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Theorem dvhvaddcomN 34463
Description: Commutativity of vector sum. (Contributed by NM, 26-Oct-2013.) (Revised by Mario Carneiro, 23-Jun-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
dvhvaddcl.h  |-  H  =  ( LHyp `  K
)
dvhvaddcl.t  |-  T  =  ( ( LTrn `  K
) `  W )
dvhvaddcl.e  |-  E  =  ( ( TEndo `  K
) `  W )
dvhvaddcl.u  |-  U  =  ( ( DVecH `  K
) `  W )
dvhvaddcl.d  |-  D  =  (Scalar `  U )
dvhvaddcl.p  |-  .+^  =  ( +g  `  D )
dvhvaddcl.a  |-  .+  =  ( +g  `  U )
Assertion
Ref Expression
dvhvaddcomN  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  ( T  X.  E
)  /\  G  e.  ( T  X.  E
) ) )  -> 
( F  .+  G
)  =  ( G 
.+  F ) )

Proof of Theorem dvhvaddcomN
Dummy variables  a 
b  c are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 454 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  ( T  X.  E
)  /\  G  e.  ( T  X.  E
) ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
2 xp1st 6605 . . . . 5  |-  ( F  e.  ( T  X.  E )  ->  ( 1st `  F )  e.  T )
32ad2antrl 722 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  ( T  X.  E
)  /\  G  e.  ( T  X.  E
) ) )  -> 
( 1st `  F
)  e.  T )
4 xp1st 6605 . . . . 5  |-  ( G  e.  ( T  X.  E )  ->  ( 1st `  G )  e.  T )
54ad2antll 723 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  ( T  X.  E
)  /\  G  e.  ( T  X.  E
) ) )  -> 
( 1st `  G
)  e.  T )
6 dvhvaddcl.h . . . . 5  |-  H  =  ( LHyp `  K
)
7 dvhvaddcl.t . . . . 5  |-  T  =  ( ( LTrn `  K
) `  W )
86, 7ltrncom 34104 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( 1st `  F
)  e.  T  /\  ( 1st `  G )  e.  T )  -> 
( ( 1st `  F
)  o.  ( 1st `  G ) )  =  ( ( 1st `  G
)  o.  ( 1st `  F ) ) )
91, 3, 5, 8syl3anc 1213 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  ( T  X.  E
)  /\  G  e.  ( T  X.  E
) ) )  -> 
( ( 1st `  F
)  o.  ( 1st `  G ) )  =  ( ( 1st `  G
)  o.  ( 1st `  F ) ) )
10 xp2nd 6606 . . . . . 6  |-  ( F  e.  ( T  X.  E )  ->  ( 2nd `  F )  e.  E )
11 xp2nd 6606 . . . . . 6  |-  ( G  e.  ( T  X.  E )  ->  ( 2nd `  G )  e.  E )
1210, 11anim12i 563 . . . . 5  |-  ( ( F  e.  ( T  X.  E )  /\  G  e.  ( T  X.  E ) )  -> 
( ( 2nd `  F
)  e.  E  /\  ( 2nd `  G )  e.  E ) )
13 dvhvaddcl.e . . . . . . 7  |-  E  =  ( ( TEndo `  K
) `  W )
14 eqid 2441 . . . . . . 7  |-  ( a  e.  E ,  b  e.  E  |->  ( c  e.  T  |->  ( ( a `  c )  o.  ( b `  c ) ) ) )  =  ( a  e.  E ,  b  e.  E  |->  ( c  e.  T  |->  ( ( a `  c )  o.  ( b `  c ) ) ) )
156, 7, 13, 14tendoplcom 34148 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( 2nd `  F
)  e.  E  /\  ( 2nd `  G )  e.  E )  -> 
( ( 2nd `  F
) ( a  e.  E ,  b  e.  E  |->  ( c  e.  T  |->  ( ( a `
 c )  o.  ( b `  c
) ) ) ) ( 2nd `  G
) )  =  ( ( 2nd `  G
) ( a  e.  E ,  b  e.  E  |->  ( c  e.  T  |->  ( ( a `
 c )  o.  ( b `  c
) ) ) ) ( 2nd `  F
) ) )
16153expb 1183 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( 2nd `  F )  e.  E  /\  ( 2nd `  G
)  e.  E ) )  ->  ( ( 2nd `  F ) ( a  e.  E , 
b  e.  E  |->  ( c  e.  T  |->  ( ( a `  c
)  o.  ( b `
 c ) ) ) ) ( 2nd `  G ) )  =  ( ( 2nd `  G
) ( a  e.  E ,  b  e.  E  |->  ( c  e.  T  |->  ( ( a `
 c )  o.  ( b `  c
) ) ) ) ( 2nd `  F
) ) )
1712, 16sylan2 471 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  ( T  X.  E
)  /\  G  e.  ( T  X.  E
) ) )  -> 
( ( 2nd `  F
) ( a  e.  E ,  b  e.  E  |->  ( c  e.  T  |->  ( ( a `
 c )  o.  ( b `  c
) ) ) ) ( 2nd `  G
) )  =  ( ( 2nd `  G
) ( a  e.  E ,  b  e.  E  |->  ( c  e.  T  |->  ( ( a `
 c )  o.  ( b `  c
) ) ) ) ( 2nd `  F
) ) )
18 dvhvaddcl.u . . . . . . 7  |-  U  =  ( ( DVecH `  K
) `  W )
19 dvhvaddcl.d . . . . . . 7  |-  D  =  (Scalar `  U )
20 dvhvaddcl.p . . . . . . 7  |-  .+^  =  ( +g  `  D )
