ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  prmuloc2 Unicode version

Theorem prmuloc2 7126
Description: Positive reals are multiplicatively located. This is a variation of prmuloc 7125 which only constructs one (named) point and is therefore often easier to work with. It states that given a ratio  B, there are elements of the lower and upper cut which have exactly that ratio between them. (Contributed by Jim Kingdon, 28-Dec-2019.)
Assertion
Ref Expression
prmuloc2  |-  ( (
<. L ,  U >.  e. 
P.  /\  1Q  <Q  B )  ->  E. x  e.  L  ( x  .Q  B )  e.  U
)
Distinct variable groups:    x, B    x, L    x, U

Proof of Theorem prmuloc2
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 prmuloc 7125 . 2  |-  ( (
<. L ,  U >.  e. 
P.  /\  1Q  <Q  B )  ->  E. x  e.  Q.  E. y  e. 
Q.  ( x  e.  L  /\  y  e.  U  /\  ( y  .Q  1Q )  <Q 
( x  .Q  B
) ) )
2 nfv 1466 . . 3  |-  F/ x
( <. L ,  U >.  e.  P.  /\  1Q  <Q  B )
3 nfre1 2419 . . 3  |-  F/ x E. x  e.  L  ( x  .Q  B
)  e.  U
4 simpr1 949 . . . . . . . 8  |-  ( ( ( ( <. L ,  U >.  e.  P.  /\  1Q  <Q  B )  /\  ( x  e.  Q.  /\  y  e.  Q. )
)  /\  ( x  e.  L  /\  y  e.  U  /\  (
y  .Q  1Q ) 
<Q  ( x  .Q  B
) ) )  ->  x  e.  L )
5 simpr3 951 . . . . . . . . . 10  |-  ( ( ( ( <. L ,  U >.  e.  P.  /\  1Q  <Q  B )  /\  ( x  e.  Q.  /\  y  e.  Q. )
)  /\  ( x  e.  L  /\  y  e.  U  /\  (
y  .Q  1Q ) 
<Q  ( x  .Q  B
) ) )  -> 
( y  .Q  1Q )  <Q  ( x  .Q  B ) )
6 simplrr 503 . . . . . . . . . . 11  |-  ( ( ( ( <. L ,  U >.  e.  P.  /\  1Q  <Q  B )  /\  ( x  e.  Q.  /\  y  e.  Q. )
)  /\  ( x  e.  L  /\  y  e.  U  /\  (
y  .Q  1Q ) 
<Q  ( x  .Q  B
) ) )  -> 
y  e.  Q. )
7 mulidnq 6948 . . . . . . . . . . 11  |-  ( y  e.  Q.  ->  (
y  .Q  1Q )  =  y )
8 breq1 3848 . . . . . . . . . . 11  |-  ( ( y  .Q  1Q )  =  y  ->  (
( y  .Q  1Q )  <Q  ( x  .Q  B )  <->  y  <Q  ( x  .Q  B ) ) )
96, 7, 83syl 17 . . . . . . . . . 10  |-  ( ( ( ( <. L ,  U >.  e.  P.  /\  1Q  <Q  B )  /\  ( x  e.  Q.  /\  y  e.  Q. )
)  /\  ( x  e.  L  /\  y  e.  U  /\  (
y  .Q  1Q ) 
<Q  ( x  .Q  B
) ) )  -> 
( ( y  .Q  1Q )  <Q  (
x  .Q  B )  <-> 
y  <Q  ( x  .Q  B ) ) )
105, 9mpbid 145 . . . . . . . . 9  |-  ( ( ( ( <. L ,  U >.  e.  P.  /\  1Q  <Q  B )  /\  ( x  e.  Q.  /\  y  e.  Q. )
)  /\  ( x  e.  L  /\  y  e.  U  /\  (
y  .Q  1Q ) 
<Q  ( x  .Q  B
) ) )  -> 
y  <Q  ( x  .Q  B ) )
11 simplll 500 . . . . . . . . . 10  |-  ( ( ( ( <. L ,  U >.  e.  P.  /\  1Q  <Q  B )  /\  ( x  e.  Q.  /\  y  e.  Q. )
)  /\  ( x  e.  L  /\  y  e.  U  /\  (
y  .Q  1Q ) 
<Q  ( x  .Q  B
) ) )  ->  <. L ,  U >.  e. 
P. )
12 simpr2 950 . . . . . . . . . 10  |-  ( ( ( ( <. L ,  U >.  e.  P.  /\  1Q  <Q  B )  /\  ( x  e.  Q.  /\  y  e.  Q. )
)  /\  ( x  e.  L  /\  y  e.  U  /\  (
y  .Q  1Q ) 
<Q  ( x  .Q  B
) ) )  -> 
y  e.  U )
13 prcunqu 7044 . . . . . . . . . 10  |-  ( (
<. L ,  U >.  e. 
