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Theorem dvhopvadd 35047
Description: The vector sum operation for the constructed full vector space H. (Contributed by NM, 21-Feb-2014.) (Revised by Mario Carneiro, 6-May-2015.)
Hypotheses
Ref Expression
dvhvadd.h  |-  H  =  ( LHyp `  K
)
dvhvadd.t  |-  T  =  ( ( LTrn `  K
) `  W )
dvhvadd.e  |-  E  =  ( ( TEndo `  K
) `  W )
dvhvadd.u  |-  U  =  ( ( DVecH `  K
) `  W )
dvhvadd.f  |-  D  =  (Scalar `  U )
dvhvadd.s  |-  .+  =  ( +g  `  U )
dvhvadd.p  |-  .+^  =  ( +g  `  D )
Assertion
Ref Expression
dvhopvadd  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  Q  e.  E )  /\  ( G  e.  T  /\  R  e.  E )
)  ->  ( <. F ,  Q >.  .+  <. G ,  R >. )  =  <. ( F  o.  G ) ,  ( Q  .+^  R ) >. )

Proof of Theorem dvhopvadd
StepHypRef Expression
1 simp1 988 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  Q  e.  E )  /\  ( G  e.  T  /\  R  e.  E )
)  ->  ( K  e.  HL  /\  W  e.  H ) )
2 opelxpi 4972 . . . 4  |-  ( ( F  e.  T  /\  Q  e.  E )  -> 
<. F ,  Q >.  e.  ( T  X.  E
) )
323ad2ant2 1010 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  Q  e.  E )  /\  ( G  e.  T  /\  R  e.  E )
)  ->  <. F ,  Q >.  e.  ( T  X.  E ) )
4 opelxpi 4972 . . . 4  |-  ( ( G  e.  T  /\  R  e.  E )  -> 
<. G ,  R >.  e.  ( T  X.  E
) )
543ad2ant3 1011 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  Q  e.  E )  /\  ( G  e.  T  /\  R  e.  E )
)  ->  <. G ,  R >.  e.  ( T  X.  E ) )
6 dvhvadd.h . . . 4  |-  H  =  ( LHyp `  K
)
7 dvhvadd.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
8 dvhvadd.e . . . 4  |-  E  =  ( ( TEndo `  K
) `  W )
9 dvhvadd.u . . . 4  |-  U  =  ( ( DVecH `  K
) `  W )
10 dvhvadd.f . . . 4  |-  D  =  (Scalar `  U )
11 dvhvadd.s . . . 4  |-  .+  =  ( +g  `  U )
12 dvhvadd.p . . . 4  |-  .+^  =  ( +g  `  D )
136, 7, 8, 9, 10, 11, 12dvhvadd 35046 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( <. F ,  Q >.  e.  ( T  X.  E )  /\  <. G ,  R >.  e.  ( T  X.  E
) ) )  -> 
( <. F ,  Q >.  .+  <. G ,  R >. )  =  <. (
( 1st `  <. F ,  Q >. )  o.  ( 1st `  <. G ,  R >. )
) ,  ( ( 2nd `  <. F ,  Q >. )  .+^  ( 2nd `  <. G ,  R >. ) ) >. )
141, 3, 5, 13syl12anc 1217 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  Q  e.  E )  /\  ( G  e.  T  /\  R  e.  E )
)  ->  ( <. F ,  Q >.  .+  <. G ,  R >. )  =  <. ( ( 1st `  <. F ,  Q >. )  o.  ( 1st `  <. G ,  R >. ) ) ,  ( ( 2nd `  <. F ,  Q >. )  .+^  ( 2nd `  <. G ,  R >. )
) >. )
15 op1stg 6692 . . . . 5  |-  ( ( F  e.  T  /\  Q  e.  E )  ->  ( 1st `  <. F ,  Q >. )  =  F )
16153ad2ant2 1010 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  Q  e.  E )  /\  ( G  e.  T  /\  R  e.  E )
)  ->  ( 1st ` 
<. F ,  Q >. )  =  F )
17 op1stg 6692 . . . . 5  |-  ( ( G  e.  T  /\  R  e.  E )  ->  ( 1st `  <. G ,  R >. )  =  G )
18173ad2ant3 1011 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  Q  e.  E )  /\  ( G  e.  T  /\  R  e.  E )
)  ->  ( 1st ` 
<. G ,  R >. )  =  G )
1916, 18coeq12d 5105 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  Q  e.  E )  /\  ( G  e.  T  /\  R  e.  E )
)  ->  ( ( 1st `  <. F ,  Q >. )  o.  ( 1st `  <. G ,  R >. ) )  =  ( F  o.  G ) )
20 op2ndg 6693 . . . . 5  |-  ( ( F  e.  T  /\  Q  e.  E )  ->  ( 2nd `  <. F ,  Q >. )  =  Q )
21203ad2ant2 1010 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  Q  e.  E )  /\  ( G  e.  T  /\  R  e.  E )
)  ->  ( 2nd ` 
<. F ,  Q >. )  =  Q )
22 op2ndg 6693 . . . . 5  |-  ( ( G  e.  T  /\  R  e.  E )  ->  ( 2nd `  <. G ,  R >. )  =  R )
23223ad2ant3 1011 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  Q  e.  E )  /\  ( G  e.  T  /\  R  e.  E )
)  ->  ( 2nd ` 
<. G ,  R >. )  =  R )
2421, 23oveq12d 6211 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  Q  e.  E )  /\  ( G  e.  T  /\  R  e.  E )
)  ->  ( ( 2nd `  <. F ,  Q >. )  .+^  ( 2nd ` 
<. G ,  R >. ) )  =  ( Q 
.+^  R ) )
2519, 24opeq12d 4168 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  Q  e.  E )  /\  ( G  e.  T  /\  R  e.  E )
)  ->  <. ( ( 1st `  <. F ,  Q >. )  o.  ( 1st `  <. G ,  R >. ) ) ,  ( ( 2nd `  <. F ,  Q >. )  .+^  ( 2nd `  <. G ,  R >. )
) >.  =  <. ( F  o.  G ) ,  ( Q  .+^  R ) >. )
2614, 25eqtrd 2492 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  Q  e.  E )  /\  ( G  e.  T  /\  R  e.  E )
)  ->  ( <. F ,  Q >.  .+  <. G ,  R >. )  =  <. ( F  o.  G ) ,  ( Q  .+^  R ) >. )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   <.cop 3984    X. cxp 4939    o. ccom 4945   ` cfv 5519  (class class class)co 6193   1stc1st 6678   2ndc2nd 6679   +g cplusg 14349  Scalarcsca 14352   HLchlt 33304   LHypclh 33937   LTrncltrn 34054   TEndoctendo 34705   DVecHcdvh 35032
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-int 4230  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-om 6580  df-1st 6680  df-2nd 6681  df-recs 6935  df-rdg 6969  df-1o 7023  df-oadd 7027  df-er 7204  df-en 7414  df-dom 7415  df-sdom 7416  df-fin 7417  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-nn 10427  df-2 10484  df-3 10485  df-4 10486  df-5 10487  df-6 10488  df-n0 10684  df-z 10751  df-uz 10966  df-fz 11548  df-struct 14287  df-ndx 14288  df-slot 14289  df-base 14290  df-plusg 14362  df-mulr 14363  df-sca 14365  df-vsca 14366  df-edring 34710  df-dvech 35033
This theorem is referenced by:  dvhopvadd2  35048  dvhgrp  35061  dvh0g  35065  diblsmopel  35125  cdlemn4  35152  cdlemn6  35156  dihopelvalcpre  35202
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