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Theorem diclss 34526
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 2442 . 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 2441 . . . . 5  |-  ( (
TEndo `  K ) `  W )  =  ( ( TEndo `  K ) `  W )
4 diclss.u . . . . 5  |-  U  =  ( ( DVecH `  K
) `  W )
5 eqid 2441 . . . . 5  |-  (Scalar `  U )  =  (Scalar `  U )
6 eqid 2441 . . . . 5  |-  ( Base `  (Scalar `  U )
)  =  ( Base `  (Scalar `  U )
)
72, 3, 4, 5, 6dvhbase 34416 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( Base `  (Scalar `  U ) )  =  ( ( TEndo `  K
) `  W )
)
87eqcomd 2446 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( TEndo `  K
) `  W )  =  ( Base `  (Scalar `  U ) ) )
98adantr 462 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( ( TEndo `  K
) `  W )  =  ( Base `  (Scalar `  U ) ) )
10 eqid 2441 . . . . 5  |-  ( (
LTrn `  K ) `  W )  =  ( ( LTrn `  K
) `  W )
11 eqid 2441 . . . . 5  |-  ( Base `  U )  =  (
Base `  U )
122, 10, 3, 4, 11dvhvbase 34420 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( Base `  U
)  =  ( ( ( LTrn `  K
) `  W )  X.  ( ( TEndo `  K
) `  W )
) )
1312eqcomd 2446 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( ( LTrn `  K ) `  W
)  X.  ( (
TEndo `  K ) `  W ) )  =  ( Base `  U
) )
1413adantr 462 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( ( ( LTrn `  K ) `  W
)  X.  ( (
TEndo `  K ) `  W ) )  =  ( Base `  U
) )
15 eqidd 2442 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( +g  `  U )  =  ( +g  `  U
) )
16 eqidd 2442 . 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 34519 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( I `  Q
)  C_  ( Base `  U ) )
2322, 14sseqtr4d 3390 . 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 34525 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( I `  Q
)  =/=  (/) )
25 simpll 748 . . 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 749 . . 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 989 . . . 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 990 . . . 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 2441 . . . . 5  |-  ( .s
`  U )  =  ( .s `  U
)
3019, 20, 2, 3, 4, 21, 29dicvscacl 34524 . . . 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 1217 . . 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 991 . . 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 2441 . . . 4  |-  ( +g  `  U )  =  ( +g  `  U )
3419, 20, 2, 4, 21, 33dicvaddcl 34523 . . 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 1217 . 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 16995 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( I `  Q
)  e.  S )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761   class class class wbr 4289    X. cxp 4834   ` cfv 5415  (class class class)co 6090   Basecbs 14170   +g cplusg 14234  Scalarcsca 14237   .scvsca 14238   lecple 14241   LSubSpclss 16991   Atomscatm 32596   HLchlt 32683   LHypclh 33316   LTrncltrn 33433   TEndoctendo 34084   DVecHcdvh 34411   DIsoCcdic 34505
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-riotaBAD 32292
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-iin 4171  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-undef 6788  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-map 7212  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-n0 10576  df-z 10643  df-uz 10858  df-fz 11434  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-plusg 14247  df-mulr 14248  df-sca 14250  df-vsca 14251  df-poset 15112  df-plt 15124  df-lub 15140  df-glb 15141  df-join 15142  df-meet 15143  df-p0 15205  df-p1 15206  df-lat 15212  df-clat 15274  df-lss 16992  df-oposet 32509  df-ol 32511  df-oml 32512  df-covers 32599  df-ats 32600  df-atl 32631  df-cvlat 32655  df-hlat 32684  df-llines 32830  df-lplanes 32831  df-lvols 32832  df-lines 32833  df-psubsp 32835  df-pmap 32836  df-padd 33128  df-lhyp 33320  df-laut 33321  df-ldil 33436  df-ltrn 33437  df-trl 33491  df-tendo 34087  df-edring 34089  df-dvech 34412  df-dic 34506
This theorem is referenced by:  cdlemn5pre  34533  cdlemn11c  34542  dihjustlem  34549  dihord1  34551  dihord2a  34552  dihord2b  34553  dihord11c  34557  dihlsscpre  34567  dihvalcqat  34572  dihopelvalcpre  34581  dihord6apre  34589  dihord5b  34592  dihord5apre  34595
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