ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  addsubeq4 Unicode version

Theorem addsubeq4 7390
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 2084 . . 3  |-  ( ( C  -  A )  =  ( B  -  D )  <->  ( B  -  D )  =  ( C  -  A ) )
2 subcl 7374 . . . . . 6  |-  ( ( C  e.  CC  /\  A  e.  CC )  ->  ( C  -  A
)  e.  CC )
32ancoms 264 . . . . 5  |-  ( ( A  e.  CC  /\  C  e.  CC )  ->  ( C  -  A
)  e.  CC )
4 subadd 7378 . . . . . . 7  |-  ( ( B  e.  CC  /\  D  e.  CC  /\  ( C  -  A )  e.  CC )  ->  (
( B  -  D
)  =  ( C  -  A )  <->  ( D  +  ( C  -  A ) )  =  B ) )
543expa 1139 . . . . . 6  |-  ( ( ( B  e.  CC  /\  D  e.  CC )  /\  ( C  -  A )  e.  CC )  ->  ( ( B  -  D )  =  ( C  -  A
)  <->  ( D  +  ( C  -  A
) )  =  B ) )
65ancoms 264 . . . . 5  |-  ( ( ( C  -  A
)  e.  CC  /\  ( B  e.  CC  /\  D  e.  CC ) )  ->  ( ( B  -  D )  =  ( C  -  A )  <->  ( D  +  ( C  -  A ) )  =  B ) )
73, 6sylan 277 . . . 4  |-  ( ( ( A  e.  CC  /\  C  e.  CC )  /\  ( B  e.  CC  /\  D  e.  CC ) )  -> 
( ( B  -  D )  =  ( C  -  A )  <-> 
( D  +  ( C  -  A ) )  =  B ) )
87an4s 553 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( ( B  -  D )  =  ( C  -  A )  <-> 
( D  +  ( C  -  A ) )  =  B ) )
91, 8syl5bb 190 . 2  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( ( C  -  A )  =  ( B  -  D )  <-> 
( D  +  ( C  -  A ) )  =  B ) )
10 addcom 7312 . . . . . . 7  |-  ( ( C  e.  CC  /\  D  e.  CC )  ->  ( C  +  D
)  =  ( D  +  C ) )
1110adantl 271 . . . . . 6  |-  ( ( A  e.  CC  /\  ( C  e.  CC  /\  D  e.  CC ) )  ->  ( C  +  D )  =  ( D  +  C ) )
1211oveq1d 5558 . . . . 5  |-  ( ( A  e.  CC  /\  ( C  e.  CC  /\  D  e.  CC ) )  ->  ( ( C  +  D )  -  A )  =  ( ( D  +  C
)  -  A ) )
13 addsubass 7385 . . . . . . . 8  |-  ( ( D  e.  CC  /\  C  e.  CC  /\  A  e.  CC )  ->  (
( D  +  C
)  -  A )  =  ( D  +  ( C  -  A
) ) )
14133com12 1143 . . . . . . 7  |-  ( ( C  e.  CC  /\  D  e.  CC  /\  A  e.  CC )  ->  (
( D  +  C
)  -  A )  =  ( D  +  ( C  -  A
) ) )
15143expa 1139 . . . . . 6  |-  ( ( ( C  e.  CC  /\  D  e.  CC )  /\  A  e.  CC )  ->  ( ( D  +  C )  -  A )  =  ( D  +  ( C  -  A ) ) )
1615ancoms 264 . . . . 5  |-  ( ( A  e.  CC  /\  ( C  e.  CC  /\  D  e.  CC ) )  ->  ( ( D  +  C )  -  A )  =  ( D  +  ( C  -  A ) ) )
1712, 16eqtrd 2114 . . . 4  |-  ( ( A  e.  CC  /\  ( C  e.  CC  /\  D  e.  CC ) )  ->  ( ( C  +  D )  -  A )  =  ( D  +  ( C  -  A ) ) )
1817adantlr 461 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( ( C  +  D )  -  A
)  =  ( D  +  ( C  -  A ) ) )
1918eqeq1d 2090 . 2  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( ( ( C  +  D )  -  A )  =  B  <-> 
( D  +  ( C  -  A ) )  =  B ) )
20 addcl 7160 . . 3  |-  ( ( C  e.  CC  /\  D  e.  CC )  ->  ( C  +  D
)  e.  CC )
21 subadd 7378 . . . . 5  |-  ( ( ( C  +  D
)  e.  CC  /\  A  e.  CC  /\  B  e.  CC )  ->  (
( ( C  +  D )  -  A
)  =  B  <->  ( A  +  B )  =  ( C  +  D ) ) )
22213expb 1140 . . . 4  |-  ( ( ( C  +  D
)  e.  CC  /\  ( A  e.  CC  /\  B  e.  CC ) )  ->  ( (
( C  +  D
)  -  A )  =  B  <->  ( A  +  B )  =  ( C  +  D ) ) )
2322ancoms 264 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  +  D )  e.  CC )  ->  ( ( ( C  +  D )  -  A )  =  B  <->  ( A  +  B )  =  ( C  +  D ) ) )
2420, 23sylan2 280 . 2  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( ( ( C  +  D )  -  A )  =  B  <-> 
( A  +  B
)  =  ( C  +  D ) ) )
259, 19, 243bitr2rd 215 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    /\ wa 102    <-> wb 103    = wceq 1285    e. wcel 1434  (class class class)co 5543   CCcc 7041    + caddc 7046    - cmin 7346
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972  ax-setind 4288  ax-resscn 7130  ax-1cn 7131  ax-icn 7133  ax-addcl 7134  ax-addrcl 7135  ax-mulcl 7136  ax-addcom 7138  ax-addass 7140  ax-distr 7142  ax-i2m1 7143  ax-0id 7146  ax-rnegex 7147  ax-cnre 7149
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-opab 3848  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-iota 4897  df-fun 4934  df-fv 4940  df-riota 5499  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-sub 7348
This theorem is referenced by:  subcan  7430  addsubeq4d  7537
  Copyright terms: Public domain W3C validator