Theorem rebtwn2z 10025

Description: A real number can be bounded by integers above and below which are two apart.

The proof starts by finding two integers which are less than and greater than the given real number. Then this range can be shrunk by choosing an integer in between the endpoints of the range and then deciding which half of the range to keep based on weak linearity, and iterating until the range consists of integers which are two apart. (Contributed by Jim Kingdon, 13-Oct-2021.)

Assertion

Ref	Expression
rebtwn2z	|- ( A e. RR -> E. x e. ZZ ( x < A /\ A < ( x + 2 ) ) )

Distinct variable group:   x, A

Proof of Theorem rebtwn2z

Dummy variables m n y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		btwnz 9163	. . 3 |- ( A e. RR -> ( E. m e. ZZ m < A /\ E. n e. ZZ A < n ) )
2		reeanv 2598	. . 3 |- ( E. m e. ZZ E. n e. ZZ ( m < A /\ A < n ) <-> ( E. m e. ZZ m < A /\ E. n e. ZZ A < n ) )
3	1, 2	sylibr 133	. 2 |- ( A e. RR -> E. m e. ZZ E. n e. ZZ ( m < A /\ A < n ) )
4		simpll 518	. . . . 5 |- ( ( ( A e. RR /\ ( m e. ZZ /\ n e. ZZ ) ) /\ ( m < A /\ A < n ) ) -> A e. RR )
5		simplrl 524	. . . . . . . . 9 |- ( ( ( A e. RR /\ ( m e. ZZ /\ n e. ZZ ) ) /\ ( m < A /\ A < n ) ) -> m e. ZZ )
6	5	zred 9166	. . . . . . . 8 |- ( ( ( A e. RR /\ ( m e. ZZ /\ n e. ZZ ) ) /\ ( m < A /\ A < n ) ) -> m e. RR )
7		simplrr 525	. . . . . . . . 9 |- ( ( ( A e. RR /\ ( m e. ZZ /\ n e. ZZ ) ) /\ ( m < A /\ A < n ) ) -> n e. ZZ )
8	7	zred 9166	. . . . . . . 8 |- ( ( ( A e. RR /\ ( m e. ZZ /\ n e. ZZ ) ) /\ ( m < A /\ A < n ) ) -> n e. RR )
9		simprl 520	. . . . . . . 8 |- ( ( ( A e. RR /\ ( m e. ZZ /\ n e. ZZ ) ) /\ ( m < A /\ A < n ) ) -> m < A )
10		simprr 521	. . . . . . . 8 |- ( ( ( A e. RR /\ ( m e. ZZ /\ n e. ZZ ) ) /\ ( m < A /\ A < n ) ) -> A < n )
11	6, 4, 8, 9, 10	lttrd 7881	. . . . . . 7 |- ( ( ( A e. RR /\ ( m e. ZZ /\ n e. ZZ ) ) /\ ( m < A /\ A < n ) ) -> m < n )
12		znnsub 9098	. . . . . . . 8 |- ( ( m e. ZZ /\ n e. ZZ ) -> ( m < n <-> ( n - m ) e. NN ) )
13	12	ad2antlr 480	. . . . . . 7 |- ( ( ( A e. RR /\ ( m e. ZZ /\ n e. ZZ ) ) /\ ( m < A /\ A < n ) ) -> ( m < n <-> ( n - m ) e. NN ) )
14	11, 13	mpbid 146	. . . . . 6 |- ( ( ( A e. RR /\ ( m e. ZZ /\ n e. ZZ ) ) /\ ( m < A /\ A < n ) ) -> ( n - m ) e. NN )
15		elnnuz 9355	. . . . . . . 8 |- ( ( n - m ) e. NN <-> ( n - m ) e. ( ZZ>= ` 1 ) )
