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Theorem ltapr 9433
Description: Ordering property of addition. Proposition 9-3.5(v) of [Gleason] p. 123. (Contributed by NM, 8-Apr-1996.) (New usage is discouraged.)
Assertion
Ref Expression
ltapr  |-  ( C  e.  P.  ->  ( A  <P  B  <->  ( C  +P.  A )  <P  ( C  +P.  B ) ) )

Proof of Theorem ltapr
StepHypRef Expression
1 dmplp 9400 . 2  |-  dom  +P.  =  ( P.  X.  P. )
2 ltrelpr 9386 . 2  |-  <P  C_  ( P.  X.  P. )
3 0npr 9380 . 2  |-  -.  (/)  e.  P.
4 ltaprlem 9432 . . . . . 6  |-  ( C  e.  P.  ->  ( A  <P  B  ->  ( C  +P.  A )  <P 
( C  +P.  B
) ) )
54adantr 465 . . . . 5  |-  ( ( C  e.  P.  /\  ( B  e.  P.  /\  A  e.  P. )
)  ->  ( A  <P  B  ->  ( C  +P.  A )  <P  ( C  +P.  B ) ) )
6 olc 384 . . . . . . . . 9  |-  ( ( C  +P.  A ) 
<P  ( C  +P.  B
)  ->  ( ( C  +P.  B )  =  ( C  +P.  A
)  \/  ( C  +P.  A )  <P 
( C  +P.  B
) ) )
7 ltaprlem 9432 . . . . . . . . . . . 12  |-  ( C  e.  P.  ->  ( B  <P  A  ->  ( C  +P.  B )  <P 
( C  +P.  A
) ) )
87adantr 465 . . . . . . . . . . 11  |-  ( ( C  e.  P.  /\  ( B  e.  P.  /\  A  e.  P. )
)  ->  ( B  <P  A  ->  ( C  +P.  B )  <P  ( C  +P.  A ) ) )
9 ltsopr 9420 . . . . . . . . . . . . 13  |-  <P  Or  P.
10 sotric 4831 . . . . . . . . . . . . 13  |-  ( ( 
<P  Or  P.  /\  ( B  e.  P.  /\  A  e.  P. ) )  -> 
( B  <P  A  <->  -.  ( B  =  A  \/  A  <P  B ) ) )
119, 10mpan 670 . . . . . . . . . . . 12  |-  ( ( B  e.  P.  /\  A  e.  P. )  ->  ( B  <P  A  <->  -.  ( B  =  A  \/  A  <P  B ) ) )
1211adantl 466 . . . . . . . . . . 11  |-  ( ( C  e.  P.  /\  ( B  e.  P.  /\  A  e.  P. )
)  ->  ( B  <P  A  <->  -.  ( B  =  A  \/  A  <P  B ) ) )
13 addclpr 9406 . . . . . . . . . . . . 13  |-  ( ( C  e.  P.  /\  B  e.  P. )  ->  ( C  +P.  B
)  e.  P. )
14 addclpr 9406 . . . . . . . . . . . . 13  |-  ( ( C  e.  P.  /\  A  e.  P. )  ->  ( C  +P.  A
)  e.  P. )
1513, 14anim12dan 835 . . . . . . . . . . . 12  |-  ( ( C  e.  P.  /\  ( B  e.  P.  /\  A  e.  P. )
)  ->  ( ( C  +P.  B )  e. 
