Theorem rebtwn2zlemshrink 10613

Description: Lemma for rebtwn2z 10614. 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 1025	. 2 |- ( ( A e. RR /\ J e. ( ZZ>= ` 2 ) /\ E. m e. ZZ ( m < A /\ A < ( m + J ) ) ) -> J e. ( ZZ>= ` 2 ) )
2		3simpb 1022	. 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 6058	. . . . . . . 8 |- ( w = 2 -> ( m + w ) = ( m + 2 ) )
4	3	breq2d 4121	. . . . . . 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 2543	. . . . 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 6058	. . . . . . . 8 |- ( w = k -> ( m + w ) = ( m + k ) )
10	9	breq2d 4121	. . . . . . 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 2543	. . . . 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 6058	. . . . . . . 8 |- ( w = ( k + 1 ) -> ( m + w ) = ( m + ( k + 1 ) ) )
16	15	breq2d 4121	. . . . . . 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 2543	. . . . 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 6058	. . . . . . . 8 |- ( w = J -> ( m + w ) = ( m + J ) )
22	21	breq2d 4121	. . . . . . 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 2543	. . . . 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 4112	. . . . . . 7 |- ( m = x -> ( m < A <-> x < A ) )
28		oveq1 6057	. . . . . . . 8 |- ( m = x -> ( m + 2 ) = ( x + 2 ) )
29	28	breq2d 4121	. . . . . . 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 2779	. . . . 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 10612	. . . . . 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 1232	. . . . 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 9924	. 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 1005   = wceq 1398   e. wcel 2203   E.wrex 2521   class class class wbr 4109   ` cfv 5352  (class class class)co 6050   RRcr 8126   1c1 8128   + caddc 8130   < clt 8308   2c2 9288   ZZcz 9577   ZZ>=cuz 9853

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-addass 8229  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-0id 8235  ax-rnegex 8236  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-ltadd 8243

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-inn 9238  df-2 9296  df-n0 9497  df-z 9578  df-uz 9854

This theorem is referenced by:  rebtwn2z  10614