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Theorem rebtwn2zlemshrink 9108
Description: Lemma for rebtwn2z 9109. 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 905 . 2  |-  ( ( A  e.  RR  /\  J  e.  ( ZZ>= ` 
2 )  /\  E. m  e.  ZZ  (
m  <  A  /\  A  <  ( m  +  J ) ) )  ->  J  e.  (
ZZ>= `  2 ) )
2 3simpb 902 . 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 8273 . . 3  |-  2  e.  ZZ
4 oveq2 5520 . . . . . . . 8  |-  ( w  =  2  ->  (
m  +  w )  =  ( m  + 
2 ) )
54breq2d 3776 . . . . . . 7  |-  ( w  =  2  ->  ( A  <  ( m  +  w )  <->  A  <  ( m  +  2 ) ) )
65anbi2d 437 . . . . . 6  |-  ( w  =  2  ->  (
( m  <  A  /\  A  <  ( m  +  w ) )  <-> 
( m  <  A  /\  A  <  ( m  +  2 ) ) ) )
76rexbidv 2327 . . . . 5  |-  ( w  =  2  ->  ( E. m  e.  ZZ  ( m  <  A  /\  A  <  ( m  +  w ) )  <->  E. m  e.  ZZ  ( m  < 
A  /\  A  <  ( m  +  2 ) ) ) )
87anbi2d 437 . . . 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 220 . . 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 5520 . . . . . . . 8  |-  ( w  =  k  ->  (
m  +  w )  =  ( m  +  k ) )
1110breq2d 3776 . . . . . . 7  |-  ( w  =  k  ->  ( A  <  ( m  +  w )  <->  A  <  ( m  +  k ) ) )
1211anbi2d 437 . . . . . 6  |-  ( w  =  k  ->  (
( m  <  A  /\  A  <  ( m  +  w ) )  <-> 
( m  <  A  /\  A  <  ( m  +  k ) ) ) )
1312rexbidv 2327 . . . . 5  |-  ( w  =  k  ->  ( E. m  e.  ZZ  ( m  <  A  /\  A  <  ( m  +  w ) )  <->  E. m  e.  ZZ  ( m  < 
A  /\  A  <  ( m  +  k ) ) ) )
1413anbi2d 437 . . . 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 220 . . 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 5520 . . . . . . . 8  |-  ( w  =  ( k  +  1 )  ->  (
m  +  w )  =  ( m  +  ( k  +  1 ) ) )
1716breq2d 3776 . . . . . . 7  |-  ( w  =  ( k  +  1 )  ->  ( A  <  ( m  +  w )  <->  A  <  ( m  +  ( k  +  1 ) ) ) )
1817anbi2d 437 . . . . . 6  |-  ( w  =  ( k  +  1 )  ->  (
( m  <  A  /\  A  <  ( m  +  w ) )  <-> 
( m  <  A  /\  A  <  ( m  +  ( k  +  1 ) ) ) ) )
1918rexbidv 2327 . . . . 5  |-  ( w  =  ( k  +  1 )  ->  ( E. m  e.  ZZ  ( m  <  A  /\  A  <  ( m  +  w ) )  <->  E. m  e.  ZZ  ( m  < 
A  /\  A  <  ( m  +  ( k  +  1 ) ) ) ) )
2019anbi2d 437 . . . 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 220 . . 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 5520 . . . . . . . 8  |-  ( w  =  J  ->  (
m  +  w )  =  ( m  +  J ) )
2322breq2d 3776 . . . . . . 7  |-  ( w  =  J  ->  ( A  <  ( m  +  w )  <->  A  <  ( m  +  J ) ) )
2423anbi2d 437 . . . . . 6  |-  ( w  =  J  ->  (
( m  <  A  /\  A  <  ( m  +  w ) )  <-> 
( m  <  A  /\  A  <  ( m  +  J ) ) ) )
2524rexbidv 2327 . . . . 5  |-  ( w  =  J  ->  ( E. m  e.  ZZ  ( m  <  A  /\  A  <  ( m  +  w ) )  <->  E. m  e.  ZZ  ( m  < 
A  /\  A  <  ( m  +  J ) ) ) )
