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Theorem addcanpr 8887
Description: Addition cancellation law for positive reals. Proposition 9-3.5(vi) of [Gleason] p. 123. (Contributed by NM, 9-Apr-1996.) (New usage is discouraged.)
Assertion
Ref Expression
addcanpr  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( A  +P.  B )  =  ( A  +P.  C )  ->  B  =  C )
)

Proof of Theorem addcanpr
StepHypRef Expression
1 addclpr 8859 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  +P.  B
)  e.  P. )
2 eleq1 2472 . . . . 5  |-  ( ( A  +P.  B )  =  ( A  +P.  C )  ->  ( ( A  +P.  B )  e. 
P. 
<->  ( A  +P.  C
)  e.  P. )
)
3 dmplp 8853 . . . . . 6  |-  dom  +P.  =  ( P.  X.  P. )
4 0npr 8833 . . . . . 6  |-  -.  (/)  e.  P.
53, 4ndmovrcl 6200 . . . . 5  |-  ( ( A  +P.  C )  e.  P.  ->  ( A  e.  P.  /\  C  e.  P. ) )
62, 5syl6bi 220 . . . 4  |-  ( ( A  +P.  B )  =  ( A  +P.  C )  ->  ( ( A  +P.  B )  e. 
P.  ->  ( A  e. 
P.  /\  C  e.  P. ) ) )
71, 6syl5com 28 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( A  +P.  B )  =  ( A  +P.  C )  -> 
( A  e.  P.  /\  C  e.  P. )
) )
8 ltapr 8886 . . . . . . . 8  |-  ( A  e.  P.  ->  ( B  <P  C  <->  ( A  +P.  B )  <P  ( A  +P.  C ) ) )
9 ltapr 8886 . . . . . . . 8  |-  ( A  e.  P.  ->  ( C  <P  B  <->  ( A  +P.  C )  <P  ( A  +P.  B ) ) )
108, 9orbi12d 691 . . . . . . 7  |-  ( A  e.  P.  ->  (
( B  <P  C  \/  C  <P  B )  <->  ( ( A  +P.  B )  <P 
( A  +P.  C
)  \/  ( A  +P.  C )  <P 
( A  +P.  B
) ) ) )
1110notbid 286 . . . . . 6  |-  ( A  e.  P.  ->  ( -.  ( B  <P  C  \/  C  <P  B )  <->  -.  (
( A  +P.  B
)  <P  ( A  +P.  C )  \/  ( A  +P.  C )  <P 
( A  +P.  B
) ) ) )
1211ad2antrr 707 . . . . 5  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( A  e.  P.  /\  C  e.  P. )
)  ->  ( -.  ( B  <P  C  \/  C  <P  B )  <->  -.  (
( A  +P.  B
)  <P  ( A  +P.  C )  \/  ( A  +P.  C )  <P 
( A  +P.  B
) ) ) )
13 ltsopr 8873 . . . . . . 7  |-  <P  Or  P.
14 sotrieq 4498 . . . . . . 7  |-  ( ( 
<P  Or  P.  /\  ( B  e.  P.  /\  C  e.  P. ) )  -> 
( B  =  C  <->  -.  ( B  <P  C  \/  C  <P  B ) ) )
1513, 14mpan 652 . . . . . 6  |-  ( ( B  e.  P.  /\  C  e.  P. )  ->  ( B  =  C  <->  -.  ( B  <P  C  \/  C  <P  B ) ) )
1615ad2ant2l 727 . . . . 5  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( A  e.  P.  /\  C  e.  P. )
)  ->  ( B  =  C  <->  -.  ( B  <P  C  \/  C  <P  B ) ) )
17 addclpr 8859 . . . . . 6  |-  ( ( A  e.  P.  /\  C  e.  P. )  ->  ( A  +P.  C
)  e.  P. )
18 sotrieq 4498 . . . . . . 7  |-  ( ( 
<P  Or  P.  /\  (
( A  +P.  B
)  e.  P.  /\  ( A  +P.  C )  e.  P. ) )  ->  ( ( A  +P.  B )  =  ( A  +P.  C
)  <->  -.  ( ( A  +P.  B )  <P 
( A  +P.  C
)  \/  ( A  +P.  C )  <P 
( A  +P.  B
) ) ) )
1913, 18mpan 652 . . . . . 6  |-  ( ( ( A  +P.  B
)  e.  P.  /\  ( A  +P.  C )  e.  P. )  -> 
( ( A  +P.  B )  =  ( A  +P.  C )  <->  -.  (
( A  +P.  B
)  <P  ( A  +P.  C )  \/  ( A  +P.  C )  <P 
( A  +P.  B
) ) ) )
201, 17, 19syl2an 464 . . . . 5  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( A  e.  P.  /\  C  e.  P. )
)  ->  ( ( A  +P.  B )  =  ( A  +P.  C
)  <->  -.  ( ( A  +P.  B )  <P 
( A  +P.  C
)  \/  ( A  +P.  C )  <P 
( A  +P.  B
) ) ) )
2112, 16, 203bitr4d 277 . . . 4  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( A  e.  P.  /\  C  e.  P. )
)  ->  ( B  =  C  <->  ( A  +P.  B )  =  ( A  +P.  C ) ) )
2221exbiri 606 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( A  e. 
P.  /\  C  e.  P. )  ->  ( ( A  +P.  B )  =  ( A  +P.  C )  ->  B  =  C ) ) )
237, 22syld 42 . 2  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( A  +P.  B )  =  ( A  +P.  C )  -> 
( ( A  +P.  B )  =  ( A  +P.  C )  ->  B  =  C )
) )
2423pm2.43d 46 1  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( A  +P.  B )  =  ( A  +P.  C )  ->  B  =  C )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1649    e. wcel 1721   class class class wbr 4180    Or wor 4470  (class class class)co 6048   P.cnp 8698    +P. cpp 8700    <P cltp 8702
This theorem is referenced by:  enrer  8907  mulcmpblnr  8913  mulgt0sr  8944
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-inf2 7560
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-recs 6600  df-rdg 6635  df-1o 6691  df-oadd 6695  df-omul 6696  df-er 6872  df-ni 8713  df-pli 8714  df-mi 8715  df-lti 8716  df-plpq 8749  df-mpq 8750  df-ltpq 8751  df-enq 8752  df-nq 8753  df-erq 8754  df-plq 8755  df-mq 8756  df-1nq 8757  df-rq 8758  df-ltnq 8759  df-np 8822  df-plp 8824  df-ltp 8826
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