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Theorem tendoplcom 31276
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 957 . 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 31275 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E  /\  V  e.  E
)  ->  ( U P V )  e.  E
)
72, 3, 4, 5tendoplcl 31275 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  V  e.  E  /\  U  e.  E
)  ->  ( V P U )  e.  E
)
873com23 1159 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E  /\  V  e.  E
)  ->  ( V P U )  e.  E
)
9 simpl1 960 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E  /\  V  e.  E )  /\  g  e.  T )  ->  ( K  e.  HL  /\  W  e.  H ) )
10 simpl2 961 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E  /\  V  e.  E )  /\  g  e.  T )  ->  U  e.  E )
11 simpr 448 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E  /\  V  e.  E )  /\  g  e.  T )  ->  g  e.  T )
122, 3, 4tendocl 31261 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E  /\  g  e.  T
)  ->  ( U `  g )  e.  T
)
139, 10, 11, 12syl3anc 1184 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E  /\  V  e.  E )  /\  g  e.  T )  ->  ( U `  g )  e.  T )
14 simpl3 962 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E  /\  V  e.  E )  /\  g  e.  T )  ->  V  e.  E )
152, 3, 4tendocl 31261 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  V  e.  E  /\  g  e.  T
)  ->  ( V `  g )  e.  T
)
169, 14, 11, 15syl3anc 1184 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E  /\  V  e.  E )  /\  g  e.  T )  ->  ( V `  g )  e.  T )
172, 3ltrncom 31232 . . . . 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 1184 . . . 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 31271 . . . . 5  |-  ( ( U  e.  E  /\  V  e.  E  /\  g  e.  T )  ->  ( ( U P V ) `  g
)  =  ( ( U `  g )  o.  ( V `  g ) ) )
2010, 14, 11, 19syl3anc 1184 . . . 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 31271 . . . . 5  |-  ( ( V  e.  E  /\  U  e.  E  /\  g  e.  T )  ->  ( ( V P U ) `  g
)  =  ( ( V `  g )  o.  ( U `  g ) ) )
2214, 10, 11, 21syl3anc 1184 . . . 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 2454 . . 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 2757 . 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 31258 . 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 1189 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E  /\  V  e.  E
)  ->  ( U P V )  =  ( V P U ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   A.wral 2674    e. cmpt 4234    o. ccom 4849   ` cfv 5421  (class class class)co 6048    e. cmpt2 6050   HLchlt 29845   LHypclh 30478   LTrncltrn 30595   TEndoctendo 31246
This theorem is referenced by:  tendo0plr  31286  tendoipl2  31292  erngdvlem2N  31483  erngdvlem2-rN  31491  dvhvaddcomN  31591
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-iin 4064  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-undef 6510  df-riota 6516  df-map 6987  df-poset 14366  df-plt 14378  df-lub 14394  df-glb 14395  df-join 14396  df-meet 14397  df-p0 14431  df-p1 14432  df-lat 14438  df-clat 14500  df-oposet 29671  df-ol 29673  df-oml 29674  df-covers 29761  df-ats 29762  df-atl 29793  df-cvlat 29817  df-hlat 29846  df-llines 29992  df-lplanes 29993  df-lvols 29994  df-lines 29995  df-psubsp 29997  df-pmap 29998  df-padd 30290  df-lhyp 30482  df-laut 30483  df-ldil 30598  df-ltrn 30599  df-trl 30653  df-tendo 31249
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