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Theorem add4 7794
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 7791 . . . . 5  |-  ( ( B  e.  CC  /\  C  e.  CC  /\  D  e.  CC )  ->  ( B  +  ( C  +  D ) )  =  ( C  +  ( B  +  D ) ) )
213expb 1150 . . . 4  |-  ( ( B  e.  CC  /\  ( C  e.  CC  /\  D  e.  CC ) )  ->  ( B  +  ( C  +  D ) )  =  ( C  +  ( B  +  D ) ) )
32oveq2d 5722 . . 3  |-  ( ( B  e.  CC  /\  ( C  e.  CC  /\  D  e.  CC ) )  ->  ( A  +  ( B  +  ( C  +  D
) ) )  =  ( A  +  ( C  +  ( B  +  D ) ) ) )
43adantll 463 . 2  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( A  +  ( B  +  ( C  +  D ) ) )  =  ( A  +  ( C  +  ( B  +  D
) ) ) )
5 addcl 7617 . . 3  |-  ( ( C  e.  CC  /\  D  e.  CC )  ->  ( C  +  D
)  e.  CC )
6 addass 7622 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  +  D )  e.  CC )  ->  (
( A  +  B
)  +  ( C  +  D ) )  =  ( A  +  ( B  +  ( C  +  D )
) ) )
763expa 1149 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  +  D )  e.  CC )  ->  ( ( A  +  B )  +  ( C  +  D
) )  =  ( A  +  ( B  +  ( C  +  D ) ) ) )
85, 7sylan2 282 . 2  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( ( A  +  B )  +  ( C  +  D ) )  =  ( A  +  ( B  +  ( C  +  D
) ) ) )
9 addcl 7617 . . . 4  |-  ( ( B  e.  CC  /\  D  e.  CC )  ->  ( B  +  D
)  e.  CC )
10 addass 7622 . . . . 5  |-  ( ( A  e.  CC  /\  C  e.  CC  /\  ( B  +  D )  e.  CC )  ->  (
( A  +  C
)  +  ( B  +  D ) )  =  ( A  +  ( C  +  ( B  +  D )
) ) )
11103expa 1149 . . . 4  |-  ( ( ( A  e.  CC  /\  C  e.  CC )  /\  ( B  +  D )  e.  CC )  ->  ( ( A  +  C )  +  ( B  +  D
) )  =  ( A  +  ( C  +  ( B  +  D ) ) ) )
129, 11sylan2 282 . . 3  |-  ( ( ( A  e.  CC  /\  C  e.  CC )  /\  ( B  e.  CC  /\  D  e.  CC ) )  -> 
( ( A  +  C )  +  ( B  +  D ) )  =  ( A  +  ( C  +  ( B  +  D
) ) ) )
1312an4s 558 . 2  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( ( A  +  C )  +  ( B  +  D ) )  =  ( A  +  ( C  +  ( B  +  D
) ) ) )
144, 8, 133eqtr4d 2142 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 1299    e. wcel 1448  (class class class)co 5706   CCcc 7498    + caddc 7503
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-addcl 7591  ax-addcom 7595  ax-addass 7597
This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-rex 2381  df-v 2643  df-un 3025  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-br 3876  df-iota 5024  df-fv 5067  df-ov 5709
This theorem is referenced by:  add42  7795  add4i  7798  add4d  7802  3dvds2dec  11358  opoe  11387
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