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Theorem addsubeq4 9791
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 2411 . . 3  |-  ( ( C  -  A )  =  ( B  -  D )  <->  ( B  -  D )  =  ( C  -  A ) )
2 subcl 9775 . . . . . 6  |-  ( ( C  e.  CC  /\  A  e.  CC )  ->  ( C  -  A
)  e.  CC )
32ancoms 451 . . . . 5  |-  ( ( A  e.  CC  /\  C  e.  CC )  ->  ( C  -  A
)  e.  CC )
4 subadd 9779 . . . . . . 7  |-  ( ( B  e.  CC  /\  D  e.  CC  /\  ( C  -  A )  e.  CC )  ->  (
( B  -  D
)  =  ( C  -  A )  <->  ( D  +  ( C  -  A ) )  =  B ) )
543expa 1197 . . . . . 6  |-  ( ( ( B  e.  CC  /\  D  e.  CC )  /\  ( C  -  A )  e.  CC )  ->  ( ( B  -  D )  =  ( C  -  A
)  <->  ( D  +  ( C  -  A
) )  =  B ) )
65ancoms 451 . . . . 5  |-  ( ( ( C  -  A
)  e.  CC  /\  ( B  e.  CC  /\  D  e.  CC ) )  ->  ( ( B  -  D )  =  ( C  -  A )  <->  ( D  +  ( C  -  A ) )  =  B ) )
73, 6sylan 469 . . . 4  |-  ( ( ( A  e.  CC  /\  C  e.  CC )  /\  ( B  e.  CC  /\  D  e.  CC ) )  -> 
( ( B  -  D )  =  ( C  -  A )  <-> 
( D  +  ( C  -  A ) )  =  B ) )
87an4s 827 . . 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 9720 . . . . . . 7  |-  ( ( C  e.  CC  /\  D  e.  CC )  ->  ( C  +  D
)  =  ( D  +  C ) )
1110adantl 464 . . . . . 6  |-  ( ( A  e.  CC  /\  ( C  e.  CC  /\  D  e.  CC ) )  ->  ( C  +  D )  =  ( D  +  C ) )
1211oveq1d 6249 . . . . 5  |-  ( ( A  e.  CC  /\  ( C  e.  CC  /\  D  e.  CC ) )  ->  ( ( C  +  D )  -  A )  =  ( ( D  +  C
)  -  A ) )
13 addsubass 9786 . . . . . . . 8  |-  ( ( D  e.  CC  /\  C  e.  CC  /\  A  e.  CC )  ->  (
( D  +  C
)  -  A )  =  ( D  +  ( C  -  A
) ) )
14133com12 1201 . . . . . . 7  |-  ( ( C  e.  CC  /\  D  e.  CC  /\  A  e.  CC )  ->  (
( D  +  C
)  -  A )  =  ( D  +  ( C  -  A
) ) )
15143expa 1197 . . . . . 6  |-  ( ( ( C  e.  CC  /\  D  e.  CC )  /\  A  e.  CC )  ->  ( ( D  +  C )  -  A )  =  ( D  +  ( C  -  A ) ) )
1615ancoms 451 . . . . 5  |-  ( ( A  e.  CC  /\  ( C  e.  CC  /\  D  e.  CC ) )  ->  ( ( D  +  C )  -  A )  =  ( D  +  ( C  -  A ) ) )
1712, 16eqtrd 2443 . . . 4  |-  ( ( A  e.  CC  /\  ( C  e.  CC  /\  D  e.  CC ) )  ->  ( ( C  +  D )  -  A )  =  ( D  +  ( C  -  A ) ) )
1817adantlr 713 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( ( C  +  D )  -  A
)  =  ( D  +  ( C  -  A ) ) )
1918eqeq1d 2404 . 2  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( ( ( C  +  D )  -  A )  =  B  <-> 
( D  +  ( C  -  A ) )  =  B ) )
20 addcl 9524 . . 3  |-  ( ( C  e.  CC  /\  D  e.  CC )  ->  ( C  +  D
)  e.  CC )
21 subadd 9779 . . . . 5  |-  ( ( ( C  +  D
)  e.  CC  /\  A  e.  CC  /\  B  e.  CC )  ->  (
( ( C  +  D )  -  A
)  =  B  <->  ( A  +  B )  =  ( C  +  D ) ) )
22213expb 1198 . . . 4  |-  ( ( ( C  +  D
)  e.  CC  /\  ( A  e.  CC  /\  B  e.  CC ) )  ->  ( (
( C  +  D
)  -  A )  =  B  <->  ( A  +  B )  =  ( C  +  D ) ) )
2322ancoms 451 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  +  D )  e.  CC )  ->  ( ( ( C  +  D )  -  A )  =  B  <->  ( A  +  B )  =  ( C  +  D ) ) )
2420, 23sylan2 472 . 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 367    = wceq 1405    e. wcel 1842  (class class class)co 6234   CCcc 9440    + caddc 9445    - cmin 9761
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530  ax-resscn 9499  ax-1cn 9500  ax-icn 9501  ax-addcl 9502  ax-addrcl 9503  ax-mulcl 9504  ax-mulrcl 9505  ax-mulcom 9506  ax-addass 9507  ax-mulass 9508  ax-distr 9509  ax-i2m1 9510  ax-1ne0 9511  ax-1rid 9512  ax-rnegex 9513  ax-rrecex 9514  ax-cnre 9515  ax-pre-lttri 9516  ax-pre-lttrn 9517  ax-pre-ltadd 9518
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-po 4743  df-so 4744  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6196  df-ov 6237  df-oprab 6238  df-mpt2 6239  df-er 7268  df-en 7475  df-dom 7476  df-sdom 7477  df-pnf 9580  df-mnf 9581  df-ltxr 9583  df-sub 9763
This theorem is referenced by:  subcan  9830  addsubeq4d  9938  dvsqrt  23304  dvcnsqrt  23306
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