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Theorem exbtwnzlemex 10282
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 9403 . . . 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 2660 . . 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 134 . 2  |-  ( ph  ->  E. m  e.  ZZ  E. j  e.  ZZ  (
m  <  A  /\  A  <  j ) )
6 simplrl 535 . . . . . 6  |-  ( ( ( ph  /\  (
m  e.  ZZ  /\  j  e.  ZZ )
)  /\  ( m  <  A  /\  A  < 
j ) )  ->  m  e.  ZZ )
76zred 9406 . . . . . . 7  |-  ( ( ( ph  /\  (
m  e.  ZZ  /\  j  e.  ZZ )
)  /\  ( m  <  A  /\  A  < 
j ) )  ->  m  e.  RR )
81ad2antrr 488 . . . . . . 7  |-  ( ( ( ph  /\  (
m  e.  ZZ  /\  j  e.  ZZ )
)  /\  ( m  <  A  /\  A  < 
j ) )  ->  A  e.  RR )
9 simprl 529 . . . . . . 7  |-  ( ( ( ph  /\  (
m  e.  ZZ  /\  j  e.  ZZ )
)  /\  ( m  <  A  /\  A  < 
j ) )  ->  m  <  A )
107, 8, 9ltled 8107 . . . . . 6  |-  ( ( ( ph  /\  (
m  e.  ZZ  /\  j  e.  ZZ )
)  /\  ( m  <  A  /\  A  < 
j ) )  ->  m  <_  A )
11 simprr 531 . . . . . . 7  |-  ( ( ( ph  /\  (
m  e.  ZZ  /\  j  e.  ZZ )
)  /\  ( m  <  A  /\  A  < 
j ) )  ->  A  <  j )
126zcnd 9407 . . . . . . . 8  |-  ( ( ( ph  /\  (
m  e.  ZZ  /\  j  e.  ZZ )
)  /\  ( m  <  A  /\  A  < 
j ) )  ->  m  e.  CC )
13 simplrr 536 . . . . . . . . 9  |-  ( ( ( ph  /\  (
m  e.  ZZ  /\  j  e.  ZZ )
)  /\  ( m  <  A  /\  A  < 
j ) )  -> 
j  e.  ZZ )
1413zcnd 9407 . . . . . . . 8  |-  ( ( ( ph  /\  (
m  e.  ZZ  /\  j  e.  ZZ )
)  /\  ( m  <  A  /\  A  < 
j ) )  -> 
j  e.  CC )
1512, 14pncan3d 8302 . . . . . . 7  |-  ( ( ( ph  /\  (
m  e.  ZZ  /\  j  e.  ZZ )
)  /\  ( m  <  A  /\  A  < 
j ) )  -> 
( m  +  ( j  -  m ) )  =  j )
1611, 15breqtrrd 4046 . . . . . 6  |-  ( ( ( ph  /\  (
m  e.  ZZ  /\  j  e.  ZZ )
)  /\  ( m  <  A  /\  A  < 
j ) )  ->  A  <  ( m  +  ( j  -  m
) ) )
17 breq1 4021 . . . . . . . 8  |-  ( y  =  m  ->  (
y  <_  A  <->  m  <_  A ) )
18 oveq1 5904 . . . . . . . . 9  |-  ( y  =  m  ->  (
y  +  ( j  -  m ) )  =  ( m  +  ( j  -  m
) ) )
1918breq2d 4030 . . . . . . . 8  |-  ( y  =  m  ->  ( A  <  ( y  +  ( j  -  m
) )  <->  A  <  ( m  +  ( j  -  m ) ) ) )
2017, 19anbi12d 473 . . . . . . 7  |-  ( y  =  m  ->  (
( y  <_  A  /\  A  <  ( y  +  ( j  -  m ) ) )  <-> 
( m  <_  A  /\  A  <  ( m  +  ( j  -  m ) ) ) ) )
2120rspcev 2856 . . . . . 6  |-  ( ( m  e.  ZZ  /\  ( m  <_  A  /\  A  <  ( m  +  ( j  -  m
) ) ) )  ->  E. y  e.  ZZ  ( y  <_  A  /\  A  <  ( y  +  ( j  -  m ) ) ) )
226, 10, 16, 21syl12anc 1247 . . . . 5  |-  ( ( ( ph  /\  (
m  e.  ZZ  /\  j  e.  ZZ )
)  /\  ( m  <  A  /\  A  < 
j ) )  ->  E. y  e.  ZZ  ( y  <_  A  /\  A  <  ( y  +  ( j  -  m ) ) ) )
2313zred 9406 . . . . . . . 8  |-  ( ( ( ph  /\  (
m  e.  ZZ  /\  j  e.  ZZ )
)  /\  ( m  <  A  /\  A  < 
j ) )  -> 
j  e.  RR )
247, 8, 23, 9, 11lttrd 8114 . . . . . . 7  |-  ( ( ( ph  /\  (
m  e.  ZZ  /\  j  e.  ZZ )
)  /\  ( m  <  A  /\  A  < 