P.  /\  ( C  +P.  A )  e.  P. ) )
16 sotric 4831 . . . . . . . . . . . 12  |-  ( ( 
<P  Or  P.  /\  (
( C  +P.  B
)  e.  P.  /\  ( C  +P.  A )  e.  P. ) )  ->  ( ( C  +P.  B )  <P 
( C  +P.  A
)  <->  -.  ( ( C  +P.  B )  =  ( C  +P.  A
)  \/  ( C  +P.  A )  <P 
( C  +P.  B
) ) ) )
179, 15, 16sylancr 663 . . . . . . . . . . 11  |-  ( ( C  e.  P.  /\  ( B  e.  P.  /\  A  e.  P. )
)  ->  ( ( C  +P.  B )  <P 
( C  +P.  A
)  <->  -.  ( ( C  +P.  B )  =  ( C  +P.  A
)  \/  ( C  +P.  A )  <P 
( C  +P.  B
) ) ) )
188, 12, 173imtr3d 267 . . . . . . . . . 10  |-  ( ( C  e.  P.  /\  ( B  e.  P.  /\  A  e.  P. )
)  ->  ( -.  ( B  =  A  \/  A  <P  B )  ->  -.  ( ( C  +P.  B )  =  ( C  +P.  A
)  \/  ( C  +P.  A )  <P 
( C  +P.  B
) ) ) )
1918con4d 105 . . . . . . . . 9  |-  ( ( C  e.  P.  /\  ( B  e.  P.  /\  A  e.  P. )
)  ->  ( (
( C  +P.  B
)  =  ( C  +P.  A )  \/  ( C  +P.  A
)  <P  ( C  +P.  B ) )  ->  ( B  =  A  \/  A  <P  B ) ) )
206, 19syl5 32 . . . . . . . 8  |-  ( ( C  e.  P.  /\  ( B  e.  P.  /\  A  e.  P. )
)  ->  ( ( C  +P.  A )  <P 
( C  +P.  B
)  ->  ( B  =  A  \/  A  <P  B ) ) )
21 df-or 370 . . . . . . . 8  |-  ( ( B  =  A  \/  A  <P  B )  <->  ( -.  B  =  A  ->  A 
<P  B ) )
2220, 21syl6ib 226 . . . . . . 7  |-  ( ( C  e.  P.  /\  ( B  e.  P.  /\  A  e.  P. )
)  ->  ( ( C  +P.  A )  <P 
( C  +P.  B
)  ->  ( -.  B  =  A  ->  A 
<P  B ) ) )
2322com23 78 . . . . . 6  |-  ( ( C  e.  P.  /\  ( B  e.  P.  /\  A  e.  P. )
)  ->  ( -.  B  =  A  ->  ( ( C  +P.  A
)  <P  ( C  +P.  B )  ->  A  <P  B ) ) )
249, 2soirri 5398 . . . . . . . 8  |-  -.  ( C  +P.  A )  <P 
( C  +P.  A
)
25 oveq2 6302 . . . . . . . . 9  |-  ( B  =  A  ->  ( C  +P.  B )  =  ( C  +P.  A
) )
2625breq2d 4464 . . . . . . . 8  |-  ( B  =  A  ->  (
( C  +P.  A
)  <P  ( C  +P.  B )  <->  ( C  +P.  A )  <P  ( C  +P.  A ) ) )
2724, 26mtbiri 303 . . . . . . 7  |-  ( B  =  A  ->  -.  ( C  +P.  A ) 
<P  ( C  +P.  B
) )
2827pm2.21d 106 . . . . . 6  |-  ( B  =  A  ->  (
( C  +P.  A
)  <P  ( C  +P.  B )  ->  A  <P  B ) )
2923, 28pm2.61d2 160 . . . . 5  |-  ( ( C  e.  P.  /\  ( B  e.  P.  /\  A  e.  P. )
)  ->  ( ( C  +P.  A )  <P 
( C  +P.  B
)  ->  A  <P  B ) )
305, 29impbid 191 . . . 4  |-  ( ( C  e.  P.  /\  ( B  e.  P.  /\  A  e.  P. )
)  ->  ( A  <P  B  <->  ( C  +P.  A )  <P  ( C  +P.  B ) ) )
31303impb 1192 . . 3  |-  ( ( C  e.  P.  /\  B  e.  P.  /\  A  e.  P. )  ->  ( A  <P  B  <->  ( C  +P.  A )  <P  ( C  +P.  B ) ) )
32313com13 1201 . 2  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( A  <P  B  <->  ( C  +P.  A )  <P  ( C  +P.  B ) ) )
331, 2, 3, 32ndmovord 6459 1  |-  ( C  e.  P.  ->  ( A  <P  B  <->  ( C  +P.  A )  <P  ( C  +P.  B ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1379    e. wcel 1767   class class class wbr 4452    Or wor 4804  (class class class)co 6294   P.cnp 9247    +P. cpp 9249    <P cltp 9251
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6586  ax-inf2 8068
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4251  df-int 4288  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-tr 4546  df-eprel 4796  df-id 4800  df-po 4805  df-so 4806  df-fr 4843  df-we 4845  df-ord 4886  df-on 4887  df-lim 4888  df-suc 4889  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-ov 6297  df-oprab 6298  df-mpt2 6299  df-om 6695  df-1st 6794  df-2nd 6795  df-recs 7052  df-rdg 7086  df-1o 7140  df-oadd 7144  df-omul 7145  df-er 7321  df-ni 9260  df-pli 9261  df-mi 9262  df-lti 9263  df-plpq 9296  df-mpq 9297  df-ltpq 9298  df-enq 9299  df-nq 9300  df-erq 9301  df-plq 9302  df-mq 9303  df-1nq 9304  df-rq 9305  df-ltnq 9306  df-np 9369  df-plp 9371  df-ltp 9373
This theorem is referenced by:  addcanpr  9434  ltsrpr  9464  gt0srpr  9465  ltsosr  9481  ltasr  9487  ltpsrpr  9496  map2psrpr  9497
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