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Theorem tendoconid 31640
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 31379 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. g  e.  T  g  =/=  (  _I  |`  B ) )
543ad2ant1 976 . 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 958 . . . . . . . 8  |-  ( ( ( ( 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 1010 . . . . . . . 8  |-  ( ( ( ( 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 . . . . . . . . 9  |-  E  =  ( ( TEndo `  K
) `  W )
92, 3, 8tendof 31574 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  V  e.  E
)  ->  V : T
--> T )
106, 7, 9syl2anc 642 . . . . . . 7  |-  ( ( ( ( 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 732 . . . . . . 7  |-  ( ( ( ( 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 5612 . . . . . . 7  |-  ( ( V : T --> T  /\  g  e.  T )  ->  ( ( U  o.  V ) `  g
)  =  ( U `
 ( V `  g ) ) )
1310, 11, 12syl2anc 642 . . . . . 6  |-  ( ( ( ( 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 1009 . . . . . . 7  |-  ( ( ( ( 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 1008 . . . . . . . . 9  |-  ( ( ( ( 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 31578 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  V  e.  E  /\  g  e.  T
)  ->  ( V `  g )  e.  T
)
176, 7, 11, 16syl3anc 1182 . . . . . . . . 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 )  e.  T )
18 simpl3r 1011 . . . . . . . . . 10  |-  ( ( ( ( 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 447 . . . . . . . . . . . 12  |-  ( ( ( ( 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 . . . . . . . . . . . . 13  |-  O  =  ( f  e.  T  |->  (  _I  |`  B ) )
211, 2, 3, 8, 20tendoid0 31636 . . . . . . . . . . . 12  |-  ( ( ( 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 1182 . . . . . . . . . . 11  |-  ( ( ( ( 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 2494 . . . . . . . . . 10  |-  ( ( ( ( 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 223 . . . . . . . . 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 ) )
251, 2, 3, 8, 20tendoid0 31636 . . . . . . . . 9  |-  ( ( ( 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 1186 . . . . . . . 8  |-  ( ( ( ( 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 2494 . . . . . . 7  |-  ( ( ( ( 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 223 . . . . . 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 ) )
2913, 28eqnetrd 2477 . . . . 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
) `  g )  =/=  (  _I  |`  B ) )
302, 8tendococl 31583 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E  /\  V  e.  E
)  ->  ( U  o.  V )  e.  E
)
316, 15, 7, 30syl3anc 1182 . . . . . . 7  |-  ( ( ( ( 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 31636 . . . . . . 7  |-  ( ( ( 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 1182 . . . . . 6  |-  ( ( ( ( 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 2494 . . . . 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 ) `  g
)  =/=  (  _I  |`  B )  <->  ( U  o.  V )  =/=  O
) )
3529, 34mpbid 201 . . . 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 )  =/=  O )
3635exp32 588 . . 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 )  =/=  O
) ) )
3736rexlimdv 2679 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  U  =/= 
O )  /\  ( V  e.  E  /\  V  =/=  O ) )  ->  ( E. g  e.  T  g  =/=  (  _I  |`  B )  ->  ( U  o.  V )  =/=  O
) )
385, 37mpd 14 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 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   E.wrex 2557    e. cmpt 4093    _I cid 4320    |` cres 4707    o. ccom 4709   -->wf 5267   ` cfv 5271   Basecbs 13164   HLchlt 30162   LHypclh 30795   LTrncltrn 30912   TEndoctendo 31563
This theorem is referenced by:  erngdvlem4  31802  erngdvlem4-rN  31810
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-3or 935  df-3an 936  df-tru 1310  df-fal 1311  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-rmo 2564  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-iin 3924  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-llines 30309  df-lplanes 30310  df-lvols 30311  df-lines 30312  df-psubsp 30314  df-pmap 30315  df-padd 30607  df-lhyp 30799  df-laut 30800  df-ldil 30915  df-ltrn 30916  df-trl 30970  df-tendo 31566
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