Theorem rebtwn2z 10063

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 9194	. . 3 |- ( A e. RR -> ( E. m e. ZZ m < A /\ E. n e. ZZ A < n ) )
2		reeanv 2603	. . 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 519	. . . . 5 |- ( ( ( A e. RR /\ ( m e. ZZ /\ n e. ZZ ) ) /\ ( m < A /\ A < n ) ) -> A e. RR )
5		simplrl 525	. . . . . . . . 9 |- ( ( ( A e. RR /\ ( m e. ZZ /\ n e. ZZ ) ) /\ ( m < A /\ A < n ) ) -> m e. ZZ )
6	5	zred 9197	. . . . . . . 8 |- ( ( ( A e. RR /\ ( m e. ZZ /\ n e. ZZ ) ) /\ ( m < A /\ A < n ) ) -> m e. RR )
7		simplrr 526	. . . . . . . . 9 |- ( ( ( A e. RR /\ ( m e. ZZ /\ n e. ZZ ) ) /\ ( m < A /\ A < n ) ) -> n e. ZZ )
8	7	zred 9197	. . . . . . . 8 |- ( ( ( A e. RR /\ ( m e. ZZ /\ n e. ZZ ) ) /\ ( m < A /\ A < n ) ) -> n e. RR )
9		simprl 521	. . . . . . . 8 |- ( ( ( A e. RR /\ ( m e. ZZ /\ n e. ZZ ) ) /\ ( m < A /\ A < n ) ) -> m < A )
10		simprr 522	. . . . . . . 8 |- ( ( ( A e. RR /\ ( m e. ZZ /\ n e. ZZ ) ) /\ ( m < A /\ A < n ) ) -> A < n )
11	6, 4, 8, 9, 10	lttrd 7912	. . . . . . 7 |- ( ( ( A e. RR /\ ( m e. ZZ /\ n e. ZZ ) ) /\ ( m < A /\ A < n ) ) -> m < n )
12		znnsub 9129	. . . . . . . 8 |- ( ( m e. ZZ /\ n e. ZZ ) -> ( m < n <-> ( n - m ) e. NN ) )
13	12	ad2antlr 481	. . . . . . 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 9386	. . . . . . . 8 |- ( ( n - m ) e. NN <-> ( n - m ) e. ( ZZ>= ` 1 ) )
16		eluzp1p1 9375	. . . . . . . 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 8803	. . . . . . . 8 |- 2 = ( 1 + 1 )
19	18	fveq2i 5432	. . . . . . 7 |- ( ZZ>= ` 2 ) = ( ZZ>= ` ( 1 + 1 ) )
20	17, 19	eleqtrrdi 2234	. . . . . 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 9198	. . . . . . . . 9 |- ( ( ( A e. RR /\ ( m e. ZZ /\ n e. ZZ ) ) /\ ( m < A /\ A < n ) ) -> m e. CC )
23	7	zcnd 9198	. . . . . . . . 9 |- ( ( ( A e. RR /\ ( m e. ZZ /\ n e. ZZ ) ) /\ ( m < A /\ A < n ) ) -> n e. CC )
24	22, 23	pncan3d 8100	. . . . . . . 8 |- ( ( ( A e. RR /\ ( m e. ZZ /\ n e. ZZ ) ) /\ ( m < A /\ A < n ) ) -> ( m + ( n - m ) ) = n )
25	24, 8	eqeltrd 2217	. . . . . . 7 |- ( ( ( A e. RR /\ ( m e. ZZ /\ n e. ZZ ) ) /\ ( m < A /\ A < n ) ) -> ( m + ( n - m ) ) e. RR )
26	8, 6	resubcld 8167	. . . . . . . . 9 |- ( ( ( A e. RR /\ ( m e. ZZ /\ n e. ZZ ) ) /\ ( m < A /\ A < n ) ) -> ( n - m ) e. RR )
27		1red 7805	. . . . . . . . 9 |- ( ( ( A e. RR /\ ( m e. ZZ /\ n e. ZZ ) ) /\ ( m < A /\ A < n ) ) -> 1 e. RR )
28	26, 27	readdcld 7819	. . . . . . . 8 |- ( ( ( A e. RR /\ ( m e. ZZ /\ n e. ZZ ) ) /\ ( m < A /\ A < n ) ) -> ( ( n - m ) + 1 ) e. RR )
29	6, 28	readdcld 7819	. . . . . . 7 |- ( ( ( A e. RR /\ ( m e. ZZ /\ n e. ZZ ) ) /\ ( m < A /\ A < n ) ) -> ( m + ( ( n - m ) + 1 ) ) e. RR )
30	10, 24	breqtrrd 3964	. . . . . . 7 |- ( ( ( A e. RR /\ ( m e. ZZ /\ n e. ZZ ) ) /\ ( m < A /\ A < n ) ) -> A < ( m + ( n - m ) ) )
31	26	ltp1d 8712	. . . . . . . 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 8208	. . . . . . 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 7912	. . . . . 6 |- ( ( ( A e. RR /\ ( m e. ZZ /\ n e. ZZ ) ) /\ ( m < A /\ A < n ) ) -> A < ( m + ( ( n - m ) + 1 ) ) )
34		breq1 3940	. . . . . . . 8 |- ( y = m -> ( y < A <-> m < A ) )
35		oveq1 5789	. . . . . . . . 9 |- ( y = m -> ( y + ( ( n - m ) + 1 ) ) = ( m + ( ( n - m ) + 1 ) ) )
36	35	breq2d 3949	. . . . . . . 8 |- ( y = m -> ( A < ( y + ( ( n - m ) + 1 ) ) <-> A < ( m + ( ( n - m ) + 1 ) ) ) )
37	34, 36	anbi12d 465	. . . . . . 7 |- ( y = m -> ( ( y < A /\ A < ( y + ( ( n - m ) + 1 ) ) ) <-> ( m < A /\ A < ( m + ( ( n - m ) + 1 ) ) ) ) )
38	37	rspcev 2793	. . . . . 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 1215	. . . . 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 10062	. . . . 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 1217	. . . 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 2560	. 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 1481   E.wrex 2418   class class class wbr 3937   ` cfv 5131  (class class class)co 5782   RRcr 7643   1c1 7645   + caddc 7647   < clt 7824   - cmin 7957   NNcn 8744   2c2 8795   ZZcz 9078   ZZ>=cuz 9350

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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-cnex 7735  ax-resscn 7736  ax-1cn 7737  ax-1re 7738  ax-icn 7739  ax-addcl 7740  ax-addrcl 7741  ax-mulcl 7742  ax-addcom 7744  ax-addass 7746  ax-distr 7748  ax-i2m1 7749  ax-0lt1 7750  ax-0id 7752  ax-rnegex 7753  ax-cnre 7755  ax-pre-ltirr 7756  ax-pre-ltwlin 7757  ax-pre-lttrn 7758  ax-pre-ltadd 7760  ax-arch 7763

This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2914  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-br 3938  df-opab 3998  df-mpt 3999  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-fv 5139  df-riota 5738  df-ov 5785  df-oprab 5786  df-mpo 5787  df-pnf 7826  df-mnf 7827  df-xr 7828  df-ltxr 7829  df-le 7830  df-sub 7959  df-neg 7960  df-inn 8745  df-2 8803  df-n0 9002  df-z 9079  df-uz 9351

This theorem is referenced by:  qbtwnre  10065