j ) )  ->  m  <  j )
25 znnsub 9335 . . . . . . . 8  |-  ( ( m  e.  ZZ  /\  j  e.  ZZ )  ->  ( m  <  j  <->  ( j  -  m )  e.  NN ) )
2625ad2antlr 489 . . . . . . 7  |-  ( ( ( ph  /\  (
m  e.  ZZ  /\  j  e.  ZZ )
)  /\  ( m  <  A  /\  A  < 
j ) )  -> 
( m  <  j  <->  ( j  -  m )  e.  NN ) )
2724, 26mpbid 147 . . . . . 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 2563 . . . . . . . . 9  |-  ( ph  ->  A. n  e.  ZZ  ( n  <_  A  \/  A  <  n ) )
30 breq1 4021 . . . . . . . . . . 11  |-  ( n  =  a  ->  (
n  <_  A  <->  a  <_  A ) )
31 breq2 4022 . . . . . . . . . . 11  |-  ( n  =  a  ->  ( A  <  n  <->  A  <  a ) )
3230, 31orbi12d 794 . . . . . . . . . 10  |-  ( n  =  a  ->  (
( n  <_  A  \/  A  <  n )  <-> 
( a  <_  A  \/  A  <  a ) ) )
3332cbvralv 2718 . . . . . . . . 9  |-  ( A. n  e.  ZZ  (
n  <_  A  \/  A  <  n )  <->  A. a  e.  ZZ  ( a  <_  A  \/  A  <  a ) )
3429, 33sylib 122 . . . . . . . 8  |-  ( ph  ->  A. a  e.  ZZ  ( a  <_  A  \/  A  <  a ) )
3534ad2antrr 488 . . . . . . 7  |-  ( ( ( ph  /\  (
m  e.  ZZ  /\  j  e.  ZZ )
)  /\  ( m  <  A  /\  A  < 
j ) )  ->  A. a  e.  ZZ  ( a  <_  A  \/  A  <  a ) )
3635r19.21bi 2578 . . . . . 6  |-  ( ( ( ( ph  /\  ( m  e.  ZZ  /\  j  e.  ZZ ) )  /\  ( m  <  A  /\  A  <  j ) )  /\  a  e.  ZZ )  ->  ( a  <_  A  \/  A  <  a ) )
3727, 8, 36exbtwnzlemshrink 10281 . . . . 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 421 . . . 4  |-  ( ( ( ph  /\  (
m  e.  ZZ  /\  j  e.  ZZ )
)  /\  ( m  <  A  /\  A  < 
j ) )  ->  E. x  e.  ZZ  ( x  <_  A  /\  A  <  ( x  + 
1 ) ) )
3938ex 115 . . 3  |-  ( (
ph  /\  ( m  e.  ZZ  /\  j  e.  ZZ ) )  -> 
( ( m  < 
A  /\  A  <  j )  ->  E. x  e.  ZZ  ( x  <_  A  /\  A  <  (
x  +  1 ) ) ) )
4039rexlimdvva 2615 . 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 104    <-> wb 105    \/ wo 709    e. wcel 2160   A.wral 2468   E.wrex 2469   class class class wbr 4018  (class class class)co 5897   RRcr 7841   1c1 7843    + caddc 7845    < clt 8023    <_ cle 8024    - cmin 8159   NNcn 8950   ZZcz 9284
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-cnex 7933  ax-resscn 7934  ax-1cn 7935  ax-1re 7936  ax-icn 7937  ax-addcl 7938  ax-addrcl 7939  ax-mulcl 7940  ax-addcom 7942  ax-addass 7944  ax-distr 7946  ax-i2m1 7947  ax-0lt1 7948  ax-0id 7950  ax-rnegex 7951  ax-cnre 7953  ax-pre-ltirr 7954  ax-pre-ltwlin 7955  ax-pre-lttrn 7956  ax-pre-ltadd 7958  ax-arch 7961
This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-br 4019  df-opab 4080  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-iota 5196  df-fun 5237  df-fv 5243  df-riota 5852  df-ov 5900  df-oprab 5901  df-mpo 5902  df-pnf 8025  df-mnf 8026  df-xr 8027  df-ltxr 8028  df-le 8029  df-sub 8161  df-neg 8162  df-inn 8951  df-n0 9208  df-z 9285
This theorem is referenced by:  qbtwnz  10284  apbtwnz  10307
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