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Theorem add4 8148
Description: Rearrangement of 4 terms in a sum. (Contributed by NM, 13-Nov-1999.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
Assertion
Ref Expression
add4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( ( A  +  B )  +  ( C  +  D ) )  =  ( ( A  +  C )  +  ( B  +  D ) ) )

Proof of Theorem add4
StepHypRef Expression
1 add12 8145 . . . . 5  |-  ( ( B  e.  CC  /\  C  e.  CC  /\  D  e.  CC )  ->  ( B  +  ( C  +  D ) )  =  ( C  +  ( B  +  D ) ) )
213expb 1206 . . . 4  |-  ( ( B  e.  CC  /\  ( C  e.  CC  /\  D  e.  CC ) )  ->  ( B  +  ( C  +  D ) )  =  ( C  +  ( B  +  D ) ) )
32oveq2d 5912 . . 3  |-  ( ( B  e.  CC  /\  ( C  e.  CC  /\  D  e.  CC ) )  ->  ( A  +  ( B  +  ( C  +  D
) ) )  =  ( A  +  ( C  +  ( B  +  D ) ) ) )
43adantll 476 . 2  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( A  +  ( B  +  ( C  +  D ) ) )  =  ( A  +  ( C  +  ( B  +  D
) ) ) )
5 addcl 7966 . . 3  |-  ( ( C  e.  CC  /\  D  e.  CC )  ->  ( C  +  D
)  e.  CC )
6 addass 7971 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  +  D )  e.  CC )  ->  (
( A  +  B
)  +  ( C  +  D ) )  =  ( A  +  ( B  +  ( C  +  D )
) ) )
763expa 1205 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  +  D )  e.  CC )  ->  ( ( A  +  B )  +  ( C  +  D
) )  =  ( A  +  ( B  +  ( C  +  D ) ) ) )
85, 7sylan2 286 . 2  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( ( A  +  B )  +  ( C  +  D ) )  =  ( A  +  ( B  +  ( C  +  D
) ) ) )
9 addcl 7966 . . . 4  |-  ( ( B  e.  CC  /\  D  e.  CC )  ->  ( B  +  D
)  e.  CC )
10 addass 7971 . . . . 5  |-  ( ( A  e.  CC  /\  C  e.  CC  /\  ( B  +  D )  e.  CC )  ->  (
( A  +  C
)  +  ( B  +  D ) )  =  ( A  +  ( C  +  ( B  +  D )
) ) )
11103expa 1205 . . . 4  |-  ( ( ( A  e.  CC  /\  C  e.  CC )  /\  ( B  +  D )  e.  CC )  ->  ( ( A  +  C )  +  ( B  +  D
) )  =  ( A  +  ( C  +  ( B  +  D ) ) ) )
129, 11sylan2 286 . . 3  |-  ( ( ( A  e.  CC  /\  C  e.  CC )  /\  ( B  e.  CC  /\  D  e.  CC ) )  -> 
( ( A  +  C )  +  ( B  +  D ) )  =  ( A  +  ( C  +  ( B  +  D
) ) ) )
1312an4s 588 . 2  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( ( A  +  C )  +  ( B  +  D ) )  =  ( A  +  ( C  +  ( B  +  D
) ) ) )
144, 8, 133eqtr4d 2232 1  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( ( A  +  B )  +  ( C  +  D ) )  =  ( ( A  +  C )  +  ( B  +  D ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1364    e. wcel 2160  (class class class)co 5896   CCcc 7839    + caddc 7844
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171  ax-addcl 7937  ax-addcom 7941  ax-addass 7943
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-rex 2474  df-v 2754  df-un 3148  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-iota 5196  df-fv 5243  df-ov 5899
This theorem is referenced by:  add42  8149  add4i  8152  add4d  8156  3dvds2dec  11903  opoe  11932  ptolemy  14702
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