2625anbi2d 437 . . . 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 220 . . 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 3767 . . . . . . 7  |-  ( m  =  x  ->  (
m  <  A  <->  x  <  A ) )
29 oveq1 5519 . . . . . . . 8  |-  ( m  =  x  ->  (
m  +  2 )  =  ( x  + 
2 ) )
3029breq2d 3776 . . . . . . 7  |-  ( m  =  x  ->  ( A  <  ( m  + 
2 )  <->  A  <  ( x  +  2 ) ) )
3128, 30anbi12d 442 . . . . . 6  |-  ( m  =  x  ->  (
( m  <  A  /\  A  <  ( m  +  2 ) )  <-> 
( x  <  A  /\  A  <  ( x  +  2 ) ) ) )
3231cbvrexv 2534 . . . . 5  |-  ( E. m  e.  ZZ  (
m  <  A  /\  A  <  ( m  + 
2 ) )  <->  E. x  e.  ZZ  ( x  < 
A  /\  A  <  ( x  +  2 ) ) )
3332biimpi 113 . . . 4  |-  ( E. m  e.  ZZ  (
m  <  A  /\  A  <  ( m  + 
2 ) )  ->  E. x  e.  ZZ  ( x  <  A  /\  A  <  ( x  + 
2 ) ) )
3433adantl 262 . . 3  |-  ( ( A  e.  RR  /\  E. m  e.  ZZ  (
m  <  A  /\  A  <  ( m  + 
2 ) ) )  ->  E. x  e.  ZZ  ( x  <  A  /\  A  <  ( x  + 
2 ) ) )
35 rebtwn2zlemstep 9107 . . . . . 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 1106 . . . . 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 422 . . . 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 69 . . 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 8535 . 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 56 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 97    /\ w3a 885    = wceq 1243    e. wcel 1393   E.wrex 2307   class class class wbr 3764   ` cfv 4902  (class class class)co 5512   RRcr 6888   1c1 6890    + caddc 6892    < clt 7060   2c2 7964   ZZcz 8245   ZZ>=cuz 8473
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311  ax-cnex 6975  ax-resscn 6976  ax-1cn 6977  ax-1re 6978  ax-icn 6979  ax-addcl 6980  ax-addrcl 6981  ax-mulcl 6982  ax-addcom 6984  ax-addass 6986  ax-distr 6988  ax-i2m1 6989  ax-0id 6992  ax-rnegex 6993  ax-cnre 6995  ax-pre-ltirr 6996  ax-pre-ltwlin 6997  ax-pre-lttrn 6998  ax-pre-ltadd 7000
This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-nel 2207  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-riota 5468  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-1o 6001  df-2o 6002  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-pli 6403  df-mi 6404  df-lti 6405  df-plpq 6442  df-mpq 6443  df-enq 6445  df-nqqs 6446  df-plqqs 6447  df-mqqs 6448  df-1nqqs 6449  df-rq 6450  df-ltnqqs 6451  df-enq0 6522  df-nq0 6523  df-0nq0 6524  df-plq0 6525  df-mq0 6526  df-inp 6564  df-i1p 6565  df-iplp 6566  df-iltp 6568  df-enr 6811  df-nr 6812  df-ltr 6815  df-0r 6816  df-1r 6817  df-0 6896  df-1 6897  df-r 6899  df-lt 6902  df-pnf 7062  df-mnf 7063  df-xr 7064  df-ltxr 7065  df-le 7066  df-sub 7184  df-neg 7185  df-inn 7915  df-2 7973  df-n0 8182  df-z 8246  df-uz 8474
This theorem is referenced by:  rebtwn2z  9109
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