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Theorem diclss 31993
Description: The value of partial isomorphism C is a subspace of partial vector space H. (Contributed by NM, 16-Feb-2014.)
Hypotheses
Ref Expression
diclss.l  |-  .<_  =  ( le `  K )
diclss.a  |-  A  =  ( Atoms `  K )
diclss.h  |-  H  =  ( LHyp `  K
)
diclss.u  |-  U  =  ( ( DVecH `  K
) `  W )
diclss.i  |-  I  =  ( ( DIsoC `  K
) `  W )
diclss.s  |-  S  =  ( LSubSp `  U )
Assertion
Ref Expression
diclss  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( I `  Q
)  e.  S )

Proof of Theorem diclss
Dummy variables  a 
b  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqidd 2439 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
(Scalar `  U )  =  (Scalar `  U )
)
2 diclss.h . . . . 5  |-  H  =  ( LHyp `  K
)
3 eqid 2438 . . . . 5  |-  ( (
TEndo `  K ) `  W )  =  ( ( TEndo `  K ) `  W )
4 diclss.u . . . . 5  |-  U  =  ( ( DVecH `  K
) `  W )
5 eqid 2438 . . . . 5  |-  (Scalar `  U )  =  (Scalar `  U )
6 eqid 2438 . . . . 5  |-  ( Base `  (Scalar `  U )
)  =  ( Base `  (Scalar `  U )
)
72, 3, 4, 5, 6dvhbase 31883 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( Base `  (Scalar `  U ) )  =  ( ( TEndo `  K
) `  W )
)
87eqcomd 2443 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( TEndo `  K
) `  W )  =  ( Base `  (Scalar `  U ) ) )
98adantr 453 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( ( TEndo `  K
) `  W )  =  ( Base `  (Scalar `  U ) ) )
10 eqid 2438 . . . . 5  |-  ( (
LTrn `  K ) `  W )  =  ( ( LTrn `  K
) `  W )
11 eqid 2438 . . . . 5  |-  ( Base `  U )  =  (
Base `  U )
122, 10, 3, 4, 11dvhvbase 31887 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( Base `  U
)  =  ( ( ( LTrn `  K
) `  W )  X.  ( ( TEndo `  K
) `  W )
) )
1312eqcomd 2443 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( ( LTrn `  K ) `  W
)  X.  ( (
TEndo `  K ) `  W ) )  =  ( Base `  U
) )
1413adantr 453 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( ( ( LTrn `  K ) `  W
)  X.  ( (
TEndo `  K ) `  W ) )  =  ( Base `  U
) )
15 eqidd 2439 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( +g  `  U )  =  ( +g  `  U
) )
16 eqidd 2439 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( .s `  U
)  =  ( .s
`  U ) )
17 diclss.s . . 3  |-  S  =  ( LSubSp `  U )
1817a1i 11 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  S  =  ( LSubSp `  U ) )
19 diclss.l . . . 4  |-  .<_  =  ( le `  K )
20 diclss.a . . . 4  |-  A  =  ( Atoms `  K )
21 diclss.i . . . 4  |-  I  =  ( ( DIsoC `  K
) `  W )
2219, 20, 2, 21, 4, 11dicssdvh 31986 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( I `  Q
)  C_  ( Base `  U ) )
2322, 14sseqtr4d 3387 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( I `  Q
)  C_  ( (
( LTrn `  K ) `  W )  X.  (
( TEndo `  K ) `  W ) ) )
2419, 20, 2, 21dicn0 31992 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( I `  Q
)  =/=  (/) )
25 simpll 732 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( x  e.  ( ( TEndo `  K ) `  W
)  /\  a  e.  ( I `  Q
)  /\  b  e.  ( I `  Q
) ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
26 simplr 733 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( x  e.  ( ( TEndo `  K ) `  W
)  /\  a  e.  ( I `  Q
)  /\  b  e.  ( I `  Q
) ) )  -> 
( Q  e.  A  /\  -.  Q  .<_  W ) )
