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Theorem tendoplcom 35455
Description: The endomorphism sum operation is commutative. (Contributed by NM, 11-Jun-2013.)
Hypotheses
Ref Expression
tendopl.h  |-  H  =  ( LHyp `  K
)
tendopl.t  |-  T  =  ( ( LTrn `  K
) `  W )
tendopl.e  |-  E  =  ( ( TEndo `  K
) `  W )
tendopl.p  |-  P  =  ( s  e.  E ,  t  e.  E  |->  ( f  e.  T  |->  ( ( s `  f )  o.  (
t `  f )
) ) )
Assertion
Ref Expression
tendoplcom  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E  /\  V  e.  E
)  ->  ( U P V )  =  ( V P U ) )
Distinct variable groups:    t, s, E    f, s, t, T   
f, W, s, t
Allowed substitution hints:    P( t, f, s)    U( t, f, s)    E( f)    H( t, f, s)    K( t, f, s)    V( t, f, s)

Proof of Theorem tendoplcom
Dummy variable  g is distinct from all other variables.
StepHypRef Expression
1 simp1 991 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E  /\  V  e.  E
)  ->  ( K  e.  HL  /\  W  e.  H ) )
2 tendopl.h . . 3  |-  H  =  ( LHyp `  K
)
3 tendopl.t . . 3  |-  T  =  ( ( LTrn `  K
) `  W )
4 tendopl.e . . 3  |-  E  =  ( ( TEndo `  K
) `  W )
5 tendopl.p . . 3  |-  P  =  ( s  e.  E ,  t  e.  E  |->  ( f  e.  T  |->  ( ( s `  f )  o.  (
t `  f )
) ) )
62, 3, 4, 5tendoplcl 35454 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E  /\  V  e.  E
)  ->  ( U P V )  e.  E
)
72, 3, 4, 5tendoplcl 35454 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  V  e.  E  /\  U  e.  E
)  ->  ( V P U )  e.  E
)
873com23 1197 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E  /\  V  e.  E
)  ->  ( V P U )  e.  E
)
9 simpl1 994 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E  /\  V  e.  E )  /\  g  e.  T )  ->  ( K  e.  HL  /\  W  e.  H ) )
10 simpl2 995 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E  /\  V  e.  E )  /\  g  e.  T )  ->  U  e.  E )
11 simpr 461 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E  /\  V  e.  E )  /\  g  e.  T )  ->  g  e.  T )
122, 3, 4tendocl 35440 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E  /\  g  e.  T
)  ->  ( U `  g )  e.  T
)
139, 10, 11, 12syl3anc 1223 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E  /\  V  e.  E )  /\  g  e.  T )  ->  ( U `  g )  e.  T )
14 simpl3 996 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E  /\  V  e.  E )  /\  g  e.  T )  ->  V  e.  E )
152, 3, 4tendocl 35440 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  V  e.  E  /\  g  e.  T
)  ->  ( V `  g )  e.  T
)
169, 14, 11, 15syl3anc 1223 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E  /\  V  e.  E )  /\  g  e.  T )  ->  ( V `  g )  e.  T )
172, 3ltrncom 35411 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U `  g )  e.  T  /\  ( V `  g
)  e.  T )  ->  ( ( U `
 g )  o.  ( V `  g
) )  =  ( ( V `  g
)  o.  ( U `
 g ) ) )
189, 13, 16, 17syl3anc 1223 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E  /\  V  e.  E )  /\  g  e.  T )  ->  (
( U `  g
)  o.  ( V `
 g ) )  =  ( ( V `
 g )  o.  ( U `  g
) ) )
195, 3tendopl2 35450 . . . . 5  |-  ( ( U  e.  E  /\  V  e.  E  /\  g  e.  T )  ->  ( ( U P V ) `  g
)  =  ( ( U `  g )  o.  ( V `  g ) ) )
2010, 14, 11, 19syl3anc 1223 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E  /\  V  e.  E )  /\  g  e.  T )  ->  (
( U P V ) `  g )  =  ( ( U `
 g )  o.  ( V `  g
) ) )
215, 3tendopl2 35450 . . . . 5  |-  ( ( V  e.  E  /\  U  e.  E  /\  g  e.  T )  ->  ( ( V P U ) `  g
)  =  ( ( V `  g )  o.  ( U `  g ) ) )
2214, 10, 11, 21syl3anc 1223 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E  /\  V  e.  E )  /\  g  e.  T )  ->  (
( V P U ) `  g )  =  ( ( V `
 g )  o.  ( U `  g
) ) )
2318, 20, 223eqtr4d 2513 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E  /\  V  e.  E )  /\  g  e.  T )  ->  (
( U P V ) `  g )  =  ( ( V P U ) `  g ) )
2423ralrimiva 2873 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E  /\  V  e.  E
)  ->  A. g  e.  T  ( ( U P V ) `  g )  =  ( ( V P U ) `  g ) )
252, 3, 4tendoeq1 35437 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( U P V )  e.  E  /\  ( V P U )  e.  E )  /\  A. g  e.  T  (
( U P V ) `  g )  =  ( ( V P U ) `  g ) )  -> 
( U P V )  =  ( V P U ) )
261, 6, 8, 24, 25syl121anc 1228 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E  /\  V  e.  E
)  ->  ( U P V )  =  ( V P U ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762   A.wral 2809    |-> cmpt 4500    o. ccom 4998   ` cfv 5581  (class class class)co 6277    |-> cmpt2 6279   HLchlt 34024   LHypclh 34657   LTrncltrn 34774   TEndoctendo 35425
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-riotaBAD 33633
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-iun 4322  df-iin 4323  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-1st 6776  df-2nd 6777  df-undef 6994  df-map 7414  df-poset 15424  df-plt 15436  df-lub 15452  df-glb 15453  df-join 15454  df-meet 15455  df-p0 15517  df-p1 15518  df-lat 15524  df-clat 15586  df-oposet 33850  df-ol 33852  df-oml 33853  df-covers 33940  df-ats 33941  df-atl 33972  df-cvlat 33996  df-hlat 34025  df-llines 34171  df-lplanes 34172  df-lvols 34173  df-lines 34174  df-psubsp 34176  df-pmap 34177  df-padd 34469  df-lhyp 34661  df-laut 34662  df-ldil 34777  df-ltrn 34778  df-trl 34832  df-tendo 35428
This theorem is referenced by:  tendo0plr  35465  tendoipl2  35471  erngdvlem2N  35662  erngdvlem2-rN  35670  dvhvaddcomN  35770
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