Theorem dihintcl 36633

Description: The intersection of closed subspaces (the range of isomorphism H) is a closed subspace. (Contributed by NM, 14-Apr-2014.)

Hypotheses

Ref	Expression
dihintcl.h	|- H = ( LHyp ` K )
dihintcl.i	|- I = ( ( DIsoH ` K ) ` W )

Assertion

Ref	Expression
dihintcl	|- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> |^| S e. ran I )

Proof of Theorem dihintcl

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

Step	Hyp	Ref	Expression
1		eqid 2622	. . . . . . . 8 |- ( Base ` K ) = ( Base ` K )
2		dihintcl.h	. . . . . . . 8 |- H = ( LHyp ` K )
3		dihintcl.i	. . . . . . . 8 |- I = ( ( DIsoH ` K ) ` W )
4	1, 2, 3	dihfn 36557	. . . . . . 7 |- ( ( K e. HL /\ W e. H ) -> I Fn ( Base ` K ) )
5	1, 2, 3	dihdm 36558	. . . . . . . 8 |- ( ( K e. HL /\ W e. H ) -> dom I = ( Base ` K ) )
6	5	fneq2d 5982	. . . . . . 7 |- ( ( K e. HL /\ W e. H ) -> ( I Fn dom I <-> I Fn ( Base ` K ) ) )
7	4, 6	mpbird 247	. . . . . 6 |- ( ( K e. HL /\ W e. H ) -> I Fn dom I )
8	7	adantr 481	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> I Fn dom I )
9		cnvimass 5485	. . . . 5 |- ( `' I " S ) C_ dom I
10		fnssres 6004	. . . . 5 |- ( ( I Fn dom I /\ ( `' I " S ) C_ dom I ) -> ( I |` ( `' I " S ) ) Fn ( `' I " S ) )
11	8, 9, 10	sylancl 694	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> ( I |` ( `' I " S ) ) Fn ( `' I " S ) )
12		fniinfv 6257	. . . 4 |- ( ( I |` ( `' I " S ) ) Fn ( `' I " S ) -> |^|_ y e. ( `' I " S ) ( ( I |` ( `' I " S ) ) ` y ) = |^| ran ( I |` ( `' I " S ) ) )
13	11, 12	syl 17	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> |^|_ y e. ( `' I " S ) ( ( I |` ( `' I " S ) ) ` y ) = |^| ran ( I |` ( `' I " S ) ) )
14		df-ima 5127	. . . . 5 |- ( I " ( `' I " S ) ) = ran ( I |` ( `' I " S ) )
15	4	adantr 481	. . . . . . 7 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> I Fn ( Base ` K ) )
16		dffn4 6121	. . . . . . 7 |- ( I Fn ( Base ` K ) <-> I : ( Base ` K ) -onto-> ran I )
17	15, 16	sylib 208	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> I : ( Base ` K ) -onto-> ran I )
18		simprl 794	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> S C_ ran I )
19		foimacnv 6154	. . . . . 6 |- ( ( I : ( Base ` K ) -onto-> ran I /\ S C_ ran I ) -> ( I " ( `' I " S ) ) = S )
20	17, 18, 19	syl2anc 693	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> ( I " ( `' I " S ) ) = S )
21	14, 20	syl5eqr 2670	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> ran ( I |` ( `' I " S ) ) = S )
22	21	inteqd 4480	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> |^| ran ( I |` ( `' I " S ) ) = |^| S )
23	13, 22	eqtrd 2656	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> |^|_ y e. ( `' I " S ) ( ( I |` ( `' I " S ) ) ` y ) = |^| S )
24		simpl 473	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> ( K e. HL /\ W e. H ) )
25	5	adantr 481	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> dom I = ( Base ` K ) )
26	9, 25	syl5sseq 3653	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> ( `' I " S ) C_ ( Base ` K ) )
27		simprr 796	. . . . . . 7 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> S =/= (/) )
28		n0 3931	. . . . . . 7 |- ( S =/= (/) <-> E. y y e. S )
29	27, 28	sylib 208	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> E. y y e. S )
30	18	sselda 3603	. . . . . . . 8 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) /\ y e. S ) -> y e. ran I )
31	25	fneq2d 5982	. . . . . . . . . . 11 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> ( I Fn dom I <-> I Fn ( Base ` K ) ) )
32	15, 31	mpbird 247	. . . . . . . . . 10 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> I Fn dom I )
33	32	adantr 481	. . . . . . . . 9 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) /\ y e. S ) -> I Fn dom I )
34		fvelrnb 6243	. . . . . . . . 9 |- ( I Fn dom I -> ( y e. ran I <-> E. x e. dom I ( I ` x ) = y ) )
35	33, 34	syl 17	. . . . . . . 8 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) /\ y e. S ) -> ( y e. ran I <-> E. x e. dom I ( I ` x ) = y ) )
36	30, 35	mpbid 222	. . . . . . 7 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) /\ y e. S ) -> E. x e. dom I ( I ` x ) = y )
37		fnfun 5988	. . . . . . . . . . . . . . 15 |- ( I Fn ( Base ` K ) -> Fun I )
38	15, 37	syl 17	. . . . . . . . . . . . . 14 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> Fun I )
