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

Theorem ltaddnq 8614
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 19 . . 3  |-  ( x  =  A  ->  x  =  A )
2 oveq1 5881 . . 3  |-  ( x  =  A  ->  (
x  +Q  y )  =  ( A  +Q  y ) )
31, 2breq12d 4052 . 2  |-  ( x  =  A  ->  (
x  <Q  ( x  +Q  y )  <->  A  <Q  ( A  +Q  y ) ) )
4 oveq2 5882 . . 3  |-  ( y  =  B  ->  ( A  +Q  y )  =  ( A  +Q  B
) )
54breq2d 4051 . 2  |-  ( y  =  B  ->  ( A  <Q  ( A  +Q  y )  <->  A  <Q  ( A  +Q  B ) ) )
6 1lt2nq 8613 . . . . . . . 8  |-  1Q  <Q  ( 1Q  +Q  1Q )
7 ltmnq 8612 . . . . . . . 8  |-  ( y  e.  Q.  ->  ( 1Q  <Q  ( 1Q  +Q  1Q )  <->  ( y  .Q  1Q )  <Q  (
y  .Q  ( 1Q 
+Q  1Q ) ) ) )
86, 7mpbii 202 . . . . . . 7  |-  ( y  e.  Q.  ->  (
y  .Q  1Q ) 
<Q  ( y  .Q  ( 1Q  +Q  1Q ) ) )
9 mulidnq 8603 . . . . . . 7  |-  ( y  e.  Q.  ->  (
y  .Q  1Q )  =  y )
10 distrnq 8601 . . . . . . . 8  |-  ( y  .Q  ( 1Q  +Q  1Q ) )  =  ( ( y  .Q  1Q )  +Q  ( y  .Q  1Q ) )
119, 9oveq12d 5892 . . . . . . . 8  |-  ( y  e.  Q.  ->  (
( y  .Q  1Q )  +Q  ( y  .Q  1Q ) )  =  ( y  +Q  y
) )
1210, 11syl5eq 2340 . . . . . . 7  |-  ( y  e.  Q.  ->  (
y  .Q  ( 1Q 
+Q  1Q ) )  =  ( y  +Q  y ) )
138, 9, 123brtr3d 4068 . . . . . 6  |-  ( y  e.  Q.  ->  y  <Q  ( y  +Q  y
) )
14 ltanq 8611 . . . . . 6  |-  ( x  e.  Q.  ->  (
y  <Q  ( y  +Q  y )  <->  ( x  +Q  y )  <Q  (
x  +Q  ( y  +Q  y ) ) ) )
1513, 14syl5ib 210 . . . . 5  |-  ( x  e.  Q.  ->  (
y  e.  Q.  ->  ( x  +Q  y ) 
<Q  ( x  +Q  (
y  +Q  y ) ) ) )
1615imp 418 . . . 4  |-  ( ( x  e.  Q.  /\  y  e.  Q. )  ->  ( x  +Q  y
)  <Q  ( x  +Q  ( y  +Q  y
) ) )
17 addcomnq 8591 . . . 4  |-  ( x  +Q  y )  =  ( y  +Q  x
)
18 vex 2804 . . . . 5  |-  x  e. 
_V
19 vex 2804 . . . . 5  |-  y  e. 
_V
20 addcomnq 8591 . . . . 5  |-  ( r  +Q  s )  =  ( s  +Q  r
)
21 addassnq 8598 . . . . 5  |-  ( ( r  +Q  s )  +Q  t )  =  ( r  +Q  (
s  +Q  t ) )
2218, 19, 19, 20, 21caov12 6064 . . . 4  |-  ( x  +Q  ( y  +Q  y ) )  =  ( y  +Q  (
x  +Q  y ) )
2316, 17, 223brtr3g 4070 . . 3  |-  ( ( x  e.  Q.  /\  y  e.  Q. )  ->  ( y  +Q  x
)  <Q  ( y  +Q  ( x  +Q  y
) ) )
24 ltanq 8611 . . . 4  |-  ( y  e.  Q.  ->  (
x  <Q  ( x  +Q  y )  <->  ( y  +Q  x )  <Q  (
y  +Q  ( x  +Q  y ) ) ) )
2524adantl 452 . . 3  |-  ( ( x  e.  Q.  /\  y  e.  Q. )  ->  ( x  <Q  (
x  +Q  y )  <-> 
( y  +Q  x
)  <Q  ( y  +Q  ( x  +Q  y
) ) ) )
2623, 25mpbird 223 . 2  |-  ( ( x  e.  Q.  /\  y  e.  Q. )  ->  x  <Q  ( x  +Q  y ) )
273, 5, 26vtocl2ga 2864 1  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  A  <Q  ( A  +Q  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   class class class wbr 4039  (class class class)co 5874   Q.cnq 8490   1Qc1q 8491    +Q cplq 8493    .Q cmq 8494    <Q cltq 8496
This theorem is referenced by:  ltexnq  8615  nsmallnq  8617  ltbtwnnq  8618  prlem934  8673  ltaddpr  8674  ltexprlem2  8677  ltexprlem4  8679
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-omul 6500  df-er 6676  df-ni 8512  df-pli 8513  df-mi 8514  df-lti 8515  df-plpq 8548  df-mpq 8549  df-ltpq 8550  df-enq 8551  df-nq 8552  df-erq 8553  df-plq 8554  df-mq 8555  df-1nq 8556  df-ltnq 8558
  Copyright terms: Public domain W3C validator