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Theorem addcnsrec 8781
Description: Technical trick to permit re-use of some equivalence class lemmas for operation laws. See dfcnqs 8780 and mulcnsrec 8782. (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 8773 . 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 4253 . . . 4  |-  <. A ,  B >.  e.  _V
32ecid 6740 . . 3  |-  [ <. A ,  B >. ] `'  _E  =  <. A ,  B >.
4 opex 4253 . . . 4  |-  <. C ,  D >.  e.  _V
54ecid 6740 . . 3  |-  [ <. C ,  D >. ] `'  _E  =  <. C ,  D >.
63, 5oveq12i 5886 . 2  |-  ( [
<. A ,  B >. ] `'  _E  +  [ <. C ,  D >. ] `'  _E  )  =  ( <. A ,  B >.  + 
<. C ,  D >. )
7 opex 4253 . . 3  |-  <. ( A  +R  C ) ,  ( B  +R  D
) >.  e.  _V
87ecid 6740 . 2  |-  [ <. ( A  +R  C ) ,  ( B  +R  D ) >. ] `'  _E  =  <. ( A  +R  C ) ,  ( B  +R  D
) >.
91, 6, 83eqtr4g 2353 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 358    = wceq 1632    e. wcel 1696   <.cop 3656    _E cep 4319   `'ccnv 4704  (class class class)co 5874   [cec 6674   R.cnr 8505    +R cplr 8509    + caddc 8756
This theorem is referenced by:  axaddass  8794  axdistr  8796
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-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-eprel 4321  df-id 4325  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-fv 5279  df-ov 5877  df-oprab 5878  df-ec 6678  df-c 8759  df-add 8764
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