39		fvimacnv 6332	. . . . . . . . . . . . . 14 |- ( ( Fun I /\ x e. dom I ) -> ( ( I ` x ) e. S <-> x e. ( `' I " S ) ) )
40	38, 39	sylan 488	. . . . . . . . . . . . 13 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) /\ x e. dom I ) -> ( ( I ` x ) e. S <-> x e. ( `' I " S ) ) )
41		ne0i 3921	. . . . . . . . . . . . 13 |- ( x e. ( `' I " S ) -> ( `' I " S ) =/= (/) )
42	40, 41	syl6bi 243	. . . . . . . . . . . 12 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) /\ x e. dom I ) -> ( ( I ` x ) e. S -> ( `' I " S ) =/= (/) ) )
43	42	ex 450	. . . . . . . . . . 11 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> ( x e. dom I -> ( ( I ` x ) e. S -> ( `' I " S ) =/= (/) ) ) )
44		eleq1 2689	. . . . . . . . . . . . 13 |- ( ( I ` x ) = y -> ( ( I ` x ) e. S <-> y e. S ) )
45	44	biimprd 238	. . . . . . . . . . . 12 |- ( ( I ` x ) = y -> ( y e. S -> ( I ` x ) e. S ) )
46	45	imim1d 82	. . . . . . . . . . 11 |- ( ( I ` x ) = y -> ( ( ( I ` x ) e. S -> ( `' I " S ) =/= (/) ) -> ( y e. S -> ( `' I " S ) =/= (/) ) ) )
47	43, 46	syl9 77	. . . . . . . . . 10 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> ( ( I ` x ) = y -> ( x e. dom I -> ( y e. S -> ( `' I " S ) =/= (/) ) ) ) )
48	47	com24 95	. . . . . . . . 9 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> ( y e. S -> ( x e. dom I -> ( ( I ` x ) = y -> ( `' I " S ) =/= (/) ) ) ) )
49	48	imp 445	. . . . . . . 8 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) /\ y e. S ) -> ( x e. dom I -> ( ( I ` x ) = y -> ( `' I " S ) =/= (/) ) ) )
50	49	rexlimdv 3030	. . . . . . 7 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) /\ y e. S ) -> ( E. x e. dom I ( I ` x ) = y -> ( `' I " S ) =/= (/) ) )
51	36, 50	mpd 15	. . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) /\ y e. S ) -> ( `' I " S ) =/= (/) )
52	29, 51	exlimddv 1863	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> ( `' I " S ) =/= (/) )
53		eqid 2622	. . . . . 6 |- ( glb ` K ) = ( glb ` K )
54	1, 53, 2, 3	dihglb 36630	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( `' I " S ) C_ ( Base ` K ) /\ ( `' I " S ) =/= (/) ) ) -> ( I ` ( ( glb ` K ) ` ( `' I " S ) ) ) = |^|_ y e. ( `' I " S ) ( I ` y ) )
55	24, 26, 52, 54	syl12anc 1324	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> ( I ` ( ( glb ` K ) ` ( `' I " S ) ) ) = |^|_ y e. ( `' I " S ) ( I ` y ) )
56		fvres 6207	. . . . 5 |- ( y e. ( `' I " S ) -> ( ( I |` ( `' I " S ) ) ` y ) = ( I ` y ) )
57	56	iineq2i 4540	. . . 4 |- |^|_ y e. ( `' I " S ) ( ( I |` ( `' I " S ) ) ` y ) = |^|_ y e. ( `' I " S ) ( I ` y )
58	55, 57	syl6eqr 2674	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> ( I ` ( ( glb ` K ) ` ( `' I " S ) ) ) = |^|_ y e. ( `' I " S ) ( ( I |` ( `' I " S ) ) ` y ) )
59		hlclat 34645	. . . . . 6 |- ( K e. HL -> K e. CLat )
60	59	ad2antrr 762	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> K e. CLat )
61	1, 53	clatglbcl 17114	. . . . 5 |- ( ( K e. CLat /\ ( `' I " S ) C_ ( Base ` K ) ) -> ( ( glb ` K ) ` ( `' I " S ) ) e. ( Base ` K ) )
62	60, 26, 61	syl2anc 693	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> ( ( glb ` K ) ` ( `' I " S ) ) e. ( Base ` K ) )
63	1, 2, 3	dihcl 36559	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( glb ` K ) ` ( `' I " S ) ) e. ( Base ` K ) ) -> ( I ` ( ( glb ` K ) ` ( `' I " S ) ) ) e. ran I )
64	62, 63	syldan 487	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> ( I ` ( ( glb ` K ) ` ( `' I " S ) ) ) e. ran I )
65	58, 64	eqeltrrd 2702	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> |^|_ y e. ( `' I " S ) ( ( I |` ( `' I " S ) ) ` y ) e. ran I )
66	23, 65	eqeltrrd 2702	1 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> |^| S e. ran I )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   E.wrex 2913   C_ wss 3574   (/)c0 3915   |^|cint 4475   |^|_ciin 4521   `'ccnv 5113   dom cdm 5114   ran crn 5115   |` cres 5116   "cima 5117   Fun wfun 5882   Fn wfn 5883   -onto->wfo 5886   ` cfv 5888   Basecbs 15857   glbcglb 16943   CLatccla 17107   HLchlt 34637   LHypclh 35270   DIsoHcdih 36517

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-lsatoms 34263  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

This theorem is referenced by:  doch2val2  36653  dochocss  36655