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Theorem dihintcl 31831
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.
StepHypRef Expression
1 eqid 2408 . . . . . . . 8  |-  ( Base `  K )  =  (
Base `  K )
2 dihintcl.h . . . . . . . 8  |-  H  =  ( LHyp `  K
)
3 dihintcl.i . . . . . . . 8  |-  I  =  ( ( DIsoH `  K
) `  W )
41, 2, 3dihfn 31755 . . . . . . 7  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  I  Fn  ( Base `  K ) )
51, 2, 3dihdm 31756 . . . . . . . 8  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  dom  I  =  (
Base `  K )
)
65fneq2d 5500 . . . . . . 7  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( I  Fn  dom  I 
<->  I  Fn  ( Base `  K ) ) )
74, 6mpbird 224 . . . . . 6  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  I  Fn  dom  I
)
87adantr 452 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  I  Fn  dom  I )
9 cnvimass 5187 . . . . 5  |-  ( `' I " S ) 
C_  dom  I
10 fnssres 5521 . . . . 5  |-  ( ( I  Fn  dom  I  /\  ( `' I " S )  C_  dom  I )  ->  (
I  |`  ( `' I " S ) )  Fn  ( `' I " S ) )
118, 9, 10sylancl 644 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  (
I  |`  ( `' I " S ) )  Fn  ( `' I " S ) )
12 fniinfv 5748 . . . 4  |-  ( ( I  |`  ( `' I " S ) )  Fn  ( `' I " S )  ->  |^|_ y  e.  ( `' I " S ) ( ( I  |`  ( `' I " S ) ) `
 y )  = 
|^| ran  ( I  |`  ( `' I " S ) ) )
1311, 12syl 16 . . 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 4854 . . . . 5  |-  ( I
" ( `' I " S ) )  =  ran  ( I  |`  ( `' I " S ) )
154adantr 452 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  I  Fn  ( Base `  K
) )
16 dffn4 5622 . . . . . . 7  |-  ( I  Fn  ( Base `  K
)  <->  I : (
Base `  K ) -onto-> ran  I )
1715, 16sylib 189 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  I : ( Base `  K
) -onto-> ran  I )
18 simprl 733 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  S  C_ 
ran  I )
19 foimacnv 5655 . . . . . 6  |-  ( ( I : ( Base `  K ) -onto-> ran  I  /\  S  C_  ran  I
)  ->  ( I " ( `' I " S ) )  =  S )
2017, 18, 19syl2anc 643 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  (
I " ( `' I " S ) )  =  S )
2114, 20syl5eqr 2454 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  ran  ( I  |`  ( `' I " S ) )  =  S )
2221inteqd 4019 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  |^| ran  ( I  |`  ( `' I " S ) )  =  |^| S
)
2313, 22eqtrd 2440 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  |^|_ y  e.  ( `' I " S ) ( ( I  |`  ( `' I " S ) ) `
 y )  = 
|^| S )
24 simpl 444 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
255adantr 452 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  dom  I  =  ( Base `  K ) )
269, 25syl5sseq 3360 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  ( `' I " S ) 
C_  ( Base `  K
) )
27 simprr 734 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  S  =/=  (/) )
28 n0 3601 . . . . . . 7  |-  ( S  =/=  (/)  <->  E. y  y  e.  S )
2927, 28sylib 189 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  E. y 
y  e.  S )
3018sselda 3312 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  /\  y  e.  S )  ->  y  e.  ran  I
)
3125fneq2d 5500 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  (
I  Fn  dom  I  <->  I  Fn  ( Base `  K
) ) )
3215, 31mpbird 224 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  I  Fn  dom  I )
3332adantr 452 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  /\  y  e.  S )  ->  I  Fn  dom  I
)
34 fvelrnb 5737 . . . . . . . . 9  |-  ( I  Fn  dom  I  -> 
( y  e.  ran  I 
<->  E. x  e.  dom  I ( I `  x )  =  y ) )
