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Theorem addsubeq4 9358
Description: Relation between sums and differences. (Contributed by Jeff Madsen, 17-Jun-2010.)
Assertion
Ref Expression
addsubeq4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( ( A  +  B )  =  ( C  +  D )  <-> 
( C  -  A
)  =  ( B  -  D ) ) )

Proof of Theorem addsubeq4
StepHypRef Expression
1 eqcom 2445 . . 3  |-  ( ( C  -  A )  =  ( B  -  D )  <->  ( B  -  D )  =  ( C  -  A ) )
2 subcl 9343 . . . . . 6  |-  ( ( C  e.  CC  /\  A  e.  CC )  ->  ( C  -  A
)  e.  CC )
32ancoms 441 . . . . 5  |-  ( ( A  e.  CC  /\  C  e.  CC )  ->  ( C  -  A
)  e.  CC )
4 subadd 9346 . . . . . . 7  |-  ( ( B  e.  CC  /\  D  e.  CC  /\  ( C  -  A )  e.  CC )  ->  (
( B  -  D
)  =  ( C  -  A )  <->  ( D  +  ( C  -  A ) )  =  B ) )
543expa 1154 . . . . . 6  |-  ( ( ( B  e.  CC  /\  D  e.  CC )  /\  ( C  -  A )  e.  CC )  ->  ( ( B  -  D )  =  ( C  -  A
)  <->  ( D  +  ( C  -  A
) )  =  B ) )
65ancoms 441 . . . . 5  |-  ( ( ( C  -  A
)  e.  CC  /\  ( B  e.  CC  /\  D  e.  CC ) )  ->  ( ( B  -  D )  =  ( C  -  A )  <->  ( D  +  ( C  -  A ) )  =  B ) )
73, 6sylan 459 . . . 4  |-  ( ( ( A  e.  CC  /\  C  e.  CC )  /\  ( B  e.  CC  /\  D  e.  CC ) )  -> 
( ( B  -  D )  =  ( C  -  A )  <-> 
( D  +  ( C  -  A ) )  =  B ) )
87an4s 801 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( ( B  -  D )  =  ( C  -  A )  <-> 
( D  +  ( C  -  A ) )  =  B ) )
91, 8syl5bb 250 . 2  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( ( C  -  A )  =  ( B  -  D )  <-> 
( D  +  ( C  -  A ) )  =  B ) )
10 addcom 9290 . . . . . . 7  |-  ( ( C  e.  CC  /\  D  e.  CC )  ->  ( C  +  D
)  =  ( D  +  C ) )
1110adantl 454 . . . . . 6  |-  ( ( A  e.  CC  /\  ( C  e.  CC  /\  D  e.  CC ) )  ->  ( C  +  D )  =  ( D  +  C ) )
1211oveq1d 6132 . . . . 5  |-  ( ( A  e.  CC  /\  ( C  e.  CC  /\  D  e.  CC ) )  ->  ( ( C  +  D )  -  A )  =  ( ( D  +  C
)  -  A ) )
13 addsubass 9353 . . . . . . . 8  |-  ( ( D  e.  CC  /\  C  e.  CC  /\  A  e.  CC )  ->  (
( D  +  C
)  -  A )  =  ( D  +  ( C  -  A
) ) )
14133com12 1158 . . . . . . 7  |-  ( ( C  e.  CC  /\  D  e.  CC  /\  A  e.  CC )  ->  (
( D  +  C
)  -  A )  =  ( D  +  ( C  -  A
) ) )
15143expa 1154 . . . . . 6  |-  ( ( ( C  e.  CC  /\  D  e.  CC )  /\  A  e.  CC )  ->  ( ( D  +  C )  -  A )  =  ( D  +  ( C  -  A ) ) )
1615ancoms 441 . . . . 5  |-  ( ( A  e.  CC  /\  ( C  e.  CC  /\  D  e.  CC ) )  ->  ( ( D  +  C )  -  A )  =  ( D  +  ( C  -  A ) ) )
1712, 16eqtrd 2475 . . . 4  |-  ( ( A  e.  CC  /\  ( C  e.  CC  /\  D  e.  CC ) )  ->  ( ( C  +  D )  -  A )  =  ( D  +  ( C  -  A ) ) )
1817adantlr 697 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( ( C  +  D )  -  A
)  =  ( D  +  ( C  -  A ) ) )
1918eqeq1d 2451 . 2  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( ( ( C  +  D )  -  A )  =  B  <-> 
( D  +  ( C  -  A ) )  =  B ) )
20 addcl 9110 . . 3  |-  ( ( C  e.  CC  /\  D  e.  CC )  ->  ( C  +  D
)  e.  CC )
21 subadd 9346 . . . . 5  |-  ( ( ( C  +  D
)  e.  CC  /\  A  e.  CC  /\  B  e.  CC )  ->  (
( ( C  +  D )  -  A
)  =  B  <->  ( A  +  B )  =  ( C  +  D ) ) )
22213expb 1155 . . . 4  |-  ( ( ( C  +  D
)  e.  CC  /\  ( A  e.  CC  /\  B  e.  CC ) )  ->  ( (
( C  +  D
)  -  A )  =  B  <->  ( A  +  B )  =  ( C  +  D ) ) )
2322ancoms 441 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  +  D )  e.  CC )  ->  ( ( ( C  +  D )  -  A )  =  B  <->  ( A  +  B )  =  ( C  +  D ) ) )
2420, 23sylan2 462 . 2  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( ( ( C  +  D )  -  A )  =  B  <-> 
( A  +  B
)  =  ( C  +  D ) ) )
259, 19, 243bitr2rd 275 1  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( ( A  +  B )  =  ( C  +  D )  <-> 
( C  -  A
)  =  ( B  -  D ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1654    e. wcel 1728  (class class class)co 6117   CCcc 9026    + caddc 9031    - cmin 9329
This theorem is referenced by:  subcan  9394  addsubeq4d  9500  dvsqr  20666
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-13 1730  ax-14 1732  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424  ax-sep 4361  ax-nul 4369  ax-pow 4412  ax-pr 4438  ax-un 4736  ax-resscn 9085  ax-1cn 9086  ax-icn 9087  ax-addcl 9088  ax-addrcl 9089  ax-mulcl 9090  ax-mulrcl 9091  ax-mulcom 9092  ax-addass 9093  ax-mulass 9094  ax-distr 9095  ax-i2m1 9096  ax-1ne0 9097  ax-1rid 9098  ax-rnegex 9099  ax-rrecex 9100  ax-cnre 9101  ax-pre-lttri 9102  ax-pre-lttrn 9103  ax-pre-ltadd 9104
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1661  df-eu 2292  df-mo 2293  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2717  df-rex 2718  df-reu 2719  df-rab 2721  df-v 2967  df-sbc 3171  df-csb 3271  df-dif 3312  df-un 3314  df-in 3316  df-ss 3323  df-nul 3617  df-if 3768  df-pw 3830  df-sn 3849  df-pr 3850  df-op 3852  df-uni 4045  df-br 4244  df-opab 4298  df-mpt 4299  df-id 4533  df-po 4538  df-so 4539  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5453  df-fun 5491  df-fn 5492  df-f 5493  df-f1 5494  df-fo 5495  df-f1o 5496  df-fv 5497  df-ov 6120  df-oprab 6121  df-mpt2 6122  df-riota 6585  df-er 6941  df-en 7146  df-dom 7147  df-sdom 7148  df-pnf 9160  df-mnf 9161  df-ltxr 9163  df-sub 9331
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