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Theorem readdcan 8413
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 8236 . . . 4 |- ( C e. RR -> E. x e. RR ( C + x ) = 0 )
213ad2ant3 1047 . . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> E. x e. RR ( C + x ) = 0 )
3 oveq2 6058 . . . . . . 7 |- ( ( C + A ) = ( C + B ) -> ( x + ( C + A ) ) = ( x + ( C + B ) ) )
43adantl 277 . . . . . 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 531 . . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> x e. RR )
65recnd 8302 . . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> x e. CC )
7 simpl3 1029 . . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> C e. RR )
87recnd 8302 . . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> C e. CC )
9 simpl1 1027 . . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> A e. RR )
109recnd 8302 . . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> A e. CC )
116, 8, 10addassd 8296 . . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> ( ( x + C ) + A ) = ( x + ( C + A ) ) )
12 simpl2 1028 . . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> B e. RR )
1312recnd 8302 . . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> B e. CC )
146, 8, 13addassd 8296 . . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> ( ( x + C ) + B ) = ( x + ( C + B ) ) )
1511, 14eqeq12d 2247 . . . . . . 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 276 . . . . . 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 167 . . . . 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 276 . . . . . . . . 9 |- ( ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) /\ ( C + A ) = ( C + B ) ) -> C e. CC )
196adantr 276 . . . . . . . . 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 8410 . . . . . . . . 9 |- ( ( C e. CC /\ x e. CC ) -> ( C + x ) = ( x + C ) )
2118, 19, 20syl2anc 411 . . . . . . . 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 538 . . . . . . . 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 2267 . . . . . . 7 |- ( ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) /\ ( C + A ) = ( C + B ) ) -> ( x + C ) = 0 )
2423oveq1d 6065 . . . . . 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 276 . . . . . . 7 |- ( ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) /\ ( C + A ) = ( C + B ) ) -> A e. CC )
26 addlid 8412 . . . . . . 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 2265 . . . . 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 6065 . . . . . 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 276 . . . . . . 7 |- ( ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) /\ ( C + A ) = ( C + B ) ) -> B e. CC )
31 addlid 8412 . . . . . . 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 2265 . . . . 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 2273 . . . 4 |- ( ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) /\ ( C + A ) = ( C + B ) ) -> A = B )
3534ex 115 . . 3 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> ( ( C + A ) = ( C + B ) -> A = B ) )
362, 35rexlimddv 2665 . 2 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( C + A ) = ( C + B ) -> A = B ) )
37 oveq2 6058 . 2 |- ( A = B -> ( C + A ) = ( C + B ) )
3836, 37impbid1 142 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 104   <-> wb 105   /\ w3a 1005   = wceq 1398   e. wcel 2203   E.wrex 2521  (class class class)co 6050   CCcc 8125   RRcr 8126   0cc0 8127   + caddc 8130
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214  ax-resscn 8219  ax-1cn 8220  ax-icn 8222  ax-addcl 8223  ax-mulcl 8225  ax-addcom 8227  ax-addass 8229  ax-i2m1 8232  ax-0id 8235  ax-rnegex 8236
This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-iota 5312  df-fv 5360  df-ov 6053
This theorem is referenced by: (None)
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