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Theorem addcanpr 8550
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 8522 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  +P.  B
)  e.  P. )
2 eleq1 2313 . . . . 5  |-  ( ( A  +P.  B )  =  ( A  +P.  C )  ->  ( ( A  +P.  B )  e. 
P. 
<->  ( A  +P.  C
)  e.  P. )
)
3 dmplp 8516 . . . . . 6  |-  dom  +P.  =  ( P.  X.  P. )
4 0npr 8496 . . . . . 6  |-  -.  (/)  e.  P.
53, 4ndmovrcl 5858 . . . . 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 28 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( A  +P.  B )  =  ( A  +P.  C )  -> 
( A  e.  P.  /\  C  e.  P. )
) )
8 ltapr 8549 . . . . . . . 8  |-  ( A  e.  P.  ->  ( B  <P  C  <->  ( A  +P.  B )  <P  ( A  +P.  C ) ) )
9 ltapr 8549 . . . . . . . 8  |-  ( A  e.  P.  ->  ( C  <P  B  <->  ( A  +P.  C )  <P  ( A  +P.  B ) ) )
108, 9orbi12d 693 . . . . . . 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 709 . . . . 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 8536 . . . . . . 7  |-  <P  Or  P.
14 sotrieq 4234 . . . . . . 7  |-  ( ( 
<P  Or  P.  /\  ( B  e.  P.  /\  C  e.  P. ) )  -> 
( B  =  C  <->  -.  ( B  <P  C  \/  C  <P  B ) ) )
1513, 14mpan 654 . . . . . 6  |-  ( ( B  e.  P.  /\  C  e.  P. )  ->  ( B  =  C  <->  -.  ( B  <P  C  \/  C  <P  B ) ) )
1615ad2ant2l 729 . . . . 5  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( A  e.  P.  /\  C  e.  P. )
)  ->  ( B  =  C  <->  -.  ( B  <P  C  \/  C  <P  B ) ) )
17 addclpr 8522 . . . . . 6  |-  ( ( A  e.  P.  /\  C  e.  P. )  ->  ( A  +P.  C
)  e.  P. )
18 sotrieq 4234 . . . . . . 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 654 . . . . . 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 608 . . 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 5    -> wi 6    <-> wb 178    \/ wo 359    /\ wa 360    = wceq 1619    e. wcel 1621   class class class wbr 3920    Or wor 4206  (class class class)co 5710   P.cnp 8361    +P. cpp 8363    <P cltp 8365
This theorem is referenced by:  enrer  8570  mulcmpblnr  8576  mulgt0sr  8607
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-inf2 7226
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-int 3761  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-om 4548  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-recs 6274  df-rdg 6309  df-1o 6365  df-oadd 6369  df-omul 6370  df-er 6546  df-ni 8376  df-pli 8377  df-mi 8378  df-lti 8379  df-plpq 8412  df-mpq 8413  df-ltpq 8414  df-enq 8415  df-nq 8416  df-erq 8417  df-plq 8418  df-mq 8419  df-1nq 8420  df-rq 8421  df-ltnq 8422  df-np 8485  df-plp 8487  df-ltp 8489
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