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Theorem lt2addnq 7219
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 7215 . . . . . 6  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( A  <Q  B  <->  ( C  +Q  A )  <Q  ( C  +Q  B ) ) )
213expa 1181 . . . . 5  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  C  e.  Q. )  ->  ( A  <Q  B  <-> 
( C  +Q  A
)  <Q  ( C  +Q  B ) ) )
32adantrr 470 . . . 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 7196 . . . . . . 7  |-  ( ( C  e.  Q.  /\  A  e.  Q. )  ->  ( C  +Q  A
)  =  ( A  +Q  C ) )
54ancoms 266 . . . . . 6  |-  ( ( A  e.  Q.  /\  C  e.  Q. )  ->  ( C  +Q  A
)  =  ( A  +Q  C ) )
65ad2ant2r 500 . . . . 5  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  ( C  e.  Q.  /\  D  e.  Q. )
)  ->  ( C  +Q  A )  =  ( A  +Q  C ) )
7 addcomnqg 7196 . . . . . . 7  |-  ( ( C  e.  Q.  /\  B  e.  Q. )  ->  ( C  +Q  B
)  =  ( B  +Q  C ) )
87ancoms 266 . . . . . 6  |-  ( ( B  e.  Q.  /\  C  e.  Q. )  ->  ( C  +Q  B
)  =  ( B  +Q  C ) )
98ad2ant2lr 501 . . . . 5  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  ( C  e.  Q.  /\  D  e.  Q. )
)  ->  ( C  +Q  B )  =  ( B  +Q  C ) )
106, 9breq12d 3942 . . . 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 187 . . 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 7215 . . . . . 6  |-  ( ( C  e.  Q.  /\  D  e.  Q.  /\  B  e.  Q. )  ->  ( C  <Q  D  <->  ( B  +Q  C )  <Q  ( B  +Q  D ) ) )
13123expa 1181 . . . . 5  |-  ( ( ( C  e.  Q.  /\  D  e.  Q. )  /\  B  e.  Q. )  ->  ( C  <Q  D  <-> 
( B  +Q  C
)  <Q  ( B  +Q  D ) ) )
1413ancoms 266 . . . 4  |-  ( ( B  e.  Q.  /\  ( C  e.  Q.  /\  D  e.  Q. )
)  ->  ( C  <Q  D  <->  ( B  +Q  C )  <Q  ( B  +Q  D ) ) )
1514adantll 467 . . 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 464 . 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 7213 . . 3  |-  <Q  Or  Q.
18 ltrelnq 7180 . . 3  |-  <Q  C_  ( Q.  X.  Q. )
1917, 18sotri 4934 . 2  |-  ( ( ( A  +Q  C
)  <Q  ( B  +Q  C )  /\  ( B  +Q  C )  <Q 
( B  +Q  D
) )  ->  ( A  +Q  C )  <Q 
( B  +Q  D
) )
2016, 19syl6bi 162 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 103    <-> wb 104    = wceq 1331    e. wcel 1480   class class class wbr 3929  (class class class)co 5774   Q.cnq 7095    +Q cplq 7097    <Q cltq 7100
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-eprel 4211  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-oadd 6317  df-omul 6318  df-er 6429  df-ec 6431  df-qs 6435  df-ni 7119  df-pli 7120  df-mi 7121  df-lti 7122  df-plpq 7159  df-enq 7162  df-nqqs 7163  df-plqqs 7164  df-ltnqqs 7168
This theorem is referenced by:  addlocprlemeqgt  7347  addnqprlemrl  7372  addnqprlemru  7373  cauappcvgprlemladdfl  7470  caucvgprlemloc  7490  caucvgprprlemloccalc  7499
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