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Theorem ltaprg 6717
Description: Ordering property of addition. Proposition 9-3.5(v) of [Gleason] p. 123. (Contributed by Jim Kingdon, 26-Dec-2019.)
Assertion
Ref Expression
ltaprg  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( A  <P  B  <->  ( C  +P.  A )  <P  ( C  +P.  B ) ) )

Proof of Theorem ltaprg
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 ltaprlem 6716 . . 3  |-  ( C  e.  P.  ->  ( A  <P  B  ->  ( C  +P.  A )  <P 
( C  +P.  B
) ) )
213ad2ant3 927 . 2  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( A  <P  B  ->  ( C  +P.  A )  <P 
( C  +P.  B
) ) )
3 ltexpri 6711 . . . . 5  |-  ( ( C  +P.  A ) 
<P  ( C  +P.  B
)  ->  E. x  e.  P.  ( ( C  +P.  A )  +P.  x )  =  ( C  +P.  B ) )
43adantl 262 . . . 4  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( C  +P.  A
)  <P  ( C  +P.  B ) )  ->  E. x  e.  P.  ( ( C  +P.  A )  +P.  x )  =  ( C  +P.  B ) )
5 simpl1 907 . . . . . . 7  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( x  e.  P.  /\  ( ( C  +P.  A )  +P.  x )  =  ( C  +P.  B ) ) )  ->  A  e.  P. )
6 simprl 483 . . . . . . 7  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( x  e.  P.  /\  ( ( C  +P.  A )  +P.  x )  =  ( C  +P.  B ) ) )  ->  x  e.  P. )
7 ltaddpr 6695 . . . . . . 7  |-  ( ( A  e.  P.  /\  x  e.  P. )  ->  A  <P  ( A  +P.  x ) )
85, 6, 7syl2anc 391 . . . . . 6  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( x  e.  P.  /\  ( ( C  +P.  A )  +P.  x )  =  ( C  +P.  B ) ) )  ->  A  <P  ( A  +P.  x ) )
9 addassprg 6677 . . . . . . . . . . . 12  |-  ( ( C  e.  P.  /\  A  e.  P.  /\  x  e.  P. )  ->  (
( C  +P.  A
)  +P.  x )  =  ( C  +P.  ( A  +P.  x ) ) )
1093com12 1108 . . . . . . . . . . 11  |-  ( ( A  e.  P.  /\  C  e.  P.  /\  x  e.  P. )  ->  (
( C  +P.  A
)  +P.  x )  =  ( C  +P.  ( A  +P.  x ) ) )
11103expa 1104 . . . . . . . . . 10  |-  ( ( ( A  e.  P.  /\  C  e.  P. )  /\  x  e.  P. )  ->  ( ( C  +P.  A )  +P.  x )  =  ( C  +P.  ( A  +P.  x ) ) )
1211adantrr 448 . . . . . . . . 9  |-  ( ( ( A  e.  P.  /\  C  e.  P. )  /\  ( x  e.  P.  /\  ( ( C  +P.  A )  +P.  x )  =  ( C  +P.  B ) ) )  -> 
( ( C  +P.  A )  +P.  x )  =  ( C  +P.  ( A  +P.  x ) ) )
13 simprr 484 . . . . . . . . 9  |-  ( ( ( A  e.  P.  /\  C  e.  P. )  /\  ( x  e.  P.  /\  ( ( C  +P.  A )  +P.  x )  =  ( C  +P.  B ) ) )  -> 
( ( C  +P.  A )  +P.  x )  =  ( C  +P.  B ) )
1412, 13eqtr3d 2074 . . . . . . . 8  |-  ( ( ( A  e.  P.  /\  C  e.  P. )  /\  ( x  e.  P.  /\  ( ( C  +P.  A )  +P.  x )  =  ( C  +P.  B ) ) )  -> 
( C  +P.  ( A  +P.  x ) )  =  ( C  +P.  B ) )
15143adantl2 1061 . . . . . . 7  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( x  e.  P.  /\  ( ( C  +P.  A )  +P.  x )  =  ( C  +P.  B ) ) )  -> 
( C  +P.  ( A  +P.  x ) )  =  ( C  +P.  B ) )
16 simpl3 909 . . . . . . . 8  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( x  e.  P.  /\  ( ( C  +P.  A )  +P.  x )  =  ( C  +P.  B ) ) )  ->  C  e.  P. )
