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Theorem dvhvaddcomN 31894
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 444 . . . 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 6376 . . . . 5  |-  ( F  e.  ( T  X.  E )  ->  ( 1st `  F )  e.  T )
32ad2antrl 709 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  ( T  X.  E
)  /\  G  e.  ( T  X.  E
) ) )  -> 
( 1st `  F
)  e.  T )
4 xp1st 6376 . . . . 5  |-  ( G  e.  ( T  X.  E )  ->  ( 1st `  G )  e.  T )
54ad2antll 710 . . . 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 31535 . . . 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 1184 . . 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 6377 . . . . . 6  |-  ( F  e.  ( T  X.  E )  ->  ( 2nd `  F )  e.  E )
11 xp2nd 6377 . . . . . 6  |-  ( G  e.  ( T  X.  E )  ->  ( 2nd `  G )  e.  E )
1210, 11anim12i 550 . . . . 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 2436 . . . . . . 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 31579 . . . . . 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 1154 . . . . 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 461 . . . 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 31882 . . . . . 6  |-  ( ( K  e.  HL  /\  W  e.  H )  -> 
.+^  =  ( a  e.  E ,  b  e.  E  |->  ( c  e.  T  |->  ( ( a `  c )  o.  ( b `  c ) ) ) ) )
2221adantr 452 . . . . 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 6098 . . . 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 6098 . . . 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 2478 . . 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 3992 . 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 31890 . 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 31890 . . 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 778 . 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 2478 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 set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   <.cop 3817    e. cmpt 4266    X. cxp 4876    o. ccom 4882   ` cfv 5454  (class class class)co 6081    e. cmpt2 6083   1stc1st 6347   2ndc2nd 6348   +g cplusg 13529  Scalarcsca 13532   HLchlt 30148   LHypclh 30781   LTrncltrn 30898   TEndoctendo 31549   DVecHcdvh 31876
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 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-iin 4096  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-undef 6543  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-map 7020  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-3 10059  df-4 10060  df-5 10061  df-6 10062  df-n0 10222  df-z 10283  df-uz 10489  df-fz 11044  df-struct 13471  df-ndx 13472  df-slot 13473  df-base 13474  df-plusg 13542  df-mulr 13543  df-sca 13545  df-vsca 13546  df-poset 14403  df-plt 14415  df-lub 14431  df-glb 14432  df-join 14433  df-meet 14434  df-p0 14468  df-p1 14469  df-lat 14475  df-clat 14537  df-oposet 29974  df-ol 29976  df-oml 29977  df-covers 30064  df-ats 30065  df-atl 30096  df-cvlat 30120  df-hlat 30149  df-llines 30295  df-lplanes 30296  df-lvols 30297  df-lines 30298  df-psubsp 30300  df-pmap 30301  df-padd 30593  df-lhyp 30785  df-laut 30786  df-ldil 30901  df-ltrn 30902  df-trl 30956  df-tendo 31552  df-edring 31554  df-dvech 31877
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