ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  lt2addnq Unicode version

Theorem lt2addnq 7417
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 7413 . . . . . 6  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( A  <Q  B  <->  ( C  +Q  A )  <Q  ( C  +Q  B ) ) )
213expa 1204 . . . . 5  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  C  e.  Q. )  ->  ( A  <Q  B  <-> 
( C  +Q  A
)  <Q  ( C  +Q  B ) ) )
32adantrr 479 . . . 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 7394 . . . . . . 7  |-  ( ( C  e.  Q.  /\  A  e.  Q. )  ->  ( C  +Q  A
)  =  ( A  +Q  C ) )
54ancoms 268 . . . . . 6  |-  ( ( A  e.  Q.  /\  C  e.  Q. )  ->  ( C  +Q  A
)  =  ( A  +Q  C ) )
65ad2ant2r 509 . . . . 5  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  ( C  e.  Q.  /\  D  e.  Q. )
)  ->  ( C  +Q  A )  =  ( A  +Q  C ) )
7 addcomnqg 7394 . . . . . . 7  |-  ( ( C  e.  Q.  /\  B  e.  Q. )  ->  ( C  +Q  B
)  =  ( B  +Q  C ) )
87ancoms 268 . . . . . 6  |-  ( ( B  e.  Q.  /\  C  e.  Q. )  ->  ( C  +Q  B
)  =  ( B  +Q  C ) )
98ad2ant2lr 510 . . . . 5  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  ( C  e.  Q.  /\  D  e.  Q. )
)  ->  ( C  +Q  B )  =  ( B  +Q  C ) )
106, 9breq12d 4028 . . . 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 188 . . 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 7413 . . . . . 6  |-  ( ( C  e.  Q.  /\  D  e.  Q.  /\  B  e.  Q. )  ->  ( C  <Q  D  <->  ( B  +Q  C )  <Q  ( B  +Q  D ) ) )
13123expa 1204 . . . . 5  |-  ( ( ( C  e.  Q.  /\  D  e.  Q. )  /\  B  e.  Q. )  ->  ( C  <Q  D  <-> 
( B  +Q  C
)  <Q  ( B  +Q  D ) ) )
1413ancoms 268 . . . 4  |-  ( ( B  e.  Q.  /\  ( C  e.  Q.  /\  D  e.  Q. )
)  ->  ( C  <Q  D  <->  ( B  +Q  C )  <Q  ( B  +Q  D ) ) )
1514adantll 476 . . 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 473 . 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 7411 . . 3  |-  <Q  Or  Q.
18 ltrelnq 7378 . . 3  |-  <Q  C_  ( Q.  X.  Q. )
1917, 18sotri 5036 . 2  |-  ( ( ( A  +Q  C
)  <Q  ( B  +Q  C )  /\  ( B  +Q  C )  <Q 
( B  +Q  D
) )  ->  ( A  +Q  C )  <Q 
( B  +Q  D
) )
2016, 19biimtrdi 163 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 104    <-> wb 105    = wceq 1363    e. wcel 2158   class class class wbr 4015  (class class class)co 5888   Q.cnq 7293    +Q cplq 7295    <Q cltq 7298
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-iinf 4599
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-ral 2470  df-rex 2471  df-reu 2472  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-eprel 4301  df-id 4305  df-po 4308  df-iso 4309  df-iord 4378  df-on 4380  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6155  df-2nd 6156  df-recs 6320  df-irdg 6385  df-oadd 6435  df-omul 6436  df-er 6549  df-ec 6551  df-qs 6555  df-ni 7317  df-pli 7318  df-mi 7319  df-lti 7320  df-plpq 7357  df-enq 7360  df-nqqs 7361  df-plqqs 7362  df-ltnqqs 7366
This theorem is referenced by:  addlocprlemeqgt  7545  addnqprlemrl  7570  addnqprlemru  7571  cauappcvgprlemladdfl  7668  caucvgprlemloc  7688  caucvgprprlemloccalc  7697
  Copyright terms: Public domain W3C validator