216, 7, 13, 18, 19, 14, 20dvhfplusr 34451 . . . . . 6  |-  ( ( K  e.  HL  /\  W  e.  H )  -> 
.+^  =  ( a  e.  E ,  b  e.  E  |->  ( c  e.  T  |->  ( ( a `  c )  o.  ( b `  c ) ) ) ) )
2221adantr 462 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  ( T  X.  E
)  /\  G  e.  ( T  X.  E
) ) )  ->  .+^  =  ( a  e.  E ,  b  e.  E  |->  ( c  e.  T  |->  ( ( a `
 c )  o.  ( b `  c
) ) ) ) )
2322oveqd 6107 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  ( T  X.  E
)  /\  G  e.  ( T  X.  E
) ) )  -> 
( ( 2nd `  F
)  .+^  ( 2nd `  G
) )  =  ( ( 2nd `  F
) ( a  e.  E ,  b  e.  E  |->  ( c  e.  T  |->  ( ( a `
 c )  o.  ( b `  c
) ) ) ) ( 2nd `  G
) ) )
2422oveqd 6107 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  ( T  X.  E
)  /\  G  e.  ( T  X.  E
) ) )  -> 
( ( 2nd `  G
)  .+^  ( 2nd `  F
) )  =  ( ( 2nd `  G
) ( a  e.  E ,  b  e.  E  |->  ( c  e.  T  |->  ( ( a `
 c )  o.  ( b `  c
) ) ) ) ( 2nd `  F
) ) )
2517, 23, 243eqtr4d 2483 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  ( T  X.  E
)  /\  G  e.  ( T  X.  E
) ) )  -> 
( ( 2nd `  F
)  .+^  ( 2nd `  G
) )  =  ( ( 2nd `  G
)  .+^  ( 2nd `  F
) ) )
269, 25opeq12d 4064 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  ( T  X.  E
)  /\  G  e.  ( T  X.  E
) ) )  ->  <. ( ( 1st `  F
)  o.  ( 1st `  G ) ) ,  ( ( 2nd `  F
)  .+^  ( 2nd `  G
) ) >.  =  <. ( ( 1st `  G
)  o.  ( 1st `  F ) ) ,  ( ( 2nd `  G
)  .+^  ( 2nd `  F
) ) >. )
27 dvhvaddcl.a . . 3  |-  .+  =  ( +g  `  U )
286, 7, 13, 18, 19, 27, 20dvhvadd 34459 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  ( T  X.  E
)  /\  G  e.  ( T  X.  E
) ) )  -> 
( F  .+  G
)  =  <. (
( 1st `  F
)  o.  ( 1st `  G ) ) ,  ( ( 2nd `  F
)  .+^  ( 2nd `  G
) ) >. )
296, 7, 13, 18, 19, 27, 20dvhvadd 34459 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( G  e.  ( T  X.  E
)  /\  F  e.  ( T  X.  E
) ) )  -> 
( G  .+  F
)  =  <. (
( 1st `  G
)  o.  ( 1st `  F ) ) ,  ( ( 2nd `  G
)  .+^  ( 2nd `  F
) ) >. )
3029ancom2s 795 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  ( T  X.  E
)  /\  G  e.  ( T  X.  E
) ) )  -> 
( G  .+  F
)  =  <. (
( 1st `  G
)  o.  ( 1st `  F ) ) ,  ( ( 2nd `  G
)  .+^  ( 2nd `  F
) ) >. )
3126, 28, 303eqtr4d 2483 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  ( T  X.  E
)  /\  G  e.  ( T  X.  E
) ) )  -> 
( F  .+  G
)  =  ( G 
.+  F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1364    e. wcel 1761   <.cop 3880    e. cmpt 4347    X. cxp 4834    o. ccom 4840   ` cfv 5415  (class class class)co 6090    e. cmpt2 6092   1stc1st 6574   2ndc2nd 6575   +g cplusg 14234  Scalarcsca 14237   HLchlt 32717   LHypclh 33350   LTrncltrn 33467   TEndoctendo 34118   DVecHcdvh 34445
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 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-riotaBAD 32326
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-iin 4171  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-undef 6788  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-map 7212  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-n0 10576  df-z 10643  df-uz 10858  df-fz 11434  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-plusg 14247  df-mulr 14248  df-sca 14250  df-vsca 14251  df-poset 15112  df-plt 15124  df-lub 15140  df-glb 15141  df-join 15142  df-meet 15143  df-p0 15205  df-p1 15206  df-lat 15212  df-clat 15274  df-oposet 32543  df-ol 32545  df-oml 32546  df-covers 32633  df-ats 32634  df-atl 32665  df-cvlat 32689  df-hlat 32718  df-llines 32864  df-lplanes 32865  df-lvols 32866  df-lines 32867  df-psubsp 32869  df-pmap 32870  df-padd 33162  df-lhyp 33354  df-laut 33355  df-ldil 33470  df-ltrn 33471  df-trl 33525  df-tendo 34121  df-edring 34123  df-dvech 34446
This theorem is referenced by: (None)
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