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Theorem tendo0tp 31600
Description: Trace-preserving property of endomorphism additive identity. (Contributed by NM, 11-Jun-2013.)
Hypotheses
Ref Expression
tendo0.b  |-  B  =  ( Base `  K
)
tendo0.h  |-  H  =  ( LHyp `  K
)
tendo0.t  |-  T  =  ( ( LTrn `  K
) `  W )
tendo0.e  |-  E  =  ( ( TEndo `  K
) `  W )
tendo0.o  |-  O  =  ( f  e.  T  |->  (  _I  |`  B ) )
tendo0tp.l  |-  .<_  =  ( le `  K )
tendo0tp.r  |-  R  =  ( ( trL `  K
) `  W )
Assertion
Ref Expression
tendo0tp  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( R `  ( O `  F
) )  .<_  ( R `
 F ) )
Distinct variable groups:    B, f    T, f
Allowed substitution hints:    R( f)    E( f)    F( f)    H( f)    K( f)    .<_ ( f)    O( f)    W( f)

Proof of Theorem tendo0tp
StepHypRef Expression
1 tendo0.o . . . . . 6  |-  O  =  ( f  e.  T  |->  (  _I  |`  B ) )
2 tendo0.b . . . . . 6  |-  B  =  ( Base `  K
)
31, 2tendo02 31598 . . . . 5  |-  ( F  e.  T  ->  ( O `  F )  =  (  _I  |`  B ) )
43adantl 452 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( O `  F )  =  (  _I  |`  B )
)
54fveq2d 5545 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( R `  ( O `  F
) )  =  ( R `  (  _I  |`  B ) ) )
6 eqid 2296 . . . . 5  |-  ( 0.
`  K )  =  ( 0. `  K
)
7 tendo0.h . . . . 5  |-  H  =  ( LHyp `  K
)
8 tendo0tp.r . . . . 5  |-  R  =  ( ( trL `  K
) `  W )
92, 6, 7, 8trlid0 30987 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( R `  (  _I  |`  B ) )  =  ( 0. `  K ) )
109adantr 451 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( R `  (  _I  |`  B ) )  =  ( 0.
`  K ) )
115, 10eqtrd 2328 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( R `  ( O `  F
) )  =  ( 0. `  K ) )
12 hlop 30174 . . . 4  |-  ( K  e.  HL  ->  K  e.  OP )
1312ad2antrr 706 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  K  e.  OP )
14 tendo0.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
152, 7, 14, 8trlcl 30975 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( R `  F )  e.  B
)
16 tendo0tp.l . . . 4  |-  .<_  =  ( le `  K )
172, 16, 6op0le 29998 . . 3  |-  ( ( K  e.  OP  /\  ( R `  F )  e.  B )  -> 
( 0. `  K
)  .<_  ( R `  F ) )
1813, 15, 17syl2anc 642 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( 0. `  K )  .<_  ( R `
 F ) )
1911, 18eqbrtrd 4059 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( R `  ( O `  F
) )  .<_  ( R `
 F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   class class class wbr 4039    e. cmpt 4093    _I cid 4320    |` cres 4707   ` cfv 5271   Basecbs 13164   lecple 13231   0.cp0 14159   OPcops 29984   HLchlt 30162   LHypclh 30795   LTrncltrn 30912   trLctrl 30969   TEndoctendo 31563
This theorem is referenced by:  tendo0cl  31601
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-map 6790  df-poset 14096  df-plt 14108  df-lub 14124  df-glb 14125  df-join 14126  df-meet 14127  df-p0 14161  df-p1 14162  df-lat 14168  df-clat 14230  df-oposet 29988  df-ol 29990  df-oml 29991  df-covers 30078  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163  df-lhyp 30799  df-laut 30800  df-ldil 30915  df-ltrn 30916  df-trl 30970
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