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Theorem readdcan 7620
Description: Cancellation law for addition over the reals. (Contributed by Scott Fenton, 3-Jan-2013.)
Assertion
Ref Expression
readdcan  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( C  +  A
)  =  ( C  +  B )  <->  A  =  B ) )

Proof of Theorem readdcan
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 ax-rnegex 7452 . . . 4  |-  ( C  e.  RR  ->  E. x  e.  RR  ( C  +  x )  =  0 )
213ad2ant3 966 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  E. x  e.  RR  ( C  +  x )  =  0 )
3 oveq2 5660 . . . . . . 7  |-  ( ( C  +  A )  =  ( C  +  B )  ->  (
x  +  ( C  +  A ) )  =  ( x  +  ( C  +  B
) ) )
43adantl 271 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  (
x  e.  RR  /\  ( C  +  x
)  =  0 ) )  /\  ( C  +  A )  =  ( C  +  B
) )  ->  (
x  +  ( C  +  A ) )  =  ( x  +  ( C  +  B
) ) )
5 simprl 498 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( x  e.  RR  /\  ( C  +  x
)  =  0 ) )  ->  x  e.  RR )
65recnd 7514 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( x  e.  RR  /\  ( C  +  x
)  =  0 ) )  ->  x  e.  CC )
7 simpl3 948 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( x  e.  RR  /\  ( C  +  x
)  =  0 ) )  ->  C  e.  RR )
87recnd 7514 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( x  e.  RR  /\  ( C  +  x
)  =  0 ) )  ->  C  e.  CC )
9 simpl1 946 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( x  e.  RR  /\  ( C  +  x
)  =  0 ) )  ->  A  e.  RR )
109recnd 7514 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( x  e.  RR  /\  ( C  +  x
)  =  0 ) )  ->  A  e.  CC )
116, 8, 10addassd 7508 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( x  e.  RR  /\  ( C  +  x
)  =  0 ) )  ->  ( (
x  +  C )  +  A )  =  ( x  +  ( C  +  A ) ) )
12 simpl2 947 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( x  e.  RR  /\  ( C  +  x
)  =  0 ) )  ->  B  e.  RR )
1312recnd 7514 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( x  e.  RR  /\  ( C  +  x
)  =  0 ) )  ->  B  e.  CC )
146, 8, 13addassd 7508 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( x  e.  RR  /\  ( C  +  x
)  =  0 ) )  ->  ( (
x  +  C )  +  B )  =  ( x  +  ( C  +  B ) ) )
1511, 14eqeq12d 2102 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( x  e.  RR  /\  ( C  +  x
)  =  0 ) )  ->  ( (
( x  +  C
)  +  A )  =  ( ( x  +  C )  +  B )  <->  ( x  +  ( C  +  A ) )  =  ( x  +  ( C  +  B ) ) ) )
1615adantr 270 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  (
x  e.  RR  /\  ( C  +  x
)  =  0 ) )  /\  ( C  +  A )  =  ( C  +  B
) )  ->  (
( ( x  +  C )  +  A
)  =  ( ( x  +  C )  +  B )  <->  ( x  +  ( C  +  A ) )  =  ( x  +  ( C  +  B ) ) ) )
174, 16mpbird 165 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  (
x  e.  RR  /\  ( C  +  x
)  =  0 ) )  /\  ( C  +  A )  =  ( C  +  B
) )  ->  (
( x  +  C
)  +  A )  =  ( ( x  +  C )  +  B ) )
188adantr 270 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  (
x  e.  RR  /\  ( C  +  x
)  =  0 ) )  /\  ( C  +  A )  =  ( C  +  B
) )  ->  C  e.  CC )
196adantr 270 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  (
x  e.  RR  /\  ( C  +  x
)  =  0 ) )  /\  ( C  +  A )  =  ( C  +  B
) )  ->  x  e.  CC )
20 addcom 7617 . . . . . . . . 9  |-  ( ( C  e.  CC  /\  x  e.  CC )  ->  ( C  +  x
)  =  ( x  +  C ) )
