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Theorem ltaddnq 8598
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 5865 . . 3  |-  ( x  =  A  ->  (
x  +Q  y )  =  ( A  +Q  y ) )
31, 2breq12d 4036 . 2  |-  ( x  =  A  ->  (
x  <Q  ( x  +Q  y )  <->  A  <Q  ( A  +Q  y ) ) )
4 oveq2 5866 . . 3  |-  ( y  =  B  ->  ( A  +Q  y )  =  ( A  +Q  B
) )
54breq2d 4035 . 2  |-  ( y  =  B  ->  ( A  <Q  ( A  +Q  y )  <->  A  <Q  ( A  +Q  B ) ) )
6 1lt2nq 8597 . . . . . . . 8  |-  1Q  <Q  ( 1Q  +Q  1Q )
7 ltmnq 8596 . . . . . . . 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 8587 . . . . . . 7  |-  ( y  e.  Q.  ->  (
y  .Q  1Q )  =  y )
10 distrnq 8585 . . . . . . . 8  |-  ( y  .Q  ( 1Q  +Q  1Q ) )  =  ( ( y  .Q  1Q )  +Q  ( y  .Q  1Q ) )
119, 9oveq12d 5876 . . . . . . . 8  |-  ( y  e.  Q.  ->  (
( y  .Q  1Q )  +Q  ( y  .Q  1Q ) )  =  ( y  +Q  y
) )
1210, 11syl5eq 2327 . . . . . . 7  |-  ( y  e.  Q.  ->  (
y  .Q  ( 1Q 
+Q  1Q ) )  =  ( y  +Q  y ) )
138, 9, 123brtr3d 4052 . . . . . 6  |-  ( y  e.  Q.  ->  y  <Q  ( y  +Q  y
) )
14 ltanq 8595 . . . . . 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 8575 . . . 4  |-  ( x  +Q  y )  =  ( y  +Q  x
)
18 vex 2791 . . . . 5  |-  x  e. 
_V
19 vex 2791 . . . . 5  |-  y  e. 
_V
20 addcomnq 8575 . . . . 5  |-  ( r  +Q  s )  =  ( s  +Q  r
)
21 addassnq 8582 . . . . 5  |-  ( ( r  +Q  s )  +Q  t )  =  ( r  +Q  (
s  +Q  t ) )
2218, 19, 19, 20, 21caov12 6048 . . . 4  |-  ( x  +Q  ( y  +Q  y ) )  =  ( y  +Q  (
x  +Q  y ) )
2316, 17, 223brtr3g 4054 . . 3  |-  ( ( x  e.  Q.  /\  y  e.  Q. )  ->  ( y  +Q  x
)  <Q  ( y  +Q  ( x  +Q  y
) ) )
24 ltanq 8595 . . . 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 2851 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 1623    e. wcel 1684   class class class wbr 4023  (class class class)co 5858   Q.cnq 8474   1Qc1q 8475    +Q cplq 8477    .Q cmq 8478    <Q cltq 8480
This theorem is referenced by:  ltexnq  8599  nsmallnq  8601  ltbtwnnq  8602  prlem934  8657  ltaddpr  8658  ltexprlem2  8661  ltexprlem4  8663
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-omul 6484  df-er 6660  df-ni 8496  df-pli 8497  df-mi 8498  df-lti 8499  df-plpq 8532  df-mpq 8533  df-ltpq 8534  df-enq 8535  df-nq 8536  df-erq 8537  df-plq 8538  df-mq 8539  df-1nq 8540  df-ltnq 8542
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