Theorem rebtwn2zlemshrink 10240

Description: Lemma for rebtwn2z 10241. 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.

Step	Hyp	Ref	Expression
1		simp2 998	. 2 |- ( ( A e. RR /\ J e. ( ZZ>= ` 2 ) /\ E. m e. ZZ ( m < A /\ A < ( m + J ) ) ) -> J e. ( ZZ>= ` 2 ) )
2		3simpb 995	. 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 5877	. . . . . . . 8 |- ( w = 2 -> ( m + w ) = ( m + 2 ) )
4	3	breq2d 4012	. . . . . . 7 |- ( w = 2 -> ( A < ( m + w ) <-> A < ( m + 2 ) ) )
5	4	anbi2d 464	. . . . . 6 |- ( w = 2 -> ( ( m < A /\ A < ( m + w ) ) <-> ( m < A /\ A < ( m + 2 ) ) ) )
6	5	rexbidv 2478	. . . . 5 |- ( w = 2 -> ( E. m e. ZZ ( m < A /\ A < ( m + w ) ) <-> E. m e. ZZ ( m < A /\ A < ( m + 2 ) ) ) )
7	6	anbi2d 464	. . . 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 ) ) ) ) )
8	7	imbi1d 231	. . 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 5877	. . . . . . . 8 |- ( w = k -> ( m + w ) = ( m + k ) )
10	9	breq2d 4012	. . . . . . 7 |- ( w = k -> ( A < ( m + w ) <-> A < ( m + k ) ) )
11	10	anbi2d 464	. . . . . 6 |- ( w = k -> ( ( m < A /\ A < ( m + w ) ) <-> ( m < A /\ A < ( m + k ) ) ) )
12	11	rexbidv 2478	. . . . 5 |- ( w = k -> ( E. m e. ZZ ( m < A /\ A < ( m + w ) ) <-> E. m e. ZZ ( m < A /\ A < ( m + k ) ) ) )
13	12	anbi2d 464	. . . 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 ) ) ) ) )
14	13	imbi1d 231	. . 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 5877	. . . . . . . 8 |- ( w = ( k + 1 ) -> ( m + w ) = ( m + ( k + 1 ) ) )
16	15	breq2d 4012	. . . . . . 7 |- ( w = ( k + 1 ) -> ( A < ( m + w ) <-> A < ( m + ( k + 1 ) ) ) )
17	16	anbi2d 464	. . . . . 6 |- ( w = ( k + 1 ) -> ( ( m < A /\ A < ( m + w ) ) <-> ( m < A /\ A < ( m + ( k + 1 ) ) ) ) )
18	17	rexbidv 2478	. . . . 5 |- ( w = ( k + 1 ) -> ( E. m e. ZZ ( m < A /\ A < ( m + w ) ) <-> E. m e. ZZ ( m < A /\ A < ( m + ( k + 1 ) ) ) ) )
19	18	anbi2d 464	. . . 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 ) ) ) ) ) )
20	19	imbi1d 231	. . 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 5877	. . . . . . . 8 |- ( w = J -> ( m + w ) = ( m + J ) )
22	21	breq2d 4012	. . . . . . 7 |- ( w = J -> ( A < ( m + w ) <-> A < ( m + J ) ) )
23	22	anbi2d 464	. . . . . 6 |- ( w = J -> ( ( m < A /\ A < ( m + w ) ) <-> ( m < A /\ A < ( m + J ) ) ) )
24	23	rexbidv 2478	. . . . 5 |- ( w = J -> ( E. m e. ZZ ( m < A /\ A < ( m + w ) ) <-> E. m e. ZZ ( m < A /\ A < ( m + J ) ) ) )
25	24	anbi2d 464	. . . 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 ) ) ) ) )
26	25	imbi1d 231	. . 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 4003	. . . . . . 7 |- ( m = x -> ( m < A <-> x < A ) )
28		oveq1 5876	. . . . . . . 8 |- ( m = x -> ( m + 2 ) = ( x + 2 ) )
29	28	breq2d 4012	. . . . . . 7 |- ( m = x -> ( A < ( m + 2 ) <-> A < ( x + 2 ) ) )
30	27, 29	anbi12d 473	. . . . . 6 |- ( m = x -> ( ( m < A /\ A < ( m + 2 ) ) <-> ( x < A /\ A < ( x + 2 ) ) ) )
31	30	cbvrexv 2704	. . . . 5 |- ( E. m e. ZZ ( m < A /\ A < ( m + 2 ) ) <-> E. x e. ZZ ( x < A /\ A < ( x + 2 ) ) )
32	31	biimpi 120	. . . 4 |- ( E. m e. ZZ ( m < A /\ A < ( m + 2 ) ) -> E. x e. ZZ ( x < A /\ A < ( x + 2 ) ) )
33	32	adantl 277	. . 3 |- ( ( A e. RR /\ E. m e. ZZ ( m < A /\ A < ( m + 2 ) ) ) -> E. x e. ZZ ( x < A /\ A < ( x + 2 ) ) )
34		rebtwn2zlemstep 10239	. . . . . 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 ) ) )
35	34	3expia 1205	. . . . 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 ) ) ) )
36	35	imdistanda 448	. . . 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 ) ) ) ) )
37	36	imim1d 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 ) ) ) ) )
38	8, 14, 20, 26, 33, 37	uzind4i 9581	. 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 ) ) ) )
39	1, 2, 38	sylc 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 ) ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 978   = wceq 1353   e. wcel 2148   E.wrex 2456   class class class wbr 4000   ` cfv 5212  (class class class)co 5869   RRcr 7801   1c1 7803   + caddc 7805   < clt 7982   2c2 8959   ZZcz 9242   ZZ>=cuz 9517

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-addcom 7902  ax-addass 7904  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-0id 7910  ax-rnegex 7911  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-ltadd 7918

This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-fv 5220  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-inn 8909  df-2 8967  df-n0 9166  df-z 9243  df-uz 9518

This theorem is referenced by:  rebtwn2z  10241