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Theorem addcnsrec 9002
Description: Technical trick to permit re-use of some equivalence class lemmas for operation laws. See dfcnqs 9001 and mulcnsrec 9003. (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 8994 . 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 4414 . . . 4  |-  <. A ,  B >.  e.  _V
32ecid 6955 . . 3  |-  [ <. A ,  B >. ] `'  _E  =  <. A ,  B >.
4 opex 4414 . . . 4  |-  <. C ,  D >.  e.  _V
54ecid 6955 . . 3  |-  [ <. C ,  D >. ] `'  _E  =  <. C ,  D >.
63, 5oveq12i 6079 . 2  |-  ( [
<. A ,  B >. ] `'  _E  +  [ <. C ,  D >. ] `'  _E  )  =  ( <. A ,  B >.  + 
<. C ,  D >. )
7 opex 4414 . . 3  |-  <. ( A  +R  C ) ,  ( B  +R  D
) >.  e.  _V
87ecid 6955 . 2  |-  [ <. ( A  +R  C ) ,  ( B  +R  D ) >. ] `'  _E  =  <. ( A  +R  C ) ,  ( B  +R  D
) >.
91, 6, 83eqtr4g 2487 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 359    = wceq 1652    e. wcel 1725   <.cop 3804    _E cep 4479   `'ccnv 4863  (class class class)co 6067   [cec 6889   R.cnr 8726    +R cplr 8730    + caddc 8977
This theorem is referenced by:  axaddass  9015  axdistr  9017
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2411  ax-sep 4317  ax-nul 4325  ax-pr 4390
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2417  df-cleq 2423  df-clel 2426  df-nfc 2555  df-ne 2595  df-ral 2697  df-rex 2698  df-rab 2701  df-v 2945  df-sbc 3149  df-dif 3310  df-un 3312  df-in 3314  df-ss 3321  df-nul 3616  df-if 3727  df-sn 3807  df-pr 3808  df-op 3810  df-uni 4003  df-br 4200  df-opab 4254  df-eprel 4481  df-id 4485  df-xp 4870  df-rel 4871  df-cnv 4872  df-co 4873  df-dm 4874  df-rn 4875  df-res 4876  df-ima 4877  df-iota 5404  df-fun 5442  df-fv 5448  df-ov 6070  df-oprab 6071  df-ec 6893  df-c 8980  df-add 8985
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