3533, 34syl 16 . . . . . . . 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 ) )
3630, 35mpbid 202 . . . . . . 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 5505 . . . . . . . . . . . . . . 15  |-  ( I  Fn  ( Base `  K
)  ->  Fun  I )
3815, 37syl 16 . . . . . . . . . . . . . 14  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  Fun  I )
39 fvimacnv 5808 . . . . . . . . . . . . . 14  |-  ( ( Fun  I  /\  x  e.  dom  I )  -> 
( ( I `  x )  e.  S  <->  x  e.  ( `' I " S ) ) )
4038, 39sylan 458 . . . . . . . . . . . . 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 3598 . . . . . . . . . . . . 13  |-  ( x  e.  ( `' I " S )  ->  ( `' I " S )  =/=  (/) )
4240, 41syl6bi 220 . . . . . . . . . . . 12  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  /\  x  e.  dom  I )  ->  ( ( I `
 x )  e.  S  ->  ( `' I " S )  =/=  (/) ) )
4342ex 424 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  (
x  e.  dom  I  ->  ( ( I `  x )  e.  S  ->  ( `' I " S )  =/=  (/) ) ) )
44 eleq1 2468 . . . . . . . . . . . . 13  |-  ( ( I `  x )  =  y  ->  (
( I `  x
)  e.  S  <->  y  e.  S ) )
4544biimprd 215 . . . . . . . . . . . 12  |-  ( ( I `  x )  =  y  ->  (
y  e.  S  -> 
( I `  x
)  e.  S ) )
4645imim1d 71 . . . . . . . . . . 11  |-  ( ( I `  x )  =  y  ->  (
( ( I `  x )  e.  S  ->  ( `' I " S )  =/=  (/) )  -> 
( y  e.  S  ->  ( `' I " S )  =/=  (/) ) ) )
4743, 46syl9 68 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  (
( I `  x
)  =  y  -> 
( x  e.  dom  I  ->  ( y  e.  S  ->  ( `' I " S )  =/=  (/) ) ) ) )
4847com24 83 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  (
y  e.  S  -> 
( x  e.  dom  I  ->  ( ( I `
 x )  =  y  ->  ( `' I " S )  =/=  (/) ) ) ) )
4948imp 419 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  /\  y  e.  S )  ->  ( x  e.  dom  I  ->  ( ( I `
 x )  =  y  ->  ( `' I " S )  =/=  (/) ) ) )
5049rexlimdv 2793 . . . . . . 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 )  =/=  (/) ) )
5136, 50mpd 15 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  /\  y  e.  S )  ->  ( `' I " S )  =/=  (/) )
5229, 51exlimddv 1645 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  ( `' I " S )  =/=  (/) )
53 eqid 2408 . . . . . 6  |-  ( glb `  K )  =  ( glb `  K )
541, 53, 2, 3dihglb 31828 . . . . 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 ) )
5524, 26, 52, 54syl12anc 1182 . . . 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 5708 . . . . 5  |-  ( y  e.  ( `' I " S )  ->  (
( I  |`  ( `' I " S ) ) `  y )  =  ( I `  y ) )
5756iineq2i 4076 . . . 4  |-  |^|_ y  e.  ( `' I " S ) ( ( I  |`  ( `' I " S ) ) `
 y )  = 
|^|_ y  e.  ( `' I " S ) ( I `  y
)
5855, 57syl6eqr 2458 . . 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 29845 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  CLat )
6059ad2antrr 707 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  K  e.  CLat )
611, 53clatglbcl 14500 . . . . 5  |-  ( ( K  e.  CLat  /\  ( `' I " S ) 
C_  ( Base `  K
) )  ->  (
( glb `  K
) `  ( `' I " S ) )  e.  ( Base `  K
) )
6260, 26, 61syl2anc 643 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  (
( glb `  K
) `  ( `' I " S ) )  e.  ( Base `  K
) )
631, 2, 3dihcl 31757 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( glb `  K ) `  ( `' I " S ) )  e.  ( Base `  K ) )  -> 
( I `  (
( glb `  K
) `  ( `' I " S ) ) )  e.  ran  I
)
6462, 63syldan 457 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  (
I `  ( ( glb `  K ) `  ( `' I " S ) ) )  e.  ran  I )
6558, 64eqeltrrd 2483 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  |^|_ y  e.  ( `' I " S ) ( ( I  |`  ( `' I " S ) ) `
 y )  e. 