27 simpr1 964 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( x  e.  ( ( TEndo `  K ) `  W
)  /\  a  e.  ( I `  Q
)  /\  b  e.  ( I `  Q
) ) )  ->  x  e.  ( ( TEndo `  K ) `  W ) )
28 simpr2 965 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( x  e.  ( ( TEndo `  K ) `  W
)  /\  a  e.  ( I `  Q
)  /\  b  e.  ( I `  Q
) ) )  -> 
a  e.  ( I `
 Q ) )
29 eqid 2438 . . . . 5  |-  ( .s
`  U )  =  ( .s `  U
)
3019, 20, 2, 3, 4, 21, 29dicvscacl 31991 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  (
x  e.  ( (
TEndo `  K ) `  W )  /\  a  e.  ( I `  Q
) ) )  -> 
( x ( .s
`  U ) a )  e.  ( I `
 Q ) )
3125, 26, 27, 28, 30syl112anc 1189 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( x  e.  ( ( TEndo `  K ) `  W
)  /\  a  e.  ( I `  Q
)  /\  b  e.  ( I `  Q
) ) )  -> 
( x ( .s
`  U ) a )  e.  ( I `
 Q ) )
32 simpr3 966 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( x  e.  ( ( TEndo `  K ) `  W
)  /\  a  e.  ( I `  Q
)  /\  b  e.  ( I `  Q
) ) )  -> 
b  e.  ( I `
 Q ) )
33 eqid 2438 . . . 4  |-  ( +g  `  U )  =  ( +g  `  U )
3419, 20, 2, 4, 21, 33dicvaddcl 31990 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  (
( x ( .s
`  U ) a )  e.  ( I `
 Q )  /\  b  e.  ( I `  Q ) ) )  ->  ( ( x ( .s `  U
) a ) ( +g  `  U ) b )  e.  ( I `  Q ) )
3525, 26, 31, 32, 34syl112anc 1189 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( x  e.  ( ( TEndo `  K ) `  W
)  /\  a  e.  ( I `  Q
)  /\  b  e.  ( I `  Q
) ) )  -> 
( ( x ( .s `  U ) a ) ( +g  `  U ) b )  e.  ( I `  Q ) )
361, 9, 14, 15, 16, 18, 23, 24, 35islssd 16014 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( I `  Q
)  e.  S )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   class class class wbr 4214    X. cxp 4878   ` cfv 5456  (class class class)co 6083   Basecbs 13471   +g cplusg 13531  Scalarcsca 13534   .scvsca 13535   lecple 13538   LSubSpclss 16010   Atomscatm 30063   HLchlt 30150   LHypclh 30783   LTrncltrn 30900   TEndoctendo 31551   DVecHcdvh 31878   DIsoCcdic 31972
This theorem is referenced by:  cdlemn5pre  32000  cdlemn11c  32009  dihjustlem  32016  dihord1  32018  dihord2a  32019  dihord2b  32020  dihord11c  32024  dihlsscpre  32034  dihvalcqat  32039  dihopelvalcpre  32048  dihord6apre  32056  dihord5b  32059  dihord5apre  32062
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-iin 4098  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-undef 6545  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-oadd 6730  df-er 6907  df-map 7022  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-nn 10003  df-2 10060  df-3 10061  df-4 10062  df-5 10063  df-6 10064  df-n0 10224  df-z 10285  df-uz 10491  df-fz 11046  df-struct 13473  df-ndx 13474  df-slot 13475  df-base 13476  df-plusg 13544  df-mulr 13545  df-sca 13547  df-vsca 13548  df-poset 14405  df-plt 14417  df-lub 14433  df-glb 14434  df-join 14435  df-meet 14436  df-p0 14470  df-p1 14471  df-lat 14477  df-clat 14539  df-lss 16011  df-oposet 29976  df-ol 29978  df-oml 29979  df-covers 30066  df-ats 30067  df-atl 30098  df-cvlat 30122  df-hlat 30151  df-llines 30297  df-lplanes 30298  df-lvols 30299  df-lines 30300  df-psubsp 30302  df-pmap 30303  df-padd 30595  df-lhyp 30787  df-laut 30788  df-ldil 30903  df-ltrn 30904  df-trl 30958  df-tendo 31554  df-edring 31556  df-dvech 31879  df-dic 31973
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