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Theorem rebtwn2zlemshrink 10189
Description: Lemma for rebtwn2z 10190. 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 988 . 2 |- ( ( A e. RR /\ J e. ( ZZ>= ` 2 ) /\ E. m e. ZZ ( m < A /\ A < ( m + J ) ) ) -> J e. ( ZZ>= ` 2 ) )
2 3simpb 985 . 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 5850 . . . . . . . 8 |- ( w = 2 -> ( m + w ) = ( m + 2 ) )
43breq2d 3994 . . . . . . 7 |- ( w = 2 -> ( A < ( m + w ) <-> A < ( m + 2 ) ) )
54anbi2d 460 . . . . . 6 |- ( w = 2 -> ( ( m < A /\ A < ( m + w ) ) <-> ( m < A /\ A < ( m + 2 ) ) ) )
65rexbidv 2467 . . . . 5 |- ( w = 2 -> ( E. m e. ZZ ( m < A /\ A < ( m + w ) ) <-> E. m e. ZZ ( m < A /\ A < ( m + 2 ) ) ) )
76anbi2d 460 . . . 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 5850 . . . . . . . 8 |- ( w = k -> ( m + w ) = ( m + k ) )
109breq2d 3994 . . . . . . 7 |- ( w = k -> ( A < ( m + w ) <-> A < ( m + k ) ) )
1110anbi2d 460 . . . . . 6 |- ( w = k -> ( ( m < A /\ A < ( m + w ) ) <-> ( m < A /\ A < ( m + k ) ) ) )
1211rexbidv 2467 . . . . 5 |- ( w = k -> ( E. m e. ZZ ( m < A /\ A < ( m + w ) ) <-> E. m e. ZZ ( m < A /\ A < ( m + k ) ) ) )
1312anbi2d 460 . . . 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 5850 . . . . . . . 8 |- ( w = ( k + 1 ) -> ( m + w ) = ( m + ( k + 1 ) ) )
1615breq2d 3994 . . . . . . 7 |- ( w = ( k + 1 ) -> ( A < ( m + w ) <-> A < ( m + ( k + 1 ) ) ) )
1716anbi2d 460 . . . . . 6 |- ( w = ( k + 1 ) -> ( ( m < A /\ A < ( m + w ) ) <-> ( m < A /\ A < ( m + ( k + 1 ) ) ) ) )
1817rexbidv 2467 . . . . 5 |- ( w = ( k + 1 ) -> ( E. m e. ZZ ( m < A /\ A < ( m + w ) ) <-> E. m e. ZZ ( m < A /\ A < ( m + ( k + 1 ) ) ) ) )
1918anbi2d 460 . . . 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 5850 . . . . . . . 8 |- ( w = J -> ( m + w ) = ( m + J ) )
2221breq2d 3994 . . . . . . 7 |- ( w = J -> ( A < ( m + w ) <-> A < ( m + J ) ) )
2322anbi2d 460 . . . . . 6 |- ( w = J -> ( ( m < A /\ A < ( m + w ) ) <-> ( m < A /\ A < ( m + J ) ) ) )
2423rexbidv 2467 . . . . 5 |- ( w = J -> ( E. m e. ZZ ( m < A /\ A < ( m + w ) ) <-> E. m e. ZZ ( m < A /\ A < ( m + J ) ) ) )
2524anbi2d 460 . . . 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 3985 . . . . . . 7 |- ( m = x -> ( m < A <-> x < A ) )
28 oveq1 5849 . . . . . . . 8 |- ( m = x -> ( m + 2 ) = ( x + 2 ) )
2928breq2d 3994 . . . . . . 7 |- ( m = x -> ( A < ( m + 2 ) <-> A < ( x + 2 ) ) )
3027, 29anbi12d 465 . . . . . 6 |- ( m = x -> ( ( m < A /\ A < ( m + 2 ) ) <-> ( x < A /\ A < ( x + 2 ) ) ) )
3130cbvrexv 2693 . . . . 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 10188 . . . . . 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 1195 . . . . 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 445 . . . 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 9530 . 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 968   = wceq 1343   e. wcel 2136   E.wrex 2445   class class class wbr 3982   ` cfv 5188  (class class class)co 5842   RRcr 7752   1c1 7754   + caddc 7756   < clt 7933   2c2 8908   ZZcz 9191   ZZ>=cuz 9466
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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-addcom 7853  ax-addass 7855  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-0id 7861  ax-rnegex 7862  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-ltadd 7869
This theorem depends on definitions:  df-bi 116  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-inn 8858  df-2 8916  df-n0 9115  df-z 9192  df-uz 9467
This theorem is referenced by:  rebtwn2z  10190
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