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Theorem diclss 34193
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 2403 . 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 2402 . . . . 5  |-  ( (
TEndo `  K ) `  W )  =  ( ( TEndo `  K ) `  W )
4 diclss.u . . . . 5  |-  U  =  ( ( DVecH `  K
) `  W )
5 eqid 2402 . . . . 5  |-  (Scalar `  U )  =  (Scalar `  U )
6 eqid 2402 . . . . 5  |-  ( Base `  (Scalar `  U )
)  =  ( Base `  (Scalar `  U )
)
72, 3, 4, 5, 6dvhbase 34083 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( Base `  (Scalar `  U ) )  =  ( ( TEndo `  K
) `  W )
)
87eqcomd 2410 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( TEndo `  K
) `  W )  =  ( Base `  (Scalar `  U ) ) )
98adantr 463 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( ( TEndo `  K
) `  W )  =  ( Base `  (Scalar `  U ) ) )
10 eqid 2402 . . . . 5  |-  ( (
LTrn `  K ) `  W )  =  ( ( LTrn `  K
) `  W )
11 eqid 2402 . . . . 5  |-  ( Base `  U )  =  (
Base `  U )
122, 10, 3, 4, 11dvhvbase 34087 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( Base `  U
)  =  ( ( ( LTrn `  K
) `  W )  X.  ( ( TEndo `  K
) `  W )
) )
1312eqcomd 2410 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( ( LTrn `  K ) `  W
)  X.  ( (
TEndo `  K ) `  W ) )  =  ( Base `  U
) )
1413adantr 463 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( ( ( LTrn `  K ) `  W
)  X.  ( (
TEndo `  K ) `  W ) )  =  ( Base `  U
) )
15 eqidd 2403 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( +g  `  U )  =  ( +g  `  U
) )
16 eqidd 2403 . 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 34186 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( I `  Q
)  C_  ( Base `  U ) )
2322, 14sseqtr4d 3478 . 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 34192 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( I `  Q
)  =/=  (/) )
25 simpll 752 . . 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 754 . . 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 1003 . . . 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 1004 . . . 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 2402 . . . . 5  |-  ( .s
`  U )  =  ( .s `  U
)
3019, 20, 2, 3, 4, 21, 29dicvscacl 34191 . . . 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 1234 . . 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 1005 . . 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 2402 . . . 4  |-  ( +g  `  U )  =  ( +g  `  U )
3419, 20, 2, 4, 21, 33dicvaddcl 34190 . . 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 1234 . 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 17900 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 367    /\ w3a 974    = wceq 1405    e. wcel 1842   class class class wbr 4394    X. cxp 4820   ` cfv 5568  (class class class)co 6277   Basecbs 14839   +g cplusg 14907  Scalarcsca 14910   .scvsca 14911   lecple 14914   LSubSpclss 17896   Atomscatm 32261   HLchlt 32348   LHypclh 32981   LTrncltrn 33098   TEndoctendo 33751   DVecHcdvh 34078   DIsoCcdic 34172
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-cnex 9577  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-mulcom 9585  ax-addass 9586  ax-mulass 9587  ax-distr 9588  ax-i2m1 9589  ax-1ne0 9590  ax-1rid 9591  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596  ax-pre-ltadd 9597  ax-pre-mulgt0 9598  ax-riotaBAD 31957
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-iin 4273  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6683  df-1st 6783  df-2nd 6784  df-undef 7004  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-1o 7166  df-oadd 7170  df-er 7347  df-map 7458  df-en 7554  df-dom 7555  df-sdom 7556  df-fin 7557  df-pnf 9659  df-mnf 9660  df-xr 9661  df-ltxr 9662  df-le 9663  df-sub 9842  df-neg 9843  df-nn 10576  df-2 10634  df-3 10635  df-4 10636  df-5 10637  df-6 10638  df-n0 10836  df-z 10905  df-uz 11127  df-fz 11725  df-struct 14841  df-ndx 14842  df-slot 14843  df-base 14844  df-plusg 14920  df-mulr 14921  df-sca 14923  df-vsca 14924  df-preset 15879  df-poset 15897  df-plt 15910  df-lub 15926  df-glb 15927  df-join 15928  df-meet 15929  df-p0 15991  df-p1 15992  df-lat 15998  df-clat 16060  df-lss 17897  df-oposet 32174  df-ol 32176  df-oml 32177  df-covers 32264  df-ats 32265  df-atl 32296  df-cvlat 32320  df-hlat 32349  df-llines 32495  df-lplanes 32496  df-lvols 32497  df-lines 32498  df-psubsp 32500  df-pmap 32501  df-padd 32793  df-lhyp 32985  df-laut 32986  df-ldil 33101  df-ltrn 33102  df-trl 33157  df-tendo 33754  df-edring 33756  df-dvech 34079  df-dic 34173
This theorem is referenced by:  cdlemn5pre  34200  cdlemn11c  34209  dihjustlem  34216  dihord1  34218  dihord2a  34219  dihord2b  34220  dihord11c  34224  dihlsscpre  34234  dihvalcqat  34239  dihopelvalcpre  34248  dihord6apre  34256  dihord5b  34259  dihord5apre  34262
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