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Theorem rebtwn2zlemshrink 9329
Description: Lemma for rebtwn2z 9330. 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 940 . 2  |-  ( ( A  e.  RR  /\  J  e.  ( ZZ>= ` 
2 )  /\  E. m  e.  ZZ  (
m  <  A  /\  A  <  ( m  +  J ) ) )  ->  J  e.  (
ZZ>= `  2 ) )
2 3simpb 937 . 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 2z 8449 . . 3  |-  2  e.  ZZ
4 oveq2 5545 . . . . . . . 8  |-  ( w  =  2  ->  (
m  +  w )  =  ( m  + 
2 ) )
54breq2d 3799 . . . . . . 7  |-  ( w  =  2  ->  ( A  <  ( m  +  w )  <->  A  <  ( m  +  2 ) ) )
65anbi2d 452 . . . . . 6  |-  ( w  =  2  ->  (
( m  <  A  /\  A  <  ( m  +  w ) )  <-> 
( m  <  A  /\  A  <  ( m  +  2 ) ) ) )
76rexbidv 2370 . . . . 5  |-  ( w  =  2  ->  ( E. m  e.  ZZ  ( m  <  A  /\  A  <  ( m  +  w ) )  <->  E. m  e.  ZZ  ( m  < 
A  /\  A  <  ( m  +  2 ) ) ) )
87anbi2d 452 . . . 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 ) ) ) ) )
98imbi1d 229 . . 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 ) ) ) ) )
10 oveq2 5545 . . . . . . . 8  |-  ( w  =  k  ->  (
m  +  w )  =  ( m  +  k ) )
1110breq2d 3799 . . . . . . 7  |-  ( w  =  k  ->  ( A  <  ( m  +  w )  <->  A  <  ( m  +  k ) ) )
1211anbi2d 452 . . . . . 6  |-  ( w  =  k  ->  (
( m  <  A  /\  A  <  ( m  +  w ) )  <-> 
( m  <  A  /\  A  <  ( m  +  k ) ) ) )
1312rexbidv 2370 . . . . 5  |-  ( w  =  k  ->  ( E. m  e.  ZZ  ( m  <  A  /\  A  <  ( m  +  w ) )  <->  E. m  e.  ZZ  ( m  < 
A  /\  A  <  ( m  +  k ) ) ) )
1413anbi2d 452 . . . 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 ) ) ) ) )
1514imbi1d 229 . . 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 ) ) ) ) )
16 oveq2 5545 . . . . . . . 8  |-  ( w  =  ( k  +  1 )  ->  (
m  +  w )  =  ( m  +  ( k  +  1 ) ) )
1716breq2d 3799 . . . . . . 7  |-  ( w  =  ( k  +  1 )  ->  ( A  <  ( m  +  w )  <->  A  <  ( m  +  ( k  +  1 ) ) ) )
1817anbi2d 452 . . . . . 6  |-  ( w  =  ( k  +  1 )  ->  (
( m  <  A  /\  A  <  ( m  +  w ) )  <-> 
( m  <  A  /\  A  <  ( m  +  ( k  +  1 ) ) ) ) )
1918rexbidv 2370 . . . . 5  |-  ( w  =  ( k  +  1 )  ->  ( E. m  e.  ZZ  ( m  <  A  /\  A  <  ( m  +  w ) )  <->  E. m  e.  ZZ  ( m  < 
A  /\  A  <  ( m  +  ( k  +  1 ) ) ) ) )
2019anbi2d 452 . . . 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 ) ) ) ) ) )
2120imbi1d 229 . . 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 ) ) ) ) )
22 oveq2 5545 . . . . . . . 8  |-  ( w  =  J  ->  (
m  +  w )  =  ( m  +  J ) )
2322breq2d 3799 . . . . . . 7  |-  ( w  =  J  ->  ( A  <  ( m  +  w )  <->  A  <  ( m  +  J ) ) )
2423anbi2d 452 . . . . . 6  |-  ( w  =  J  ->  (
( m  <  A  /\  A  <  ( m  +  w ) )  <-> 
( m  <  A  /\  A  <  ( m  +  J ) ) ) )
2524rexbidv 2370 . . . . 5  |-  ( w  =  J  ->  ( E. m  e.  ZZ  ( m  <  A  /\  A  <  ( m  +  w ) )  <->  E. m  e.  ZZ  ( m  < 
