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Theorem tendoplcom 31506
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 31505 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E  /\  V  e.  E
)  ->  ( U P V )  e.  E
)
72, 3, 4, 5tendoplcl 31505 . . 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 31491 . . . . . 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 31491 . . . . . 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 31462 . . . . 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 31501 . . . . 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 31501 . . . . 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 2477 . . 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 2781 . 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 31488 . 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 1652    e. wcel 1725   A.wral 2697    e. cmpt 4258    o. ccom 4874   ` cfv 5446  (class class class)co 6073    e. cmpt2 6075   HLchlt 30075   LHypclh 30708   LTrncltrn 30825   TEndoctendo 31476
This theorem is referenced by:  tendo0plr  31516  tendoipl2  31522  erngdvlem2N  31713  erngdvlem2-rN  31721  dvhvaddcomN  31821
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-undef 6535  df-riota 6541  df-map 7012  df-poset 14395  df-plt 14407  df-lub 14423  df-glb 14424  df-join 14425  df-meet 14426  df-p0 14460  df-p1 14461  df-lat 14467  df-clat 14529  df-oposet 29901  df-ol 29903  df-oml 29904  df-covers 29991  df-ats 29992  df-atl 30023  df-cvlat 30047  df-hlat 30076  df-llines 30222  df-lplanes 30223  df-lvols 30224  df-lines 30225  df-psubsp 30227  df-pmap 30228  df-padd 30520  df-lhyp 30712  df-laut 30713  df-ldil 30828  df-ltrn 30829  df-trl 30883  df-tendo 31479
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