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Theorem tendoconid 34836
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 34575 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. g  e.  T  g  =/=  (  _I  |`  B ) )
543ad2ant1 1009 . 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 991 . . . . . 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 1043 . . . . . 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 34770 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  V  e.  E
)  ->  V : T
--> T )
106, 7, 9syl2anc 661 . . . . 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 755 . . . . 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 5880 . . . . 5  |-  ( ( V : T --> T  /\  g  e.  T )  ->  ( ( U  o.  V ) `  g
)  =  ( U `
 ( V `  g ) ) )
1310, 11, 12syl2anc 661 . . . 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 1042 . . . . 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 1041 . . . . . . 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 34774 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  V  e.  E  /\  g  e.  T
)  ->  ( V `  g )  e.  T
)
176, 7, 11, 16syl3anc 1219 . . . . . . 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 1044 . . . . . . . 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 461 . . . . . . . . . 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 34832 . . . . . . . . . 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 1219 . . . . . . . . 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 2710 . . . . . . . 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 232 . . . . . . 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 34832 . . . . . . 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 1223 . . . . . 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 2710 . . . . 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 232 . . . 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 2745 . . 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 34779 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E  /\  V  e.  E
)  ->  ( U  o.  V )  e.  E
)
316, 15, 7, 30syl3anc 1219 . . . . 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 34832 . . . . 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 1219 . . . 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 2710 . . 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 210 . 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 2951 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2648   E.wrex 2800    |-> cmpt 4461    _I cid 4742    |` cres 4953    o. ccom 4955   -->wf 5525   ` cfv 5529   Basecbs 14296   HLchlt 33358   LHypclh 33991   LTrncltrn 34108   TEndoctendo 34759
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-riotaBAD 32967
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-iin 4285  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-1st 6690  df-2nd 6691  df-undef 6905  df-map 7329  df-poset 15239  df-plt 15251  df-lub 15267  df-glb 15268  df-join 15269  df-meet 15270  df-p0 15332  df-p1 15333  df-lat 15339  df-clat 15401  df-oposet 33184  df-ol 33186  df-oml 33187  df-covers 33274  df-ats 33275  df-atl 33306  df-cvlat 33330  df-hlat 33359  df-llines 33505  df-lplanes 33506  df-lvols 33507  df-lines 33508  df-psubsp 33510  df-pmap 33511  df-padd 33803  df-lhyp 33995  df-laut 33996  df-ldil 34111  df-ltrn 34112  df-trl 34166  df-tendo 34762
This theorem is referenced by:  erngdvlem4  34998  erngdvlem4-rN  35006
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