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Theorem tendoconid 30943
Description: The composition (product) of trace-preserving endormorphisms is nonzero when each argument is nonzero. (Contributed by NM, 8-Aug-2013.)
Hypotheses
Ref Expression
tendoid0.b  |-  B  =  ( Base `  K
)
tendoid0.h  |-  H  =  ( LHyp `  K
)
tendoid0.t  |-  T  =  ( ( LTrn `  K
) `  W )
tendoid0.e  |-  E  =  ( ( TEndo `  K
) `  W )
tendoid0.o  |-  O  =  ( f  e.  T  |->  (  _I  |`  B ) )
Assertion
Ref Expression
tendoconid  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  U  =/= 
O )  /\  ( V  e.  E  /\  V  =/=  O ) )  ->  ( U  o.  V )  =/=  O
)
Distinct variable groups:    B, f    T, f
Allowed substitution hints:    U( f)    E( f)    H( f)    K( f)    O( f)    V( f)    W( f)

Proof of Theorem tendoconid
Dummy variable  g is distinct from all other variables.
StepHypRef Expression
1 tendoid0.b . . . 4  |-  B  =  ( Base `  K
)
2 tendoid0.h . . . 4  |-  H  =  ( LHyp `  K
)
3 tendoid0.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
41, 2, 3cdlemftr0 30682 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. g  e.  T  g  =/=  (  _I  |`  B ) )
543ad2ant1 978 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  U  =/= 
O )  /\  ( V  e.  E  /\  V  =/=  O ) )  ->  E. g  e.  T  g  =/=  (  _I  |`  B ) )
6 simpl1 960 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  U  =/=  O )  /\  ( V  e.  E  /\  V  =/=  O
) )  /\  (
g  e.  T  /\  g  =/=  (  _I  |`  B ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
7 simpl3l 1012 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  U  =/=  O )  /\  ( V  e.  E  /\  V  =/=  O
) )  /\  (
g  e.  T  /\  g  =/=  (  _I  |`  B ) ) )  ->  V  e.  E )
8 tendoid0.e . . . . . . 7  |-  E  =  ( ( TEndo `  K
) `  W )
92, 3, 8tendof 30877 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  V  e.  E
)  ->  V : T
--> T )
106, 7, 9syl2anc 643 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  U  =/=  O )  /\  ( V  e.  E  /\  V  =/=  O
) )  /\  (
g  e.  T  /\  g  =/=  (  _I  |`  B ) ) )  ->  V : T --> T )
11 simprl 733 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  U  =/=  O )  /\  ( V  e.  E  /\  V  =/=  O
) )  /\  (
g  e.  T  /\  g  =/=  (  _I  |`  B ) ) )  ->  g  e.  T )
12 fvco3 5739 . . . . 5  |-  ( ( V : T --> T  /\  g  e.  T )  ->  ( ( U  o.  V ) `  g
)  =  ( U `
 ( V `  g ) ) )
1310, 11, 12syl2anc 643 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  U  =/=  O )  /\  ( V  e.  E  /\  V  =/=  O
) )  /\  (
g  e.  T  /\  g  =/=  (  _I  |`  B ) ) )  ->  (
( U  o.  V
) `  g )  =  ( U `  ( V `  g ) ) )
14 simpl2r 1011 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  U  =/=  O )  /\  ( V  e.  E  /\  V  =/=  O
) )  /\  (
g  e.  T  /\  g  =/=  (  _I  |`  B ) ) )  ->  U  =/=  O )
15 simpl2l 1010 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  U  =/=  O )  /\  ( V  e.  E  /\  V  =/=  O
) )  /\  (
g  e.  T  /\  g  =/=  (  _I  |`  B ) ) )  ->  U  e.  E )
162, 3, 8tendocl 30881 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  V  e.  E  /\  g  e.  T
)  ->  ( V `  g )  e.  T
)
176, 7, 11, 16syl3anc 1184 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  U  =/=  O )  /\  ( V  e.  E  /\  V  =/=  O
) )  /\  (
g  e.  T  /\  g  =/=  (  _I  |`  B ) ) )  ->  ( V `  g )  e.  T )
18 simpl3r 1013 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  U  =/=  O )  /\  ( V  e.  E  /\  V  =/=  O
) )  /\  (
g  e.  T  /\  g  =/=  (  _I  |`  B ) ) )  ->  V  =/=  O )
19 simpr 448 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  U  =/=  O )  /\  ( V  e.  E  /\  V  =/=  O
) )  /\  (
g  e.  T  /\  g  =/=  (  _I  |`  B ) ) )  ->  (
g  e.  T  /\  g  =/=  (  _I  |`  B ) ) )
20 tendoid0.o . . . . . . . . . . 11  |-  O  =  ( f  e.  T  |->  (  _I  |`  B ) )
