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Theorem tendopltp 30894
Description: Trace-preserving property of endomorphism sum operation  P, based on theorem trlco 30841. Part of remark in [Crawley] p. 118, 2nd line, "it is clear from the second part of G (our trlco 30841) that Delta is a subring of E." (In our development, we will bypass their E and go directly to their Delta, whose base set is our  ( TEndo `  K
) `  W.) (Contributed by NM, 9-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 )
) ) )
tendopltp.l  |-  .<_  =  ( le `  K )
tendopltp.r  |-  R  =  ( ( trL `  K
) `  W )
Assertion
Ref Expression
tendopltp  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  F  e.  T )  ->  ( R `  ( ( U P V ) `  F ) )  .<_  ( R `  F ) )
Distinct variable groups:    t, s, E    f, s, t, T   
f, W, s, t
Allowed substitution hints:    P( t, f, s)    R( t, f, s)    U( t, f, s)    E( f)    F( t, f, s)    H( t, f, s)    K( t, f, s)    .<_ ( t, f, s)    V( t, f, s)

Proof of Theorem tendopltp
StepHypRef Expression
1 eqid 2387 . 2  |-  ( Base `  K )  =  (
Base `  K )
2 tendopltp.l . 2  |-  .<_  =  ( le `  K )
3 simp1l 981 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  F  e.  T )  ->  K  e.  HL )
4 hllat 29478 . . 3  |-  ( K  e.  HL  ->  K  e.  Lat )
53, 4syl 16 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  F  e.  T )  ->  K  e.  Lat )
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 . . . 4  |-  H  =  ( LHyp `  K
)
8 tendopl.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
9 tendopl.e . . . 4  |-  E  =  ( ( TEndo `  K
) `  W )
10 tendopl.p . . . 4  |-  P  =  ( s  e.  E ,  t  e.  E  |->  ( f  e.  T  |->  ( ( s `  f )  o.  (
t `  f )
) ) )
117, 8, 9, 10tendoplcl2 30892 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  F  e.  T )  ->  (
( U P V ) `  F )  e.  T )
12 tendopltp.r . . . 4  |-  R  =  ( ( trL `  K
) `  W )
131, 7, 8, 12trlcl 30278 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( U P V ) `  F )  e.  T
)  ->  ( R `  ( ( U P V ) `  F
) )  e.  (
Base `  K )
)
146, 11, 13syl2anc 643 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  F  e.  T )  ->  ( R `  ( ( U P V ) `  F ) )  e.  ( Base `  K
) )
157, 8, 9tendocl 30881 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E  /\  F  e.  T
)  ->  ( U `  F )  e.  T
)
16153adant2r 1179 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  F  e.  T )  ->  ( U `  F )  e.  T )
171, 7, 8, 12trlcl 30278 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U `  F )  e.  T
)  ->  ( R `  ( U `  F
) )  e.  (
Base `  K )
)
186, 16, 17syl2anc 643 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  F  e.  T )  ->  ( R `  ( U `  F ) )  e.  ( Base `  K
) )
197, 8, 9tendocl 30881 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  V  e.  E  /\  F  e.  T
)  ->  ( V `  F )  e.  T
)
20193adant2l 1178 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  F  e.  T )  ->  ( V `  F )  e.  T )
211, 7, 8, 12trlcl 30278 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( V `  F )  e.  T
)  ->  ( R `  ( V `  F
) )  e.  (
Base `  K )
)
226, 20, 21syl2anc 643 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  F  e.  T )  ->  ( R `  ( V `  F ) )  e.  ( Base `  K
) )
23 eqid 2387 . . . 4  |-  ( join `  K )  =  (
join `  K )
241, 23latjcl 14406 . . 3  |-  ( ( K  e.  Lat  /\  ( R `  ( U `
 F ) )  e.  ( Base `  K
)  /\  ( R `  ( V `  F
) )  e.  (
Base `  K )
)  ->  ( ( R `  ( U `  F ) ) (
join `  K )
( R `  ( V `  F )
) )  e.  (
Base `  K )
)
255, 18, 22, 24syl3anc 1184 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  F  e.  T )  ->  (
( R `  ( U `  F )
) ( join `  K
) ( R `  ( V `  F ) ) )  e.  (
Base `  K )
)
26 simp3 959 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  F  e.  T )  ->  F  e.  T )
271, 7, 8, 12trlcl 30278 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( R `  F )  e.  (
Base `  K )
)
286, 26, 27syl2anc 643 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  F  e.  T )  ->  ( R `  F )  e.  ( Base `  K
