Theorem rebtwn2zlemshrink 10328

Description: Lemma for rebtwn2z 10329. 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 1000	. 2 |- ( ( A e. RR /\ J e. ( ZZ>= ` 2 ) /\ E. m e. ZZ ( m < A /\ A < ( m + J ) ) ) -> J e. ( ZZ>= ` 2 ) )
2		3simpb 997	. 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 5930	. . . . . . . 8 |- ( w = 2 -> ( m + w ) = ( m + 2 ) )
4	3	breq2d 4045	. . . . . . 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 2498	. . . . 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 5930	. . . . . . . 8 |- ( w = k -> ( m + w ) = ( m + k ) )
10	9	breq2d 4045	. . . . . . 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 2498	. . . . 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 5930	. . . . . . . 8 |- ( w = ( k + 1 ) -> ( m + w ) = ( m + ( k + 1 ) ) )
16	15	breq2d 4045	. . . . . . 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 2498	. . . . 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 5930	. . . . . . . 8 |- ( w = J -> ( m + w ) = ( m + J ) )
22	21	breq2d 4045	. . . . . . 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 2498	. . . . 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 4036	. . . . . . 7 |- ( m = x -> ( m < A <-> x < A ) )
28		oveq1 5929	. . . . . . . 8 |- ( m = x -> ( m + 2 ) = ( x + 2 ) )
29	28	breq2d 4045	. . . . . . 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 2730	. . . . 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 10327	. . . . . 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 1207	. . . . 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 9663	. 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 980   = wceq 1364   e. wcel 2167   E.wrex 2476   class class class wbr 4033   ` cfv 5258  (class class class)co 5922   RRcr 7876   1c1 7878   + caddc 7880   < clt 8059   2c2 9038   ZZcz 9323   ZZ>=cuz 9598

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7968  ax-resscn 7969  ax-1cn 7970  ax-1re 7971  ax-icn 7972  ax-addcl 7973  ax-addrcl 7974  ax-mulcl 7975  ax-addcom 7977  ax-addass 7979  ax-distr 7981  ax-i2m1 7982  ax-0lt1 7983  ax-0id 7985  ax-rnegex 7986  ax-cnre 7988  ax-pre-ltirr 7989  ax-pre-ltwlin 7990  ax-pre-lttrn 7991  ax-pre-ltadd 7993

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-pnf 8061  df-mnf 8062  df-xr 8063  df-ltxr 8064  df-le 8065  df-sub 8197  df-neg 8198  df-inn 8988  df-2 9046  df-n0 9247  df-z 9324  df-uz 9599

This theorem is referenced by:  rebtwn2z  10329