MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ltaddnq Unicode version

Theorem ltaddnq 8785
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.) (New usage is discouraged.)
Assertion
Ref Expression
ltaddnq  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  A  <Q  ( A  +Q  B ) )

Proof of Theorem ltaddnq
Dummy variables  x  y  s  r  t are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 id 20 . . 3  |-  ( x  =  A  ->  x  =  A )
2 oveq1 6028 . . 3  |-  ( x  =  A  ->  (
x  +Q  y )  =  ( A  +Q  y ) )
31, 2breq12d 4167 . 2  |-  ( x  =  A  ->  (
x  <Q  ( x  +Q  y )  <->  A  <Q  ( A  +Q  y ) ) )
4 oveq2 6029 . . 3  |-  ( y  =  B  ->  ( A  +Q  y )  =  ( A  +Q  B
) )
54breq2d 4166 . 2  |-  ( y  =  B  ->  ( A  <Q  ( A  +Q  y )  <->  A  <Q  ( A  +Q  B ) ) )
6 1lt2nq 8784 . . . . . . . 8  |-  1Q  <Q  ( 1Q  +Q  1Q )
7 ltmnq 8783 . . . . . . . 8  |-  ( y  e.  Q.  ->  ( 1Q  <Q  ( 1Q  +Q  1Q )  <->  ( y  .Q  1Q )  <Q  (
y  .Q  ( 1Q 
+Q  1Q ) ) ) )
86, 7mpbii 203 . . . . . . 7  |-  ( y  e.  Q.  ->  (
y  .Q  1Q ) 
<Q  ( y  .Q  ( 1Q  +Q  1Q ) ) )
9 mulidnq 8774 . . . . . . 7  |-  ( y  e.  Q.  ->  (
y  .Q  1Q )  =  y )
10 distrnq 8772 . . . . . . . 8  |-  ( y  .Q  ( 1Q  +Q  1Q ) )  =  ( ( y  .Q  1Q )  +Q  ( y  .Q  1Q ) )
119, 9oveq12d 6039 . . . . . . . 8  |-  ( y  e.  Q.  ->  (
( y  .Q  1Q )  +Q  ( y  .Q  1Q ) )  =  ( y  +Q  y
) )
1210, 11syl5eq 2432 . . . . . . 7  |-  ( y  e.  Q.  ->  (
y  .Q  ( 1Q 
+Q  1Q ) )  =  ( y  +Q  y ) )
138, 9, 123brtr3d 4183 . . . . . 6  |-  ( y  e.  Q.  ->  y  <Q  ( y  +Q  y
) )
14 ltanq 8782 . . . . . 6  |-  ( x  e.  Q.  ->  (
y  <Q  ( y  +Q  y )  <->  ( x  +Q  y )  <Q  (
x  +Q  ( y  +Q  y ) ) ) )
1513, 14syl5ib 211 . . . . 5  |-  ( x  e.  Q.  ->  (
y  e.  Q.  ->  ( x  +Q  y ) 
<Q  ( x  +Q  (
y  +Q  y ) ) ) )
1615imp 419 . . . 4  |-  ( ( x  e.  Q.  /\  y  e.  Q. )  ->  ( x  +Q  y
)  <Q  ( x  +Q  ( y  +Q  y
) ) )
17 addcomnq 8762 . . . 4  |-  ( x  +Q  y )  =  ( y  +Q  x
)
18 vex 2903 . . . . 5  |-  x  e. 
_V
19 vex 2903 . . . . 5  |-  y  e. 
_V
20 addcomnq 8762 . . . . 5  |-  ( r  +Q  s )  =  ( s  +Q  r
)
21 addassnq 8769 . . . . 5  |-  ( ( r  +Q  s )  +Q  t )  =  ( r  +Q  (
s  +Q  t ) )
2218, 19, 19, 20, 21caov12 6215 . . . 4  |-  ( x  +Q  ( y  +Q  y ) )  =  ( y  +Q  (
x  +Q  y ) )
2316, 17, 223brtr3g 4185 . . 3  |-  ( ( x  e.  Q.  /\  y  e.  Q. )  ->  ( y  +Q  x
)  <Q  ( y  +Q  ( x  +Q  y
) ) )
24 ltanq 8782 . . . 4  |-  ( y  e.  Q.  ->  (
x  <Q  ( x  +Q  y )  <->  ( y  +Q  x )  <Q  (
y  +Q  ( x  +Q  y ) ) ) )
2524adantl 453 . . 3  |-  ( ( x  e.  Q.  /\  y  e.  Q. )  ->  ( x  <Q  (
x  +Q  y )  <-> 
( y  +Q  x
)  <Q  ( y  +Q  ( x  +Q  y
) ) ) )
2623, 25mpbird 224 . 2  |-  ( ( x  e.  Q.  /\  y  e.  Q. )  ->  x  <Q  ( x  +Q  y ) )
273, 5, 26vtocl2ga 2963 1  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  A  <Q  ( A  +Q  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   class class class wbr 4154  (class class class)co 6021   Q.cnq 8661   1Qc1q 8662    +Q cplq 8664    .Q cmq 8665    <Q cltq 8667
This theorem is referenced by:  ltexnq  8786  nsmallnq  8788  ltbtwnnq  8789  prlem934  8844  ltaddpr  8845  ltexprlem2  8848  ltexprlem4  8850
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-recs 6570  df-rdg 6605  df-1o 6661  df-oadd 6665  df-omul 6666  df-er 6842  df-ni 8683  df-pli 8684  df-mi 8685  df-lti 8686  df-plpq 8719  df-mpq 8720  df-ltpq 8721  df-enq 8722  df-nq 8723  df-erq 8724  df-plq 8725  df-mq 8726  df-1nq 8727  df-ltnq 8729
  Copyright terms: Public domain W3C validator