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Theorem prmuloc2 7343
Description: Positive reals are multiplicatively located. This is a variation of prmuloc 7342 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 7342 . 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 1493 . . 3  |-  F/ x
( <. L ,  U >.  e.  P.  /\  1Q  <Q  B )
3 nfre1 2453 . . 3  |-  F/ x E. x  e.  L  ( x  .Q  B
)  e.  U
4 simpr1 972 . . . . . . . 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 974 . . . . . . . . . 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 510 . . . . . . . . . . 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 7165 . . . . . . . . . . 11  |-  ( y  e.  Q.  ->  (
y  .Q  1Q )  =  y )
8 breq1 3902 . . . . . . . . . . 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 507 . . . . . . . . . 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 973 . . . . . . . . . 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 7261 . . . . . . . . . 10  |-  ( (
<. L ,  U >.  e. 
P.  /\  y  e.  U )  ->  (
y  <Q  ( x  .Q  B )  ->  (
x  .Q  B )  e.  U ) )
1411, 12, 13syl2anc 408 . . . . . . . . 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 2458 . . . . . . . 8  |-  ( ( x  e.  L  /\  ( x  .Q  B
)  e.  U )  ->  E. x  e.  L  ( x  .Q  B
)  e.  U )
174, 15, 16syl2anc 408 . . . . . . 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 397 . . . . 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 2526 . . . 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 2523 . 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 947    = wceq 1316    e. wcel 1465   E.wrex 2394   <.cop 3500   class class class wbr 3899  (class class class)co 5742   Q.cnq 7056   1Qc1q 7057    .Q cmq 7059    <Q cltq 7061   P.cnp 7067
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-eprel 4181  df-id 4185  df-po 4188  df-iso 4189  df-iord 4258  df-on 4260  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-recs 6170  df-irdg 6235  df-1o 6281  df-2o 6282  df-oadd 6285  df-omul 6286  df-er 6397  df-ec 6399  df-qs 6403  df-ni 7080  df-pli 7081  df-mi 7082  df-lti 7083  df-plpq 7120  df-mpq 7121  df-enq 7123  df-nqqs 7124  df-plqqs 7125  df-mqqs 7126  df-1nqqs 7127  df-rq 7128  df-ltnqqs 7129  df-enq0 7200  df-nq0 7201  df-0nq0 7202  df-plq0 7203  df-mq0 7204  df-inp 7242
This theorem is referenced by:  recexprlem1ssl  7409  recexprlem1ssu  7410
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