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Theorem addsubeq4 9245
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 2382 . . 3  |-  ( ( C  -  A )  =  ( B  -  D )  <->  ( B  -  D )  =  ( C  -  A ) )
2 subcl 9230 . . . . . 6  |-  ( ( C  e.  CC  /\  A  e.  CC )  ->  ( C  -  A
)  e.  CC )
32ancoms 440 . . . . 5  |-  ( ( A  e.  CC  /\  C  e.  CC )  ->  ( C  -  A
)  e.  CC )
4 subadd 9233 . . . . . . 7  |-  ( ( B  e.  CC  /\  D  e.  CC  /\  ( C  -  A )  e.  CC )  ->  (
( B  -  D
)  =  ( C  -  A )  <->  ( D  +  ( C  -  A ) )  =  B ) )
543expa 1153 . . . . . 6  |-  ( ( ( B  e.  CC  /\  D  e.  CC )  /\  ( C  -  A )  e.  CC )  ->  ( ( B  -  D )  =  ( C  -  A
)  <->  ( D  +  ( C  -  A
) )  =  B ) )
65ancoms 440 . . . . 5  |-  ( ( ( C  -  A
)  e.  CC  /\  ( B  e.  CC  /\  D  e.  CC ) )  ->  ( ( B  -  D )  =  ( C  -  A )  <->  ( D  +  ( C  -  A ) )  =  B ) )
73, 6sylan 458 . . . 4  |-  ( ( ( A  e.  CC  /\  C  e.  CC )  /\  ( B  e.  CC  /\  D  e.  CC ) )  -> 
( ( B  -  D )  =  ( C  -  A )  <-> 
( D  +  ( C  -  A ) )  =  B ) )
87an4s 800 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( ( B  -  D )  =  ( C  -  A )  <-> 
( D  +  ( C  -  A ) )  =  B ) )
91, 8syl5bb 249 . 2  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( ( C  -  A )  =  ( B  -  D )  <-> 
( D  +  ( C  -  A ) )  =  B ) )
10 addcom 9177 . . . . . . 7  |-  ( ( C  e.  CC  /\  D  e.  CC )  ->  ( C  +  D
)  =  ( D  +  C ) )
1110adantl 453 . . . . . 6  |-  ( ( A  e.  CC  /\  ( C  e.  CC  /\  D  e.  CC ) )  ->  ( C  +  D )  =  ( D  +  C ) )
1211oveq1d 6028 . . . . 5  |-  ( ( A  e.  CC  /\  ( C  e.  CC  /\  D  e.  CC ) )  ->  ( ( C  +  D )  -  A )  =  ( ( D  +  C
)  -  A ) )
13 addsubass 9240 . . . . . . . 8  |-  ( ( D  e.  CC  /\  C  e.  CC  /\  A  e.  CC )  ->  (
( D  +  C
)  -  A )  =  ( D  +  ( C  -  A
) ) )
14133com12 1157 . . . . . . 7  |-  ( ( C  e.  CC  /\  D  e.  CC  /\  A  e.  CC )  ->  (
( D  +  C
)  -  A )  =  ( D  +  ( C  -  A
) ) )
15143expa 1153 . . . . . 6  |-  ( ( ( C  e.  CC  /\  D  e.  CC )  /\  A  e.  CC )  ->  ( ( D  +  C )  -  A )  =  ( D  +  ( C  -  A ) ) )
1615ancoms 440 . . . . 5  |-  ( ( A  e.  CC  /\  ( C  e.  CC  /\  D  e.  CC ) )  ->  ( ( D  +  C )  -  A )  =  ( D  +  ( C  -  A ) ) )
1712, 16eqtrd 2412 . . . 4  |-  ( ( A  e.  CC  /\  ( C  e.  CC  /\  D  e.  CC ) )  ->  ( ( C  +  D )  -  A )  =  ( D  +  ( C  -  A ) ) )
1817adantlr 696 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( ( C  +  D )  -  A
)  =  ( D  +  ( C  -  A ) ) )
1918eqeq1d 2388 . 2  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( ( ( C  +  D )  -  A )  =  B  <-> 
( D  +  ( C  -  A ) )  =  B ) )
20 addcl 8998 . . 3  |-  ( ( C  e.  CC  /\  D  e.  CC )  ->  ( C  +  D
)  e.  CC )
21 subadd 9233 . . . . 5  |-  ( ( ( C  +  D
)  e.  CC  /\  A  e.  CC  /\  B  e.  CC )  ->  (
( ( C  +  D )  -  A
)  =  B  <->  ( A  +  B )  =  ( C  +  D ) ) )
22213expb 1154 . . . 4  |-  ( ( ( C  +  D
)  e.  CC  /\  ( A  e.  CC  /\  B  e.  CC ) )  ->  ( (
( C  +  D
)  -  A )  =  B  <->  ( A  +  B )  =  ( C  +  D ) ) )
2322ancoms 440 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  +  D )  e.  CC )  ->  ( ( ( C  +  D )  -  A )  =  B  <->  ( A  +  B )  =  ( C  +  D ) ) )
2420, 23sylan2 461 . 2  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( ( ( C  +  D )  -  A )  =  B  <-> 
( A  +  B
)  =  ( C  +  D ) ) )
259, 19, 243bitr2rd 274 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 177    /\ wa 359    = wceq 1649    e. wcel 1717  (class class class)co 6013   CCcc 8914    + caddc 8919    - cmin 9216
This theorem is referenced by:  subcan  9281  addsubeq4d  9387  dvsqr  20488
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-po 4437  df-so 4438  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-riota 6478  df-er 6834  df-en 7039  df-dom 7040  df-sdom 7041  df-pnf 9048  df-mnf 9049  df-ltxr 9051  df-sub 9218
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