P.  /\  y  e.  U )  ->  (
y  <Q  ( x  .Q  B )  ->  (
x  .Q  B )  e.  U ) )
1411, 12, 13syl2anc 403 . . . . . . . . 9  |-  ( ( ( ( <. L ,  U >.  e.  P.  /\  1Q  <Q  B )  /\  ( x  e.  Q.  /\  y  e.  Q. )
)  /\  ( x  e.  L  /\  y  e.  U  /\  (
y  .Q  1Q ) 
<Q  ( x  .Q  B
) ) )  -> 
( y  <Q  (
x  .Q  B )  ->  ( x  .Q  B )  e.  U
) )
1510, 14mpd 13 . . . . . . . 8  |-  ( ( ( ( <. L ,  U >.  e.  P.  /\  1Q  <Q  B )  /\  ( x  e.  Q.  /\  y  e.  Q. )
)  /\  ( x  e.  L  /\  y  e.  U  /\  (
y  .Q  1Q ) 
<Q  ( x  .Q  B
) ) )  -> 
( x  .Q  B
)  e.  U )
16 rspe 2424 . . . . . . . 8  |-  ( ( x  e.  L  /\  ( x  .Q  B
)  e.  U )  ->  E. x  e.  L  ( x  .Q  B
)  e.  U )
174, 15, 16syl2anc 403 . . . . . . 7  |-  ( ( ( ( <. L ,  U >.  e.  P.  /\  1Q  <Q  B )  /\  ( x  e.  Q.  /\  y  e.  Q. )
)  /\  ( x  e.  L  /\  y  e.  U  /\  (
y  .Q  1Q ) 
<Q  ( x  .Q  B
) ) )  ->  E. x  e.  L  ( x  .Q  B
)  e.  U )
1817ex 113 . . . . . 6  |-  ( ( ( <. L ,  U >.  e.  P.  /\  1Q  <Q  B )  /\  (
x  e.  Q.  /\  y  e.  Q. )
)  ->  ( (
x  e.  L  /\  y  e.  U  /\  ( y  .Q  1Q )  <Q  ( x  .Q  B ) )  ->  E. x  e.  L  ( x  .Q  B
)  e.  U ) )
1918anassrs 392 . . . . 5  |-  ( ( ( ( <. L ,  U >.  e.  P.  /\  1Q  <Q  B )  /\  x  e.  Q. )  /\  y  e.  Q. )  ->  ( ( x  e.  L  /\  y  e.  U  /\  (
y  .Q  1Q ) 
<Q  ( x  .Q  B
) )  ->  E. x  e.  L  ( x  .Q  B )  e.  U
) )
2019rexlimdva 2489 . . . 4  |-  ( ( ( <. L ,  U >.  e.  P.  /\  1Q  <Q  B )  /\  x  e.  Q. )  ->  ( E. y  e.  Q.  ( x  e.  L  /\  y  e.  U  /\  ( y  .Q  1Q )  <Q  ( x  .Q  B ) )  ->  E. x  e.  L  ( x  .Q  B
)  e.  U ) )
2120ex 113 . . 3  |-  ( (
<. L ,  U >.  e. 
P.  /\  1Q  <Q  B )  ->  ( x  e.  Q.  ->  ( E. y  e.  Q.  (
x  e.  L  /\  y  e.  U  /\  ( y  .Q  1Q )  <Q  ( x  .Q  B ) )  ->  E. x  e.  L  ( x  .Q  B
)  e.  U ) ) )
222, 3, 21rexlimd 2486 . 2  |-  ( (
<. L ,  U >.  e. 
P.  /\  1Q  <Q  B )  ->  ( E. x  e.  Q.  E. y  e.  Q.  ( x  e.  L  /\  y  e.  U  /\  ( y  .Q  1Q )  <Q 
( x  .Q  B
) )  ->  E. x  e.  L  ( x  .Q  B )  e.  U
) )
231, 22mpd 13 1  |-  ( (
<. L ,  U >.  e. 
P.  /\  1Q  <Q  B )  ->  E. x  e.  L  ( x  .Q  B )  e.  U
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    /\ w3a 924    = wceq 1289    e. wcel 1438   E.wrex 2360   <.cop 3449   class class class wbr 3845  (class class class)co 5652   Q.cnq 6839   1Qc1q 6840    .Q cmq 6842    <Q cltq 6844   P.cnp 6850
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3954  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-iinf 4403
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-tr 3937  df-eprel 4116  df-id 4120  df-po 4123  df-iso 4124  df-iord 4193  df-on 4195  df-suc 4198  df-iom 4406  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-1st 5911  df-2nd 5912  df-recs 6070  df-irdg 6135  df-1o 6181  df-2o 6182  df-oadd 6185  df-omul 6186  df-er 6292  df-ec 6294  df-qs 6298  df-ni 6863  df-pli 6864  df-mi 6865  df-lti 6866  df-plpq 6903  df-mpq 6904  df-enq 6906  df-nqqs 6907  df-plqqs 6908  df-mqqs 6909  df-1nqqs 6910  df-rq 6911  df-ltnqqs 6912  df-enq0 6983  df-nq0 6984  df-0nq0 6985  df-plq0 6986  df-mq0 6987  df-inp 7025
This theorem is referenced by:  recexprlem1ssl  7192  recexprlem1ssu  7193
  Copyright terms: Public domain W3C validator