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Theorem addcanpr 8672
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 8644 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  +P.  B
)  e.  P. )
2 eleq1 2345 . . . . 5  |-  ( ( A  +P.  B )  =  ( A  +P.  C )  ->  ( ( A  +P.  B )  e. 
P. 
<->  ( A  +P.  C
)  e.  P. )
)
3 dmplp 8638 . . . . . 6  |-  dom  +P.  =  ( P.  X.  P. )
4 0npr 8618 . . . . . 6  |-  -.  (/)  e.  P.
53, 4ndmovrcl 6008 . . . . 5  |-  ( ( A  +P.  C )  e.  P.  ->  ( A  e.  P.  /\  C  e.  P. ) )
62, 5syl6bi 219 . . . 4  |-  ( ( A  +P.  B )  =  ( A  +P.  C )  ->  ( ( A  +P.  B )  e. 
P.  ->  ( A  e. 
P.  /\  C  e.  P. ) ) )
71, 6syl5com 26 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( A  +P.  B )  =  ( A  +P.  C )  -> 
( A  e.  P.  /\  C  e.  P. )
) )
8 ltapr 8671 . . . . . . . 8  |-  ( A  e.  P.  ->  ( B  <P  C  <->  ( A  +P.  B )  <P  ( A  +P.  C ) ) )
9 ltapr 8671 . . . . . . . 8  |-  ( A  e.  P.  ->  ( C  <P  B  <->  ( A  +P.  C )  <P  ( A  +P.  B ) ) )
108, 9orbi12d 690 . . . . . . 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 285 . . . . . 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 706 . . . . 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 8658 . . . . . . 7  |-  <P  Or  P.
14 sotrieq 4343 . . . . . . 7  |-  ( ( 
<P  Or  P.  /\  ( B  e.  P.  /\  C  e.  P. ) )  -> 
( B  =  C  <->  -.  ( B  <P  C  \/  C  <P  B ) ) )
1513, 14mpan 651 . . . . . 6  |-  ( ( B  e.  P.  /\  C  e.  P. )  ->  ( B  =  C  <->  -.  ( B  <P  C  \/  C  <P  B ) ) )
1615ad2ant2l 726 . . . . 5  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( A  e.  P.  /\  C  e.  P. )
)  ->  ( B  =  C  <->  -.  ( B  <P  C  \/  C  <P  B ) ) )
17 addclpr 8644 . . . . . 6  |-  ( ( A  e.  P.  /\  C  e.  P. )  ->  ( A  +P.  C
)  e.  P. )
18 sotrieq 4343 . . . . . . 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 651 . . . . . 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 463 . . . . 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 276 . . . 4  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( A  e.  P.  /\  C  e.  P. )
)  ->  ( B  =  C  <->  ( A  +P.  B )  =  ( A  +P.  C ) ) )
2221exbiri 605 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( A  e. 
P.  /\  C  e.  P. )  ->  ( ( A  +P.  B )  =  ( A  +P.  C )  ->  B  =  C ) ) )
237, 22syld 40 . 2  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( A  +P.  B )  =  ( A  +P.  C )  -> 
( ( A  +P.  B )  =  ( A  +P.  C )  ->  B  =  C )
) )
2423pm2.43d 44 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 176    \/ wo 357    /\ wa 358    = wceq 1625    e. wcel 1686   class class class wbr 4025    Or wor 4315  (class class class)co 5860   P.cnp 8483    +P. cpp 8485    <P cltp 8487
This theorem is referenced by:  enrer  8692  mulcmpblnr  8698  mulgt0sr  8729
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-inf2 7344
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 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-int 3865  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-1st 6124  df-2nd 6125  df-recs 6390  df-rdg 6425  df-1o 6481  df-oadd 6485  df-omul 6486  df-er 6662  df-ni 8498  df-pli 8499  df-mi 8500  df-lti 8501  df-plpq 8534  df-mpq 8535  df-ltpq 8536  df-enq 8537  df-nq 8538  df-erq 8539  df-plq 8540  df-mq 8541  df-1nq 8542  df-rq 8543  df-ltnq 8544  df-np 8607  df-plp 8609  df-ltp 8611
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