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Theorem rebtwn2z 10269
Description: A real number can be bounded by integers above and below which are two apart.

The proof starts by finding two integers which are less than and greater than the given real number. 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 weak linearity, and iterating until the range consists of integers which are two apart. (Contributed by Jim Kingdon, 13-Oct-2021.)

Assertion
Ref Expression
rebtwn2z  |-  ( A  e.  RR  ->  E. x  e.  ZZ  ( x  < 
A  /\  A  <  ( x  +  2 ) ) )
Distinct variable group:    x, A

Proof of Theorem rebtwn2z
Dummy variables  m  n  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 btwnz 9386 . . 3  |-  ( A  e.  RR  ->  ( E. m  e.  ZZ  m  <  A  /\  E. n  e.  ZZ  A  <  n ) )
2 reeanv 2657 . . 3  |-  ( E. m  e.  ZZ  E. n  e.  ZZ  (
m  <  A  /\  A  <  n )  <->  ( E. m  e.  ZZ  m  <  A  /\  E. n  e.  ZZ  A  <  n
) )
31, 2sylibr 134 . 2  |-  ( A  e.  RR  ->  E. m  e.  ZZ  E. n  e.  ZZ  ( m  < 
A  /\  A  <  n ) )
4 simpll 527 . . . . 5  |-  ( ( ( A  e.  RR  /\  ( m  e.  ZZ  /\  n  e.  ZZ ) )  /\  ( m  <  A  /\  A  <  n ) )  ->  A  e.  RR )
5 simplrl 535 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  ( m  e.  ZZ  /\  n  e.  ZZ ) )  /\  ( m  <  A  /\  A  <  n ) )  ->  m  e.  ZZ )
65zred 9389 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  ( m  e.  ZZ  /\  n  e.  ZZ ) )  /\  ( m  <  A  /\  A  <  n ) )  ->  m  e.  RR )
7 simplrr 536 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  ( m  e.  ZZ  /\  n  e.  ZZ ) )  /\  ( m  <  A  /\  A  <  n ) )  ->  n  e.  ZZ )
87zred 9389 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  ( m  e.  ZZ  /\  n  e.  ZZ ) )  /\  ( m  <  A  /\  A  <  n ) )  ->  n  e.  RR )
9 simprl 529 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  ( m  e.  ZZ  /\  n  e.  ZZ ) )  /\  ( m  <  A  /\  A  <  n ) )  ->  m  <  A )
10 simprr 531 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  ( m  e.  ZZ  /\  n  e.  ZZ ) )  /\  ( m  <  A  /\  A  <  n ) )  ->  A  <  n )
116, 4, 8, 9, 10lttrd 8097 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  ( m  e.  ZZ  /\  n  e.  ZZ ) )  /\  ( m  <  A  /\  A  <  n ) )  ->  m  <  n )
12 znnsub 9318 . . . . . . . 8  |-  ( ( m  e.  ZZ  /\  n  e.  ZZ )  ->  ( m  <  n  <->  ( n  -  m )  e.  NN ) )
1312ad2antlr 489 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  ( m  e.  ZZ  /\  n  e.  ZZ ) )  /\  ( m  <  A  /\  A  <  n ) )  -> 
( m  <  n  <->  ( n  -  m )  e.  NN ) )
1411, 13mpbid 147 . . . . . 6  |-  ( ( ( A  e.  RR  /\  ( m  e.  ZZ  /\  n  e.  ZZ ) )  /\  ( m  <  A  /\  A  <  n ) )  -> 
( n  -  m
)  e.  NN )
15 elnnuz 9578 . . . . . . . 8  |-  ( ( n  -  m )  e.  NN  <->  ( n  -  m )  e.  (
ZZ>= `  1 ) )
16 eluzp1p1 9567 . . . . . . . 8  |-  ( ( n  -  m )  e.  ( ZZ>= `  1
