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Theorem exbtwnzlemex 9967
Description: Existence of an integer so that a given real number is between the integer and its successor. The real number must satisfy the  n  <_  A  \/  A  <  n hypothesis. For example either a rational number or a number which is irrational (in the sense of being apart from any rational number) will meet this condition.

The proof starts by finding two integers which are less than and greater than  A. Then this range can be shrunk by choosing an integer in between the endpoints of the range and then deciding which half of the range to keep based on the  n  <_  A  \/  A  <  n hypothesis, and iterating until the range consists of two consecutive integers. (Contributed by Jim Kingdon, 8-Oct-2021.)

Hypotheses
Ref Expression
exbtwnzlemex.a  |-  ( ph  ->  A  e.  RR )
exbtwnzlemex.tri  |-  ( (
ph  /\  n  e.  ZZ )  ->  ( n  <_  A  \/  A  <  n ) )
Assertion
Ref Expression
exbtwnzlemex  |-  ( ph  ->  E. x  e.  ZZ  ( x  <_  A  /\  A  <  ( x  + 
1 ) ) )
Distinct variable groups:    A, n    x, A    ph, n
Allowed substitution hint:    ph( x)

Proof of Theorem exbtwnzlemex
Dummy variables  a  j  m  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 exbtwnzlemex.a . . . 4  |-  ( ph  ->  A  e.  RR )
2 btwnz 9121 . . . 4  |-  ( A  e.  RR  ->  ( E. m  e.  ZZ  m  <  A  /\  E. j  e.  ZZ  A  <  j ) )
31, 2syl 14 . . 3  |-  ( ph  ->  ( E. m  e.  ZZ  m  <  A  /\  E. j  e.  ZZ  A  <  j ) )
4 reeanv 2575 . . 3  |-  ( E. m  e.  ZZ  E. j  e.  ZZ  (
m  <  A  /\  A  <  j )  <->  ( E. m  e.  ZZ  m  <  A  /\  E. j  e.  ZZ  A  <  j
) )
53, 4sylibr 133 . 2  |-  ( ph  ->  E. m  e.  ZZ  E. j  e.  ZZ  (
m  <  A  /\  A  <  j ) )
6 simplrl 507 . . . . . 6  |-  ( ( ( ph  /\  (
m  e.  ZZ  /\  j  e.  ZZ )
)  /\  ( m  <  A  /\  A  < 
j ) )  ->  m  e.  ZZ )
76zred 9124 . . . . . . 7  |-  ( ( ( ph  /\  (
m  e.  ZZ  /\  j  e.  ZZ )
)  /\  ( m  <  A  /\  A  < 
j ) )  ->  m  e.  RR )
81ad2antrr 477 . . . . . . 7  |-  ( ( ( ph  /\  (
m  e.  ZZ  /\  j  e.  ZZ )
)  /\  ( m  <  A  /\  A  < 
j ) )  ->  A  e.  RR )
9 simprl 503 . . . . . . 7  |-  ( ( ( ph  /\  (
m  e.  ZZ  /\  j  e.  ZZ )
)  /\  ( m  <  A  /\  A  < 
j ) )  ->  m  <  A )
107, 8, 9ltled 7845 . . . . . 6  |-  ( ( ( ph  /\  (
m  e.  ZZ  /\  j  e.  ZZ )
)  /\  ( m  <  A  /\  A  < 
j ) )  ->  m  <_  A )
11 simprr 504 . . . . . . 7  |-  ( ( ( ph  /\  (
m  e.  ZZ  /\  j  e.  ZZ )
)  /\  ( m  <  A  /\  A  < 
j ) )  ->  A  <  j )
126zcnd 9125 . . . . . . . 8  |-  ( ( ( ph  /\  (
m  e.  ZZ  /\  j  e.  ZZ )
)  /\  ( m  <  A  /\  A  < 
j ) )  ->  m  e.  CC )
