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Theorem readdcan 7367
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 7199 . . . 4  |-  ( C  e.  RR  ->  E. x  e.  RR  ( C  +  x )  =  0 )
213ad2ant3 962 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  E. x  e.  RR  ( C  +  x )  =  0 )
3 oveq2 5571 . . . . . . 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 7261 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( x  e.  RR  /\  ( C  +  x
)  =  0 ) )  ->  x  e.  CC )
7 simpl3 944 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( x  e.  RR  /\  ( C  +  x
)  =  0 ) )  ->  C  e.  RR )
87recnd 7261 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( x  e.  RR  /\  ( C  +  x
)  =  0 ) )  ->  C  e.  CC )
9 simpl1 942 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( x  e.  RR  /\  ( C  +  x
)  =  0 ) )  ->  A  e.  RR )
109recnd 7261 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( x  e.  RR  /\  ( C  +  x
)  =  0 ) )  ->  A  e.  CC )
116, 8, 10addassd 7255 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( x  e.  RR  /\  ( C  +  x
)  =  0 ) )  ->  ( (
x  +  C )  +  A )  =  ( x  +  ( C  +  A ) ) )
12 simpl2 943 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( x  e.  RR  /\  ( C  +  x
)  =  0 ) )  ->  B  e.  RR )
1312recnd 7261 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( x  e.  RR  /\  ( C  +  x
)  =  0 ) )  ->  B  e.  CC )
146, 8, 13addassd 7255 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( x  e.  RR  /\  ( C  +  x
)  =  0 ) )  ->  ( (
x  +  C )  +  B )  =  ( x  +  ( C  +  B ) ) )
1511, 14eqeq12d 2097 . . . . . . 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 7364 . . . . . . . . 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 2117 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  (
x  e.  RR  /\  ( C  +  x
)  =  0 ) )  /\  ( C  +  A )  =  ( C  +  B
) )  ->  (
x  +  C )  =  0 )
2423oveq1d 5578 . . . . . 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 7366 . . . . . . 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 2115 . . . . 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 5578 . . . . . 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 7366 . . . . . . 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 2115 . . . . 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 2123 . . . 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 2486 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( C  +  A
)  =  ( C  +  B )  ->  A  =  B )
)
37 oveq2 5571 . 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 920    = wceq 1285    e. wcel 1434   E.wrex 2354  (class class class)co 5563   CCcc 7093   RRcr 7094   0cc0 7095    + caddc 7098
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 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-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-resscn 7182  ax-1cn 7183  ax-icn 7185  ax-addcl 7186  ax-mulcl 7188  ax-addcom 7190  ax-addass 7192  ax-i2m1 7195  ax-0id 7198  ax-rnegex 7199
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-un 2986  df-in 2988  df-ss 2995  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-br 3806  df-iota 4917  df-fv 4960  df-ov 5566
This theorem is referenced by: (None)
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