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Theorem rebtwn2zlemshrink 10043
Description: Lemma for rebtwn2z 10044. Shrinking the range around the given real number. (Contributed by Jim Kingdon, 13-Oct-2021.)
Assertion
Ref Expression
rebtwn2zlemshrink  |-  ( ( A  e.  RR  /\  J  e.  ( ZZ>= ` 
2 )  /\  E. m  e.  ZZ  (
m  <  A  /\  A  <  ( m  +  J ) ) )  ->  E. x  e.  ZZ  ( x  <  A  /\  A  <  ( x  + 
2 ) ) )
Distinct variable groups:    A, m, x   
m, J
Allowed substitution hint:    J( x)

Proof of Theorem rebtwn2zlemshrink
Dummy variables  k  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp2 982 . 2  |-  ( ( A  e.  RR  /\  J  e.  ( ZZ>= ` 
2 )  /\  E. m  e.  ZZ  (
m  <  A  /\  A  <  ( m  +  J ) ) )  ->  J  e.  (
ZZ>= `  2 ) )
2 3simpb 979 . 2  |-  ( ( A  e.  RR  /\  J  e.  ( ZZ>= ` 
2 )  /\  E. m  e.  ZZ  (
m  <  A  /\  A  <  ( m  +  J ) ) )  ->  ( A  e.  RR  /\  E. m  e.  ZZ  ( m  < 
A  /\  A  <  ( m  +  J ) ) ) )
3 oveq2 5782 . . . . . . . 8  |-  ( w  =  2  ->  (
m  +  w )  =  ( m  + 
2 ) )
43breq2d 3941 . . . . . . 7  |-  ( w  =  2  ->  ( A  <  ( m  +  w )  <->  A  <  ( m  +  2 ) ) )
54anbi2d 459 . . . . . 6  |-  ( w  =  2  ->  (
( m  <  A  /\  A  <  ( m  +  w ) )  <-> 
( m  <  A  /\  A  <  ( m  +  2 ) ) ) )
65rexbidv 2438 . . . . 5  |-  ( w  =  2  ->  ( E. m  e.  ZZ  ( m  <  A  /\  A  <  ( m  +  w ) )  <->  E. m  e.  ZZ  ( m  < 
A  /\  A  <  ( m  +  2 ) ) ) )
76anbi2d 459 . . . 4  |-  ( w  =  2  ->  (
( A  e.  RR  /\ 
E. m  e.  ZZ  ( m  <  A  /\  A  <  ( m  +  w ) ) )  <-> 
( A  e.  RR  /\ 
E. m  e.  ZZ  ( m  <  A  /\  A  <  ( m  + 
2 ) ) ) ) )
87imbi1d 230 . . 3  |-  ( w  =  2  ->  (
( ( A  e.  RR  /\  E. m  e.  ZZ  ( m  < 
A  /\  A  <  ( m  +  w ) ) )  ->  E. x  e.  ZZ  ( x  < 
A  /\  A  <  ( x  +  2 ) ) )  <->  ( ( A  e.  RR  /\  E. m  e.  ZZ  (
m  <  A  /\  A  <  ( m  + 
2 ) ) )  ->  E. x  e.  ZZ  ( x  <  A  /\  A  <  ( x  + 
2 ) ) ) ) )
9 oveq2 5782 . . . . . . . 8  |-  ( w  =  k  ->  (
m  +  w )  =  ( m  +  k ) )
109breq2d 3941 . . . . . . 7  |-  ( w  =  k  ->  ( A  <  ( m  +  w )  <->  A  <  ( m  +  k ) ) )
1110anbi2d 459 . . . . . 6  |-  ( w  =  k  ->  (
( m  <  A  /\  A  <  ( m  +  w ) )  <-> 
( m  <  A  /\  A  <  ( m  +  k ) ) ) )
1211rexbidv 2438 . . . . 5  |-  ( w  =  k  ->  ( E. m  e.  ZZ  ( m  <  A  /\  A  <  ( m  +  w ) )  <->  E. m  e.  ZZ  ( m  < 
A  /\  A  <  ( m  +  k ) ) ) )
1312anbi2d 459 . . . 4  |-  ( w  =  k  ->  (
( A  e.  RR  /\ 
E. m  e.  ZZ  ( m  <  A  /\  A  <  ( m  +  w ) ) )  <-> 
( A  e.  RR  /\ 
E. m  e.  ZZ  ( m  <  A  /\  A  <  ( m  +  k ) ) ) ) )
1413imbi1d 230 . . 3  |-  ( w  =  k  ->  (
( ( A  e.  RR  /\  E. m  e.  ZZ  ( m  < 
