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Theorem dvhvaddcomN 31104
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 443 . . . 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 6191 . . . . 5  |-  ( F  e.  ( T  X.  E )  ->  ( 1st `  F )  e.  T )
32ad2antrl 708 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  ( T  X.  E
)  /\  G  e.  ( T  X.  E
) ) )  -> 
( 1st `  F
)  e.  T )
4 xp1st 6191 . . . . 5  |-  ( G  e.  ( T  X.  E )  ->  ( 1st `  G )  e.  T )
54ad2antll 709 . . . 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 30745 . . . 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 1182 . . 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 6192 . . . . . 6  |-  ( F  e.  ( T  X.  E )  ->  ( 2nd `  F )  e.  E )
11 xp2nd 6192 . . . . . 6  |-  ( G  e.  ( T  X.  E )  ->  ( 2nd `  G )  e.  E )
1210, 11anim12i 549 . . . . 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 2316 . . . . . . 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 30789 . . . . . 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 1152 . . . . 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 460 . . . 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 31092 . . . . . 6  |-  ( ( K  e.  HL  /\  W  e.  H )  -> 
.+^  =  ( a  e.  E ,  b  e.  E  |->  ( c  e.  T  |->  ( ( a `  c )  o.  ( b `  c ) ) ) ) )
2221adantr 451 . . . . 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 5917 . . . 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 5917 . . . 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 2358 . . 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 3841 . 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 31100 . 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 31100 . . 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 777 . 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 2358 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 358    = wceq 1633    e. wcel 1701   <.cop 3677    e. cmpt 4114    X. cxp 4724    o. ccom 4730   ` cfv 5292  (class class class)co 5900    e. cmpt2 5902   1stc1st 6162   2ndc2nd 6163   +g cplusg 13255  Scalarcsca 13258   HLchlt 29358   LHypclh 29991   LTrncltrn 30108   TEndoctendo 30759   DVecHcdvh 31086
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-cnex 8838  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-mulcom 8846  ax-addass 8847  ax-mulass 8848  ax-distr 8849  ax-i2m1 8850  ax-1ne0 8851  ax-1rid 8852  ax-rnegex 8853  ax-rrecex 8854  ax-cnre 8855  ax-pre-lttri 8856  ax-pre-lttrn 8857  ax-pre-ltadd 8858  ax-pre-mulgt0 8859
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-int 3900  df-iun 3944  df-iin 3945  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-undef 6340  df-riota 6346  df-recs 6430  df-rdg 6465  df-1o 6521  df-oadd 6525  df-er 6702  df-map 6817  df-en 6907  df-dom 6908  df-sdom 6909  df-fin 6910  df-pnf 8914  df-mnf 8915  df-xr 8916  df-ltxr 8917  df-le 8918  df-sub 9084  df-neg 9085  df-nn 9792  df-2 9849  df-3 9850  df-4 9851  df-5 9852  df-6 9853  df-n0 10013  df-z 10072  df-uz 10278  df-fz 10830  df-struct 13197  df-ndx 13198  df-slot 13199  df-base 13200  df-plusg 13268  df-mulr 13269  df-sca 13271  df-vsca 13272  df-poset 14129  df-plt 14141  df-lub 14157  df-glb 14158  df-join 14159  df-meet 14160  df-p0 14194  df-p1 14195  df-lat 14201  df-clat 14263  df-oposet 29184  df-ol 29186  df-oml 29187  df-covers 29274  df-ats 29275  df-atl 29306  df-cvlat 29330  df-hlat 29359  df-llines 29505  df-lplanes 29506  df-lvols 29507  df-lines 29508  df-psubsp 29510  df-pmap 29511  df-padd 29803  df-lhyp 29995  df-laut 29996  df-ldil 30111  df-ltrn 30112  df-trl 30166  df-tendo 30762  df-edring 30764  df-dvech 31087
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