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Theorem addcnsrec 9056
Description: Technical trick to permit re-use of some equivalence class lemmas for operation laws. See dfcnqs 9055 and mulcnsrec 9057. (Contributed by NM, 13-Aug-1995.) (New usage is discouraged.)
Assertion
Ref Expression
addcnsrec  |-  ( ( ( A  e.  R.  /\  B  e.  R. )  /\  ( C  e.  R.  /\  D  e.  R. )
)  ->  ( [ <. A ,  B >. ] `'  _E  +  [ <. C ,  D >. ] `'  _E  )  =  [ <. ( A  +R  C
) ,  ( B  +R  D ) >. ] `'  _E  )

Proof of Theorem addcnsrec
StepHypRef Expression
1 addcnsr 9048 . 2  |-  ( ( ( A  e.  R.  /\  B  e.  R. )  /\  ( C  e.  R.  /\  D  e.  R. )
)  ->  ( <. A ,  B >.  +  <. C ,  D >. )  =  <. ( A  +R  C ) ,  ( B  +R  D )
>. )
2 opex 4462 . . . 4  |-  <. A ,  B >.  e.  _V
32ecid 7005 . . 3  |-  [ <. A ,  B >. ] `'  _E  =  <. A ,  B >.
4 opex 4462 . . . 4  |-  <. C ,  D >.  e.  _V
54ecid 7005 . . 3  |-  [ <. C ,  D >. ] `'  _E  =  <. C ,  D >.
63, 5oveq12i 6129 . 2  |-  ( [
<. A ,  B >. ] `'  _E  +  [ <. C ,  D >. ] `'  _E  )  =  ( <. A ,  B >.  + 
<. C ,  D >. )
7 opex 4462 . . 3  |-  <. ( A  +R  C ) ,  ( B  +R  D
) >.  e.  _V
87ecid 7005 . 2  |-  [ <. ( A  +R  C ) ,  ( B  +R  D ) >. ] `'  _E  =  <. ( A  +R  C ) ,  ( B  +R  D
) >.
91, 6, 83eqtr4g 2500 1  |-  ( ( ( A  e.  R.  /\  B  e.  R. )  /\  ( C  e.  R.  /\  D  e.  R. )
)  ->  ( [ <. A ,  B >. ] `'  _E  +  [ <. C ,  D >. ] `'  _E  )  =  [ <. ( A  +R  C
) ,  ( B  +R  D ) >. ] `'  _E  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1654    e. wcel 1728   <.cop 3846    _E cep 4527   `'ccnv 4912  (class class class)co 6117   [cec 6939   R.cnr 8780    +R cplr 8784    + caddc 9031
This theorem is referenced by:  axaddass  9069  axdistr  9071
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-14 1732  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424  ax-sep 4361  ax-nul 4369  ax-pr 4438
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1661  df-eu 2292  df-mo 2293  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2717  df-rex 2718  df-rab 2721  df-v 2967  df-sbc 3171  df-dif 3312  df-un 3314  df-in 3316  df-ss 3323  df-nul 3617  df-if 3768  df-sn 3849  df-pr 3850  df-op 3852  df-uni 4045  df-br 4244  df-opab 4298  df-eprel 4529  df-id 4533  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5453  df-fun 5491  df-fv 5497  df-ov 6120  df-oprab 6121  df-ec 6943  df-c 9034  df-add 9039
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