A  /\  A  <  ( m  +  w ) ) )  ->  E. x  e.  ZZ  ( x  < 
A  /\  A  <  ( x  +  2 ) ) )  <->  ( ( A  e.  RR  /\  E. m  e.  ZZ  (
m  <  A  /\  A  <  ( m  +  k ) ) )  ->  E. x  e.  ZZ  ( x  <  A  /\  A  <  ( x  + 
2 ) ) ) ) )
15 oveq2 5782 . . . . . . . 8  |-  ( w  =  ( k  +  1 )  ->  (
m  +  w )  =  ( m  +  ( k  +  1 ) ) )
1615breq2d 3941 . . . . . . 7  |-  ( w  =  ( k  +  1 )  ->  ( A  <  ( m  +  w )  <->  A  <  ( m  +  ( k  +  1 ) ) ) )
1716anbi2d 459 . . . . . 6  |-  ( w  =  ( k  +  1 )  ->  (
( m  <  A  /\  A  <  ( m  +  w ) )  <-> 
( m  <  A  /\  A  <  ( m  +  ( k  +  1 ) ) ) ) )
1817rexbidv 2438 . . . . 5  |-  ( w  =  ( k  +  1 )  ->  ( E. m  e.  ZZ  ( m  <  A  /\  A  <  ( m  +  w ) )  <->  E. m  e.  ZZ  ( m  < 
A  /\  A  <  ( m  +  ( k  +  1 ) ) ) ) )
1918anbi2d 459 . . . 4  |-  ( w  =  ( k  +  1 )  ->  (
( A  e.  RR  /\ 
E. m  e.  ZZ  ( m  <  A  /\  A  <  ( m  +  w ) ) )  <-> 
( A  e.  RR  /\ 
E. m  e.  ZZ  ( m  <  A  /\  A  <  ( m  +  ( k  +  1 ) ) ) ) ) )
2019imbi1d 230 . . 3  |-  ( w  =  ( k  +  1 )  ->  (
( ( A  e.  RR  /\  E. m  e.  ZZ  ( m  < 
A  /\  A  <  ( m  +  w ) ) )  ->  E. x  e.  ZZ  ( x  < 
A  /\  A  <  ( x  +  2 ) ) )  <->  ( ( A  e.  RR  /\  E. m  e.  ZZ  (
m  <  A  /\  A  <  ( m  +  ( k  +  1 ) ) ) )  ->  E. x  e.  ZZ  ( x  <  A  /\  A  <  ( x  + 
2 ) ) ) ) )
21 oveq2 5782 . . . . . . . 8  |-  ( w  =  J  ->  (
m  +  w )  =  ( m  +  J ) )
2221breq2d 3941 . . . . . . 7  |-  ( w  =  J  ->  ( A  <  ( m  +  w )  <->  A  <  ( m  +  J ) ) )
2322anbi2d 459 . . . . . 6  |-  ( w  =  J  ->  (
( m  <  A  /\  A  <  ( m  +  w ) )  <-> 
( m  <  A  /\  A  <  ( m  +  J ) ) ) )
2423rexbidv 2438 . . . . 5  |-  ( w  =  J  ->  ( E. m  e.  ZZ  ( m  <  A  /\  A  <  ( m  +  w ) )  <->  E. m  e.  ZZ  ( m  < 
A  /\  A  <  ( m  +  J ) ) ) )
2524anbi2d 459 . . . 4  |-  ( w  =  J  ->  (
( A  e.  RR  /\ 
E. m  e.  ZZ  ( m  <  A  /\  A  <  ( m  +  w ) ) )  <-> 
( A  e.  RR  /\ 
E. m  e.  ZZ  ( m  <  A  /\  A  <  ( m  +  J ) ) ) ) )
2625imbi1d 230 . . 3  |-  ( w  =  J  ->  (
( ( A  e.  RR  /\  E. m  e.  ZZ  ( m  < 
A  /\  A  <  ( m  +  w ) ) )  ->  E. x  e.  ZZ  ( x  < 
A  /\  A  <  ( x  +  2 ) ) )  <->  ( ( A  e.  RR  /\  E. m  e.  ZZ  (
m  <  A  /\  A  <  ( m  +  J ) ) )  ->  E. x  e.  ZZ  ( x  <  A  /\  A  <  ( x  + 
2 ) ) ) ) )
27 breq1 3932 . . . . . . 7  |-  ( m  =  x  ->  (
m  <  A  <->  x  <  A ) )
28 oveq1 5781 . . . . . . . 8  |-  ( m  =  x  ->  (
m  +  2 )  =  ( x  + 
2 ) )
2928breq2d 3941 . . . . . . 7  |-  ( m  =  x  ->  ( A  <  ( m  + 
2 )  <->  A  <  ( x  +  2 ) ) )
3027, 29anbi12d 464 . . . . . 6  |-  ( m  =  x  ->  (
( m  <  A  /\  A  <  ( m  +  2 ) )  <-> 
( x  <  A  /\  A  <  ( x  +  2 ) ) ) )
