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Theorem lt2addnq 6502
Description: Ordering property of addition for positive fractions. (Contributed by Jim Kingdon, 7-Dec-2019.)
Assertion
Ref Expression
lt2addnq  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  ( C  e.  Q.  /\  D  e.  Q. )
)  ->  ( ( A  <Q  B  /\  C  <Q  D )  ->  ( A  +Q  C )  <Q 
( B  +Q  D
) ) )

Proof of Theorem lt2addnq
StepHypRef Expression
1 ltanqg 6498 . . . . . 6  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( A  <Q  B  <->  ( C  +Q  A )  <Q  ( C  +Q  B ) ) )
213expa 1104 . . . . 5  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  C  e.  Q. )  ->  ( A  <Q  B  <-> 
( C  +Q  A
)  <Q  ( C  +Q  B ) ) )
32adantrr 448 . . . 4  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  ( C  e.  Q.  /\  D  e.  Q. )
)  ->  ( A  <Q  B  <->  ( C  +Q  A )  <Q  ( C  +Q  B ) ) )
4 addcomnqg 6479 . . . . . . 7  |-  ( ( C  e.  Q.  /\  A  e.  Q. )  ->  ( C  +Q  A
)  =  ( A  +Q  C ) )
54ancoms 255 . . . . . 6  |-  ( ( A  e.  Q.  /\  C  e.  Q. )  ->  ( C  +Q  A
)  =  ( A  +Q  C ) )
65ad2ant2r 478 . . . . 5  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  ( C  e.  Q.  /\  D  e.  Q. )
)  ->  ( C  +Q  A )  =  ( A  +Q  C ) )
7 addcomnqg 6479 . . . . . . 7  |-  ( ( C  e.  Q.  /\  B  e.  Q. )  ->  ( C  +Q  B
)  =  ( B  +Q  C ) )
87ancoms 255 . . . . . 6  |-  ( ( B  e.  Q.  /\  C  e.  Q. )  ->  ( C  +Q  B
)  =  ( B  +Q  C ) )
98ad2ant2lr 479 . . . . 5  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  ( C  e.  Q.  /\  D  e.  Q. )
)  ->  ( C  +Q  B )  =  ( B  +Q  C ) )
106, 9breq12d 3777 . . . 4  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  ( C  e.  Q.  /\  D  e.  Q. )
)  ->  ( ( C  +Q  A )  <Q 
( C  +Q  B
)  <->  ( A  +Q  C )  <Q  ( B  +Q  C ) ) )
113, 10bitrd 177 . . 3  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  ( C  e.  Q.  /\  D  e.  Q. )
)  ->  ( A  <Q  B  <->  ( A  +Q  C )  <Q  ( B  +Q  C ) ) )
12 ltanqg 6498 . . . . . 6  |-  ( ( C  e.  Q.  /\  D  e.  Q.  /\  B  e.  Q. )  ->  ( C  <Q  D  <->  ( B  +Q  C )  <Q  ( B  +Q  D ) ) )
13123expa 1104 . . . . 5  |-  ( ( ( C  e.  Q.  /\  D  e.  Q. )  /\  B  e.  Q. )  ->  ( C  <Q  D  <-> 
( B  +Q  C
)  <Q  ( B  +Q  D ) ) )
1413ancoms 255 . . . 4  |-  ( ( B  e.  Q.  /\  ( C  e.  Q.  /\  D  e.  Q. )
)  ->  ( C  <Q  D  <->  ( B  +Q  C )  <Q  ( B  +Q  D ) ) )
1514adantll 445 . . 3  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  ( C  e.  Q.  /\  D  e.  Q. )
)  ->  ( C  <Q  D  <->  ( B  +Q  C )  <Q  ( B  +Q  D ) ) )
1611, 15anbi12d 442 . 2  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  ( C  e.  Q.  /\  D  e.  Q. )
)  ->  ( ( A  <Q  B  /\  C  <Q  D )  <->  ( ( A  +Q  C )  <Q 
( B  +Q  C
)  /\  ( B  +Q  C )  <Q  ( B  +Q  D ) ) ) )
17 ltsonq 6496 . . 3  |-  <Q  Or  Q.
18 ltrelnq 6463 . . 3  |-  <Q  C_  ( Q.  X.  Q. )
1917, 18sotri 4720 . 2  |-  ( ( ( A  +Q  C
)  <Q  ( B  +Q  C )  /\  ( B  +Q  C )  <Q 
( B  +Q  D
) )  ->  ( A  +Q  C )  <Q 
( B  +Q  D
) )
2016, 19syl6bi 152 1  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  ( C  e.  Q.  /\  D  e.  Q. )
)  ->  ( ( A  <Q  B  /\  C  <Q  D )  ->  ( A  +Q  C )  <Q 
( B  +Q  D
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    <-> wb 98    = wceq 1243    e. wcel 1393   class class class wbr 3764  (class class class)co 5512   Q.cnq 6378    +Q cplq 6380    <Q cltq 6383
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-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-enq 6445  df-nqqs 6446  df-plqqs 6447  df-ltnqqs 6451
This theorem is referenced by:  addlocprlemeqgt  6630  addnqprlemrl  6655  addnqprlemru  6656  cauappcvgprlemladdfl  6753  caucvgprlemloc  6773  caucvgprprlemloccalc  6782
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