)  ->  ( (
n  -  m )  +  1 )  e.  ( ZZ>= `  ( 1  +  1 ) ) )
1715, 16sylbi 121 . . . . . . 7  |-  ( ( n  -  m )  e.  NN  ->  (
( n  -  m
)  +  1 )  e.  ( ZZ>= `  (
1  +  1 ) ) )
18 df-2 8992 . . . . . . . 8  |-  2  =  ( 1  +  1 )
1918fveq2i 5530 . . . . . . 7  |-  ( ZZ>= ` 
2 )  =  (
ZZ>= `  ( 1  +  1 ) )
2017, 19eleqtrrdi 2281 . . . . . 6  |-  ( ( n  -  m )  e.  NN  ->  (
( n  -  m
)  +  1 )  e.  ( ZZ>= `  2
) )
2114, 20syl 14 . . . . 5  |-  ( ( ( A  e.  RR  /\  ( m  e.  ZZ  /\  n  e.  ZZ ) )  /\  ( m  <  A  /\  A  <  n ) )  -> 
( ( n  -  m )  +  1 )  e.  ( ZZ>= ` 
2 ) )
225zcnd 9390 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  ( m  e.  ZZ  /\  n  e.  ZZ ) )  /\  ( m  <  A  /\  A  <  n ) )  ->  m  e.  CC )
237zcnd 9390 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  ( m  e.  ZZ  /\  n  e.  ZZ ) )  /\  ( m  <  A  /\  A  <  n ) )  ->  n  e.  CC )
2422, 23pncan3d 8285 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  ( m  e.  ZZ  /\  n  e.  ZZ ) )  /\  ( m  <  A  /\  A  <  n ) )  -> 
( m  +  ( n  -  m ) )  =  n )
2524, 8eqeltrd 2264 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  ( m  e.  ZZ  /\  n  e.  ZZ ) )  /\  ( m  <  A  /\  A  <  n ) )  -> 
( m  +  ( n  -  m ) )  e.  RR )
268, 6resubcld 8352 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  ( m  e.  ZZ  /\  n  e.  ZZ ) )  /\  ( m  <  A  /\  A  <  n ) )  -> 
( n  -  m
)  e.  RR )
27 1red 7986 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  ( m  e.  ZZ  /\  n  e.  ZZ ) )  /\  ( m  <  A  /\  A  <  n ) )  -> 
1  e.  RR )
2826, 27readdcld 8001 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  ( m  e.  ZZ  /\  n  e.  ZZ ) )  /\  ( m  <  A  /\  A  <  n ) )  -> 
( ( n  -  m )  +  1 )  e.  RR )
296, 28readdcld 8001 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  ( m  e.  ZZ  /\  n  e.  ZZ ) )  /\  ( m  <  A  /\  A  <  n ) )  -> 
( m  +  ( ( n  -  m
)  +  1 ) )  e.  RR )
3010, 24breqtrrd 4043 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  ( m  e.  ZZ  /\  n  e.  ZZ ) )  /\  ( m  <  A  /\  A  <  n ) )  ->  A  <  ( m  +  ( n  -  m
) ) )
3126ltp1d 8901 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  ( m  e.  ZZ  /\  n  e.  ZZ ) )  /\  ( m  <  A  /\  A  <  n ) )  -> 
( n  -  m
)  <  ( (
n  -  m )  +  1 ) )
3226, 28, 6, 31ltadd2dd 8393 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  ( m  e.  ZZ  /\  n  e.  ZZ ) )  /\  ( m  <  A  /\  A  <  n ) )  -> 
( m  +  ( n  -  m ) )  <  ( m  +  ( ( n  -  m )  +  1 ) ) )
334, 25, 29, 30, 32lttrd 8097 . . . . . 6  |-  ( ( ( A  e.  RR  /\  ( m  e.  ZZ  /\  n  e.  ZZ ) )  /\  ( m  <  A  /\  A  <  n ) )  ->  A  <  ( m  +  ( ( n  -  m )  +  1 ) ) )
34 breq1 4018 . . . . . . . 8  |-  ( y  =  m  ->  (
y  <  A  <->  m  <  A ) )
35 oveq1 5895 . . . . . . . . 9  |-  ( y  =  m  ->  (
