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Theorem readdcan 8986
Description: Cancellation law for addition over the reals. (Contributed by Scott Fenton, 3-Jan-2013.) (Proof shortened by Mario Carneiro, 27-May-2016.)
Assertion
Ref Expression
readdcan  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( C  +  A
)  =  ( C  +  B )  <->  A  =  B ) )

Proof of Theorem readdcan
StepHypRef Expression
1 ltadd2 8924 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <  B  <->  ( C  +  A )  <  ( C  +  B )
) )
21notbid 285 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( -.  A  <  B  <->  -.  ( C  +  A )  <  ( C  +  B
) ) )
3 simp2 956 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  B  e.  RR )
4 simp1 955 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  A  e.  RR )
5 simp3 957 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  C  e.  RR )
63, 4, 5ltadd2d 8972 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( B  <  A  <->  ( C  +  B )  <  ( C  +  A )
) )
76notbid 285 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( -.  B  <  A  <->  -.  ( C  +  B )  <  ( C  +  A
) ) )
82, 7anbi12d 691 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( -.  A  < 
B  /\  -.  B  <  A )  <->  ( -.  ( C  +  A
)  <  ( C  +  B )  /\  -.  ( C  +  B
)  <  ( C  +  A ) ) ) )
94, 3lttri3d 8959 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  =  B  <->  ( -.  A  <  B  /\  -.  B  <  A ) ) )
105, 4readdcld 8862 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( C  +  A )  e.  RR )
115, 3readdcld 8862 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( C  +  B )  e.  RR )
1210, 11lttri3d 8959 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( C  +  A
)  =  ( C  +  B )  <->  ( -.  ( C  +  A
)  <  ( C  +  B )  /\  -.  ( C  +  B
)  <  ( C  +  A ) ) ) )
138, 9, 123bitr4rd 277 1  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( C  +  A
)  =  ( C  +  B )  <->  A  =  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   class class class wbr 4023  (class class class)co 5858   RRcr 8736    + caddc 8740    < clt 8867
This theorem is referenced by:  00id  8987  mul02lem2  8989  addid1  8992  rpnnen2lem10  12502
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-resscn 8794  ax-addrcl 8798  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-ltxr 8872
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