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Theorem addsubeq4 9737
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 2463 . . 3  |-  ( ( C  -  A )  =  ( B  -  D )  <->  ( B  -  D )  =  ( C  -  A ) )
2 subcl 9721 . . . . . 6  |-  ( ( C  e.  CC  /\  A  e.  CC )  ->  ( C  -  A
)  e.  CC )
32ancoms 453 . . . . 5  |-  ( ( A  e.  CC  /\  C  e.  CC )  ->  ( C  -  A
)  e.  CC )
4 subadd 9725 . . . . . . 7  |-  ( ( B  e.  CC  /\  D  e.  CC  /\  ( C  -  A )  e.  CC )  ->  (
( B  -  D
)  =  ( C  -  A )  <->  ( D  +  ( C  -  A ) )  =  B ) )
543expa 1188 . . . . . 6  |-  ( ( ( B  e.  CC  /\  D  e.  CC )  /\  ( C  -  A )  e.  CC )  ->  ( ( B  -  D )  =  ( C  -  A
)  <->  ( D  +  ( C  -  A
) )  =  B ) )
65ancoms 453 . . . . 5  |-  ( ( ( C  -  A
)  e.  CC  /\  ( B  e.  CC  /\  D  e.  CC ) )  ->  ( ( B  -  D )  =  ( C  -  A )  <->  ( D  +  ( C  -  A ) )  =  B ) )
73, 6sylan 471 . . . 4  |-  ( ( ( A  e.  CC  /\  C  e.  CC )  /\  ( B  e.  CC  /\  D  e.  CC ) )  -> 
( ( B  -  D )  =  ( C  -  A )  <-> 
( D  +  ( C  -  A ) )  =  B ) )
87an4s 822 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( ( B  -  D )  =  ( C  -  A )  <-> 
( D  +  ( C  -  A ) )  =  B ) )
91, 8syl5bb 257 . 2  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( ( C  -  A )  =  ( B  -  D )  <-> 
( D  +  ( C  -  A ) )  =  B ) )
10 addcom 9667 . . . . . . 7  |-  ( ( C  e.  CC  /\  D  e.  CC )  ->  ( C  +  D
)  =  ( D  +  C ) )
1110adantl 466 . . . . . 6  |-  ( ( A  e.  CC  /\  ( C  e.  CC  /\  D  e.  CC ) )  ->  ( C  +  D )  =  ( D  +  C ) )
1211oveq1d 6216 . . . . 5  |-  ( ( A  e.  CC  /\  ( C  e.  CC  /\  D  e.  CC ) )  ->  ( ( C  +  D )  -  A )  =  ( ( D  +  C
)  -  A ) )
13 addsubass 9732 . . . . . . . 8  |-  ( ( D  e.  CC  /\  C  e.  CC  /\  A  e.  CC )  ->  (
( D  +  C
)  -  A )  =  ( D  +  ( C  -  A
) ) )
14133com12 1192 . . . . . . 7  |-  ( ( C  e.  CC  /\  D  e.  CC  /\  A  e.  CC )  ->  (
( D  +  C
)  -  A )  =  ( D  +  ( C  -  A
) ) )
15143expa 1188 . . . . . 6  |-  ( ( ( C  e.  CC  /\  D  e.  CC )  /\  A  e.  CC )  ->  ( ( D  +  C )  -  A )  =  ( D  +  ( C  -  A ) ) )
1615ancoms 453 . . . . 5  |-  ( ( A  e.  CC  /\  ( C  e.  CC  /\  D  e.  CC ) )  ->  ( ( D  +  C )  -  A )  =  ( D  +  ( C  -  A ) ) )
1712, 16eqtrd 2495 . . . 4  |-  ( ( A  e.  CC  /\  ( C  e.  CC  /\  D  e.  CC ) )  ->  ( ( C  +  D )  -  A )  =  ( D  +  ( C  -  A ) ) )
1817adantlr 714 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( ( C  +  D )  -  A
)  =  ( D  +  ( C  -  A ) ) )
1918eqeq1d 2456 . 2  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( ( ( C  +  D )  -  A )  =  B  <-> 
( D  +  ( C  -  A ) )  =  B ) )
20 addcl 9476 . . 3  |-  ( ( C  e.  CC  /\  D  e.  CC )  ->  ( C  +  D
)  e.  CC )
21 subadd 9725 . . . . 5  |-  ( ( ( C  +  D
)  e.  CC  /\  A  e.  CC  /\  B  e.  CC )  ->  (
( ( C  +  D )  -  A
)  =  B  <->  ( A  +  B )  =  ( C  +  D ) ) )
22213expb 1189 . . . 4  |-  ( ( ( C  +  D
)  e.  CC  /\  ( A  e.  CC  /\  B  e.  CC ) )  ->  ( (
( C  +  D
)  -  A )  =  B  <->  ( A  +  B )  =  ( C  +  D ) ) )
2322ancoms 453 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  +  D )  e.  CC )  ->  ( ( ( C  +  D )  -  A )  =  B  <->  ( A  +  B )  =  ( C  +  D ) ) )
2420, 23sylan2 474 . 2  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( ( ( C  +  D )  -  A )  =  B  <-> 
( A  +  B
)  =  ( C  +  D ) ) )
259, 19, 243bitr2rd 282 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758  (class class class)co 6201   CCcc 9392    + caddc 9397    - cmin 9707
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-mulcom 9458  ax-addass 9459  ax-mulass 9460  ax-distr 9461  ax-i2m1 9462  ax-1ne0 9463  ax-1rid 9464  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469  ax-pre-ltadd 9470
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-po 4750  df-so 4751  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-er 7212  df-en 7422  df-dom 7423  df-sdom 7424  df-pnf 9532  df-mnf 9533  df-ltxr 9535  df-sub 9709
This theorem is referenced by:  subcan  9776  addsubeq4d  9882  dvsqr  22316  dvcnsqr  28627
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