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Theorem lt2addnq 7180
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 7176 . . . . . 6  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( A  <Q  B  <->  ( C  +Q  A )  <Q  ( C  +Q  B ) ) )
213expa 1166 . . . . 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 7157 . . . . . . 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 7157 . . . . . . 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 3912 . . . 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 7176 . . . . . 6  |-  ( ( C  e.  Q.  /\  D  e.  Q.  /\  B  e.  Q. )  ->  ( C  <Q  D  <->  ( B  +Q  C )  <Q  ( B  +Q  D ) ) )
13123expa 1166 . . . . 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 7174 . . 3  |-  <Q  Or  Q.
18 ltrelnq 7141 . . 3  |-  <Q  C_  ( Q.  X.  Q. )
1917, 18sotri 4904 . 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 1316    e. wcel 1465   class class class wbr 3899  (class class class)co 5742   Q.cnq 7056    +Q cplq 7058    <Q cltq 7061
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-eprel 4181  df-id 4185  df-po 4188  df-iso 4189  df-iord 4258  df-on 4260  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-recs 6170  df-irdg 6235  df-oadd 6285  df-omul 6286  df-er 6397  df-ec 6399  df-qs 6403  df-ni 7080  df-pli 7081  df-mi 7082  df-lti 7083  df-plpq 7120  df-enq 7123  df-nqqs 7124  df-plqqs 7125  df-ltnqqs 7129
This theorem is referenced by:  addlocprlemeqgt  7308  addnqprlemrl  7333  addnqprlemru  7334  cauappcvgprlemladdfl  7431  caucvgprlemloc  7451  caucvgprprlemloccalc  7460
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