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Theorem ltapr 8664
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 8631 . 2  |-  dom  +P.  =  ( P.  X.  P. )
2 ltrelpr 8617 . 2  |-  <P  C_  ( P.  X.  P. )
3 0npr 8611 . 2  |-  -.  (/)  e.  P.
4 ltaprlem 8663 . . . . . 6  |-  ( C  e.  P.  ->  ( A  <P  B  ->  ( C  +P.  A )  <P 
( C  +P.  B
) ) )
54adantr 453 . . . . 5  |-  ( ( C  e.  P.  /\  ( B  e.  P.  /\  A  e.  P. )
)  ->  ( A  <P  B  ->  ( C  +P.  A )  <P  ( C  +P.  B ) ) )
6 olc 375 . . . . . . . . 9  |-  ( ( C  +P.  A ) 
<P  ( C  +P.  B
)  ->  ( ( C  +P.  B )  =  ( C  +P.  A
)  \/  ( C  +P.  A )  <P 
( C  +P.  B
) ) )
7 ltaprlem 8663 . . . . . . . . . . . 12  |-  ( C  e.  P.  ->  ( B  <P  A  ->  ( C  +P.  B )  <P 
( C  +P.  A
) ) )
87adantr 453 . . . . . . . . . . 11  |-  ( ( C  e.  P.  /\  ( B  e.  P.  /\  A  e.  P. )
)  ->  ( B  <P  A  ->  ( C  +P.  B )  <P  ( C  +P.  A ) ) )
9 ltsopr 8651 . . . . . . . . . . . . 13  |-  <P  Or  P.
10 sotric 4339 . . . . . . . . . . . . 13  |-  ( ( 
<P  Or  P.  /\  ( B  e.  P.  /\  A  e.  P. ) )  -> 
( B  <P  A  <->  -.  ( B  =  A  \/  A  <P  B ) ) )
119, 10mpan 653 . . . . . . . . . . . 12  |-  ( ( B  e.  P.  /\  A  e.  P. )  ->  ( B  <P  A  <->  -.  ( B  =  A  \/  A  <P  B ) ) )
1211adantl 454 . . . . . . . . . . 11  |-  ( ( C  e.  P.  /\  ( B  e.  P.  /\  A  e.  P. )
)  ->  ( B  <P  A  <->  -.  ( B  =  A  \/  A  <P  B ) ) )
13 addclpr 8637 . . . . . . . . . . . . 13  |-  ( ( C  e.  P.  /\  B  e.  P. )  ->  ( C  +P.  B
)  e.  P. )
14 addclpr 8637 . . . . . . . . . . . . 13  |-  ( ( C  e.  P.  /\  A  e.  P. )  ->  ( C  +P.  A
)  e.  P. )
1513, 14anim12dan 812 . . . . . . . . . . . 12  |-  ( ( C  e.  P.  /\  ( B  e.  P.  /\  A  e.  P. )
)  ->  ( ( C  +P.  B )  e. 
P.  /\  ( C  +P.  A )  e.  P. ) )
16 sotric 4339 . . . . . . . . . . . 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 646 . . . . . . . . . . 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 260 . . . . . . . . . 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 361 . . . . . . . 8  |-  ( ( B  =  A  \/  A  <P  B )  <->  ( -.  B  =  A  ->  A 
<P  B ) )
2220, 21syl6ib 219 . . . . . . 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 5068 . . . . . . . 8  |-  -.  ( C  +P.  A )  <P 
( C  +P.  A
)
25 oveq2 5827 . . . . . . . . 9  |-  ( B  =  A  ->  ( C  +P.  B )  =  ( C  +P.  A
) )
2625breq2d 4036 . . . . . . . 8  |-  ( B  =  A  ->  (
( C  +P.  A
)  <P  ( C  +P.  B )  <->  ( C  +P.  A )  <P  ( C  +P.  A ) ) )
2724, 26mtbiri 296 . . . . . . 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 185 . . . 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 5971 1  |-  ( C  e.  P.  ->  ( A  <P  B  <->  ( C  +P.  A )  <P  ( C  +P.  B ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    \/ wo 359    /\ wa 360    = wceq 1624    e. wcel 1685   class class class wbr 4024    Or wor 4312  (class class class)co 5819   P.cnp 8476    +P. cpp 8478    <P cltp 8480
This theorem is referenced by:  addcanpr  8665  ltsrpr  8694  gt0srpr  8695  ltsosr  8711  ltasr  8717  ltpsrpr  8726  map2psrpr  8727
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-inf2 7337
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-int 3864  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-1st 6083  df-2nd 6084  df-recs 6383  df-rdg 6418  df-1o 6474  df-oadd 6478  df-omul 6479  df-er 6655  df-ni 8491  df-pli 8492  df-mi 8493  df-lti 8494  df-plpq 8527  df-mpq 8528  df-ltpq 8529  df-enq 8530  df-nq 8531  df-erq 8532  df-plq 8533  df-mq 8534  df-1nq 8535  df-rq 8536  df-ltnq 8537  df-np 8600  df-plp 8602  df-ltp 8604
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