ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  subhalfnqq Unicode version

Theorem subhalfnqq 7426
Description: There is a number which is less than half of any positive fraction. The case where  A is one is Lemma 11.4 of [BauerTaylor], p. 50, and they use the word "approximate half" for such a number (since there may be constructions, for some structures other than the rationals themselves, which rely on such an approximate half but do not require division by two as seen at halfnqq 7422). (Contributed by Jim Kingdon, 25-Nov-2019.)
Assertion
Ref Expression
subhalfnqq  |-  ( A  e.  Q.  ->  E. x  e.  Q.  ( x  +Q  x )  <Q  A )
Distinct variable group:    x, A

Proof of Theorem subhalfnqq
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 halfnqq 7422 . . . . . 6  |-  ( A  e.  Q.  ->  E. y  e.  Q.  ( y  +Q  y )  =  A )
2 df-rex 2471 . . . . . . 7  |-  ( E. y  e.  Q.  (
y  +Q  y )  =  A  <->  E. y
( y  e.  Q.  /\  ( y  +Q  y
)  =  A ) )
3 halfnqq 7422 . . . . . . . . . 10  |-  ( y  e.  Q.  ->  E. x  e.  Q.  ( x  +Q  x )  =  y )
43adantr 276 . . . . . . . . 9  |-  ( ( y  e.  Q.  /\  ( y  +Q  y
)  =  A )  ->  E. x  e.  Q.  ( x  +Q  x
)  =  y )
54ancli 323 . . . . . . . 8  |-  ( ( y  e.  Q.  /\  ( y  +Q  y
)  =  A )  ->  ( ( y  e.  Q.  /\  (
y  +Q  y )  =  A )  /\  E. x  e.  Q.  (
x  +Q  x )  =  y ) )
65eximi 1610 . . . . . . 7  |-  ( E. y ( y  e. 
Q.  /\  ( y  +Q  y )  =  A )  ->  E. y
( ( y  e. 
Q.  /\  ( y  +Q  y )  =  A )  /\  E. x  e.  Q.  ( x  +Q  x )  =  y ) )
72, 6sylbi 121 . . . . . 6  |-  ( E. y  e.  Q.  (
y  +Q  y )  =  A  ->  E. y
( ( y  e. 
Q.  /\  ( y  +Q  y )  =  A )  /\  E. x  e.  Q.  ( x  +Q  x )  =  y ) )
81, 7syl 14 . . . . 5  |-  ( A  e.  Q.  ->  E. y
( ( y  e. 
Q.  /\  ( y  +Q  y )  =  A )  /\  E. x  e.  Q.  ( x  +Q  x )  =  y ) )
9 df-rex 2471 . . . . . . 7  |-  ( E. x  e.  Q.  (
x  +Q  x )  =  y  <->  E. x
( x  e.  Q.  /\  ( x  +Q  x
)  =  y ) )
109anbi2i 457 . . . . . 6  |-  ( ( ( y  e.  Q.  /\  ( y  +Q  y
)  =  A )  /\  E. x  e. 
Q.  ( x  +Q  x )  =  y )  <->  ( ( y  e.  Q.  /\  (
y  +Q  y )  =  A )  /\  E. x ( x  e. 
Q.  /\  ( x  +Q  x )  =  y ) ) )
1110exbii 1615 . . . . 5  |-  ( E. y ( ( y  e.  Q.  /\  (
y  +Q  y )  =  A )  /\  E. x  e.  Q.  (
x  +Q  x )  =  y )  <->  E. y
( ( y  e. 
Q.  /\  ( y  +Q  y )  =  A )  /\  E. x
( x  e.  Q.  /\  ( x  +Q  x
)  =  y ) ) )
128, 11sylib 122 . . . 4  |-  ( A  e.  Q.  ->  E. y
( ( y  e. 
Q.  /\  ( y  +Q  y )  =  A )  /\  E. x
( x  e.  Q.  /\  ( x  +Q  x
)  =  y ) ) )
13 exdistr 1919 . . . 4  |-  ( E. y E. x ( ( y  e.  Q.  /\  ( y  +Q  y
)  =  A )  /\  ( x  e. 
Q.  /\  ( x  +Q  x )  =  y ) )  <->  E. y
( ( y  e. 
Q.  /\  ( y  +Q  y )  =  A )  /\  E. x
( x  e.  Q.  /\  ( x  +Q  x
)  =  y ) ) )
1412, 13sylibr 134 . . 3  |-  ( A  e.  Q.  ->  E. y E. x ( ( y  e.  Q.  /\  (
y  +Q  y )  =  A )  /\  ( x  e.  Q.  /\  ( x  +Q  x
)  =  y ) ) )
15 simprl 529 . . . . . 6  |-  ( ( ( y  e.  Q.  /\  ( y  +Q  y
)  =  A )  /\  ( x  e. 
