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Theorem addcnsrec 9568
Description: Technical trick to permit re-use of some equivalence class lemmas for operation laws. See dfcnqs 9567 and mulcnsrec 9569. (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 9560 . 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 4682 . . . 4  |-  <. A ,  B >.  e.  _V
32ecid 7433 . . 3  |-  [ <. A ,  B >. ] `'  _E  =  <. A ,  B >.
4 opex 4682 . . . 4  |-  <. C ,  D >.  e.  _V
54ecid 7433 . . 3  |-  [ <. C ,  D >. ] `'  _E  =  <. C ,  D >.
63, 5oveq12i 6314 . 2  |-  ( [
<. A ,  B >. ] `'  _E  +  [ <. C ,  D >. ] `'  _E  )  =  ( <. A ,  B >.  + 
<. C ,  D >. )
7 opex 4682 . . 3  |-  <. ( A  +R  C ) ,  ( B  +R  D
) >.  e.  _V
87ecid 7433 . 2  |-  [ <. ( A  +R  C ) ,  ( B  +R  D ) >. ] `'  _E  =  <. ( A  +R  C ) ,  ( B  +R  D
) >.
91, 6, 83eqtr4g 2488 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 setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1868   <.cop 4002    _E cep 4759   `'ccnv 4849  (class class class)co 6302   [cec 7366   R.cnr 9291    +R cplr 9295    + caddc 9543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543  ax-nul 4552  ax-pr 4657
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-rab 2784  df-v 3083  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-sn 3997  df-pr 3999  df-op 4003  df-uni 4217  df-br 4421  df-opab 4480  df-eprel 4761  df-id 4765  df-xp 4856  df-rel 4857  df-cnv 4858  df-co 4859  df-dm 4860  df-rn 4861  df-res 4862  df-ima 4863  df-iota 5562  df-fun 5600  df-fv 5606  df-ov 6305  df-oprab 6306  df-ec 7370  df-c 9546  df-add 9551
This theorem is referenced by:  axaddass  9581  axdistr  9583
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