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Theorem tendoconid 31018
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 30757 . . 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 30952 . . . . . . . 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 5596 . . . . . . 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 30956 . . . . . . . . . 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 31014 . . . . . . . . . . . 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 2481 . . . . . . . . . 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 31014 . . . . . . . . 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 2481 . . . . . . 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 2464 . . . . 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 30961 . . . . . . . 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 31014 . . . . . . 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 2481 . . . . 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 2666 . 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 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544    e. cmpt 4077    _I cid 4304    |` cres 4691    o. ccom 4693   -->wf 5251   ` cfv 5255   Basecbs 13148   HLchlt 29540   LHypclh 30173   LTrncltrn 30290   TEndoctendo 30941
This theorem is referenced by:  erngdvlem4  31180  erngdvlem4-rN  31188
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-map 6774  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-p1 14146  df-lat 14152  df-clat 14214  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-llines 29687  df-lplanes 29688  df-lvols 29689  df-lines 29690  df-psubsp 29692  df-pmap 29693  df-padd 29985  df-lhyp 30177  df-laut 30178  df-ldil 30293  df-ltrn 30294  df-trl 30348  df-tendo 30944
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