Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  tendoplcl2 Unicode version

Theorem tendoplcl2 30894
Description: Value of result of endomorphism sum operation. (Contributed by NM, 10-Jun-2013.)
Hypotheses
Ref Expression
tendopl.h  |-  H  =  ( LHyp `  K
)
tendopl.t  |-  T  =  ( ( LTrn `  K
) `  W )
tendopl.e  |-  E  =  ( ( TEndo `  K
) `  W )
tendopl.p  |-  P  =  ( s  e.  E ,  t  e.  E  |->  ( f  e.  T  |->  ( ( s `  f )  o.  (
t `  f )
) ) )
Assertion
Ref Expression
tendoplcl2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  F  e.  T )  ->  (
( U P V ) `  F )  e.  T )
Distinct variable groups:    t, s, E    f, s, t, T   
f, W, s, t
Allowed substitution hints:    P( t, f, s)    U( t, f, s)    E( f)    F( t, f, s)    H( t, f, s)    K( t, f, s)    V( t, f, s)

Proof of Theorem tendoplcl2
StepHypRef Expression
1 tendopl.p . . . . 5  |-  P  =  ( s  e.  E ,  t  e.  E  |->  ( f  e.  T  |->  ( ( s `  f )  o.  (
t `  f )
) ) )
2 tendopl.t . . . . 5  |-  T  =  ( ( LTrn `  K
) `  W )
31, 2tendopl2 30893 . . . 4  |-  ( ( U  e.  E  /\  V  e.  E  /\  F  e.  T )  ->  ( ( U P V ) `  F
)  =  ( ( U `  F )  o.  ( V `  F ) ) )
433expa 1153 . . 3  |-  ( ( ( U  e.  E  /\  V  e.  E
)  /\  F  e.  T )  ->  (
( U P V ) `  F )  =  ( ( U `
 F )  o.  ( V `  F
) ) )
543adant1 975 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  F  e.  T )  ->  (
( U P V ) `  F )  =  ( ( U `
 F )  o.  ( V `  F
) ) )
6 simp1 957 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  F  e.  T )  ->  ( K  e.  HL  /\  W  e.  H ) )
7 tendopl.h . . . . 5  |-  H  =  ( LHyp `  K
)
8 tendopl.e . . . . 5  |-  E  =  ( ( TEndo `  K
) `  W )
97, 2, 8tendocl 30883 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E  /\  F  e.  T
)  ->  ( U `  F )  e.  T
)
1093adant2r 1179 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  F  e.  T )  ->  ( U `  F )  e.  T )
117, 2, 8tendocl 30883 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  V  e.  E  /\  F  e.  T
)  ->  ( V `  F )  e.  T
)
12113adant2l 1178 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  F  e.  T )  ->  ( V `  F )  e.  T )
137, 2ltrnco 30835 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U `  F )  e.  T  /\  ( V `  F
)  e.  T )  ->  ( ( U `
 F )  o.  ( V `  F
) )  e.  T
)
146, 10, 12, 13syl3anc 1184 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  F  e.  T )  ->  (
( U `  F
)  o.  ( V `
 F ) )  e.  T )
155, 14eqeltrd 2463 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  F  e.  T )  ->  (
( U P V ) `  F )  e.  T )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    e. cmpt 4209    o. ccom 4824   ` cfv 5396  (class class class)co 6022    e. cmpt2 6024   HLchlt 29467   LHypclh 30100   LTrncltrn 30217   TEndoctendo 30868
This theorem is referenced by:  tendopltp  30896
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-iun 4039  df-iin 4040  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-undef 6481  df-riota 6487  df-map 6958  df-poset 14332  df-plt 14344  df-lub 14360  df-glb 14361  df-join 14362  df-meet 14363  df-p0 14397  df-p1 14398  df-lat 14404  df-clat 14466  df-oposet 29293  df-ol 29295  df-oml 29296  df-covers 29383  df-ats 29384  df-atl 29415  df-cvlat 29439  df-hlat 29468  df-llines 29614  df-lplanes 29615  df-lvols 29616  df-lines 29617  df-psubsp 29619  df-pmap 29620  df-padd 29912  df-lhyp 30104  df-laut 30105  df-ldil 30220  df-ltrn 30221  df-trl 30275  df-tendo 30871
  Copyright terms: Public domain W3C validator