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Theorem ltaddnq 9255
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 22 . . 3  |-  ( x  =  A  ->  x  =  A )
2 oveq1 6208 . . 3  |-  ( x  =  A  ->  (
x  +Q  y )  =  ( A  +Q  y ) )
31, 2breq12d 4414 . 2  |-  ( x  =  A  ->  (
x  <Q  ( x  +Q  y )  <->  A  <Q  ( A  +Q  y ) ) )
4 oveq2 6209 . . 3  |-  ( y  =  B  ->  ( A  +Q  y )  =  ( A  +Q  B
) )
54breq2d 4413 . 2  |-  ( y  =  B  ->  ( A  <Q  ( A  +Q  y )  <->  A  <Q  ( A  +Q  B ) ) )
6 1lt2nq 9254 . . . . . . . 8  |-  1Q  <Q  ( 1Q  +Q  1Q )
7 ltmnq 9253 . . . . . . . 8  |-  ( y  e.  Q.  ->  ( 1Q  <Q  ( 1Q  +Q  1Q )  <->  ( y  .Q  1Q )  <Q  (
y  .Q  ( 1Q 
+Q  1Q ) ) ) )
86, 7mpbii 211 . . . . . . 7  |-  ( y  e.  Q.  ->  (
y  .Q  1Q ) 
<Q  ( y  .Q  ( 1Q  +Q  1Q ) ) )
9 mulidnq 9244 . . . . . . 7  |-  ( y  e.  Q.  ->  (
y  .Q  1Q )  =  y )
10 distrnq 9242 . . . . . . . 8  |-  ( y  .Q  ( 1Q  +Q  1Q ) )  =  ( ( y  .Q  1Q )  +Q  ( y  .Q  1Q ) )
119, 9oveq12d 6219 . . . . . . . 8  |-  ( y  e.  Q.  ->  (
( y  .Q  1Q )  +Q  ( y  .Q  1Q ) )  =  ( y  +Q  y
) )
1210, 11syl5eq 2507 . . . . . . 7  |-  ( y  e.  Q.  ->  (
y  .Q  ( 1Q 
+Q  1Q ) )  =  ( y  +Q  y ) )
138, 9, 123brtr3d 4430 . . . . . 6  |-  ( y  e.  Q.  ->  y  <Q  ( y  +Q  y
) )
14 ltanq 9252 . . . . . 6  |-  ( x  e.  Q.  ->  (
y  <Q  ( y  +Q  y )  <->  ( x  +Q  y )  <Q  (
x  +Q  ( y  +Q  y ) ) ) )
1513, 14syl5ib 219 . . . . 5  |-  ( x  e.  Q.  ->  (
y  e.  Q.  ->  ( x  +Q  y ) 
<Q  ( x  +Q  (
y  +Q  y ) ) ) )
1615imp 429 . . . 4  |-  ( ( x  e.  Q.  /\  y  e.  Q. )  ->  ( x  +Q  y
)  <Q  ( x  +Q  ( y  +Q  y
) ) )
17 addcomnq 9232 . . . 4  |-  ( x  +Q  y )  =  ( y  +Q  x
)
18 vex 3081 . . . . 5  |-  x  e. 
_V
19 vex 3081 . . . . 5  |-  y  e. 
_V
20 addcomnq 9232 . . . . 5  |-  ( r  +Q  s )  =  ( s  +Q  r
)
21 addassnq 9239 . . . . 5  |-  ( ( r  +Q  s )  +Q  t )  =  ( r  +Q  (
s  +Q  t ) )
2218, 19, 19, 20, 21caov12 6402 . . . 4  |-  ( x  +Q  ( y  +Q  y ) )  =  ( y  +Q  (
x  +Q  y ) )
2316, 17, 223brtr3g 4432 . . 3  |-  ( ( x  e.  Q.  /\  y  e.  Q. )  ->  ( y  +Q  x
)  <Q  ( y  +Q  ( x  +Q  y
) ) )
24 ltanq 9252 . . . 4  |-  ( y  e.  Q.  ->  (
x  <Q  ( x  +Q  y )  <->  ( y  +Q  x )  <Q  (
y  +Q  ( x  +Q  y ) ) ) )
2524adantl 466 . . 3  |-  ( ( x  e.  Q.  /\  y  e.  Q. )  ->  ( x  <Q  (
x  +Q  y )  <-> 
( y  +Q  x
)  <Q  ( y  +Q  ( x  +Q  y
) ) ) )
2623, 25mpbird 232 . 2  |-  ( ( x  e.  Q.  /\  y  e.  Q. )  ->  x  <Q  ( x  +Q  y ) )
273, 5, 26vtocl2ga 3144 1  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  A  <Q  ( A  +Q  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   class class class wbr 4401  (class class class)co 6201   Q.cnq 9131   1Qc1q 9132    +Q cplq 9134    .Q cmq 9135    <Q cltq 9137
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-om 6588  df-1st 6688  df-2nd 6689  df-recs 6943  df-rdg 6977  df-1o 7031  df-oadd 7035  df-omul 7036  df-er 7212  df-ni 9153  df-pli 9154  df-mi 9155  df-lti 9156  df-plpq 9189  df-mpq 9190  df-ltpq 9191  df-enq 9192  df-nq 9193  df-erq 9194  df-plq 9195  df-mq 9196  df-1nq 9197  df-ltnq 9199
This theorem is referenced by:  ltexnq  9256  nsmallnq  9258  ltbtwnnq  9259  prlem934  9314  ltaddpr  9315  ltexprlem2  9318  ltexprlem4  9320
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