211, 2, 3, 8, 20tendoid0 30939 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  V  e.  E  /\  ( g  e.  T  /\  g  =/=  (  _I  |`  B ) ) )  ->  ( ( V `  g )  =  (  _I  |`  B )  <-> 
V  =  O ) )
226, 7, 19, 21syl3anc 1184 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  U  =/=  O )  /\  ( V  e.  E  /\  V  =/=  O
) )  /\  (
g  e.  T  /\  g  =/=  (  _I  |`  B ) ) )  ->  (
( V `  g
)  =  (  _I  |`  B )  <->  V  =  O ) )
2322necon3bid 2585 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  U  =/=  O )  /\  ( V  e.  E  /\  V  =/=  O
) )  /\  (
g  e.  T  /\  g  =/=  (  _I  |`  B ) ) )  ->  (
( V `  g
)  =/=  (  _I  |`  B )  <->  V  =/=  O ) )
2418, 23mpbird 224 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  U  =/=  O )  /\  ( V  e.  E  /\  V  =/=  O
) )  /\  (
g  e.  T  /\  g  =/=  (  _I  |`  B ) ) )  ->  ( V `  g )  =/=  (  _I  |`  B ) )
251, 2, 3, 8, 20tendoid0 30939 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E  /\  ( ( V `  g )  e.  T  /\  ( V `  g
)  =/=  (  _I  |`  B ) ) )  ->  ( ( U `
 ( V `  g ) )  =  (  _I  |`  B )  <-> 
U  =  O ) )
266, 15, 17, 24, 25syl112anc 1188 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  U  =/=  O )  /\  ( V  e.  E  /\  V  =/=  O
) )  /\  (
g  e.  T  /\  g  =/=  (  _I  |`  B ) ) )  ->  (
( U `  ( V `  g )
)  =  (  _I  |`  B )  <->  U  =  O ) )
2726necon3bid 2585 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  U  =/=  O )  /\  ( V  e.  E  /\  V  =/=  O
) )  /\  (
g  e.  T  /\  g  =/=  (  _I  |`  B ) ) )  ->  (
( U `  ( V `  g )
)  =/=  (  _I  |`  B )  <->  U  =/=  O ) )
2814, 27mpbird 224 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  U  =/=  O )  /\  ( V  e.  E  /\  V  =/=  O
) )  /\  (
g  e.  T  /\  g  =/=  (  _I  |`  B ) ) )  ->  ( U `  ( V `  g ) )  =/=  (  _I  |`  B ) )
2913, 28eqnetrd 2568 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  U  =/=  O )  /\  ( V  e.  E  /\  V  =/=  O
) )  /\  (
g  e.  T  /\  g  =/=  (  _I  |`  B ) ) )  ->  (
( U  o.  V
) `  g )  =/=  (  _I  |`  B ) )
302, 8tendococl 30886 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E  /\  V  e.  E
)  ->  ( U  o.  V )  e.  E
)
316, 15, 7, 30syl3anc 1184 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  U  =/=  O )  /\  ( V  e.  E  /\  V  =/=  O
) )  /\  (
g  e.  T  /\  g  =/=  (  _I  |`  B ) ) )  ->  ( U  o.  V )  e.  E )
321, 2, 3, 8, 20tendoid0 30939 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  o.  V )  e.  E  /\  ( g  e.  T  /\  g  =/=  (  _I  |`  B ) ) )  ->  ( (
( U  o.  V
) `  g )  =  (  _I  |`  B )  <-> 
( U  o.  V
)  =  O ) )
336, 31, 19, 32syl3anc 1184 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  U  =/=  O )  /\  ( V  e.  E  /\  V  =/=  O
) )  /\  (
g  e.  T  /\  g  =/=  (  _I  |`  B ) ) )  ->  (
( ( U  o.  V ) `  g
)  =  (  _I  |`  B )  <->  ( U  o.  V )  =  O ) )
3433necon3bid 2585 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  U  =/=  O )  /\  ( V  e.  E  /\  V  =/=  O
) )  /\  (
g  e.  T  /\  g  =/=  (  _I  |`  B ) ) )  ->  (
( ( U  o.  V ) `  g
)  =/=  (  _I  |`  B )  <->  ( U  o.  V )  =/=  O
) )
3529, 34mpbid 202 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  U  =/=  O )  /\  ( V  e.  E  /\  V  =/=  O
) )  /\  (
g  e.  T  /\  g  =/=  (  _I  |`  B ) ) )  ->  ( U  o.  V )  =/=  O )
365, 35rexlimddv 2777 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  U  =/= 
O )  /\  ( V  e.  E  /\  V  =/=  O ) )  ->  ( U  o.  V )  =/=  O
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2550   E.wrex 2650    e. cmpt 4207    _I cid 4434    |` cres 4820    o. ccom 4822   -->wf 5390   ` cfv 5394   Basecbs 13396   HLchlt 29465   LHypclh 30098   LTrncltrn 30215   TEndoctendo 30866
This theorem is referenced by:  erngdvlem4  31105  erngdvlem4-rN  31113
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-fal 1326  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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