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Theorem ltapr 8669
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 8636 . 2  |-  dom  +P.  =  ( P.  X.  P. )
2 ltrelpr 8622 . 2  |-  <P  C_  ( P.  X.  P. )
3 0npr 8616 . 2  |-  -.  (/)  e.  P.
4 ltaprlem 8668 . . . . . 6  |-  ( C  e.  P.  ->  ( A  <P  B  ->  ( C  +P.  A )  <P 
( C  +P.  B
) ) )
54adantr 451 . . . . 5  |-  ( ( C  e.  P.  /\  ( B  e.  P.  /\  A  e.  P. )
)  ->  ( A  <P  B  ->  ( C  +P.  A )  <P  ( C  +P.  B ) ) )
6 olc 373 . . . . . . . . 9  |-  ( ( C  +P.  A ) 
<P  ( C  +P.  B
)  ->  ( ( C  +P.  B )  =  ( C  +P.  A
)  \/  ( C  +P.  A )  <P 
( C  +P.  B
) ) )
7 ltaprlem 8668 . . . . . . . . . . . 12  |-  ( C  e.  P.  ->  ( B  <P  A  ->  ( C  +P.  B )  <P 
( C  +P.  A
) ) )
87adantr 451 . . . . . . . . . . 11  |-  ( ( C  e.  P.  /\  ( B  e.  P.  /\  A  e.  P. )
)  ->  ( B  <P  A  ->  ( C  +P.  B )  <P  ( C  +P.  A ) ) )
9 ltsopr 8656 . . . . . . . . . . . . 13  |-  <P  Or  P.
10 sotric 4340 . . . . . . . . . . . . 13  |-  ( ( 
<P  Or  P.  /\  ( B  e.  P.  /\  A  e.  P. ) )  -> 
( B  <P  A  <->  -.  ( B  =  A  \/  A  <P  B ) ) )
119, 10mpan 651 . . . . . . . . . . . 12  |-  ( ( B  e.  P.  /\  A  e.  P. )  ->  ( B  <P  A  <->  -.  ( B  =  A  \/  A  <P  B ) ) )
1211adantl 452 . . . . . . . . . . 11  |-  ( ( C  e.  P.  /\  ( B  e.  P.  /\  A  e.  P. )
)  ->  ( B  <P  A  <->  -.  ( B  =  A  \/  A  <P  B ) ) )
13 addclpr 8642 . . . . . . . . . . . . 13  |-  ( ( C  e.  P.  /\  B  e.  P. )  ->  ( C  +P.  B
)  e.  P. )
14 addclpr 8642 . . . . . . . . . . . . 13  |-  ( ( C  e.  P.  /\  A  e.  P. )  ->  ( C  +P.  A
)  e.  P. )
1513, 14anim12dan 810 . . . . . . . . . . . 12  |-  ( ( C  e.  P.  /\  ( B  e.  P.  /\  A  e.  P. )
)  ->  ( ( C  +P.  B )  e. 
P.  /\  ( C  +P.  A )  e.  P. ) )
16 sotric 4340 . . . . . . . . . . . 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 644 . . . . . . . . . . 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 258 . . . . . . . . . 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 97 . . . . . . . . 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 28 . . . . . . . 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 359 . . . . . . . 8  |-  ( ( B  =  A  \/  A  <P  B )  <->  ( -.  B  =  A  ->  A 
<P  B ) )
2220, 21syl6ib 217 . . . . . . 7  |-  ( ( C  e.  P.  /\  ( B  e.  P.  /\  A  e.  P. )
)  ->  ( ( C  +P.  A )  <P 
( C  +P.  B
)  ->  ( -.  B  =  A  ->  A 
<P  B ) ) )
2322com23 72 . . . . . 6  |-  ( ( C  e.  P.  /\  ( B  e.  P.  /\  A  e.  P. )
)  ->  ( -.  B  =  A  ->  ( ( C  +P.  A
)  <P  ( C  +P.  B )  ->  A  <P  B ) ) )
249, 2soirri 5069 . . . . . . . 8  |-  -.  ( C  +P.  A )  <P 
( C  +P.  A
)
25 oveq2 5866 . . . . . . . . 9  |-  ( B  =  A  ->  ( C  +P.  B )  =  ( C  +P.  A
) )
2625breq2d 4035 . . . . . . . 8  |-  ( B  =  A  ->  (
( C  +P.  A
)  <P  ( C  +P.  B )  <->  ( C  +P.  A )  <P  ( C  +P.  A ) ) )
2724, 26mtbiri 294 . . . . . . 7  |-  ( B  =  A  ->  -.  ( C  +P.  A ) 
<P  ( C  +P.  B
) )
2827pm2.21d 98 . . . . . 6  |-  ( B  =  A  ->  (
( C  +P.  A
)  <P  ( C  +P.  B )  ->  A  <P  B ) )
2923, 28pm2.61d2 152 . . . . 5  |-  ( ( C  e.  P.  /\  ( B  e.  P.  /\  A  e.  P. )
)  ->  ( ( C  +P.  A )  <P 
( C  +P.  B
)  ->  A  <P  B ) )
305, 29impbid 183 . . . 4  |-  ( ( C  e.  P.  /\  ( B  e.  P.  /\  A  e.  P. )
)  ->  ( A  <P  B  <->  ( C  +P.  A )  <P  ( C  +P.  B ) ) )
31303impb 1147 . . 3  |-  ( ( C  e.  P.  /\  B  e.  P.  /\  A  e.  P. )  ->  ( A  <P  B  <->  ( C  +P.  A )  <P  ( C  +P.  B ) ) )
32313com13 1156 . 2  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( A  <P  B  <->  ( C  +P.  A )  <P  ( C  +P.  B ) ) )
331, 2, 3, 32ndmovord 6010 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 176    \/ wo 357    /\ wa 358    = wceq 1623    e. wcel 1684   class class class wbr 4023    Or wor 4313  (class class class)co 5858   P.cnp 8481    +P. cpp 8483    <P cltp 8485
This theorem is referenced by:  addcanpr  8670  ltsrpr  8699  gt0srpr  8700  ltsosr  8716  ltasr  8722  ltpsrpr  8731  map2psrpr  8732
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-omul 6484  df-er 6660  df-ni 8496  df-pli 8497  df-mi 8498  df-lti 8499  df-plpq 8532  df-mpq 8533  df-ltpq 8534  df-enq 8535  df-nq 8536  df-erq 8537  df-plq 8538  df-mq 8539  df-1nq 8540  df-rq 8541  df-ltnq 8542  df-np 8605  df-plp 8607  df-ltp 8609
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