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Theorem ltapr 8922
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 8889 . 2  |-  dom  +P.  =  ( P.  X.  P. )
2 ltrelpr 8875 . 2  |-  <P  C_  ( P.  X.  P. )
3 0npr 8869 . 2  |-  -.  (/)  e.  P.
4 ltaprlem 8921 . . . . . 6  |-  ( C  e.  P.  ->  ( A  <P  B  ->  ( C  +P.  A )  <P 
( C  +P.  B
) ) )
54adantr 452 . . . . 5  |-  ( ( C  e.  P.  /\  ( B  e.  P.  /\  A  e.  P. )
)  ->  ( A  <P  B  ->  ( C  +P.  A )  <P  ( C  +P.  B ) ) )
6 olc 374 . . . . . . . . 9  |-  ( ( C  +P.  A ) 
<P  ( C  +P.  B
)  ->  ( ( C  +P.  B )  =  ( C  +P.  A
)  \/  ( C  +P.  A )  <P 
( C  +P.  B
) ) )
7 ltaprlem 8921 . . . . . . . . . . . 12  |-  ( C  e.  P.  ->  ( B  <P  A  ->  ( C  +P.  B )  <P 
( C  +P.  A
) ) )
87adantr 452 . . . . . . . . . . 11  |-  ( ( C  e.  P.  /\  ( B  e.  P.  /\  A  e.  P. )
)  ->  ( B  <P  A  ->  ( C  +P.  B )  <P  ( C  +P.  A ) ) )
9 ltsopr 8909 . . . . . . . . . . . . 13  |-  <P  Or  P.
10 sotric 4529 . . . . . . . . . . . . 13  |-  ( ( 
<P  Or  P.  /\  ( B  e.  P.  /\  A  e.  P. ) )  -> 
( B  <P  A  <->  -.  ( B  =  A  \/  A  <P  B ) ) )
119, 10mpan 652 . . . . . . . . . . . 12  |-  ( ( B  e.  P.  /\  A  e.  P. )  ->  ( B  <P  A  <->  -.  ( B  =  A  \/  A  <P  B ) ) )
1211adantl 453 . . . . . . . . . . 11  |-  ( ( C  e.  P.  /\  ( B  e.  P.  /\  A  e.  P. )
)  ->  ( B  <P  A  <->  -.  ( B  =  A  \/  A  <P  B ) ) )
13 addclpr 8895 . . . . . . . . . . . . 13  |-  ( ( C  e.  P.  /\  B  e.  P. )  ->  ( C  +P.  B
)  e.  P. )
14 addclpr 8895 . . . . . . . . . . . . 13  |-  ( ( C  e.  P.  /\  A  e.  P. )  ->  ( C  +P.  A
)  e.  P. )
1513, 14anim12dan 811 . . . . . . . . . . . 12  |-  ( ( C  e.  P.  /\  ( B  e.  P.  /\  A  e.  P. )
)  ->  ( ( C  +P.  B )  e. 
P.  /\  ( C  +P.  A )  e.  P. ) )
16 sotric 4529 . . . . . . . . . . . 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 645 . . . . . . . . . . 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 259 . . . . . . . . . 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 99 . . . . . . . . 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 30 . . . . . . . 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 360 . . . . . . . 8  |-  ( ( B  =  A  \/  A  <P  B )  <->  ( -.  B  =  A  ->  A 
<P  B ) )
2220, 21syl6ib 218 . . . . . . 7  |-  ( ( C  e.  P.  /\  ( B  e.  P.  /\  A  e.  P. )
)  ->  ( ( C  +P.  A )  <P 
( C  +P.  B
)  ->  ( -.  B  =  A  ->  A 
<P  B ) ) )
2322com23 74 . . . . . 6  |-  ( ( C  e.  P.  /\  ( B  e.  P.  /\  A  e.  P. )
)  ->  ( -.  B  =  A  ->  ( ( C  +P.  A
)  <P  ( C  +P.  B )  ->  A  <P  B ) ) )
249, 2soirri 5260 . . . . . . . 8  |-  -.  ( C  +P.  A )  <P 
( C  +P.  A
)
25 oveq2 6089 . . . . . . . . 9  |-  ( B  =  A  ->  ( C  +P.  B )  =  ( C  +P.  A
) )
2625breq2d 4224 . . . . . . . 8  |-  ( B  =  A  ->  (
( C  +P.  A
)  <P  ( C  +P.  B )  <->  ( C  +P.  A )  <P  ( C  +P.  A ) ) )
2724, 26mtbiri 295 . . . . . . 7  |-  ( B  =  A  ->  -.  ( C  +P.  A ) 
<P  ( C  +P.  B
) )
2827pm2.21d 100 . . . . . 6  |-  ( B  =  A  ->  (
( C  +P.  A
)  <P  ( C  +P.  B )  ->  A  <P  B ) )
2923, 28pm2.61d2 154 . . . . 5  |-  ( ( C  e.  P.  /\  ( B  e.  P.  /\  A  e.  P. )
)  ->  ( ( C  +P.  A )  <P 
( C  +P.  B
)  ->  A  <P  B ) )
305, 29impbid 184 . . . 4  |-  ( ( C  e.  P.  /\  ( B  e.  P.  /\  A  e.  P. )
)  ->  ( A  <P  B  <->  ( C  +P.  A )  <P  ( C  +P.  B ) ) )
31303impb 1149 . . 3  |-  ( ( C  e.  P.  /\  B  e.  P.  /\  A  e.  P. )  ->  ( A  <P  B  <->  ( C  +P.  A )  <P  ( C  +P.  B ) ) )
32313com13 1158 . 2  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( A  <P  B  <->  ( C  +P.  A )  <P  ( C  +P.  B ) ) )
331, 2, 3, 32ndmovord 6237 1  |-  ( C  e.  P.  ->  ( A  <P  B  <->  ( C  +P.  A )  <P  ( C  +P.  B ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1652    e. wcel 1725   class class class wbr 4212    Or wor 4502  (class class class)co 6081   P.cnp 8734    +P. cpp 8736    <P cltp 8738
This theorem is referenced by:  addcanpr  8923  ltsrpr  8952  gt0srpr  8953  ltsosr  8969  ltasr  8975  ltpsrpr  8984  map2psrpr  8985
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-omul 6729  df-er 6905  df-ni 8749  df-pli 8750  df-mi 8751  df-lti 8752  df-plpq 8785  df-mpq 8786  df-ltpq 8787  df-enq 8788  df-nq 8789  df-erq 8790  df-plq 8791  df-mq 8792  df-1nq 8793  df-rq 8794  df-ltnq 8795  df-np 8858  df-plp 8860  df-ltp 8862
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