ran  I )
6623, 65eqeltrrd 2483 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) )  ->  |^| S  e.  ran  I )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1721    =/= wne 2571   E.wrex 2671    C_ wss 3284   (/)c0 3592   |^|cint 4014   |^|_ciin 4058   `'ccnv 4840   dom cdm 4841   ran crn 4842    |` cres 4843   "cima 4844   Fun wfun 5411    Fn wfn 5412   -onto->wfo 5415   ` cfv 5417   Basecbs 13428   glbcglb 14359   CLatccla 14495   HLchlt 29837   LHypclh 30470   DIsoHcdih 31715
This theorem is referenced by:  doch2val2  31851  dochocss  31853
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664  ax-cnex 9006  ax-resscn 9007  ax-1cn 9008  ax-icn 9009  ax-addcl 9010  ax-addrcl 9011  ax-mulcl 9012  ax-mulrcl 9013  ax-mulcom 9014  ax-addass 9015  ax-mulass 9016  ax-distr 9017  ax-i2m1 9018  ax-1ne0 9019  ax-1rid 9020  ax-rnegex 9021  ax-rrecex 9022  ax-cnre 9023  ax-pre-lttri 9024  ax-pre-lttrn 9025  ax-pre-ltadd 9026  ax-pre-mulgt0 9027
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-fal 1326  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-int 4015  df-iun 4059  df-iin 4060  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-1st 6312  df-2nd 6313  df-tpos 6442  df-undef 6506  df-riota 6512  df-recs 6596  df-rdg 6631  df-1o 6687  df-oadd 6691  df-er 6868  df-map 6983  df-en 7073  df-dom 7074  df-sdom 7075  df-fin 7076  df-pnf 9082  df-mnf 9083  df-xr 9084  df-ltxr 9085  df-le 9086  df-sub 9253  df-neg 9254  df-nn 9961  df-2 10018  df-3 10019  df-4 10020  df-5 10021  df-6 10022  df-n0 10182  df-z 10243  df-uz 10449  df-fz 11004  df-struct 13430  df-ndx 13431  df-slot 13432  df-base 13433  df-sets 13434  df-ress 13435  df-plusg 13501  df-mulr 13502  df-sca 13504  df-vsca 13505  df-0g 13686  df-poset 14362  df-plt 14374  df-lub 14390  df-glb 14391  df-join 14392  df-meet 14393  df-p0 14427  df-p1 14428  df-lat 14434  df-clat 14496  df-mnd 14649  df-submnd 14698  df-grp 14771  df-minusg 14772  df-sbg 14773  df-subg 14900  df-cntz 15075  df-lsm 15229  df-cmn 15373  df-abl 15374  df-mgp 15608  df-rng 15622  df-ur 15624  df-oppr 15687  df-dvdsr 15705  df-unit 15706  df-invr 15736  df-dvr 15747  df-drng 15796  df-lmod 15911  df-lss 15968  df-lsp 16007  df-lvec 16134  df-lsatoms 29463  df-oposet 29663  df-ol 29665  df-oml 29666  df-covers 29753  df-ats 29754  df-atl 29785  df-cvlat 29809  df-hlat 29838  df-llines 29984  df-lplanes 29985  df-lvols 29986  df-lines 29987  df-psubsp 29989  df-pmap 29990  df-padd 30282  df-lhyp 30474  df-laut 30475  df-ldil 30590  df-ltrn 30591  df-trl 30645  df-tendo 31241  df-edring 31243  df-disoa 31516  df-dvech 31566  df-dib 31626  df-dic 31660  df-dih 31716
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