13 simplrr 508 . . . . . . . . 9  |-  ( ( ( ph  /\  (
m  e.  ZZ  /\  j  e.  ZZ )
)  /\  ( m  <  A  /\  A  < 
j ) )  -> 
j  e.  ZZ )
1413zcnd 9125 . . . . . . . 8  |-  ( ( ( ph  /\  (
m  e.  ZZ  /\  j  e.  ZZ )
)  /\  ( m  <  A  /\  A  < 
j ) )  -> 
j  e.  CC )
1512, 14pncan3d 8040 . . . . . . 7  |-  ( ( ( ph  /\  (
m  e.  ZZ  /\  j  e.  ZZ )
)  /\  ( m  <  A  /\  A  < 
j ) )  -> 
( m  +  ( j  -  m ) )  =  j )
1611, 15breqtrrd 3924 . . . . . 6  |-  ( ( ( ph  /\  (
m  e.  ZZ  /\  j  e.  ZZ )
)  /\  ( m  <  A  /\  A  < 
j ) )  ->  A  <  ( m  +  ( j  -  m
) ) )
17 breq1 3900 . . . . . . . 8  |-  ( y  =  m  ->  (
y  <_  A  <->  m  <_  A ) )
18 oveq1 5747 . . . . . . . . 9  |-  ( y  =  m  ->  (
y  +  ( j  -  m ) )  =  ( m  +  ( j  -  m
) ) )
1918breq2d 3909 . . . . . . . 8  |-  ( y  =  m  ->  ( A  <  ( y  +  ( j  -  m
) )  <->  A  <  ( m  +  ( j  -  m ) ) ) )
2017, 19anbi12d 462 . . . . . . 7  |-  ( y  =  m  ->  (
( y  <_  A  /\  A  <  ( y  +  ( j  -  m ) ) )  <-> 
( m  <_  A  /\  A  <  ( m  +  ( j  -  m ) ) ) ) )
2120rspcev 2761 . . . . . 6  |-  ( ( m  e.  ZZ  /\  ( m  <_  A  /\  A  <  ( m  +  ( j  -  m
) ) ) )  ->  E. y  e.  ZZ  ( y  <_  A  /\  A  <  ( y  +  ( j  -  m ) ) ) )
226, 10, 16, 21syl12anc 1197 . . . . 5  |-  ( ( ( ph  /\  (
m  e.  ZZ  /\  j  e.  ZZ )
)  /\  ( m  <  A  /\  A  < 
j ) )  ->  E. y  e.  ZZ  ( y  <_  A  /\  A  <  ( y  +  ( j  -  m ) ) ) )
2313zred 9124 . . . . . . . 8  |-  ( ( ( ph  /\  (
m  e.  ZZ  /\  j  e.  ZZ )
)  /\  ( m  <  A  /\  A  < 
j ) )  -> 
j  e.  RR )
247, 8, 23, 9, 11lttrd 7852 . . . . . . 7  |-  ( ( ( ph  /\  (
m  e.  ZZ  /\  j  e.  ZZ )
)  /\  ( m  <  A  /\  A  < 
j ) )  ->  m  <  j )
25 znnsub 9056 . . . . . . . 8  |-  ( ( m  e.  ZZ  /\  j  e.  ZZ )  ->  ( m  <  j  <->  ( j  -  m )  e.  NN ) )
2625ad2antlr 478 . . . . . . 7  |-  ( ( ( ph  /\  (
m  e.  ZZ  /\  j  e.  ZZ )
)  /\  ( m  <  A  /\  A  < 
j ) )  -> 
( m  <  j  <->  ( j  -  m )  e.  NN ) )
2724, 26mpbid 146 . . . . . 6  |-  ( ( ( ph  /\  (
m  e.  ZZ  /\  j  e.  ZZ )
)  /\  ( m  <  A  /\  A  < 
j ) )  -> 
( j  -  m
)  e.  NN )
28 exbtwnzlemex.tri . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  ZZ )  ->  ( n  <_  A  \/  A  <  n ) )
2928ralrimiva 2480 . . . . . . . . 9  |-  ( ph  ->  A. n  e.  ZZ  ( n  <_  A  \/  A  <  n ) )
30 breq1 3900 . . . . . . . . . . 11  |-  ( n  =  a  ->  (
n  <_  A  <->  a  <_  A ) )
31 breq2 3901 . . . . . . . . . . 11  |-  ( n  =  a  ->  ( A  <  n  <->  A  <  a ) )
3230, 31orbi12d 765 . . . . . . . . . 10  |-  ( n  =  a  ->  (
( n  <_  A  \/  A  <  n )  <-> 
( a  <_  A  \/  A  <  a ) ) )
3332cbvralv 2629 . . . . . . . . 9  |-  ( A. n  e.  ZZ  (
