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Theorem tendoplcom 34424
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 988 . 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 34423 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E  /\  V  e.  E
)  ->  ( U P V )  e.  E
)
72, 3, 4, 5tendoplcl 34423 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  V  e.  E  /\  U  e.  E
)  ->  ( V P U )  e.  E
)
873com23 1193 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E  /\  V  e.  E
)  ->  ( V P U )  e.  E
)
9 simpl1 991 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E  /\  V  e.  E )  /\  g  e.  T )  ->  ( K  e.  HL  /\  W  e.  H ) )
10 simpl2 992 . . . . . 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 34409 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E  /\  g  e.  T
)  ->  ( U `  g )  e.  T
)
139, 10, 11, 12syl3anc 1218 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E  /\  V  e.  E )  /\  g  e.  T )  ->  ( U `  g )  e.  T )
14 simpl3 993 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E  /\  V  e.  E )  /\  g  e.  T )  ->  V  e.  E )
152, 3, 4tendocl 34409 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  V  e.  E  /\  g  e.  T
)  ->  ( V `  g )  e.  T
)
169, 14, 11, 15syl3anc 1218 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E  /\  V  e.  E )  /\  g  e.  T )  ->  ( V `  g )  e.  T )
172, 3ltrncom 34380 . . . . 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 1218 . . . 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 34419 . . . . 5  |-  ( ( U  e.  E  /\  V  e.  E  /\  g  e.  T )  ->  ( ( U P V ) `  g
)  =  ( ( U `  g )  o.  ( V `  g ) ) )
2010, 14, 11, 19syl3anc 1218 . . . 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 34419 . . . . 5  |-  ( ( V  e.  E  /\  U  e.  E  /\  g  e.  T )  ->  ( ( V P U ) `  g
)  =  ( ( V `  g )  o.  ( U `  g ) ) )
2214, 10, 11, 21syl3anc 1218 . . . 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 2484 . . 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 2798 . 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 34406 . 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 1223 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 965    = wceq 1369    e. wcel 1756   A.wral 2714    e. cmpt 4349    o. ccom 4843   ` cfv 5417  (class class class)co 6090    e. cmpt2 6092   HLchlt 32993   LHypclh 33626   LTrncltrn 33743   TEndoctendo 34394
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-riotaBAD 32602
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-iun 4172  df-iin 4173  df-br 4292  df-opab 4350  df-mpt 4351  df-id 4635  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-1st 6576  df-2nd 6577  df-undef 6791  df-map 7215  df-poset 15115  df-plt 15127  df-lub 15143  df-glb 15144  df-join 15145  df-meet 15146  df-p0 15208  df-p1 15209  df-lat 15215  df-clat 15277  df-oposet 32819  df-ol 32821  df-oml 32822  df-covers 32909  df-ats 32910  df-atl 32941  df-cvlat 32965  df-hlat 32994  df-llines 33140  df-lplanes 33141  df-lvols 33142  df-lines 33143  df-psubsp 33145  df-pmap 33146  df-padd 33438  df-lhyp 33630  df-laut 33631  df-ldil 33746  df-ltrn 33747  df-trl 33801  df-tendo 34397
This theorem is referenced by:  tendo0plr  34434  tendoipl2  34440  erngdvlem2N  34631  erngdvlem2-rN  34639  dvhvaddcomN  34739
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