Theorem rebtwn2zlemshrink 9262

Description: Lemma for rebtwn2z 9263. 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 939	. 2 |- ( ( A e. RR /\ J e. ( ZZ>= ` 2 ) /\ E. m e. ZZ ( m < A /\ A < ( m + J ) ) ) -> J e. ( ZZ>= ` 2 ) )
2		3simpb 936	. 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 8379	. . 3 |- 2 e. ZZ
4		oveq2 5540	. . . . . . . 8 |- ( w = 2 -> ( m + w ) = ( m + 2 ) )
5	4	breq2d 3797	. . . . . . 7 |- ( w = 2 -> ( A < ( m + w ) <-> A < ( m + 2 ) ) )
6	5	anbi2d 451	. . . . . 6 |- ( w = 2 -> ( ( m < A /\ A < ( m + w ) ) <-> ( m < A /\ A < ( m + 2 ) ) ) )
7	6	rexbidv 2369	. . . . 5 |- ( w = 2 -> ( E. m e. ZZ ( m < A /\ A < ( m + w ) ) <-> E. m e. ZZ ( m < A /\ A < ( m + 2 ) ) ) )
8	7	anbi2d 451	. . . 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 ) ) ) ) )
9	8	imbi1d 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 5540	. . . . . . . 8 |- ( w = k -> ( m + w ) = ( m + k ) )
11	10	breq2d 3797	. . . . . . 7 |- ( w = k -> ( A < ( m + w ) <-> A < ( m + k ) ) )
12	11	anbi2d 451	. . . . . 6 |- ( w = k -> ( ( m < A /\ A < ( m + w ) ) <-> ( m < A /\ A < ( m + k ) ) ) )
13	12	rexbidv 2369	. . . . 5 |- ( w = k -> ( E. m e. ZZ ( m < A /\ A < ( m + w ) ) <-> E. m e. ZZ ( m < A /\ A < ( m + k ) ) ) )
14	13	anbi2d 451	. . . 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 ) ) ) ) )
15	14	imbi1d 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 5540	. . . . . . . 8 |- ( w = ( k + 1 ) -> ( m + w ) = ( m + ( k + 1 ) ) )
17	16	breq2d 3797	. . . . . . 7 |- ( w = ( k + 1 ) -> ( A < ( m + w ) <-> A < ( m + ( k + 1 ) ) ) )
18	17	anbi2d 451	. . . . . 6 |- ( w = ( k + 1 ) -> ( ( m < A /\ A < ( m + w ) ) <-> ( m < A /\ A < ( m + ( k + 1 ) ) ) ) )
19	18	rexbidv 2369	. . . . 5 |- ( w = ( k + 1 ) -> ( E. m e. ZZ ( m < A /\ A < ( m + w ) ) <-> E. m e. ZZ ( m < A /\ A < ( m + ( k + 1 ) ) ) ) )
20	19	anbi2d 451	. . . 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 ) ) ) ) ) )
21	20	imbi1d 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 5540	. . . . . . . 8 |- ( w = J -> ( m + w ) = ( m + J ) )
23	22	breq2d 3797	. . . . . . 7 |- ( w = J -> ( A < ( m + w ) <-> A < ( m + J ) ) )
24	23	anbi2d 451	. . . . . 6 |- ( w = J -> ( ( m < A /\ A < ( m + w ) ) <-> ( m < A /\ A < ( m + J ) ) ) )
25	24	rexbidv 2369	. . . . 5 |- ( w = J -> ( E. m e. ZZ ( m < A /\ A < ( m + w ) ) <-> E. m e. ZZ ( m < A /\ A < ( m + J ) ) ) )
26	25	anbi2d 451	. . . 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 ) ) ) ) )
27	26	imbi1d 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 3788	. . . . . . 7 |- ( m = x -> ( m < A <-> x < A ) )
29		oveq1 5539	. . . . . . . 8 |- ( m = x -> ( m + 2 ) = ( x + 2 ) )
30	29	breq2d 3797	. . . . . . 7 |- ( m = x -> ( A < ( m + 2 ) <-> A < ( x + 2 ) ) )
31	28, 30	anbi12d 456	. . . . . 6 |- ( m = x -> ( ( m < A /\ A < ( m + 2 ) ) <-> ( x < A /\ A < ( x + 2 ) ) ) )
32	31	cbvrexv 2578	. . . . 5 |- ( E. m e. ZZ ( m < A /\ A < ( m + 2 ) ) <-> E. x e. ZZ ( x < A /\ A < ( x + 2 ) ) )
33	32	biimpi 118	. . . 4 |- ( E. m e. ZZ ( m < A /\ A < ( m + 2 ) ) -> E. x e. ZZ ( x < A /\ A < ( x + 2 ) ) )
34	33	adantl 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 9261	. . . . . 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 ) ) )
36	35	3expia 1140	. . . . 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 ) ) ) )
37	36	imdistanda 436	. . . 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 ) ) ) ) )
38	37	imim1d 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 ) ) ) ) )
39	3, 9, 15, 21, 27, 34, 38	uzind4i 8680	. 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 ) ) ) )
40	1, 2, 39	sylc 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 ) ) )

Syntax hints:   -> wi 4   /\ wa 102   /\ w3a 919   = wceq 1284   e. wcel 1433   E.wrex 2349   class class class wbr 3785   ` cfv 4922  (class class class)co 5532   RRcr 6980   1c1 6982   + caddc 6984   < clt 7153   2c2 8089   ZZcz 8351   ZZ>=cuz 8619

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 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-cnex 7067  ax-resscn 7068  ax-1cn 7069  ax-1re 7070  ax-icn 7071  ax-addcl 7072  ax-addrcl 7073  ax-mulcl 7074  ax-addcom 7076  ax-addass 7078  ax-distr 7080  ax-i2m1 7081  ax-0lt1 7082  ax-0id 7084  ax-rnegex 7085  ax-cnre 7087  ax-pre-ltirr 7088  ax-pre-ltwlin 7089  ax-pre-lttrn 7090  ax-pre-ltadd 7092

This theorem depends on definitions:  df-bi 115  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-nel 2340  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-br 3786  df-opab 3840  df-mpt 3841  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-fv 4930  df-riota 5488  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-pnf 7155  df-mnf 7156  df-xr 7157  df-ltxr 7158  df-le 7159  df-sub 7281  df-neg 7282  df-inn 8040  df-2 8098  df-n0 8289  df-z 8352  df-uz 8620

This theorem is referenced by:  rebtwn2z  9263