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Theorem dicssdvh 35983
Description: The partial isomorphism C maps to a set of vectors in full vector space H. (Contributed by NM, 19-Jan-2014.)
Hypotheses
Ref Expression
dicssdvh.l  |-  .<_  =  ( le `  K )
dicssdvh.a  |-  A  =  ( Atoms `  K )
dicssdvh.h  |-  H  =  ( LHyp `  K
)
dicssdvh.i  |-  I  =  ( ( DIsoC `  K
) `  W )
dicssdvh.u  |-  U  =  ( ( DVecH `  K
) `  W )
dicssdvh.v  |-  V  =  ( Base `  U
)
Assertion
Ref Expression
dicssdvh  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( I `  Q
)  C_  V )

Proof of Theorem dicssdvh
Dummy variables  f 
g  s are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simprl 755 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( f  =  ( s `  ( iota_ g  e.  ( ( LTrn `  K
) `  W )
( g `  (
( oc `  K
) `  W )
)  =  Q ) )  /\  s  e.  ( ( TEndo `  K
) `  W )
) )  ->  f  =  ( s `  ( iota_ g  e.  ( ( LTrn `  K
) `  W )
( g `  (
( oc `  K
) `  W )
)  =  Q ) ) )
2 simpll 753 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( f  =  ( s `  ( iota_ g  e.  ( ( LTrn `  K
) `  W )
( g `  (
( oc `  K
) `  W )
)  =  Q ) )  /\  s  e.  ( ( TEndo `  K
) `  W )
) )  ->  ( K  e.  HL  /\  W  e.  H ) )
3 simprr 756 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( f  =  ( s `  ( iota_ g  e.  ( ( LTrn `  K
) `  W )
( g `  (
( oc `  K
) `  W )
)  =  Q ) )  /\  s  e.  ( ( TEndo `  K
) `  W )
) )  ->  s  e.  ( ( TEndo `  K
) `  W )
)
4 dicssdvh.l . . . . . . . . . . 11  |-  .<_  =  ( le `  K )
5 eqid 2467 . . . . . . . . . . 11  |-  ( oc
`  K )  =  ( oc `  K
)
6 dicssdvh.a . . . . . . . . . . 11  |-  A  =  ( Atoms `  K )
7 dicssdvh.h . . . . . . . . . . 11  |-  H  =  ( LHyp `  K
)
84, 5, 6, 7lhpocnel 34814 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( ( oc
`  K ) `  W )  e.  A  /\  -.  ( ( oc
`  K ) `  W )  .<_  W ) )
98ad2antrr 725 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( f  =  ( s `  ( iota_ g  e.  ( ( LTrn `  K
) `  W )
( g `  (
( oc `  K
) `  W )
)  =  Q ) )  /\  s  e.  ( ( TEndo `  K
) `  W )
) )  ->  (
( ( oc `  K ) `  W
)  e.  A  /\  -.  ( ( oc `  K ) `  W
)  .<_  W ) )
10 simplr 754 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( f  =  ( s `  ( iota_ g  e.  ( ( LTrn `  K
) `  W )
( g `  (
( oc `  K
) `  W )
)  =  Q ) )  /\  s  e.  ( ( TEndo `  K
) `  W )
) )  ->  ( Q  e.  A  /\  -.  Q  .<_  W ) )
11 eqid 2467 . . . . . . . . . 10  |-  ( (
LTrn `  K ) `  W )  =  ( ( LTrn `  K
) `  W )
12 eqid 2467 . . . . . . . . . 10  |-  ( iota_ g  e.  ( ( LTrn `  K ) `  W
) ( g `  ( ( oc `  K ) `  W
) )  =  Q )  =  ( iota_ g  e.  ( ( LTrn `  K ) `  W
) ( g `  ( ( oc `  K ) `  W
) )  =  Q )
134, 6, 7, 11, 12ltrniotacl 35375 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( ( oc `  K ) `
 W )  e.  A  /\  -.  (
( oc `  K
) `  W )  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  ( iota_ g  e.  ( ( LTrn `  K ) `  W
) ( g `  ( ( oc `  K ) `  W
) )  =  Q )  e.  ( (
LTrn `  K ) `  W ) )
142, 9, 10, 13syl3anc 1228 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( f  =  ( s `  ( iota_ g  e.  ( ( LTrn `  K
) `  W )
( g `  (
( oc `  K
) `  W )
)  =  Q ) )  /\  s  e.  ( ( TEndo `  K
) `  W )
) )  ->  ( iota_ g  e.  ( (
LTrn `  K ) `  W ) ( g `
 ( ( oc
`  K ) `  W ) )  =  Q )  e.  ( ( LTrn `  K
) `  W )
)
15 eqid 2467 . . . . . . . . 9  |-  ( (
TEndo `  K ) `  W )  =  ( ( TEndo `  K ) `  W )
167, 11, 15tendocl 35563 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  s  e.  ( ( TEndo `  K ) `  W )  /\  ( iota_ g  e.  ( (
LTrn `  K ) `  W ) ( g `
 ( ( oc
`  K ) `  W ) )  =  Q )  e.  ( ( LTrn `  K
) `  W )
)  ->  ( s `  ( iota_ g  e.  ( ( LTrn `  K
) `  W )
( g `  (
( oc `  K
) `  W )
)  =  Q ) )  e.  ( (
LTrn `  K ) `  W ) )
172, 3, 14, 16syl3anc 1228 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( f  =  ( s `  ( iota_ g  e.  ( ( LTrn `  K
) `  W )
( g `  (
( oc `  K
) `  W )
)  =  Q ) )  /\  s  e.  ( ( TEndo `  K
) `  W )
) )  ->  (
s `  ( iota_ g  e.  ( ( LTrn `  K
) `  W )
( g `  (
( oc `  K
) `  W )
)  =  Q ) )  e.  ( (
LTrn `  K ) `  W ) )