17 addclpr 6635 . . . . . . . . 9  |-  ( ( A  e.  P.  /\  x  e.  P. )  ->  ( A  +P.  x
)  e.  P. )
185, 6, 17syl2anc 391 . . . . . . . 8  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( x  e.  P.  /\  ( ( C  +P.  A )  +P.  x )  =  ( C  +P.  B ) ) )  -> 
( A  +P.  x
)  e.  P. )
19 simpl2 908 . . . . . . . 8  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( x  e.  P.  /\  ( ( C  +P.  A )  +P.  x )  =  ( C  +P.  B ) ) )  ->  B  e.  P. )
20 addcanprg 6714 . . . . . . . 8  |-  ( ( C  e.  P.  /\  ( A  +P.  x )  e.  P.  /\  B  e.  P. )  ->  (
( C  +P.  ( A  +P.  x ) )  =  ( C  +P.  B )  ->  ( A  +P.  x )  =  B ) )
2116, 18, 19, 20syl3anc 1135 . . . . . . 7  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( x  e.  P.  /\  ( ( C  +P.  A )  +P.  x )  =  ( C  +P.  B ) ) )  -> 
( ( C  +P.  ( A  +P.  x ) )  =  ( C  +P.  B )  -> 
( A  +P.  x
)  =  B ) )
2215, 21mpd 13 . . . . . 6  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( x  e.  P.  /\  ( ( C  +P.  A )  +P.  x )  =  ( C  +P.  B ) ) )  -> 
( A  +P.  x
)  =  B )
238, 22breqtrd 3788 . . . . 5  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( x  e.  P.  /\  ( ( C  +P.  A )  +P.  x )  =  ( C  +P.  B ) ) )  ->  A  <P  B )
2423adantlr 446 . . . 4  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( C  +P.  A )  <P  ( C  +P.  B ) )  /\  ( x  e.  P.  /\  ( ( C  +P.  A )  +P.  x )  =  ( C  +P.  B ) ) )  ->  A  <P  B )
254, 24rexlimddv 2437 . . 3  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( C  +P.  A
)  <P  ( C  +P.  B ) )  ->  A  <P  B )
2625ex 108 . 2  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  (
( C  +P.  A
)  <P  ( C  +P.  B )  ->  A  <P  B ) )
272, 26impbid 120 1  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( A  <P  B  <->  ( C  +P.  A )  <P  ( C  +P.  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    <-> wb 98    /\ w3a 885    = wceq 1243    e. wcel 1393   E.wrex 2307   class class class wbr 3764  (class class class)co 5512   P.cnp 6389    +P. cpp 6391    <P cltp 6393
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311
This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-1o 6001  df-2o 6002  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-pli 6403  df-mi 6404  df-lti 6405  df-plpq 6442  df-mpq 6443  df-enq 6445  df-nqqs 6446  df-plqqs 6447  df-mqqs 6448  df-1nqqs 6449  df-rq 6450  df-ltnqqs 6451  df-enq0 6522  df-nq0 6523  df-0nq0 6524  df-plq0 6525  df-mq0 6526  df-inp 6564  df-iplp 6566  df-iltp 6568
This theorem is referenced by:  prplnqu  6718  addextpr  6719  caucvgprlemcanl  6742  caucvgprprlemnkltj  6787  caucvgprprlemnbj  6791  caucvgprprlemmu  6793  caucvgprprlemloc  6801  caucvgprprlemexbt  6804  caucvgprprlemexb  6805  caucvgprprlemaddq  6806  caucvgprprlem1  6807  caucvgprprlem2  6808  ltsrprg  6832  gt0srpr  6833  lttrsr  6847  ltsosr  6849  ltasrg  6855  prsrlt  6871
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