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Theorem readdcan 8166
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 7988 . . . 4 |- ( C e. RR -> E. x e. RR ( C + x ) = 0 )
213ad2ant3 1022 . . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> E. x e. RR ( C + x ) = 0 )
3 oveq2 5930 . . . . . . 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 529 . . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> x e. RR )
65recnd 8055 . . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> x e. CC )
7 simpl3 1004 . . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> C e. RR )
87recnd 8055 . . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> C e. CC )
9 simpl1 1002 . . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> A e. RR )
109recnd 8055 . . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> A e. CC )
116, 8, 10addassd 8049 . . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> ( ( x + C ) + A ) = ( x + ( C + A ) ) )
12 simpl2 1003 . . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> B e. RR )
1312recnd 8055 . . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> B e. CC )
146, 8, 13addassd 8049 . . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> ( ( x + C ) + B ) = ( x + ( C + B ) ) )
1511, 14eqeq12d 2211 . . . . . . 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 8163 . . . . . . . . 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 536 . . . . . . . 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 2231 . . . . . . 7 |- ( ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) /\ ( C + A ) = ( C + B ) ) -> ( x + C ) = 0 )
2423oveq1d 5937 . . . . . 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 8165 . . . . . . 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 2229 . . . . 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 5937 . . . . . 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 8165 . . . . . . 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 2229 . . . . 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 2237 . . . 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 2619 . 2 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( C + A ) = ( C + B ) -> A = B ) )
37 oveq2 5930 . 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 980   = wceq 1364   e. wcel 2167   E.wrex 2476  (class class class)co 5922   CCcc 7877   RRcr 7878   0cc0 7879   + caddc 7882
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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178  ax-resscn 7971  ax-1cn 7972  ax-icn 7974  ax-addcl 7975  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-i2m1 7984  ax-0id 7987  ax-rnegex 7988
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-iota 5219  df-fv 5266  df-ov 5925
This theorem is referenced by: (None)
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