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Theorem dvhvaddcomN 35893
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 6811 . . . . 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 6811 . . . . 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 35534 . . . 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 1228 . . 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 6812 . . . . . 6  |-  ( F  e.  ( T  X.  E )  ->  ( 2nd `  F )  e.  E )
11 xp2nd 6812 . . . . . 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 2467 . . . . . . 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 35578 . . . . . 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 1197 . . . . 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 35881 . . . . . 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 6299 . . . 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 6299 . . . 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 2518 . . 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 4221 . 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 35889 . 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 35889 . . 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 2518 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 1379    e. wcel 1767   <.cop 4033    |-> cmpt 4505    X. cxp 4997    o. ccom 5003   ` cfv 5586  (class class class)co 6282    |-> cmpt2 6284   1stc1st 6779   2ndc2nd 6780   +g cplusg 14548  Scalarcsca 14551   HLchlt 34147   LHypclh 34780   LTrncltrn 34897   TEndoctendo 35548   DVecHcdvh 35875
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-riotaBAD 33756
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-undef 6999  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-map 7419  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-n0 10792  df-z 10861  df-uz 11079  df-fz 11669  df-struct 14485  df-ndx 14486  df-slot 14487  df-base 14488  df-plusg 14561  df-mulr 14562  df-sca 14564  df-vsca 14565  df-poset 15426  df-plt 15438  df-lub 15454  df-glb 15455  df-join 15456  df-meet 15457  df-p0 15519  df-p1 15520  df-lat 15526  df-clat 15588  df-oposet 33973  df-ol 33975  df-oml 33976  df-covers 34063  df-ats 34064  df-atl 34095  df-cvlat 34119  df-hlat 34148  df-llines 34294  df-lplanes 34295  df-lvols 34296  df-lines 34297  df-psubsp 34299  df-pmap 34300  df-padd 34592  df-lhyp 34784  df-laut 34785  df-ldil 34900  df-ltrn 34901  df-trl 34955  df-tendo 35551  df-edring 35553  df-dvech 35876
This theorem is referenced by: (None)
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