y  +  ( ( n  -  m )  +  1 ) )  =  ( m  +  ( ( n  -  m )  +  1 ) ) )
3635breq2d 4027 . . . . . . . 8  |-  ( y  =  m  ->  ( A  <  ( y  +  ( ( n  -  m )  +  1 ) )  <->  A  <  ( m  +  ( ( n  -  m )  +  1 ) ) ) )
3734, 36anbi12d 473 . . . . . . 7  |-  ( y  =  m  ->  (
( y  <  A  /\  A  <  ( y  +  ( ( n  -  m )  +  1 ) ) )  <-> 
( m  <  A  /\  A  <  ( m  +  ( ( n  -  m )  +  1 ) ) ) ) )
3837rspcev 2853 . . . . . 6  |-  ( ( m  e.  ZZ  /\  ( m  <  A  /\  A  <  ( m  +  ( ( n  -  m )  +  1 ) ) ) )  ->  E. y  e.  ZZ  ( y  <  A  /\  A  <  ( y  +  ( ( n  -  m )  +  1 ) ) ) )
395, 9, 33, 38syl12anc 1246 . . . . 5  |-  ( ( ( A  e.  RR  /\  ( m  e.  ZZ  /\  n  e.  ZZ ) )  /\  ( m  <  A  /\  A  <  n ) )  ->  E. y  e.  ZZ  ( y  <  A  /\  A  <  ( y  +  ( ( n  -  m )  +  1 ) ) ) )
40 rebtwn2zlemshrink 10268 . . . . 5  |-  ( ( A  e.  RR  /\  ( ( n  -  m )  +  1 )  e.  ( ZZ>= ` 
2 )  /\  E. y  e.  ZZ  (
y  <  A  /\  A  <  ( y  +  ( ( n  -  m )  +  1 ) ) ) )  ->  E. x  e.  ZZ  ( x  <  A  /\  A  <  ( x  + 
2 ) ) )
414, 21, 39, 40syl3anc 1248 . . . 4  |-  ( ( ( A  e.  RR  /\  ( m  e.  ZZ  /\  n  e.  ZZ ) )  /\  ( m  <  A  /\  A  <  n ) )  ->  E. x  e.  ZZ  ( x  <  A  /\  A  <  ( x  + 
2 ) ) )
4241ex 115 . . 3  |-  ( ( A  e.  RR  /\  ( m  e.  ZZ  /\  n  e.  ZZ ) )  ->  ( (
m  <  A  /\  A  <  n )  ->  E. x  e.  ZZ  ( x  <  A  /\  A  <  ( x  + 
2 ) ) ) )
4342rexlimdvva 2612 . 2  |-  ( A  e.  RR  ->  ( E. m  e.  ZZ  E. n  e.  ZZ  (
m  <  A  /\  A  <  n )  ->  E. x  e.  ZZ  ( x  <  A  /\  A  <  ( x  + 
2 ) ) ) )
443, 43mpd 13 1  |-  ( A  e.  RR  ->  E. x  e.  ZZ  ( x  < 
A  /\  A  <  ( x  +  2 ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    e. wcel 2158   E.wrex 2466   class class class wbr 4015   ` cfv 5228  (class class class)co 5888   RRcr 7824   1c1 7826    + caddc 7828    < clt 8006    - cmin 8142   NNcn 8933   2c2 8984   ZZcz 9267   ZZ>=cuz 9542
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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-cnex 7916  ax-resscn 7917  ax-1cn 7918  ax-1re 7919  ax-icn 7920  ax-addcl 7921  ax-addrcl 7922  ax-mulcl 7923  ax-addcom 7925  ax-addass 7927  ax-distr 7929  ax-i2m1 7930  ax-0lt1 7931  ax-0id 7933  ax-rnegex 7934  ax-cnre 7936  ax-pre-ltirr 7937  ax-pre-ltwlin 7938  ax-pre-lttrn 7939  ax-pre-ltadd 7941  ax-arch 7944
This theorem depends on definitions:  df-bi 117  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rab 2474  df-v 2751  df-sbc 2975  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-fv 5236  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-pnf 8008  df-mnf 8009  df-xr 8010  df-ltxr 8011  df-le 8012  df-sub 8144  df-neg 8145  df-inn 8934  df-2 8992  df-n0 9191  df-z 9268  df-uz 9543
This theorem is referenced by:  qbtwnre  10271
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