) )
29 simp2l 983 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  F  e.  T )  ->  U  e.  E )
30 simp2r 984 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  F  e.  T )  ->  V  e.  E )
3110, 8tendopl2 30891 . . . . 5  |-  ( ( U  e.  E  /\  V  e.  E  /\  F  e.  T )  ->  ( ( U P V ) `  F
)  =  ( ( U `  F )  o.  ( V `  F ) ) )
3229, 30, 26, 31syl3anc 1184 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  F  e.  T )  ->  (
( U P V ) `  F )  =  ( ( U `
 F )  o.  ( V `  F
) ) )
3332fveq2d 5672 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  F  e.  T )  ->  ( R `  ( ( U P V ) `  F ) )  =  ( R `  (
( U `  F
)  o.  ( V `
 F ) ) ) )
342, 23, 7, 8, 12trlco 30841 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U `  F )  e.  T  /\  ( V `  F
)  e.  T )  ->  ( R `  ( ( U `  F )  o.  ( V `  F )
) )  .<_  ( ( R `  ( U `
 F ) ) ( join `  K
) ( R `  ( V `  F ) ) ) )
356, 16, 20, 34syl3anc 1184 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  F  e.  T )  ->  ( R `  ( ( U `  F )  o.  ( V `  F
) ) )  .<_  ( ( R `  ( U `  F ) ) ( join `  K
) ( R `  ( V `  F ) ) ) )
3633, 35eqbrtrd 4173 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  F  e.  T )  ->  ( R `  ( ( U P V ) `  F ) )  .<_  ( ( R `  ( U `  F ) ) ( join `  K
) ( R `  ( V `  F ) ) ) )
372, 7, 8, 12, 9tendotp 30875 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E  /\  F  e.  T
)  ->  ( R `  ( U `  F
) )  .<_  ( R `
 F ) )
38373adant2r 1179 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  F  e.  T )  ->  ( R `  ( U `  F ) )  .<_  ( R `  F ) )
392, 7, 8, 12, 9tendotp 30875 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  V  e.  E  /\  F  e.  T
)  ->  ( R `  ( V `  F
) )  .<_  ( R `
 F ) )
40393adant2l 1178 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  F  e.  T )  ->  ( R `  ( V `  F ) )  .<_  ( R `  F ) )
411, 2, 23latjle12 14418 . . . 4  |-  ( ( K  e.  Lat  /\  ( ( R `  ( U `  F ) )  e.  ( Base `  K )  /\  ( R `  ( V `  F ) )  e.  ( Base `  K
)  /\  ( R `  F )  e.  (
Base `  K )
) )  ->  (
( ( R `  ( U `  F ) )  .<_  ( R `  F )  /\  ( R `  ( V `  F ) )  .<_  ( R `  F ) )  <->  ( ( R `
 ( U `  F ) ) (
join `  K )
( R `  ( V `  F )
) )  .<_  ( R `
 F ) ) )
425, 18, 22, 28, 41syl13anc 1186 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  F  e.  T )  ->  (
( ( R `  ( U `  F ) )  .<_  ( R `  F )  /\  ( R `  ( V `  F ) )  .<_  ( R `  F ) )  <->  ( ( R `
 ( U `  F ) ) (
join `  K )
( R `  ( V `  F )
) )  .<_  ( R `
 F ) ) )
4338, 40, 42mpbi2and 888 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  F  e.  T )  ->  (
( R `  ( U `  F )
) ( join `  K
) ( R `  ( V `  F ) ) )  .<_  ( R `
 F ) )
441, 2, 5, 14, 25, 28, 36, 43lattrd 14414 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  F  e.  T )  ->  ( R `  ( ( U P V ) `  F ) )  .<_  ( R `  F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   class class class wbr 4153    e. cmpt 4207    o. ccom 4822   ` cfv 5394  (class class class)co 6020    e. cmpt2 6022   Basecbs 13396   lecple 13463   joincjn 14328   Latclat 14401   HLchlt 29465   LHypclh 30098   LTrncltrn 30215   trLctrl 30272   TEndoctendo 30866
This theorem is referenced by:  tendoplcl  30895
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 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-iin 4038  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-undef 6479  df-riota 6485  df-map 6956  df-poset 14330  df-plt 14342  df-lub 14358  df-glb 14359  df-join 14360  df-meet 14361  df-p0 14395  df-p1 14396  df-lat 14402  df-clat 14464  df-oposet 29291  df-ol 29293  df-oml 29294  df-covers 29381  df-ats 29382  df-atl 29413  df-cvlat 29437  df-hlat 29466  df-llines 29612  df-lplanes 29613  df-lvols 29614  df-lines 29615  df-psubsp 29617  df-pmap 29618  df-padd 29910  df-lhyp 30102  df-laut 30103  df-ldil 30218  df-ltrn 30219  df-trl 30273  df-tendo 30869
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