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Theorem prarloc2 7276
Description: A Dedekind cut is arithmetically located. This is a variation of prarloc 7275 which only constructs one (named) point and is therefore often easier to work with. It states that given a tolerance  P, there are elements of the lower and upper cut which are exactly that tolerance from each other. (Contributed by Jim Kingdon, 26-Dec-2019.)
Assertion
Ref Expression
prarloc2  |-  ( (
<. L ,  U >.  e. 
P.  /\  P  e.  Q. )  ->  E. a  e.  L  ( a  +Q  P )  e.  U
)
Distinct variable groups:    L, a    P, a    U, a

Proof of Theorem prarloc2
Dummy variable  b is distinct from all other variables.
StepHypRef Expression
1 prarloc 7275 . 2  |-  ( (
<. L ,  U >.  e. 
P.  /\  P  e.  Q. )  ->  E. a  e.  L  E. b  e.  U  b  <Q  ( a  +Q  P ) )
2 prcunqu 7257 . . . . 5  |-  ( (
<. L ,  U >.  e. 
P.  /\  b  e.  U )  ->  (
b  <Q  ( a  +Q  P )  ->  (
a  +Q  P )  e.  U ) )
32rexlimdva 2524 . . . 4  |-  ( <. L ,  U >.  e. 
P.  ->  ( E. b  e.  U  b  <Q  ( a  +Q  P )  ->  ( a  +Q  P )  e.  U
) )
43reximdv 2508 . . 3  |-  ( <. L ,  U >.  e. 
P.  ->  ( E. a  e.  L  E. b  e.  U  b  <Q  ( a  +Q  P )  ->  E. a  e.  L  ( a  +Q  P
)  e.  U ) )
54adantr 272 . 2  |-  ( (
<. L ,  U >.  e. 
P.  /\  P  e.  Q. )  ->  ( E. a  e.  L  E. b  e.  U  b  <Q  ( a  +Q  P
)  ->  E. a  e.  L  ( a  +Q  P )  e.  U
) )
61, 5mpd 13 1  |-  ( (
<. L ,  U >.  e. 
P.  /\  P  e.  Q. )  ->  E. a  e.  L  ( a  +Q  P )  e.  U
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    e. wcel 1463   E.wrex 2392   <.cop 3498   class class class wbr 3897  (class class class)co 5740   Q.cnq 7052    +Q cplq 7054    <Q cltq 7057   P.cnp 7063
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-iinf 4470
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-eprel 4179  df-id 4183  df-po 4186  df-iso 4187  df-iord 4256  df-on 4258  df-suc 4261  df-iom 4473  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-ov 5743  df-oprab 5744  df-mpo 5745  df-1st 6004  df-2nd 6005  df-recs 6168  df-irdg 6233  df-1o 6279  df-2o 6280  df-oadd 6283  df-omul 6284  df-er 6395  df-ec 6397  df-qs 6401  df-ni 7076  df-pli 7077  df-mi 7078  df-lti 7079  df-plpq 7116  df-mpq 7117  df-enq 7119  df-nqqs 7120  df-plqqs 7121  df-mqqs 7122  df-1nqqs 7123  df-rq 7124  df-ltnqqs 7125  df-enq0 7196  df-nq0 7197  df-0nq0 7198  df-plq0 7199  df-mq0 7200  df-inp 7238
This theorem is referenced by:  addcanprleml  7386  addcanprlemu  7387  aptiprleml  7411  aptiprlemu  7412
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