Q.  /\  ( x  +Q  x )  =  y ) )  ->  x  e.  Q. )
16 simpll 527 . . . . . . . . 9  |-  ( ( ( y  e.  Q.  /\  ( y  +Q  y
)  =  A )  /\  ( x  e. 
Q.  /\  ( x  +Q  x )  =  y ) )  ->  y  e.  Q. )
17 ltaddnq 7419 . . . . . . . . 9  |-  ( ( y  e.  Q.  /\  y  e.  Q. )  ->  y  <Q  ( y  +Q  y ) )
1816, 16, 17syl2anc 411 . . . . . . . 8  |-  ( ( ( y  e.  Q.  /\  ( y  +Q  y
)  =  A )  /\  ( x  e. 
Q.  /\  ( x  +Q  x )  =  y ) )  ->  y  <Q  ( y  +Q  y
) )
19 breq2 4019 . . . . . . . . 9  |-  ( ( y  +Q  y )  =  A  ->  (
y  <Q  ( y  +Q  y )  <->  y  <Q  A ) )
2019ad2antlr 489 . . . . . . . 8  |-  ( ( ( y  e.  Q.  /\  ( y  +Q  y
)  =  A )  /\  ( x  e. 
Q.  /\  ( x  +Q  x )  =  y ) )  ->  (
y  <Q  ( y  +Q  y )  <->  y  <Q  A ) )
2118, 20mpbid 147 . . . . . . 7  |-  ( ( ( y  e.  Q.  /\  ( y  +Q  y
)  =  A )  /\  ( x  e. 
Q.  /\  ( x  +Q  x )  =  y ) )  ->  y  <Q  A )
22 breq1 4018 . . . . . . . 8  |-  ( ( x  +Q  x )  =  y  ->  (
( x  +Q  x
)  <Q  A  <->  y  <Q  A ) )
2322ad2antll 491 . . . . . . 7  |-  ( ( ( y  e.  Q.  /\  ( y  +Q  y
)  =  A )  /\  ( x  e. 
Q.  /\  ( x  +Q  x )  =  y ) )  ->  (
( x  +Q  x
)  <Q  A  <->  y  <Q  A ) )
2421, 23mpbird 167 . . . . . 6  |-  ( ( ( y  e.  Q.  /\  ( y  +Q  y
)  =  A )  /\  ( x  e. 
Q.  /\  ( x  +Q  x )  =  y ) )  ->  (
x  +Q  x ) 
<Q  A )
2515, 24jca 306 . . . . 5  |-  ( ( ( y  e.  Q.  /\  ( y  +Q  y
)  =  A )  /\  ( x  e. 
Q.  /\  ( x  +Q  x )  =  y ) )  ->  (
x  e.  Q.  /\  ( x  +Q  x
)  <Q  A ) )
2625eximi 1610 . . . 4  |-  ( E. x ( ( y  e.  Q.  /\  (
y  +Q  y )  =  A )  /\  ( x  e.  Q.  /\  ( x  +Q  x
)  =  y ) )  ->  E. x
( x  e.  Q.  /\  ( x  +Q  x
)  <Q  A ) )
2726exlimiv 1608 . . 3  |-  ( E. y E. x ( ( y  e.  Q.  /\  ( y  +Q  y
)  =  A )  /\  ( x  e. 
Q.  /\  ( x  +Q  x )  =  y ) )  ->  E. x
( x  e.  Q.  /\  ( x  +Q  x
)  <Q  A ) )
2814, 27syl 14 . 2  |-  ( A  e.  Q.  ->  E. x
( x  e.  Q.  /\  ( x  +Q  x
)  <Q  A ) )
29 df-rex 2471 . 2  |-  ( E. x  e.  Q.  (
x  +Q  x ) 
<Q  A  <->  E. x ( x  e.  Q.  /\  (
x  +Q  x ) 
<Q  A ) )
3028, 29sylibr 134 1  |-  ( A  e.  Q.  ->  E. x  e.  Q.  ( x  +Q  x )  <Q  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1363   E.wex 1502    e. wcel 2158   E.wrex 2466   class class class wbr 4015  (class class class)co 5888   Q.cnq 7292    +Q cplq 7294    <Q cltq 7297
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-iinf 4599
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-ral 2470  df-rex 2471  df-reu 2472  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-eprel 4301  df-id 4305  df-iord 4378  df-on 4380  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6154  df-2nd 6155  df-recs 6319  df-irdg 6384  df-1o 6430  df-oadd 6434  df-omul 6435  df-er 6548  df-ec 6550  df-qs 6554  df-ni 7316  df-pli 7317  df-mi 7318  df-lti 7319  df-plpq 7356  df-mpq 7357  df-enq 7359  df-nqqs 7360  df-plqqs 7361  df-mqqs 7362  df-1nqqs 7363  df-rq 7364  df-ltnqqs 7365
This theorem is referenced by:  prarloc  7515  cauappcvgprlemloc  7664  caucvgprlemloc  7687  caucvgprprlemml  7706  caucvgprprlemloc  7715
  Copyright terms: Public domain W3C validator