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Theorem dvhopvadd 31283
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 4721 . . . 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 4721 . . . 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 31282 . . 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 6132 . . . . 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 6132 . . . . 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 4848 . . 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 6133 . . . . 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 6133 . . . . 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 5876 . . 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 3804 . 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 2315 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 1623    e. wcel 1684   <.cop 3643    X. cxp 4687    o. ccom 4693   ` cfv 5255  (class class class)co 5858   1stc1st 6120   2ndc2nd 6121   +g cplusg 13208  Scalarcsca 13211   HLchlt 29540   LHypclh 30173   LTrncltrn 30290   TEndoctendo 30941   DVecHcdvh 31268
This theorem is referenced by:  dvhopvadd2  31284  dvhgrp  31297  dvh0g  31301  diblsmopel  31361  cdlemn4  31388  cdlemn6  31392  dihopelvalcpre  31438
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-plusg 13221  df-mulr 13222  df-sca 13224  df-vsca 13225  df-edring 30946  df-dvech 31269
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