3130cbvrexv 2655 . . . . 5  |-  ( E. m  e.  ZZ  (
m  <  A  /\  A  <  ( m  + 
2 ) )  <->  E. x  e.  ZZ  ( x  < 
A  /\  A  <  ( x  +  2 ) ) )
3231biimpi 119 . . . 4  |-  ( E. m  e.  ZZ  (
m  <  A  /\  A  <  ( m  + 
2 ) )  ->  E. x  e.  ZZ  ( x  <  A  /\  A  <  ( x  + 
2 ) ) )
3332adantl 275 . . 3  |-  ( ( A  e.  RR  /\  E. m  e.  ZZ  (
m  <  A  /\  A  <  ( m  + 
2 ) ) )  ->  E. x  e.  ZZ  ( x  <  A  /\  A  <  ( x  + 
2 ) ) )
34 rebtwn2zlemstep 10042 . . . . . 6  |-  ( ( k  e.  ( ZZ>= ` 
2 )  /\  A  e.  RR  /\  E. m  e.  ZZ  ( m  < 
A  /\  A  <  ( m  +  ( k  +  1 ) ) ) )  ->  E. m  e.  ZZ  ( m  < 
A  /\  A  <  ( m  +  k ) ) )
35343expia 1183 . . . . 5  |-  ( ( k  e.  ( ZZ>= ` 
2 )  /\  A  e.  RR )  ->  ( E. m  e.  ZZ  ( m  <  A  /\  A  <  ( m  +  ( k  +  1 ) ) )  ->  E. m  e.  ZZ  ( m  <  A  /\  A  <  ( m  +  k ) ) ) )
3635imdistanda 444 . . . 4  |-  ( k  e.  ( ZZ>= `  2
)  ->  ( ( A  e.  RR  /\  E. m  e.  ZZ  (
m  <  A  /\  A  <  ( m  +  ( k  +  1 ) ) ) )  ->  ( A  e.  RR  /\  E. m  e.  ZZ  ( m  < 
A  /\  A  <  ( m  +  k ) ) ) ) )
3736imim1d 75 . . 3  |-  ( k  e.  ( ZZ>= `  2
)  ->  ( (
( A  e.  RR  /\ 
E. m  e.  ZZ  ( m  <  A  /\  A  <  ( m  +  k ) ) )  ->  E. x  e.  ZZ  ( x  <  A  /\  A  <  ( x  + 
2 ) ) )  ->  ( ( A  e.  RR  /\  E. m  e.  ZZ  (
m  <  A  /\  A  <  ( m  +  ( k  +  1 ) ) ) )  ->  E. x  e.  ZZ  ( x  <  A  /\  A  <  ( x  + 
2 ) ) ) ) )
388, 14, 20, 26, 33, 37uzind4i 9399 . 2  |-  ( J  e.  ( ZZ>= `  2
)  ->  ( ( A  e.  RR  /\  E. m  e.  ZZ  (
m  <  A  /\  A  <  ( m  +  J ) ) )  ->  E. x  e.  ZZ  ( x  <  A  /\  A  <  ( x  + 
2 ) ) ) )
391, 2, 38sylc 62 1  |-  ( ( A  e.  RR  /\  J  e.  ( ZZ>= ` 
2 )  /\  E. m  e.  ZZ  (
m  <  A  /\  A  <  ( m  +  J ) ) )  ->  E. x  e.  ZZ  ( x  <  A  /\  A  <  ( x  + 
2 ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    /\ w3a 962    = wceq 1331    e. wcel 1480   E.wrex 2417   class class class wbr 3929   ` cfv 5123  (class class class)co 5774   RRcr 7631   1c1 7633    + caddc 7635    < clt 7812   2c2 8783   ZZcz 9066   ZZ>=cuz 9338
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7723  ax-resscn 7724  ax-1cn 7725  ax-1re 7726  ax-icn 7727  ax-addcl 7728  ax-addrcl 7729  ax-mulcl 7730  ax-addcom 7732  ax-addass 7734  ax-distr 7736  ax-i2m1 7737  ax-0lt1 7738  ax-0id 7740  ax-rnegex 7741  ax-cnre 7743  ax-pre-ltirr 7744  ax-pre-ltwlin 7745  ax-pre-lttrn 7746  ax-pre-ltadd 7748
This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-pnf 7814  df-mnf 7815  df-xr 7816  df-ltxr 7817  df-le 7818  df-sub 7947  df-neg 7948  df-inn 8733  df-2 8791  df-n0 8990  df-z 9067  df-uz 9339
This theorem is referenced by:  rebtwn2z  10044
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