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Theorem dvhvaddcomN 34741
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 457 . . . 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 6606 . . . . 5  |-  ( F  e.  ( T  X.  E )  ->  ( 1st `  F )  e.  T )
32ad2antrl 727 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  ( T  X.  E
)  /\  G  e.  ( T  X.  E
) ) )  -> 
( 1st `  F
)  e.  T )
4 xp1st 6606 . . . . 5  |-  ( G  e.  ( T  X.  E )  ->  ( 1st `  G )  e.  T )
54ad2antll 728 . . . 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 34382 . . . 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 1218 . . 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 6607 . . . . . 6  |-  ( F  e.  ( T  X.  E )  ->  ( 2nd `  F )  e.  E )
11 xp2nd 6607 . . . . . 6  |-  ( G  e.  ( T  X.  E )  ->  ( 2nd `  G )  e.  E )
1210, 11anim12i 566 . . . . 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 2443 . . . . . . 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 34426 . . . . . 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 1188 . . . . 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 474 . . . 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 34729 . . . . . 6  |-  ( ( K  e.  HL  /\  W  e.  H )  -> 
.+^  =  ( a  e.  E ,  b  e.  E  |->  ( c  e.  T  |->  ( ( a `  c )  o.  ( b `  c ) ) ) ) )
2221adantr 465 . . . . 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 6108 . . . 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 6108 . . . 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 2485 . . 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 4067 . 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 34737 . 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 34737 . . 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 800 . 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 2485 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 1369    e. wcel 1756   <.cop 3883    e. cmpt 4350    X. cxp 4838    o. ccom 4844   ` cfv 5418  (class class class)co 6091    e. cmpt2 6093   1stc1st 6575   2ndc2nd 6576   +g cplusg 14238  Scalarcsca 14241   HLchlt 32995   LHypclh 33628   LTrncltrn 33745   TEndoctendo 34396   DVecHcdvh 34723
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 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-riotaBAD 32604
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 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-iin 4174  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-undef 6792  df-recs 6832  df-rdg 6866  df-1o 6920  df-oadd 6924  df-er 7101  df-map 7216  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-nn 10323  df-2 10380  df-3 10381  df-4 10382  df-5 10383  df-6 10384  df-n0 10580  df-z 10647  df-uz 10862  df-fz 11438  df-struct 14176  df-ndx 14177  df-slot 14178  df-base 14179  df-plusg 14251  df-mulr 14252  df-sca 14254  df-vsca 14255  df-poset 15116  df-plt 15128  df-lub 15144  df-glb 15145  df-join 15146  df-meet 15147  df-p0 15209  df-p1 15210  df-lat 15216  df-clat 15278  df-oposet 32821  df-ol 32823  df-oml 32824  df-covers 32911  df-ats 32912  df-atl 32943  df-cvlat 32967  df-hlat 32996  df-llines 33142  df-lplanes 33143  df-lvols 33144  df-lines 33145  df-psubsp 33147  df-pmap 33148  df-padd 33440  df-lhyp 33632  df-laut 33633  df-ldil 33748  df-ltrn 33749  df-trl 33803  df-tendo 34399  df-edring 34401  df-dvech 34724
This theorem is referenced by: (None)
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