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Theorem dvhopvadd 31905
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 955 . . 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 4737 . . . 4  |-  ( ( F  e.  T  /\  Q  e.  E )  -> 
<. F ,  Q >.  e.  ( T  X.  E
) )
323ad2ant2 977 . . 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 4737 . . . 4  |-  ( ( G  e.  T  /\  R  e.  E )  -> 
<. G ,  R >.  e.  ( T  X.  E
) )
543ad2ant3 978 . . 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 31904 . . 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 1180 . 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 6148 . . . . 5  |-  ( ( F  e.  T  /\  Q  e.  E )  ->  ( 1st `  <. F ,  Q >. )  =  F )
16153ad2ant2 977 . . . 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 6148 . . . . 5  |-  ( ( G  e.  T  /\  R  e.  E )  ->  ( 1st `  <. G ,  R >. )  =  G )
18173ad2ant3 978 . . . 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 4864 . . 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 6149 . . . . 5  |-  ( ( F  e.  T  /\  Q  e.  E )  ->  ( 2nd `  <. F ,  Q >. )  =  Q )
21203ad2ant2 977 . . . 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 6149 . . . . 5  |-  ( ( G  e.  T  /\  R  e.  E )  ->  ( 2nd `  <. G ,  R >. )  =  R )
23223ad2ant3 978 . . . 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 5892 . . 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 3820 . 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 2328 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 set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   <.cop 3656    X. cxp 4703    o. ccom 4709   ` cfv 5271  (class class class)co 5874   1stc1st 6136   2ndc2nd 6137   +g cplusg 13224  Scalarcsca 13227   HLchlt 30162   LHypclh 30795   LTrncltrn 30912   TEndoctendo 31563   DVecHcdvh 31890
This theorem is referenced by:  dvhopvadd2  31906  dvhgrp  31919  dvh0g  31923  diblsmopel  31983  cdlemn4  32010  cdlemn6  32014  dihopelvalcpre  32060
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-plusg 13237  df-mulr 13238  df-sca 13240  df-vsca 13241  df-edring 31568  df-dvech 31891
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