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Theorem diclss 34755
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 2451 . 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 2450 . . . . 5  |-  ( (
TEndo `  K ) `  W )  =  ( ( TEndo `  K ) `  W )
4 diclss.u . . . . 5  |-  U  =  ( ( DVecH `  K
) `  W )
5 eqid 2450 . . . . 5  |-  (Scalar `  U )  =  (Scalar `  U )
6 eqid 2450 . . . . 5  |-  ( Base `  (Scalar `  U )
)  =  ( Base `  (Scalar `  U )
)
72, 3, 4, 5, 6dvhbase 34645 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( Base `  (Scalar `  U ) )  =  ( ( TEndo `  K
) `  W )
)
87eqcomd 2456 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( TEndo `  K
) `  W )  =  ( Base `  (Scalar `  U ) ) )
98adantr 467 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( ( TEndo `  K
) `  W )  =  ( Base `  (Scalar `  U ) ) )
10 eqid 2450 . . . . 5  |-  ( (
LTrn `  K ) `  W )  =  ( ( LTrn `  K
) `  W )
11 eqid 2450 . . . . 5  |-  ( Base `  U )  =  (
Base `  U )
122, 10, 3, 4, 11dvhvbase 34649 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( Base `  U
)  =  ( ( ( LTrn `  K
) `  W )  X.  ( ( TEndo `  K
) `  W )
) )
1312eqcomd 2456 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( ( LTrn `  K ) `  W
)  X.  ( (
TEndo `  K ) `  W ) )  =  ( Base `  U
) )
1413adantr 467 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( ( ( LTrn `  K ) `  W
)  X.  ( (
TEndo `  K ) `  W ) )  =  ( Base `  U
) )
15 eqidd 2451 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( +g  `  U )  =  ( +g  `  U
) )
16 eqidd 2451 . 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 34748 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( I `  Q
)  C_  ( Base `  U ) )
2322, 14sseqtr4d 3468 . 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 34754 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( I `  Q
)  =/=  (/) )
25 simpll 759 . . 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 761 . . 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 1013 . . . 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 1014 . . . 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 2450 . . . . 5  |-  ( .s
`  U )  =  ( .s `  U
)
3019, 20, 2, 3, 4, 21, 29dicvscacl 34753 . . . 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 1271 . . 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 1015 . . 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 2450 . . . 4  |-  ( +g  `  U )  =  ( +g  `  U )
3419, 20, 2, 4, 21, 33dicvaddcl 34752 . . 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 1271 . 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 18152 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 371    /\ w3a 984    = wceq 1443    e. wcel 1886   class class class wbr 4401    X. cxp 4831   ` cfv 5581  (class class class)co 6288   Basecbs 15114   +g cplusg 15183  Scalarcsca 15186   .scvsca 15187   lecple 15190   LSubSpclss 18148   Atomscatm 32823   HLchlt 32910   LHypclh 33543   LTrncltrn 33660   TEndoctendo 34313   DVecHcdvh 34640   DIsoCcdic 34734
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613  ax-riotaBAD 32519
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-int 4234  df-iun 4279  df-iin 4280  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-om 6690  df-1st 6790  df-2nd 6791  df-undef 7017  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-1o 7179  df-oadd 7183  df-er 7360  df-map 7471  df-en 7567  df-dom 7568  df-sdom 7569  df-fin 7570  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-nn 10607  df-2 10665  df-3 10666  df-4 10667  df-5 10668  df-6 10669  df-n0 10867  df-z 10935  df-uz 11157  df-fz 11782  df-struct 15116  df-ndx 15117  df-slot 15118  df-base 15119  df-plusg 15196  df-mulr 15197  df-sca 15199  df-vsca 15200  df-preset 16166  df-poset 16184  df-plt 16197  df-lub 16213  df-glb 16214  df-join 16215  df-meet 16216  df-p0 16278  df-p1 16279  df-lat 16285  df-clat 16347  df-lss 18149  df-oposet 32736  df-ol 32738  df-oml 32739  df-covers 32826  df-ats 32827  df-atl 32858  df-cvlat 32882  df-hlat 32911  df-llines 33057  df-lplanes 33058  df-lvols 33059  df-lines 33060  df-psubsp 33062  df-pmap 33063  df-padd 33355  df-lhyp 33547  df-laut 33548  df-ldil 33663  df-ltrn 33664  df-trl 33719  df-tendo 34316  df-edring 34318  df-dvech 34641  df-dic 34735
This theorem is referenced by:  cdlemn5pre  34762  cdlemn11c  34771  dihjustlem  34778  dihord1  34780  dihord2a  34781  dihord2b  34782  dihord11c  34786  dihlsscpre  34796  dihvalcqat  34801  dihopelvalcpre  34810  dihord6apre  34818  dihord5b  34821  dihord5apre  34824
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