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Theorem readdcan 7866
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 7693 . . . 4  |-  ( C  e.  RR  ->  E. x  e.  RR  ( C  +  x )  =  0 )
213ad2ant3 987 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  E. x  e.  RR  ( C  +  x )  =  0 )
3 oveq2 5748 . . . . . . 7  |-  ( ( C  +  A )  =  ( C  +  B )  ->  (
x  +  ( C  +  A ) )  =  ( x  +  ( C  +  B
) ) )
43adantl 273 . . . . . 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 503 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( x  e.  RR  /\  ( C  +  x
)  =  0 ) )  ->  x  e.  RR )
65recnd 7758 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( x  e.  RR  /\  ( C  +  x
)  =  0 ) )  ->  x  e.  CC )
7 simpl3 969 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( x  e.  RR  /\  ( C  +  x
)  =  0 ) )  ->  C  e.  RR )
87recnd 7758 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( x  e.  RR  /\  ( C  +  x
)  =  0 ) )  ->  C  e.  CC )
9 simpl1 967 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( x  e.  RR  /\  ( C  +  x
)  =  0 ) )  ->  A  e.  RR )
109recnd 7758 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( x  e.  RR  /\  ( C  +  x
)  =  0 ) )  ->  A  e.  CC )
116, 8, 10addassd 7752 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( x  e.  RR  /\  ( C  +  x
)  =  0 ) )  ->  ( (
x  +  C )  +  A )  =  ( x  +  ( C  +  A ) ) )
12 simpl2 968 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( x  e.  RR  /\  ( C  +  x
)  =  0 ) )  ->  B  e.  RR )
1312recnd 7758 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( x  e.  RR  /\  ( C  +  x
)  =  0 ) )  ->  B  e.  CC )
146, 8, 13addassd 7752 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( x  e.  RR  /\  ( C  +  x
)  =  0 ) )  ->  ( (
x  +  C )  +  B )  =  ( x  +  ( C  +  B ) ) )
1511, 14eqeq12d 2130 . . . . . . 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 272 . . . . . 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 166 . . . . 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 272 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  (
x  e.  RR  /\  ( C  +  x
)  =  0 ) )  /\  ( C  +  A )  =  ( C  +  B
) )  ->  C  e.  CC )
196adantr 272 . . . . . . . . 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 7863 . . . . . . . . 9  |-  ( ( C  e.  CC  /\  x  e.  CC )  ->  ( C  +  x
)  =  ( x  +  C ) )
2118, 19, 20syl2anc 406 . . . . . . . 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 508 . . . . . . . 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 2150 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  (
x  e.  RR  /\  ( C  +  x
)  =  0 ) )  /\  ( C  +  A )  =  ( C  +  B
) )  ->  (
x  +  C )  =  0 )
2423oveq1d 5755 . . . . . 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 272 . . . . . . 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 7865 . . . . . . 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 2148 . . . . 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 5755 . . . . . 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 272 . . . . . . 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 7865 . . . . . . 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 2148 . . . . 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 2156 . . . 4  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  (
x  e.  RR  /\  ( C  +  x
)  =  0 ) )  /\  ( C  +  A )  =  ( C  +  B
) )  ->  A  =  B )
3534ex 114 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( x  e.  RR  /\  ( C  +  x
)  =  0 ) )  ->  ( ( C  +  A )  =  ( C  +  B )  ->  A  =  B ) )
362, 35rexlimddv 2529 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( C  +  A
)  =  ( C  +  B )  ->  A  =  B )
)
37 oveq2 5748 . 2  |-  ( A  =  B  ->  ( C  +  A )  =  ( C  +  B ) )
3836, 37impbid1 141 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 103    <-> wb 104    /\ w3a 945    = wceq 1314    e. wcel 1463   E.wrex 2392  (class class class)co 5740   CCcc 7582   RRcr 7583   0cc0 7584    + caddc 7587
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-resscn 7676  ax-1cn 7677  ax-icn 7679  ax-addcl 7680  ax-mulcl 7682  ax-addcom 7684  ax-addass 7686  ax-i2m1 7689  ax-0id 7692  ax-rnegex 7693
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-iota 5056  df-fv 5099  df-ov 5743
This theorem is referenced by: (None)
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