n  <_  A  \/  A  <  n )  <->  A. a  e.  ZZ  ( a  <_  A  \/  A  <  a ) )
3429, 33sylib 121 . . . . . . . 8  |-  ( ph  ->  A. a  e.  ZZ  ( a  <_  A  \/  A  <  a ) )
3534ad2antrr 477 . . . . . . 7  |-  ( ( ( ph  /\  (
m  e.  ZZ  /\  j  e.  ZZ )
)  /\  ( m  <  A  /\  A  < 
j ) )  ->  A. a  e.  ZZ  ( a  <_  A  \/  A  <  a ) )
3635r19.21bi 2495 . . . . . 6  |-  ( ( ( ( ph  /\  ( m  e.  ZZ  /\  j  e.  ZZ ) )  /\  ( m  <  A  /\  A  <  j ) )  /\  a  e.  ZZ )  ->  ( a  <_  A  \/  A  <  a ) )
3727, 8, 36exbtwnzlemshrink 9966 . . . . 5  |-  ( ( ( ( ph  /\  ( m  e.  ZZ  /\  j  e.  ZZ ) )  /\  ( m  <  A  /\  A  <  j ) )  /\  E. y  e.  ZZ  (
y  <_  A  /\  A  <  ( y  +  ( j  -  m
) ) ) )  ->  E. x  e.  ZZ  ( x  <_  A  /\  A  <  ( x  + 
1 ) ) )
3822, 37mpdan 415 . . . 4  |-  ( ( ( ph  /\  (
m  e.  ZZ  /\  j  e.  ZZ )
)  /\  ( m  <  A  /\  A  < 
j ) )  ->  E. x  e.  ZZ  ( x  <_  A  /\  A  <  ( x  + 
1 ) ) )
3938ex 114 . . 3  |-  ( (
ph  /\  ( m  e.  ZZ  /\  j  e.  ZZ ) )  -> 
( ( m  < 
A  /\  A  <  j )  ->  E. x  e.  ZZ  ( x  <_  A  /\  A  <  (
x  +  1 ) ) ) )
4039rexlimdvva 2532 . 2  |-  ( ph  ->  ( E. m  e.  ZZ  E. j  e.  ZZ  ( m  < 
A  /\  A  <  j )  ->  E. x  e.  ZZ  ( x  <_  A  /\  A  <  (
x  +  1 ) ) ) )
415, 40mpd 13 1  |-  ( ph  ->  E. x  e.  ZZ  ( x  <_  A  /\  A  <  ( x  + 
1 ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 680    e. wcel 1463   A.wral 2391   E.wrex 2392   class class class wbr 3897  (class class class)co 5740   RRcr 7583   1c1 7585    + caddc 7587    < clt 7764    <_ cle 7765    - cmin 7897   NNcn 8677   ZZcz 9005
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-sep 4014  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-cnex 7675  ax-resscn 7676  ax-1cn 7677  ax-1re 7678  ax-icn 7679  ax-addcl 7680  ax-addrcl 7681  ax-mulcl 7682  ax-addcom 7684  ax-addass 7686  ax-distr 7688  ax-i2m1 7689  ax-0lt1 7690  ax-0id 7692  ax-rnegex 7693  ax-cnre 7695  ax-pre-ltirr 7696  ax-pre-ltwlin 7697  ax-pre-lttrn 7698  ax-pre-ltadd 7700  ax-arch 7703
This theorem depends on definitions:  df-bi 116  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-nel 2379  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-br 3898  df-opab 3958  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-iota 5056  df-fun 5093  df-fv 5099  df-riota 5696  df-ov 5743  df-oprab 5744  df-mpo 5745  df-pnf 7766  df-mnf 7767  df-xr 7768  df-ltxr 7769  df-le 7770  df-sub 7899  df-neg 7900  df-inn 8678  df-n0 8929  df-z 9006
This theorem is referenced by:  qbtwnz  9969  apbtwnz  9987
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