16		eluzp1p1 9344	. . . . . . . 8 |- ( ( n - m ) e. ( ZZ>= ` 1 ) -> ( ( n - m ) + 1 ) e. ( ZZ>= ` ( 1 + 1 ) ) )
17	15, 16	sylbi 120	. . . . . . 7 |- ( ( n - m ) e. NN -> ( ( n - m ) + 1 ) e. ( ZZ>= ` ( 1 + 1 ) ) )
18		df-2 8772	. . . . . . . 8 |- 2 = ( 1 + 1 )
19	18	fveq2i 5417	. . . . . . 7 |- ( ZZ>= ` 2 ) = ( ZZ>= ` ( 1 + 1 ) )
20	17, 19	eleqtrrdi 2231	. . . . . 6 |- ( ( n - m ) e. NN -> ( ( n - m ) + 1 ) e. ( ZZ>= ` 2 ) )
21	14, 20	syl 14	. . . . 5 |- ( ( ( A e. RR /\ ( m e. ZZ /\ n e. ZZ ) ) /\ ( m < A /\ A < n ) ) -> ( ( n - m ) + 1 ) e. ( ZZ>= ` 2 ) )
22	5	zcnd 9167	. . . . . . . . 9 |- ( ( ( A e. RR /\ ( m e. ZZ /\ n e. ZZ ) ) /\ ( m < A /\ A < n ) ) -> m e. CC )
23	7	zcnd 9167	. . . . . . . . 9 |- ( ( ( A e. RR /\ ( m e. ZZ /\ n e. ZZ ) ) /\ ( m < A /\ A < n ) ) -> n e. CC )
24	22, 23	pncan3d 8069	. . . . . . . 8 |- ( ( ( A e. RR /\ ( m e. ZZ /\ n e. ZZ ) ) /\ ( m < A /\ A < n ) ) -> ( m + ( n - m ) ) = n )
25	24, 8	eqeltrd 2214	. . . . . . 7 |- ( ( ( A e. RR /\ ( m e. ZZ /\ n e. ZZ ) ) /\ ( m < A /\ A < n ) ) -> ( m + ( n - m ) ) e. RR )
26	8, 6	resubcld 8136	. . . . . . . . 9 |- ( ( ( A e. RR /\ ( m e. ZZ /\ n e. ZZ ) ) /\ ( m < A /\ A < n ) ) -> ( n - m ) e. RR )
27		1red 7774	. . . . . . . . 9 |- ( ( ( A e. RR /\ ( m e. ZZ /\ n e. ZZ ) ) /\ ( m < A /\ A < n ) ) -> 1 e. RR )
28	26, 27	readdcld 7788	. . . . . . . 8 |- ( ( ( A e. RR /\ ( m e. ZZ /\ n e. ZZ ) ) /\ ( m < A /\ A < n ) ) -> ( ( n - m ) + 1 ) e. RR )
29	6, 28	readdcld 7788	. . . . . . 7 |- ( ( ( A e. RR /\ ( m e. ZZ /\ n e. ZZ ) ) /\ ( m < A /\ A < n ) ) -> ( m + ( ( n - m ) + 1 ) ) e. RR )
30	10, 24	breqtrrd 3951	. . . . . . 7 |- ( ( ( A e. RR /\ ( m e. ZZ /\ n e. ZZ ) ) /\ ( m < A /\ A < n ) ) -> A < ( m + ( n - m ) ) )
31	26	ltp1d 8681	. . . . . . . 8 |- ( ( ( A e. RR /\ ( m e. ZZ /\ n e. ZZ ) ) /\ ( m < A /\ A < n ) ) -> ( n - m ) < ( ( n - m ) + 1 ) )
32	26, 28, 6, 31	ltadd2dd 8177	. . . . . . 7 |- ( ( ( A e. RR /\ ( m e. ZZ /\ n e. ZZ ) ) /\ ( m < A /\ A < n ) ) -> ( m + ( n - m ) ) < ( m + ( ( n - m ) + 1 ) ) )
33	4, 25, 29, 30, 32	lttrd 7881	. . . . . 6 |- ( ( ( A e. RR /\ ( m e. ZZ /\ n e. ZZ ) ) /\ ( m < A /\ A < n ) ) -> A < ( m + ( ( n - m ) + 1 ) ) )
34		breq1 3927	. . . . . . . 8 |- ( y = m -> ( y < A <-> m < A ) )
35		oveq1 5774	. . . . . . . . 9 |- ( y = m -> ( y + ( ( n - m ) + 1 ) ) = ( m + ( ( n - m ) + 1 ) ) )
