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Theorem ltaddnq 8843
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 6080 . . 3  |-  ( x  =  A  ->  (
x  +Q  y )  =  ( A  +Q  y ) )
31, 2breq12d 4217 . 2  |-  ( x  =  A  ->  (
x  <Q  ( x  +Q  y )  <->  A  <Q  ( A  +Q  y ) ) )
4 oveq2 6081 . . 3  |-  ( y  =  B  ->  ( A  +Q  y )  =  ( A  +Q  B
) )
54breq2d 4216 . 2  |-  ( y  =  B  ->  ( A  <Q  ( A  +Q  y )  <->  A  <Q  ( A  +Q  B ) ) )
6 1lt2nq 8842 . . . . . . . 8  |-  1Q  <Q  ( 1Q  +Q  1Q )
7 ltmnq 8841 . . . . . . . 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 8832 . . . . . . 7  |-  ( y  e.  Q.  ->  (
y  .Q  1Q )  =  y )
10 distrnq 8830 . . . . . . . 8  |-  ( y  .Q  ( 1Q  +Q  1Q ) )  =  ( ( y  .Q  1Q )  +Q  ( y  .Q  1Q ) )
119, 9oveq12d 6091 . . . . . . . 8  |-  ( y  e.  Q.  ->  (
( y  .Q  1Q )  +Q  ( y  .Q  1Q ) )  =  ( y  +Q  y
) )
1210, 11syl5eq 2479 . . . . . . 7  |-  ( y  e.  Q.  ->  (
y  .Q  ( 1Q 
+Q  1Q ) )  =  ( y  +Q  y ) )
138, 9, 123brtr3d 4233 . . . . . 6  |-  ( y  e.  Q.  ->  y  <Q  ( y  +Q  y
) )
14 ltanq 8840 . . . . . 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 8820 . . . 4  |-  ( x  +Q  y )  =  ( y  +Q  x
)
18 vex 2951 . . . . 5  |-  x  e. 
_V
19 vex 2951 . . . . 5  |-  y  e. 
_V
20 addcomnq 8820 . . . . 5  |-  ( r  +Q  s )  =  ( s  +Q  r
)
21 addassnq 8827 . . . . 5  |-  ( ( r  +Q  s )  +Q  t )  =  ( r  +Q  (
s  +Q  t ) )
2218, 19, 19, 20, 21caov12 6267 . . . 4  |-  ( x  +Q  ( y  +Q  y ) )  =  ( y  +Q  (
x  +Q  y ) )
2316, 17, 223brtr3g 4235 . . 3  |-  ( ( x  e.  Q.  /\  y  e.  Q. )  ->  ( y  +Q  x
)  <Q  ( y  +Q  ( x  +Q  y
) ) )
24 ltanq 8840 . . . 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 3011 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 1652    e. wcel 1725   class class class wbr 4204  (class class class)co 6073   Q.cnq 8719   1Qc1q 8720    +Q cplq 8722    .Q cmq 8723    <Q cltq 8725
This theorem is referenced by:  ltexnq  8844  nsmallnq  8846  ltbtwnnq  8847  prlem934  8902  ltaddpr  8903  ltexprlem2  8906  ltexprlem4  8908
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-omul 6721  df-er 6897  df-ni 8741  df-pli 8742  df-mi 8743  df-lti 8744  df-plpq 8777  df-mpq 8778  df-ltpq 8779  df-enq 8780  df-nq 8781  df-erq 8782  df-plq 8783  df-mq 8784  df-1nq 8785  df-ltnq 8787
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