2118, 19, 20syl2anc 403 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  (
x  e.  RR  /\  ( C  +  x
)  =  0 ) )  /\  ( C  +  A )  =  ( C  +  B
) )  ->  ( C  +  x )  =  ( x  +  C ) )
22 simplrr 503 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  (
x  e.  RR  /\  ( C  +  x
)  =  0 ) )  /\  ( C  +  A )  =  ( C  +  B
) )  ->  ( C  +  x )  =  0 )
2321, 22eqtr3d 2122 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  (
x  e.  RR  /\  ( C  +  x
)  =  0 ) )  /\  ( C  +  A )  =  ( C  +  B
) )  ->  (
x  +  C )  =  0 )
2423oveq1d 5667 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  (
x  e.  RR  /\  ( C  +  x
)  =  0 ) )  /\  ( C  +  A )  =  ( C  +  B
) )  ->  (
( x  +  C
)  +  A )  =  ( 0  +  A ) )
2510adantr 270 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  (
x  e.  RR  /\  ( C  +  x
)  =  0 ) )  /\  ( C  +  A )  =  ( C  +  B
) )  ->  A  e.  CC )
26 addid2 7619 . . . . . . 7  |-  ( A  e.  CC  ->  (
0  +  A )  =  A )
2725, 26syl 14 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  (
x  e.  RR  /\  ( C  +  x
)  =  0 ) )  /\  ( C  +  A )  =  ( C  +  B
) )  ->  (
0  +  A )  =  A )
2824, 27eqtrd 2120 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  (
x  e.  RR  /\  ( C  +  x
)  =  0 ) )  /\  ( C  +  A )  =  ( C  +  B
) )  ->  (
( x  +  C
)  +  A )  =  A )
2923oveq1d 5667 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  (
x  e.  RR  /\  ( C  +  x
)  =  0 ) )  /\  ( C  +  A )  =  ( C  +  B
) )  ->  (
( x  +  C
)  +  B )  =  ( 0  +  B ) )
3013adantr 270 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  (
x  e.  RR  /\  ( C  +  x
)  =  0 ) )  /\  ( C  +  A )  =  ( C  +  B
) )  ->  B  e.  CC )
31 addid2 7619 . . . . . . 7  |-  ( B  e.  CC  ->  (
0  +  B )  =  B )
3230, 31syl 14 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  (
x  e.  RR  /\  ( C  +  x
)  =  0 ) )  /\  ( C  +  A )  =  ( C  +  B
) )  ->  (
0  +  B )  =  B )
3329, 32eqtrd 2120 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  (
x  e.  RR  /\  ( C  +  x
)  =  0 ) )  /\  ( C  +  A )  =  ( C  +  B
) )  ->  (
( x  +  C
)  +  B )  =  B )
3417, 28, 333eqtr3d 2128 . . . 4  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  (
x  e.  RR  /\  ( C  +  x
)  =  0 ) )  /\  ( C  +  A )  =  ( C  +  B
) )  ->  A  =  B )
3534ex 113 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( x  e.  RR  /\  ( C  +  x
)  =  0 ) )  ->  ( ( C  +  A )  =  ( C  +  B )  ->  A  =  B ) )
362, 35rexlimddv 2493 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( C  +  A
)  =  ( C  +  B )  ->  A  =  B )
)
37 oveq2 5660 . 2  |-  ( A  =  B  ->  ( C  +  A )  =  ( C  +  B ) )
3836, 37impbid1 140 1  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( C  +  A
)  =  ( C  +  B )  <->  A  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    /\ w3a 924    = wceq 1289    e. wcel 1438   E.wrex 2360  (class class class)co 5652   CCcc 7346   RRcr 7347   0cc0 7348    + caddc 7351
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-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-resscn 7435  ax-1cn 7436  ax-icn 7438  ax-addcl 7439  ax-mulcl 7441  ax-addcom 7443  ax-addass 7445  ax-i2m1 7448  ax-0id 7451  ax-rnegex 7452
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-iota 4980  df-fv 5023  df-ov 5655
This theorem is referenced by: (None)
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