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Theorem addcnsrec 8765
Description: Technical trick to permit re-use of some equivalence class lemmas for operation laws. See dfcnqs 8764 and mulcnsrec 8766. (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 8757 . 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 4237 . . . 4  |-  <. A ,  B >.  e.  _V
32ecid 6724 . . 3  |-  [ <. A ,  B >. ] `'  _E  =  <. A ,  B >.
4 opex 4237 . . . 4  |-  <. C ,  D >.  e.  _V
54ecid 6724 . . 3  |-  [ <. C ,  D >. ] `'  _E  =  <. C ,  D >.
63, 5oveq12i 5870 . 2  |-  ( [
<. A ,  B >. ] `'  _E  +  [ <. C ,  D >. ] `'  _E  )  =  ( <. A ,  B >.  + 
<. C ,  D >. )
7 opex 4237 . . 3  |-  <. ( A  +R  C ) ,  ( B  +R  D
) >.  e.  _V
87ecid 6724 . 2  |-  [ <. ( A  +R  C ) ,  ( B  +R  D ) >. ] `'  _E  =  <. ( A  +R  C ) ,  ( B  +R  D
) >.
91, 6, 83eqtr4g 2340 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 1623    e. wcel 1684   <.cop 3643    _E cep 4303   `'ccnv 4688  (class class class)co 5858   [cec 6658   R.cnr 8489    +R cplr 8493    + caddc 8740
This theorem is referenced by:  axaddass  8778  axdistr  8780
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-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-eprel 4305  df-id 4309  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-fv 5263  df-ov 5861  df-oprab 5862  df-ec 6662  df-c 8743  df-add 8748
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