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Theorem add4 7336
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 7333 . . . . 5  |-  ( ( B  e.  CC  /\  C  e.  CC  /\  D  e.  CC )  ->  ( B  +  ( C  +  D ) )  =  ( C  +  ( B  +  D ) ) )
213expb 1140 . . . 4  |-  ( ( B  e.  CC  /\  ( C  e.  CC  /\  D  e.  CC ) )  ->  ( B  +  ( C  +  D ) )  =  ( C  +  ( B  +  D ) ) )
32oveq2d 5559 . . 3  |-  ( ( B  e.  CC  /\  ( C  e.  CC  /\  D  e.  CC ) )  ->  ( A  +  ( B  +  ( C  +  D
) ) )  =  ( A  +  ( C  +  ( B  +  D ) ) ) )
43adantll 460 . 2  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( A  +  ( B  +  ( C  +  D ) ) )  =  ( A  +  ( C  +  ( B  +  D
) ) ) )
5 addcl 7160 . . 3  |-  ( ( C  e.  CC  /\  D  e.  CC )  ->  ( C  +  D
)  e.  CC )
6 addass 7165 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  +  D )  e.  CC )  ->  (
( A  +  B
)  +  ( C  +  D ) )  =  ( A  +  ( B  +  ( C  +  D )
) ) )
763expa 1139 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  +  D )  e.  CC )  ->  ( ( A  +  B )  +  ( C  +  D
) )  =  ( A  +  ( B  +  ( C  +  D ) ) ) )
85, 7sylan2 280 . 2  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( ( A  +  B )  +  ( C  +  D ) )  =  ( A  +  ( B  +  ( C  +  D
) ) ) )
9 addcl 7160 . . . 4  |-  ( ( B  e.  CC  /\  D  e.  CC )  ->  ( B  +  D
)  e.  CC )
10 addass 7165 . . . . 5  |-  ( ( A  e.  CC  /\  C  e.  CC  /\  ( B  +  D )  e.  CC )  ->  (
( A  +  C
)  +  ( B  +  D ) )  =  ( A  +  ( C  +  ( B  +  D )
) ) )
11103expa 1139 . . . 4  |-  ( ( ( A  e.  CC  /\  C  e.  CC )  /\  ( B  +  D )  e.  CC )  ->  ( ( A  +  C )  +  ( B  +  D
) )  =  ( A  +  ( C  +  ( B  +  D ) ) ) )
129, 11sylan2 280 . . 3  |-  ( ( ( A  e.  CC  /\  C  e.  CC )  /\  ( B  e.  CC  /\  D  e.  CC ) )  -> 
( ( A  +  C )  +  ( B  +  D ) )  =  ( A  +  ( C  +  ( B  +  D
) ) ) )
1312an4s 553 . 2  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( ( A  +  C )  +  ( B  +  D ) )  =  ( A  +  ( C  +  ( B  +  D
) ) ) )
144, 8, 133eqtr4d 2124 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 102    = wceq 1285    e. wcel 1434  (class class class)co 5543   CCcc 7041    + caddc 7046
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-addcl 7134  ax-addcom 7138  ax-addass 7140
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-rex 2355  df-v 2604  df-un 2978  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-iota 4897  df-fv 4940  df-ov 5546
This theorem is referenced by:  add42  7337  add4i  7340  add4d  7344  3dvds2dec  10410  opoe  10439
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