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Theorem prmuloc2 6819
Description: Positive reals are multiplicatively located. This is a variation of prmuloc 6818 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 6818 . 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 1462 . . 3  |-  F/ x
( <. L ,  U >.  e.  P.  /\  1Q  <Q  B )
3 nfre1 2408 . . 3  |-  F/ x E. x  e.  L  ( x  .Q  B
)  e.  U
4 simpr1 945 . . . . . . . 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 947 . . . . . . . . . 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 6641 . . . . . . . . . . 11  |-  ( y  e.  Q.  ->  (
y  .Q  1Q )  =  y )
8 breq1 3796 . . . . . . . . . . 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 946 . . . . . . . . . 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 6737 . . . . . . . . . 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 2413 . . . . . . . 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 2478 . . . 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 2475 . 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 920    = wceq 1285    e. wcel 1434   E.wrex 2350   <.cop 3409   class class class wbr 3793  (class class class)co 5543   Q.cnq 6532   1Qc1q 6533    .Q cmq 6535    <Q cltq 6537   P.cnp 6543
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3901  ax-sep 3904  ax-nul 3912  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-iinf 4337
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-tr 3884  df-eprel 4052  df-id 4056  df-po 4059  df-iso 4060  df-iord 4129  df-on 4131  df-suc 4134  df-iom 4340  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-1st 5798  df-2nd 5799  df-recs 5954  df-irdg 6019  df-1o 6065  df-2o 6066  df-oadd 6069  df-omul 6070  df-er 6172  df-ec 6174  df-qs 6178  df-ni 6556  df-pli 6557  df-mi 6558  df-lti 6559  df-plpq 6596  df-mpq 6597  df-enq 6599  df-nqqs 6600  df-plqqs 6601  df-mqqs 6602  df-1nqqs 6603  df-rq 6604  df-ltnqqs 6605  df-enq0 6676  df-nq0 6677  df-0nq0 6678  df-plq0 6679  df-mq0 6680  df-inp 6718
This theorem is referenced by:  recexprlem1ssl  6885  recexprlem1ssu  6886
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