ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  addcan2d Unicode version

Theorem addcan2d 8140
Description: Cancellation law for addition. Theorem I.1 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
Hypotheses
Ref Expression
addcand.1  |-  ( ph  ->  A  e.  CC )
addcand.2  |-  ( ph  ->  B  e.  CC )
addcand.3  |-  ( ph  ->  C  e.  CC )
Assertion
Ref Expression
addcan2d  |-  ( ph  ->  ( ( A  +  C )  =  ( B  +  C )  <-> 
A  =  B ) )

Proof of Theorem addcan2d
StepHypRef Expression
1 addcand.1 . 2  |-  ( ph  ->  A  e.  CC )
2 addcand.2 . 2  |-  ( ph  ->  B  e.  CC )
3 addcand.3 . 2  |-  ( ph  ->  C  e.  CC )
4 addcan2 8136 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  +  C
)  =  ( B  +  C )  <->  A  =  B ) )
51, 2, 3, 4syl3anc 1238 1  |-  ( ph  ->  ( ( A  +  C )  =  ( B  +  C )  <-> 
A  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105    = wceq 1353    e. wcel 2148  (class class class)co 5874   CCcc 7808    + caddc 7813
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159  ax-resscn 7902  ax-1cn 7903  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-addcom 7910  ax-addass 7912  ax-distr 7914  ax-i2m1 7915  ax-0id 7918  ax-rnegex 7919  ax-cnre 7921
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-iota 5178  df-fv 5224  df-ov 5877
This theorem is referenced by:  addcan2ad  8142  addneintr2d  8144  nn0opthd  10697
  Copyright terms: Public domain W3C validator