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Theorem addcanpr 8915
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 8887 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  +P.  B
)  e.  P. )
2 eleq1 2495 . . . . 5  |-  ( ( A  +P.  B )  =  ( A  +P.  C )  ->  ( ( A  +P.  B )  e. 
P. 
<->  ( A  +P.  C
)  e.  P. )
)
3 dmplp 8881 . . . . . 6  |-  dom  +P.  =  ( P.  X.  P. )
4 0npr 8861 . . . . . 6  |-  -.  (/)  e.  P.
53, 4ndmovrcl 6225 . . . . 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 8914 . . . . . . . 8  |-  ( A  e.  P.  ->  ( B  <P  C  <->  ( A  +P.  B )  <P  ( A  +P.  C ) ) )
9 ltapr 8914 . . . . . . . 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 8901 . . . . . . 7  |-  <P  Or  P.
14 sotrieq 4522 . . . . . . 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 8887 . . . . . 6  |-  ( ( A  e.  P.  /\  C  e.  P. )  ->  ( A  +P.  C
)  e.  P. )
18 sotrieq 4522 . . . . . . 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 1652    e. wcel 1725   class class class wbr 4204    Or wor 4494  (class class class)co 6073   P.cnp 8726    +P. cpp 8728    <P cltp 8730
This theorem is referenced by:  enrer  8935  mulcmpblnr  8941  mulgt0sr  8972
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-omul 6721  df-er 6897  df-ni 8741  df-pli 8742  df-mi 8743  df-lti 8744  df-plpq 8777  df-mpq 8778  df-ltpq 8779  df-enq 8780  df-nq 8781  df-erq 8782  df-plq 8783  df-mq 8784  df-1nq 8785  df-rq 8786  df-ltnq 8787  df-np 8850  df-plp 8852  df-ltp 8854
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