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Theorem prmuloc2 7489
Description: Positive reals are multiplicatively located. This is a variation of prmuloc 7488 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 7488 . 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 1508 . . 3  |-  F/ x
( <. L ,  U >.  e.  P.  /\  1Q  <Q  B )
3 nfre1 2500 . . 3  |-  F/ x E. x  e.  L  ( x  .Q  B
)  e.  U
4 simpr1 988 . . . . . . . 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 990 . . . . . . . . . 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 526 . . . . . . . . . . 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 7311 . . . . . . . . . . 11  |-  ( y  e.  Q.  ->  (
y  .Q  1Q )  =  y )
8 breq1 3970 . . . . . . . . . . 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 146 . . . . . . . . 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 523 . . . . . . . . . 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 989 . . . . . . . . . 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 7407 . . . . . . . . . 10  |-  ( (
<. L ,  U >.  e. 
P.  /\  y  e.  U )  ->  (
y  <Q  ( x  .Q  B )  ->  (
x  .Q  B )  e.  U ) )
1411, 12, 13syl2anc 409 . . . . . . . . 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 2506 . . . . . . . 8  |-  ( ( x  e.  L  /\  ( x  .Q  B
)  e.  U )  ->  E. x  e.  L  ( x  .Q  B
)  e.  U )
174, 15, 16syl2anc 409 . . . . . . 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 114 . . . . . 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 398 . . . . 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 2574 . . . 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 114 . . 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 2571 . 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 103    <-> wb 104    /\ w3a 963    = wceq 1335    e. wcel 2128   E.wrex 2436   <.cop 3564   class class class wbr 3967  (class class class)co 5826   Q.cnq 7202   1Qc1q 7203    .Q cmq 7205    <Q cltq 7207   P.cnp 7213
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4081  ax-sep 4084  ax-nul 4092  ax-pow 4137  ax-pr 4171  ax-un 4395  ax-setind 4498  ax-iinf 4549
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3396  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-uni 3775  df-int 3810  df-iun 3853  df-br 3968  df-opab 4028  df-mpt 4029  df-tr 4065  df-eprel 4251  df-id 4255  df-po 4258  df-iso 4259  df-iord 4328  df-on 4330  df-suc 4333  df-iom 4552  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-iota 5137  df-fun 5174  df-fn 5175  df-f 5176  df-f1 5177  df-fo 5178  df-f1o 5179  df-fv 5180  df-ov 5829  df-oprab 5830  df-mpo 5831  df-1st 6090  df-2nd 6091  df-recs 6254  df-irdg 6319  df-1o 6365  df-2o 6366  df-oadd 6369  df-omul 6370  df-er 6482  df-ec 6484  df-qs 6488  df-ni 7226  df-pli 7227  df-mi 7228  df-lti 7229  df-plpq 7266  df-mpq 7267  df-enq 7269  df-nqqs 7270  df-plqqs 7271  df-mqqs 7272  df-1nqqs 7273  df-rq 7274  df-ltnqqs 7275  df-enq0 7346  df-nq0 7347  df-0nq0 7348  df-plq0 7349  df-mq0 7350  df-inp 7388
This theorem is referenced by:  recexprlem1ssl  7555  recexprlem1ssu  7556
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