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Theorem add4 9785
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 9782 . . . . 5  |-  ( ( B  e.  CC  /\  C  e.  CC  /\  D  e.  CC )  ->  ( B  +  ( C  +  D ) )  =  ( C  +  ( B  +  D ) ) )
213expb 1195 . . . 4  |-  ( ( B  e.  CC  /\  ( C  e.  CC  /\  D  e.  CC ) )  ->  ( B  +  ( C  +  D ) )  =  ( C  +  ( B  +  D ) ) )
32oveq2d 6286 . . 3  |-  ( ( B  e.  CC  /\  ( C  e.  CC  /\  D  e.  CC ) )  ->  ( A  +  ( B  +  ( C  +  D
) ) )  =  ( A  +  ( C  +  ( B  +  D ) ) ) )
43adantll 711 . 2  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( A  +  ( B  +  ( C  +  D ) ) )  =  ( A  +  ( C  +  ( B  +  D
) ) ) )
5 addcl 9563 . . 3  |-  ( ( C  e.  CC  /\  D  e.  CC )  ->  ( C  +  D
)  e.  CC )
6 addass 9568 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  +  D )  e.  CC )  ->  (
( A  +  B
)  +  ( C  +  D ) )  =  ( A  +  ( B  +  ( C  +  D )
) ) )
763expa 1194 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  +  D )  e.  CC )  ->  ( ( A  +  B )  +  ( C  +  D
) )  =  ( A  +  ( B  +  ( C  +  D ) ) ) )
85, 7sylan2 472 . 2  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( ( A  +  B )  +  ( C  +  D ) )  =  ( A  +  ( B  +  ( C  +  D
) ) ) )
9 addcl 9563 . . . 4  |-  ( ( B  e.  CC  /\  D  e.  CC )  ->  ( B  +  D
)  e.  CC )
10 addass 9568 . . . . 5  |-  ( ( A  e.  CC  /\  C  e.  CC  /\  ( B  +  D )  e.  CC )  ->  (
( A  +  C
)  +  ( B  +  D ) )  =  ( A  +  ( C  +  ( B  +  D )
) ) )
11103expa 1194 . . . 4  |-  ( ( ( A  e.  CC  /\  C  e.  CC )  /\  ( B  +  D )  e.  CC )  ->  ( ( A  +  C )  +  ( B  +  D
) )  =  ( A  +  ( C  +  ( B  +  D ) ) ) )
129, 11sylan2 472 . . 3  |-  ( ( ( A  e.  CC  /\  C  e.  CC )  /\  ( B  e.  CC  /\  D  e.  CC ) )  -> 
( ( A  +  C )  +  ( B  +  D ) )  =  ( A  +  ( C  +  ( B  +  D
) ) ) )
1312an4s 824 . 2  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( ( A  +  C )  +  ( B  +  D ) )  =  ( A  +  ( C  +  ( B  +  D
) ) ) )
144, 8, 133eqtr4d 2505 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 setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823  (class class class)co 6270   CCcc 9479    + caddc 9484
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-po 4789  df-so 4790  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-ltxr 9622
This theorem is referenced by:  add42  9786  add4i  9789  add4d  9794  opoe  14419  ptolemy  23055
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