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Theorem readdcan 8161
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 7983 . . . 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 5927 . . . . . . 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 8050 . . . . . . . . 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 8050 . . . . . . . . 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 8050 . . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> A e. CC )
116, 8, 10addassd 8044 . . . . . . . 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 8050 . . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> B e. CC )
146, 8, 13addassd 8044 . . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> ( ( x + C ) + B ) = ( x + ( C + B ) ) )
1511, 14eqeq12d 2208 . . . . . . 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 8158 . . . . . . . . 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 2228 . . . . . . 7 |- ( ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) /\ ( C + A ) = ( C + B ) ) -> ( x + C ) = 0 )
2423oveq1d 5934 . . . . . 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 8160 . . . . . . 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 2226 . . . . 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 5934 . . . . . 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 8160 . . . . . . 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 2226 . . . . 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 2234 . . . 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 2616 . 2 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( C + A ) = ( C + B ) -> A = B ) )
37 oveq2 5927 . 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 2164   E.wrex 2473  (class class class)co 5919   CCcc 7872   RRcr 7873   0cc0 7874   + caddc 7877
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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175  ax-resscn 7966  ax-1cn 7967  ax-icn 7969  ax-addcl 7970  ax-mulcl 7972  ax-addcom 7974  ax-addass 7976  ax-i2m1 7979  ax-0id 7982  ax-rnegex 7983
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3158  df-in 3160  df-ss 3167  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-iota 5216  df-fv 5263  df-ov 5922
This theorem is referenced by: (None)
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