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Theorem ltaddnq 6505
Description: The sum of two fractions is greater than one of them. (Contributed by NM, 14-Mar-1996.) (Revised by Mario Carneiro, 10-May-2013.)
Assertion
Ref Expression
ltaddnq  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  A  <Q  ( A  +Q  B ) )

Proof of Theorem ltaddnq
Dummy variables  r  s  t are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 1lt2nq 6504 . . . . . . 7  |-  1Q  <Q  ( 1Q  +Q  1Q )
2 1nq 6464 . . . . . . . 8  |-  1Q  e.  Q.
3 addclnq 6473 . . . . . . . . 9  |-  ( ( 1Q  e.  Q.  /\  1Q  e.  Q. )  -> 
( 1Q  +Q  1Q )  e.  Q. )
42, 2, 3mp2an 402 . . . . . . . 8  |-  ( 1Q 
+Q  1Q )  e. 
Q.
5 ltmnqg 6499 . . . . . . . 8  |-  ( ( 1Q  e.  Q.  /\  ( 1Q  +Q  1Q )  e.  Q.  /\  B  e.  Q. )  ->  ( 1Q  <Q  ( 1Q  +Q  1Q )  <->  ( B  .Q  1Q )  <Q  ( B  .Q  ( 1Q  +Q  1Q ) ) ) )
62, 4, 5mp3an12 1222 . . . . . . 7  |-  ( B  e.  Q.  ->  ( 1Q  <Q  ( 1Q  +Q  1Q )  <->  ( B  .Q  1Q )  <Q  ( B  .Q  ( 1Q  +Q  1Q ) ) ) )
71, 6mpbii 136 . . . . . 6  |-  ( B  e.  Q.  ->  ( B  .Q  1Q )  <Q 
( B  .Q  ( 1Q  +Q  1Q ) ) )
8 mulidnq 6487 . . . . . 6  |-  ( B  e.  Q.  ->  ( B  .Q  1Q )  =  B )
9 distrnqg 6485 . . . . . . . 8  |-  ( ( B  e.  Q.  /\  1Q  e.  Q.  /\  1Q  e.  Q. )  ->  ( B  .Q  ( 1Q  +Q  1Q ) )  =  ( ( B  .Q  1Q )  +Q  ( B  .Q  1Q ) ) )
102, 2, 9mp3an23 1224 . . . . . . 7  |-  ( B  e.  Q.  ->  ( B  .Q  ( 1Q  +Q  1Q ) )  =  ( ( B  .Q  1Q )  +Q  ( B  .Q  1Q ) ) )
118, 8oveq12d 5530 . . . . . . 7  |-  ( B  e.  Q.  ->  (
( B  .Q  1Q )  +Q  ( B  .Q  1Q ) )  =  ( B  +Q  B ) )
1210, 11eqtrd 2072 . . . . . 6  |-  ( B  e.  Q.  ->  ( B  .Q  ( 1Q  +Q  1Q ) )  =  ( B  +Q  B ) )
137, 8, 123brtr3d 3793 . . . . 5  |-  ( B  e.  Q.  ->  B  <Q  ( B  +Q  B
) )
1413adantl 262 . . . 4  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  B  <Q  ( B  +Q  B ) )
15 simpr 103 . . . . 5  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  B  e.  Q. )
16 addclnq 6473 . . . . . . 7  |-  ( ( B  e.  Q.  /\  B  e.  Q. )  ->  ( B  +Q  B
)  e.  Q. )
1716anidms 377 . . . . . 6  |-  ( B  e.  Q.  ->  ( B  +Q  B )  e. 
Q. )
1817adantl 262 . . . . 5  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( B  +Q  B
)  e.  Q. )
19 simpl 102 . . . . 5  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  A  e.  Q. )
20 ltanqg 6498 . . . . 5  |-  ( ( B  e.  Q.  /\  ( B  +Q  B
)  e.  Q.  /\  A  e.  Q. )  ->  ( B  <Q  ( B  +Q  B )  <->  ( A  +Q  B )  <Q  ( A  +Q  ( B  +Q  B ) ) ) )
2115, 18, 19, 20syl3anc 1135 . . . 4  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( B  <Q  ( B  +Q  B )  <->  ( A  +Q  B )  <Q  ( A  +Q  ( B  +Q  B ) ) ) )
2214, 21mpbid 135 . . 3  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( A  +Q  B
)  <Q  ( A  +Q  ( B  +Q  B
) ) )
23 addcomnqg 6479 . . 3  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( A  +Q  B
)  =  ( B  +Q  A ) )
24 addcomnqg 6479 . . . . 5  |-  ( ( r  e.  Q.  /\  s  e.  Q. )  ->  ( r  +Q  s
)  =  ( s  +Q  r ) )
2524adantl 262 . . . 4  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  ( r  e.  Q.  /\  s  e.  Q. )
)  ->  ( r  +Q  s )  =  ( s  +Q  r ) )
26 addassnqg 6480 . . . . 5  |-  ( ( r  e.  Q.  /\  s  e.  Q.  /\  t  e.  Q. )  ->  (
( r  +Q  s
)  +Q  t )  =  ( r  +Q  ( s  +Q  t
) ) )
2726adantl 262 . . . 4  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  ( r  e.  Q.  /\  s  e.  Q.  /\  t  e.  Q. )
)  ->  ( (
r  +Q  s )  +Q  t )  =  ( r  +Q  (
s  +Q  t ) ) )
2819, 15, 15, 25, 27caov12d 5682 . . 3  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( A  +Q  ( B  +Q  B ) )  =  ( B  +Q  ( A  +Q  B
) ) )
2922, 23, 283brtr3d 3793 . 2  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( B  +Q  A
)  <Q  ( B  +Q  ( A  +Q  B
) ) )
30 addclnq 6473 . . 3  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( A  +Q  B
)  e.  Q. )
31 ltanqg 6498 . . 3  |-  ( ( A  e.  Q.  /\  ( A  +Q  B
)  e.  Q.  /\  B  e.  Q. )  ->  ( A  <Q  ( A  +Q  B )  <->  ( B  +Q  A )  <Q  ( B  +Q  ( A  +Q  B ) ) ) )
3219, 30, 15, 31syl3anc 1135 . 2  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( A  <Q  ( A  +Q  B )  <->  ( B  +Q  A )  <Q  ( B  +Q  ( A  +Q  B ) ) ) )
3329, 32mpbird 156 1  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  A  <Q  ( A  +Q  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    <-> wb 98    /\ w3a 885    = wceq 1243    e. wcel 1393   class class class wbr 3764  (class class class)co 5512   Q.cnq 6378   1Qc1q 6379    +Q cplq 6380    .Q cmq 6381    <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-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-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-ltnqqs 6451
This theorem is referenced by:  ltexnqq  6506  nsmallnqq  6510  subhalfnqq  6512  ltbtwnnqq  6513  prarloclemarch2  6517  ltexprlemm  6698  ltexprlemopl  6699  addcanprleml  6712  addcanprlemu  6713  recexprlemm  6722  cauappcvgprlemm  6743  cauappcvgprlemopl  6744  cauappcvgprlem2  6758  caucvgprlemnkj  6764  caucvgprlemnbj  6765  caucvgprlemm  6766  caucvgprlemopl  6767  caucvgprprlemnjltk  6789  caucvgprprlemopl  6795
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