181, 17eqeltrd 2555 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( f  =  ( s `  ( iota_ g  e.  ( ( LTrn `  K
) `  W )
( g `  (
( oc `  K
) `  W )
)  =  Q ) )  /\  s  e.  ( ( TEndo `  K
) `  W )
) )  ->  f  e.  ( ( LTrn `  K
) `  W )
)
1918, 3, 3jca31 534 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( f  =  ( s `  ( iota_ g  e.  ( ( LTrn `  K
) `  W )
( g `  (
( oc `  K
) `  W )
)  =  Q ) )  /\  s  e.  ( ( TEndo `  K
) `  W )
) )  ->  (
( f  e.  ( ( LTrn `  K
) `  W )  /\  s  e.  (
( TEndo `  K ) `  W ) )  /\  s  e.  ( ( TEndo `  K ) `  W ) ) )
2019ex 434 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( ( f  =  ( s `  ( iota_ g  e.  ( (
LTrn `  K ) `  W ) ( g `
 ( ( oc
`  K ) `  W ) )  =  Q ) )  /\  s  e.  ( ( TEndo `  K ) `  W ) )  -> 
( ( f  e.  ( ( LTrn `  K
) `  W )  /\  s  e.  (
( TEndo `  K ) `  W ) )  /\  s  e.  ( ( TEndo `  K ) `  W ) ) ) )
2120ssopab2dv 4776 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  { <. f ,  s
>.  |  ( f  =  ( s `  ( iota_ g  e.  ( ( LTrn `  K
) `  W )
( g `  (
( oc `  K
) `  W )
)  =  Q ) )  /\  s  e.  ( ( TEndo `  K
) `  W )
) }  C_  { <. f ,  s >.  |  ( ( f  e.  ( ( LTrn `  K
) `  W )  /\  s  e.  (
( TEndo `  K ) `  W ) )  /\  s  e.  ( ( TEndo `  K ) `  W ) ) } )
22 opabssxp 5072 . . 3  |-  { <. f ,  s >.  |  ( ( f  e.  ( ( LTrn `  K
) `  W )  /\  s  e.  (
( TEndo `  K ) `  W ) )  /\  s  e.  ( ( TEndo `  K ) `  W ) ) } 
C_  ( ( (
LTrn `  K ) `  W )  X.  (
( TEndo `  K ) `  W ) )
2321, 22syl6ss 3516 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  { <. f ,  s
>.  |  ( f  =  ( s `  ( iota_ g  e.  ( ( LTrn `  K
) `  W )
( g `  (
( oc `  K
) `  W )
)  =  Q ) )  /\  s  e.  ( ( TEndo `  K
) `  W )
) }  C_  (
( ( LTrn `  K
) `  W )  X.  ( ( TEndo `  K
) `  W )
) )
24 eqid 2467 . . 3  |-  ( ( oc `  K ) `
 W )  =  ( ( oc `  K ) `  W
)
25 dicssdvh.i . . 3  |-  I  =  ( ( DIsoC `  K
) `  W )
264, 6, 7, 24, 11, 15, 25dicval 35973 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( I `  Q
)  =  { <. f ,  s >.  |  ( f  =  ( s `
 ( iota_ g  e.  ( ( LTrn `  K
) `  W )
( g `  (
( oc `  K
) `  W )
)  =  Q ) )  /\  s  e.  ( ( TEndo `  K
) `  W )
) } )
27 dicssdvh.u . . . 4  |-  U  =  ( ( DVecH `  K
) `  W )
28 dicssdvh.v . . . 4  |-  V  =  ( Base `  U
)
297, 11, 15, 27, 28dvhvbase 35884 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  V  =  ( ( ( LTrn `  K
) `  W )  X.  ( ( TEndo `  K
) `  W )
) )
3029adantr 465 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  V  =  ( (
( LTrn `  K ) `  W )  X.  (
( TEndo `  K ) `  W ) ) )
3123, 26, 303sstr4d 3547 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( I `  Q
)  C_  V )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767    C_ wss 3476   class class class wbr 4447   {copab 4504    X. cxp 4997   ` cfv 5586   iota_crio 6242   Basecbs 14486   lecple 14558   occoc 14559   Atomscatm 34060   HLchlt 34147   LHypclh 34780   LTrncltrn 34897   TEndoctendo 35548   DVecHcdvh 35875   DIsoCcdic 35969
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-riotaBAD 33756
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-undef 6999  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-map 7419  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-n0 10792  df-z 10861  df-uz 11079  df-fz 11669  df-struct 14488  df-ndx 14489  df-slot 14490  df-base 14491  df-plusg 14564  df-sca 14567  df-vsca 14568  df-poset 15429  df-plt 15441  df-lub 15457  df-glb 15458  df-join 15459  df-meet 15460  df-p0 15522  df-p1 15523  df-lat 15529  df-clat 15591  df-oposet 33973  df-ol 33975  df-oml 33976  df-covers 34063  df-ats 34064  df-atl 34095  df-cvlat 34119  df-hlat 34148  df-llines 34294  df-lplanes 34295  df-lvols 34296  df-lines 34297  df-psubsp 34299  df-pmap 34300  df-padd 34592  df-lhyp 34784  df-laut 34785  df-ldil 34900  df-ltrn 34901  df-trl 34955  df-tendo 35551  df-dvech 35876  df-dic 35970
This theorem is referenced by:  dicelval1stN  35985  dicelval2nd  35986  dicvaddcl  35987  dicvscacl  35988  diclss  35990
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