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Theorem tendo0tp 31586
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 31584 . . . . 5  |-  ( F  e.  T  ->  ( O `  F )  =  (  _I  |`  B ) )
43adantl 453 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( O `  F )  =  (  _I  |`  B )
)
54fveq2d 5732 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( R `  ( O `  F
) )  =  ( R `  (  _I  |`  B ) ) )
6 eqid 2436 . . . . 5  |-  ( 0.
`  K )  =  ( 0. `  K
)
7 tendo0.h . . . . 5  |-  H  =  ( LHyp `  K
)
8 tendo0tp.r . . . . 5  |-  R  =  ( ( trL `  K
) `  W )
92, 6, 7, 8trlid0 30973 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( R `  (  _I  |`  B ) )  =  ( 0. `  K ) )
109adantr 452 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( R `  (  _I  |`  B ) )  =  ( 0.
`  K ) )
115, 10eqtrd 2468 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( R `  ( O `  F
) )  =  ( 0. `  K ) )
12 hlop 30160 . . . 4  |-  ( K  e.  HL  ->  K  e.  OP )
1312ad2antrr 707 . . 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 30961 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( R `  F )  e.  B
)
16 tendo0tp.l . . . 4  |-  .<_  =  ( le `  K )
172, 16, 6op0le 29984 . . 3  |-  ( ( K  e.  OP  /\  ( R `  F )  e.  B )  -> 
( 0. `  K
)  .<_  ( R `  F ) )
1813, 15, 17syl2anc 643 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( 0. `  K )  .<_  ( R `
 F ) )
1911, 18eqbrtrd 4232 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 359    = wceq 1652    e. wcel 1725   class class class wbr 4212    e. cmpt 4266    _I cid 4493    |` cres 4880   ` cfv 5454   Basecbs 13469   lecple 13536   0.cp0 14466   OPcops 29970   HLchlt 30148   LHypclh 30781   LTrncltrn 30898   trLctrl 30955   TEndoctendo 31549
This theorem is referenced by:  tendo0cl  31587
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-undef 6543  df-riota 6549  df-map 7020  df-poset 14403  df-plt 14415  df-lub 14431  df-glb 14432  df-join 14433  df-meet 14434  df-p0 14468  df-p1 14469  df-lat 14475  df-clat 14537  df-oposet 29974  df-ol 29976  df-oml 29977  df-covers 30064  df-ats 30065  df-atl 30096  df-cvlat 30120  df-hlat 30149  df-lhyp 30785  df-laut 30786  df-ldil 30901  df-ltrn 30902  df-trl 30956
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