) } ) ) ) )
42		simp3 1063	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ Y C_ V /\ X C_ Y ) -> X C_ Y )
43		simp2 1062	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ Y C_ V /\ X C_ Y ) -> Y C_ V )
44	42, 43	sstrd 3613	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ Y C_ V /\ X C_ Y ) -> X C_ V )
45	11, 13, 23, 32, 33, 37, 38, 39	dochval 36640	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ X C_ V ) -> ( ._|_ ` X ) = ( ( ( DIsoH ` K ) ` W ) ` ( ( oc ` K ) ` ( ( glb ` K ) ` { z e. ( Base ` K ) | X C_ ( ( ( DIsoH ` K ) ` W ) ` z ) } ) ) ) )
46	27, 44, 45	syl2anc 693	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ Y C_ V /\ X C_ Y ) -> ( ._|_ ` X ) = ( ( ( DIsoH ` K ) ` W ) ` ( ( oc ` K ) ` ( ( glb ` K ) ` { z e. ( Base ` K ) | X C_ ( ( ( DIsoH ` K ) ` W ) ` z ) } ) ) ) )
47	36, 41, 46	3sstr4d 3648	1 |- ( ( ( K e. HL /\ W e. H ) /\ Y C_ V /\ X C_ Y ) -> ( ._|_ ` Y ) C_ ( ._|_ ` X ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   {crab 2916   C_ wss 3574   class class class wbr 4653   ` cfv 5888   Basecbs 15857   lecple 15948   occoc 15949   glbcglb 16943   CLatccla 17107   OPcops 34459   HLchlt 34637   LHypclh 35270   DVecHcdvh 36367   DIsoHcdih 36517   ocHcoch 36636

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  ax-riotaBAD 34239

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  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-int 4476  df-iun 4522  df-iin 4523  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-1st 7168  df-2nd 7169  df-tpos 7352  df-undef 7399  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  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-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-sca 15957  df-vsca 15958  df-0g 16102  df-preset 16928  df-poset 16946  df-plt 16958  df-lub 16974  df-glb 16975  df-join 16976  df-meet 16977  df-p0 17039  df-p1 17040  df-lat 17046  df-clat 17108  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-grp 17425  df-minusg 17426  df-sbg 17427  df-subg 17591  df-cntz 17750  df-lsm 18051  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-ring 18549  df-oppr 18623  df-dvdsr 18641  df-unit 18642  df-invr 18672  df-dvr 18683  df-drng 18749  df-lmod 18865  df-lss 18933  df-lsp 18972  df-lvec 19103  df-oposet 34463  df-ol 34465  df-oml 34466  df-covers 34553  df-ats 34554  df-atl 34585  df-cvlat 34609  df-hlat 34638  df-llines 34784  df-lplanes 34785  df-lvols 34786  df-lines 34787  df-psubsp 34789  df-pmap 34790  df-padd 35082  df-lhyp 35274  df-laut 35275  df-ldil 35390  df-ltrn 35391  df-trl 35446  df-tendo 36043  df-edring 36045  df-disoa 36318  df-dvech 36368  df-dib 36428  df-dic 36462  df-dih 36518  df-doch 36637

This theorem is referenced by:  dochsscl  36657  dochord  36659  dihoml4  36666  dochocsp  36668  dochdmj1  36679  dochpolN  36779  lclkrlem2p  36811  lclkrslem1  36826  lclkrslem2  36827  lcfrvalsnN  36830  mapdsn  36930