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Theorem add4 7942
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 7939 . . . . 5  |-  ( ( B  e.  CC  /\  C  e.  CC  /\  D  e.  CC )  ->  ( B  +  ( C  +  D ) )  =  ( C  +  ( B  +  D ) ) )
213expb 1182 . . . 4  |-  ( ( B  e.  CC  /\  ( C  e.  CC  /\  D  e.  CC ) )  ->  ( B  +  ( C  +  D ) )  =  ( C  +  ( B  +  D ) ) )
32oveq2d 5793 . . 3  |-  ( ( B  e.  CC  /\  ( C  e.  CC  /\  D  e.  CC ) )  ->  ( A  +  ( B  +  ( C  +  D
) ) )  =  ( A  +  ( C  +  ( B  +  D ) ) ) )
43adantll 467 . 2  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( A  +  ( B  +  ( C  +  D ) ) )  =  ( A  +  ( C  +  ( B  +  D
) ) ) )
5 addcl 7764 . . 3  |-  ( ( C  e.  CC  /\  D  e.  CC )  ->  ( C  +  D
)  e.  CC )
6 addass 7769 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  +  D )  e.  CC )  ->  (
( A  +  B
)  +  ( C  +  D ) )  =  ( A  +  ( B  +  ( C  +  D )
) ) )
763expa 1181 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  +  D )  e.  CC )  ->  ( ( A  +  B )  +  ( C  +  D
) )  =  ( A  +  ( B  +  ( C  +  D ) ) ) )
85, 7sylan2 284 . 2  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( ( A  +  B )  +  ( C  +  D ) )  =  ( A  +  ( B  +  ( C  +  D
) ) ) )
9 addcl 7764 . . . 4  |-  ( ( B  e.  CC  /\  D  e.  CC )  ->  ( B  +  D
)  e.  CC )
10 addass 7769 . . . . 5  |-  ( ( A  e.  CC  /\  C  e.  CC  /\  ( B  +  D )  e.  CC )  ->  (
( A  +  C
)  +  ( B  +  D ) )  =  ( A  +  ( C  +  ( B  +  D )
) ) )
11103expa 1181 . . . 4  |-  ( ( ( A  e.  CC  /\  C  e.  CC )  /\  ( B  +  D )  e.  CC )  ->  ( ( A  +  C )  +  ( B  +  D
) )  =  ( A  +  ( C  +  ( B  +  D ) ) ) )
129, 11sylan2 284 . . 3  |-  ( ( ( A  e.  CC  /\  C  e.  CC )  /\  ( B  e.  CC  /\  D  e.  CC ) )  -> 
( ( A  +  C )  +  ( B  +  D ) )  =  ( A  +  ( C  +  ( B  +  D
) ) ) )
1312an4s 577 . 2  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( ( A  +  C )  +  ( B  +  D ) )  =  ( A  +  ( C  +  ( B  +  D
) ) ) )
144, 8, 133eqtr4d 2182 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 103    = wceq 1331    e. wcel 1480  (class class class)co 5777   CCcc 7637    + caddc 7642
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-addcl 7735  ax-addcom 7739  ax-addass 7741
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rex 2422  df-v 2688  df-un 3075  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3740  df-br 3933  df-iota 5091  df-fv 5134  df-ov 5780
This theorem is referenced by:  add42  7943  add4i  7946  add4d  7950  3dvds2dec  11586  opoe  11615  ptolemy  12939
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