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

Theorem ltaddnq 7020
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 7019 . . . . . . 7  |-  1Q  <Q  ( 1Q  +Q  1Q )
2 1nq 6979 . . . . . . . 8  |-  1Q  e.  Q.
3 addclnq 6988 . . . . . . . . 9  |-  ( ( 1Q  e.  Q.  /\  1Q  e.  Q. )  -> 
( 1Q  +Q  1Q )  e.  Q. )
42, 2, 3mp2an 418 . . . . . . . 8  |-  ( 1Q 
+Q  1Q )  e. 
Q.
5 ltmnqg 7014 . . . . . . . 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 1264 . . . . . . 7  |-  ( B  e.  Q.  ->  ( 1Q  <Q  ( 1Q  +Q  1Q )  <->  ( B  .Q  1Q )  <Q  ( B  .Q  ( 1Q  +Q  1Q ) ) ) )
71, 6mpbii 147 . . . . . 6  |-  ( B  e.  Q.  ->  ( B  .Q  1Q )  <Q 
( B  .Q  ( 1Q  +Q  1Q ) ) )
8 mulidnq 7002 . . . . . 6  |-  ( B  e.  Q.  ->  ( B  .Q  1Q )  =  B )
9 distrnqg 7000 . . . . . . . 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 1266 . . . . . . 7  |-  ( B  e.  Q.  ->  ( B  .Q  ( 1Q  +Q  1Q ) )  =  ( ( B  .Q  1Q )  +Q  ( B  .Q  1Q ) ) )
118, 8oveq12d 5684 . . . . . . 7  |-  ( B  e.  Q.  ->  (
( B  .Q  1Q )  +Q  ( B  .Q  1Q ) )  =  ( B  +Q  B ) )
1210, 11eqtrd 2121 . . . . . 6  |-  ( B  e.  Q.  ->  ( B  .Q  ( 1Q  +Q  1Q ) )  =  ( B  +Q  B ) )
137, 8, 123brtr3d 3880 . . . . 5  |-  ( B  e.  Q.  ->  B  <Q  ( B  +Q  B
) )
1413adantl 272 . . . 4  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  B  <Q  ( B  +Q  B ) )
15 simpr 109 . . . . 5  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  B  e.  Q. )
16 addclnq 6988 . . . . . . 7  |-  ( ( B  e.  Q.  /\  B  e.  Q. )  ->  ( B  +Q  B
)  e.  Q. )
1716anidms 390 . . . . . 6  |-  ( B  e.  Q.  ->  ( B  +Q  B )  e. 
Q. )
1817adantl 272 . . . . 5  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( B  +Q  B
)  e.  Q. )
19 simpl 108 . . . . 5  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  A  e.  Q. )
20 ltanqg 7013 . . . . 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 1175 . . . 4  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( B  <Q  ( B  +Q  B )  <->  ( A  +Q  B )  <Q  ( A  +Q  ( B  +Q  B ) ) ) )
2214, 21mpbid 146 . . 3  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( A  +Q  B
)  <Q  ( A  +Q  ( B  +Q  B
) ) )
23 addcomnqg 6994 . . 3  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( A  +Q  B
)  =  ( B  +Q  A ) )
24 addcomnqg 6994 . . . . 5  |-  ( ( r  e.  Q.  /\  s  e.  Q. )  ->  ( r  +Q  s
)  =  ( s  +Q  r ) )
2524adantl 272 . . . 4  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  ( r  e.  Q.  /\  s  e.  Q. )
)  ->  ( r  +Q  s )  =  ( s  +Q  r ) )
26 addassnqg 6995 . . . . 5  |-  ( ( r  e.  Q.  /\  s  e.  Q.  /\  t  e.  Q. )  ->  (
( r  +Q  s
)  +Q  t )  =  ( r  +Q  ( s  +Q  t
) ) )
2726adantl 272 . . . 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 5840 . . 3  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( A  +Q  ( B  +Q  B ) )  =  ( B  +Q  ( A  +Q  B
) ) )
2922, 23, 283brtr3d 3880 . 2  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( B  +Q  A
)  <Q  ( B  +Q  ( A  +Q  B
) ) )
30 addclnq 6988 . . 3  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( A  +Q  B
)  e.  Q. )
31 ltanqg 7013 . . 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 1175 . 2  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( A  <Q  ( A  +Q  B )  <->  ( B  +Q  A )  <Q  ( B  +Q  ( A  +Q  B ) ) ) )
3329, 32mpbird 166 1  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  A  <Q  ( A  +Q  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 925    = wceq 1290    e. wcel 1439   class class class wbr 3851  (class class class)co 5666   Q.cnq 6893   1Qc1q 6894    +Q cplq 6895    .Q cmq 6896    <Q cltq 6898
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-coll 3960  ax-sep 3963  ax-nul 3971  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366  ax-iinf 4416
This theorem depends on definitions:  df-bi 116  df-dc 782  df-3or 926  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-ral 2365  df-rex 2366  df-reu 2367  df-rab 2369  df-v 2622  df-sbc 2842  df-csb 2935  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-nul 3288  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-int 3695  df-iun 3738  df-br 3852  df-opab 3906  df-mpt 3907  df-tr 3943  df-eprel 4125  df-id 4129  df-iord 4202  df-on 4204  df-suc 4207  df-iom 4419  df-xp 4457  df-rel 4458  df-cnv 4459  df-co 4460  df-dm 4461  df-rn 4462  df-res 4463  df-ima 4464  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-f1 5033  df-fo 5034  df-f1o 5035  df-fv 5036  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-1st 5925  df-2nd 5926  df-recs 6084  df-irdg 6149  df-1o 6195  df-oadd 6199  df-omul 6200  df-er 6306  df-ec 6308  df-qs 6312  df-ni 6917  df-pli 6918  df-mi 6919  df-lti 6920  df-plpq 6957  df-mpq 6958  df-enq 6960  df-nqqs 6961  df-plqqs 6962  df-mqqs 6963  df-1nqqs 6964  df-ltnqqs 6966
This theorem is referenced by:  ltexnqq  7021  nsmallnqq  7025  subhalfnqq  7027  ltbtwnnqq  7028  prarloclemarch2  7032  ltexprlemm  7213  ltexprlemopl  7214  addcanprleml  7227  addcanprlemu  7228  recexprlemm  7237  cauappcvgprlemm  7258  cauappcvgprlemopl  7259  cauappcvgprlem2  7273  caucvgprlemnkj  7279  caucvgprlemnbj  7280  caucvgprlemm  7281  caucvgprlemopl  7282  caucvgprprlemnjltk  7304  caucvgprprlemopl  7310
  Copyright terms: Public domain W3C validator