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Theorem ltapr 9421
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 9388 . 2  |-  dom  +P.  =  ( P.  X.  P. )
2 ltrelpr 9374 . 2  |-  <P  C_  ( P.  X.  P. )
3 0npr 9368 . 2  |-  -.  (/)  e.  P.
4 ltaprlem 9420 . . . . . 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 9420 . . . . . . . . . . . 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 9408 . . . . . . . . . . . . 13  |-  <P  Or  P.
10 sotric 4812 . . . . . . . . . . . . 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 9394 . . . . . . . . . . . . 13  |-  ( ( C  e.  P.  /\  B  e.  P. )  ->  ( C  +P.  B
)  e.  P. )
14 addclpr 9394 . . . . . . . . . . . . 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 4812 . . . . . . . . . . . 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 5379 . . . . . . . 8  |-  -.  ( C  +P.  A )  <P 
( C  +P.  A
)
25 oveq2 6285 . . . . . . . . 9  |-  ( B  =  A  ->  ( C  +P.  B )  =  ( C  +P.  A
) )
2625breq2d 4445 . . . . . . . 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 1191 . . 3  |-  ( ( C  e.  P.  /\  B  e.  P.  /\  A  e.  P. )  ->  ( A  <P  B  <->  ( C  +P.  A )  <P  ( C  +P.  B ) ) )
32313com13 1200 . 2  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( A  <P  B  <->  ( C  +P.  A )  <P  ( C  +P.  B ) ) )
331, 2, 3, 32ndmovord 6446 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 1381    e. wcel 1802   class class class wbr 4433    Or wor 4785  (class class class)co 6277   P.cnp 9235    +P. cpp 9237    <P cltp 9239
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-inf2 8056
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-int 4268  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6682  df-1st 6781  df-2nd 6782  df-recs 7040  df-rdg 7074  df-1o 7128  df-oadd 7132  df-omul 7133  df-er 7309  df-ni 9248  df-pli 9249  df-mi 9250  df-lti 9251  df-plpq 9284  df-mpq 9285  df-ltpq 9286  df-enq 9287  df-nq 9288  df-erq 9289  df-plq 9290  df-mq 9291  df-1nq 9292  df-rq 9293  df-ltnq 9294  df-np 9357  df-plp 9359  df-ltp 9361
This theorem is referenced by:  addcanpr  9422  ltsrpr  9452  gt0srpr  9453  ltsosr  9469  ltasr  9475  ltpsrpr  9484  map2psrpr  9485
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