A  /\  A  <  ( m  +  J ) ) ) )
2625anbi2d 452 . . . 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 ) ) ) ) )
2726imbi1d 229 . . 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 ) ) ) ) )
28 breq1 3790 . . . . . . 7  |-  ( m  =  x  ->  (
m  <  A  <->  x  <  A ) )
29 oveq1 5544 . . . . . . . 8  |-  ( m  =  x  ->  (
m  +  2 )  =  ( x  + 
2 ) )
3029breq2d 3799 . . . . . . 7  |-  ( m  =  x  ->  ( A  <  ( m  + 
2 )  <->  A  <  ( x  +  2 ) ) )
3128, 30anbi12d 457 . . . . . 6  |-  ( m  =  x  ->  (
( m  <  A  /\  A  <  ( m  +  2 ) )  <-> 
( x  <  A  /\  A  <  ( x  +  2 ) ) ) )
3231cbvrexv 2579 . . . . 5  |-  ( E. m  e.  ZZ  (
m  <  A  /\  A  <  ( m  + 
2 ) )  <->  E. x  e.  ZZ  ( x  < 
A  /\  A  <  ( x  +  2 ) ) )
3332biimpi 118 . . . 4  |-  ( E. m  e.  ZZ  (
m  <  A  /\  A  <  ( m  + 
2 ) )  ->  E. x  e.  ZZ  ( x  <  A  /\  A  <  ( x  + 
2 ) ) )
3433adantl 271 . . 3  |-  ( ( A  e.  RR  /\  E. m  e.  ZZ  (
m  <  A  /\  A  <  ( m  + 
2 ) ) )  ->  E. x  e.  ZZ  ( x  <  A  /\  A  <  ( x  + 
2 ) ) )
35 rebtwn2zlemstep 9328 . . . . . 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 ) ) )
36353expia 1141 . . . . 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 ) ) ) )
3736imdistanda 437 . . . 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 ) ) ) ) )
3837imim1d 74 . . 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 ) ) ) ) )
393, 9, 15, 21, 27, 34, 38uzind4i 8750 . 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 ) ) ) )
401, 2, 39sylc 61 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 102    /\ w3a 920    = wceq 1285    e. wcel 1434   E.wrex 2350   class class class wbr 3787   ` cfv 4926  (class class class)co 5537   RRcr 7031   1c1 7033    + caddc 7035    < clt 7204   2c2 8145   ZZcz 8421   ZZ>=cuz 8689
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3898  ax-pow 3950  ax-pr 3966  ax-un 4190  ax-setind 4282  ax-cnex 7118  ax-resscn 7119  ax-1cn 7120  ax-1re 7121  ax-icn 7122  ax-addcl 7123  ax-addrcl 7124  ax-mulcl 7125  ax-addcom 7127  ax-addass 7129  ax-distr 7131  ax-i2m1 7132  ax-0lt1 7133  ax-0id 7135  ax-rnegex 7136  ax-cnre 7138  ax-pre-ltirr 7139  ax-pre-ltwlin 7140  ax-pre-lttrn 7141  ax-pre-ltadd 7143
This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-nel 2341  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3604  df-int 3639  df-br 3788  df-opab 3842  df-mpt 3843  df-id 4050  df-xp 4371  df-rel 4372  df-cnv 4373  df-co 4374  df-dm 4375  df-rn 4376  df-res 4377  df-ima 4378  df-iota 4891  df-fun 4928  df-fn 4929  df-f 4930  df-fv 4934  df-riota 5493  df-ov 5540  df-oprab 5541  df-mpt2 5542  df-pnf 7206  df-mnf 7207  df-xr 7208  df-ltxr 7209  df-le 7210  df-sub 7337  df-neg 7338  df-inn 8096  df-2 8154  df-n0 8345  df-z 8422  df-uz 8690
This theorem is referenced by:  rebtwn2z  9330
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