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Theorem addsub4 9336
Description: Rearrangement of 4 terms in a mixed addition and subtraction. (Contributed by NM, 4-Mar-2005.)
Assertion
Ref Expression
addsub4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( ( A  +  B )  -  ( C  +  D )
)  =  ( ( A  -  C )  +  ( B  -  D ) ) )

Proof of Theorem addsub4
StepHypRef Expression
1 simpll 731 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  ->  A  e.  CC )
2 simplr 732 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  ->  B  e.  CC )
3 simprl 733 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  ->  C  e.  CC )
4 addsub 9308 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  +  B
)  -  C )  =  ( ( A  -  C )  +  B ) )
51, 2, 3, 4syl3anc 1184 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( ( A  +  B )  -  C
)  =  ( ( A  -  C )  +  B ) )
65oveq1d 6088 . 2  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( ( ( A  +  B )  -  C )  -  D
)  =  ( ( ( A  -  C
)  +  B )  -  D ) )
71, 2addcld 9099 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( A  +  B
)  e.  CC )
8 simprr 734 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  ->  D  e.  CC )
9 subsub4 9326 . . 3  |-  ( ( ( A  +  B
)  e.  CC  /\  C  e.  CC  /\  D  e.  CC )  ->  (
( ( A  +  B )  -  C
)  -  D )  =  ( ( A  +  B )  -  ( C  +  D
) ) )
107, 3, 8, 9syl3anc 1184 . 2  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( ( ( A  +  B )  -  C )  -  D
)  =  ( ( A  +  B )  -  ( C  +  D ) ) )
11 subcl 9297 . . . 4  |-  ( ( A  e.  CC  /\  C  e.  CC )  ->  ( A  -  C
)  e.  CC )
1211ad2ant2r 728 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( A  -  C
)  e.  CC )
13 addsubass 9307 . . 3  |-  ( ( ( A  -  C
)  e.  CC  /\  B  e.  CC  /\  D  e.  CC )  ->  (
( ( A  -  C )  +  B
)  -  D )  =  ( ( A  -  C )  +  ( B  -  D
) ) )
1412, 2, 8, 13syl3anc 1184 . 2  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( ( ( A  -  C )  +  B )  -  D
)  =  ( ( A  -  C )  +  ( B  -  D ) ) )
156, 10, 143eqtr3d 2475 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 359    = wceq 1652    e. wcel 1725  (class class class)co 6073   CCcc 8980    + caddc 8985    - cmin 9283
This theorem is referenced by:  subadd4  9337  addsub4i  9388  addsub4d  9450  zaddcl  10309  sersub  11358  ppiub  20980  pntibndlem2  21277
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-po 4495  df-so 4496  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-riota 6541  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-ltxr 9117  df-sub 9285
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