36	35	breq2d 3936	. . . . . . . 8 |- ( y = m -> ( A < ( y + ( ( n - m ) + 1 ) ) <-> A < ( m + ( ( n - m ) + 1 ) ) ) )
37	34, 36	anbi12d 464	. . . . . . 7 |- ( y = m -> ( ( y < A /\ A < ( y + ( ( n - m ) + 1 ) ) ) <-> ( m < A /\ A < ( m + ( ( n - m ) + 1 ) ) ) ) )
38	37	rspcev 2784	. . . . . 6 |- ( ( m e. ZZ /\ ( m < A /\ A < ( m + ( ( n - m ) + 1 ) ) ) ) -> E. y e. ZZ ( y < A /\ A < ( y + ( ( n - m ) + 1 ) ) ) )
39	5, 9, 33, 38	syl12anc 1214	. . . . 5 |- ( ( ( A e. RR /\ ( m e. ZZ /\ n e. ZZ ) ) /\ ( m < A /\ A < n ) ) -> E. y e. ZZ ( y < A /\ A < ( y + ( ( n - m ) + 1 ) ) ) )
40		rebtwn2zlemshrink 10024	. . . . 5 |- ( ( A e. RR /\ ( ( n - m ) + 1 ) e. ( ZZ>= ` 2 ) /\ E. y e. ZZ ( y < A /\ A < ( y + ( ( n - m ) + 1 ) ) ) ) -> E. x e. ZZ ( x < A /\ A < ( x + 2 ) ) )
41	4, 21, 39, 40	syl3anc 1216	. . . 4 |- ( ( ( A e. RR /\ ( m e. ZZ /\ n e. ZZ ) ) /\ ( m < A /\ A < n ) ) -> E. x e. ZZ ( x < A /\ A < ( x + 2 ) ) )
42	41	ex 114	. . 3 |- ( ( A e. RR /\ ( m e. ZZ /\ n e. ZZ ) ) -> ( ( m < A /\ A < n ) -> E. x e. ZZ ( x < A /\ A < ( x + 2 ) ) ) )
43	42	rexlimdvva 2555	. 2 |- ( A e. RR -> ( E. m e. ZZ E. n e. ZZ ( m < A /\ A < n ) -> E. x e. ZZ ( x < A /\ A < ( x + 2 ) ) ) )
44	3, 43	mpd 13	1 |- ( A e. RR -> E. x e. ZZ ( x < A /\ A < ( x + 2 ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   e. wcel 1480   E.wrex 2415   class class class wbr 3924   ` cfv 5118  (class class class)co 5767   RRcr 7612   1c1 7614   + caddc 7616   < clt 7793   - cmin 7926   NNcn 8713   2c2 8764   ZZcz 9047   ZZ>=cuz 9319

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-cnex 7704  ax-resscn 7705  ax-1cn 7706  ax-1re 7707  ax-icn 7708  ax-addcl 7709  ax-addrcl 7710  ax-mulcl 7711  ax-addcom 7713  ax-addass 7715  ax-distr 7717  ax-i2m1 7718  ax-0lt1 7719  ax-0id 7721  ax-rnegex 7722  ax-cnre 7724  ax-pre-ltirr 7725  ax-pre-ltwlin 7726  ax-pre-lttrn 7727  ax-pre-ltadd 7729  ax-arch 7732

This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-nel 2402  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-fv 5126  df-riota 5723  df-ov 5770  df-oprab 5771  df-mpo 5772  df-pnf 7795  df-mnf 7796  df-xr 7797  df-ltxr 7798  df-le 7799  df-sub 7928  df-neg 7929  df-inn 8714  df-2 8772  df-n0 8971  df-z 9048  df-uz 9320

This theorem is referenced by:  qbtwnre  10027