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Theorem addcanpr 8928
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 8900 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  +P.  B
)  e.  P. )
2 eleq1 2498 . . . . 5  |-  ( ( A  +P.  B )  =  ( A  +P.  C )  ->  ( ( A  +P.  B )  e. 
P. 
<->  ( A  +P.  C
)  e.  P. )
)
3 dmplp 8894 . . . . . 6  |-  dom  +P.  =  ( P.  X.  P. )
4 0npr 8874 . . . . . 6  |-  -.  (/)  e.  P.
53, 4ndmovrcl 6236 . . . . 5  |-  ( ( A  +P.  C )  e.  P.  ->  ( A  e.  P.  /\  C  e.  P. ) )
62, 5syl6bi 221 . . . 4  |-  ( ( A  +P.  B )  =  ( A  +P.  C )  ->  ( ( A  +P.  B )  e. 
P.  ->  ( A  e. 
P.  /\  C  e.  P. ) ) )
71, 6syl5com 29 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( A  +P.  B )  =  ( A  +P.  C )  -> 
( A  e.  P.  /\  C  e.  P. )
) )
8 ltapr 8927 . . . . . . . 8  |-  ( A  e.  P.  ->  ( B  <P  C  <->  ( A  +P.  B )  <P  ( A  +P.  C ) ) )
9 ltapr 8927 . . . . . . . 8  |-  ( A  e.  P.  ->  ( C  <P  B  <->  ( A  +P.  C )  <P  ( A  +P.  B ) ) )
108, 9orbi12d 692 . . . . . . 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 287 . . . . . 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 708 . . . . 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 8914 . . . . . . 7  |-  <P  Or  P.
14 sotrieq 4533 . . . . . . 7  |-  ( ( 
<P  Or  P.  /\  ( B  e.  P.  /\  C  e.  P. ) )  -> 
( B  =  C  <->  -.  ( B  <P  C  \/  C  <P  B ) ) )
1513, 14mpan 653 . . . . . 6  |-  ( ( B  e.  P.  /\  C  e.  P. )  ->  ( B  =  C  <->  -.  ( B  <P  C  \/  C  <P  B ) ) )
1615ad2ant2l 728 . . . . 5  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( A  e.  P.  /\  C  e.  P. )
)  ->  ( B  =  C  <->  -.  ( B  <P  C  \/  C  <P  B ) ) )
17 addclpr 8900 . . . . . 6  |-  ( ( A  e.  P.  /\  C  e.  P. )  ->  ( A  +P.  C
)  e.  P. )
18 sotrieq 4533 . . . . . . 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 653 . . . . . 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 465 . . . . 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 278 . . . 4  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( A  e.  P.  /\  C  e.  P. )
)  ->  ( B  =  C  <->  ( A  +P.  B )  =  ( A  +P.  C ) ) )
2221exbiri 607 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( A  e. 
P.  /\  C  e.  P. )  ->  ( ( A  +P.  B )  =  ( A  +P.  C )  ->  B  =  C ) ) )
237, 22syld 43 . 2  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( A  +P.  B )  =  ( A  +P.  C )  -> 
( ( A  +P.  B )  =  ( A  +P.  C )  ->  B  =  C )
) )
2423pm2.43d 47 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 178    \/ wo 359    /\ wa 360    = wceq 1653    e. wcel 1726   class class class wbr 4215    Or wor 4505  (class class class)co 6084   P.cnp 8739    +P. cpp 8741    <P cltp 8743
This theorem is referenced by:  enrer  8948  mulcmpblnr  8954  mulgt0sr  8985
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-inf2 7599
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-recs 6636  df-rdg 6671  df-1o 6727  df-oadd 6731  df-omul 6732  df-er 6908  df-ni 8754  df-pli 8755  df-mi 8756  df-lti 8757  df-plpq 8790  df-mpq 8791  df-ltpq 8792  df-enq 8793  df-nq 8794  df-erq 8795  df-plq 8796  df-mq 8797  df-1nq 8798  df-rq 8799  df-ltnq 8800  df-np 8863  df-plp 8865  df-ltp 8867
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