Theorem rebtwn2z 9873

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 9022	. . 3 |- ( A e. RR -> ( E. m e. ZZ m < A /\ E. n e. ZZ A < n ) )
2		reeanv 2558	. . 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 499	. . . . 5 |- ( ( ( A e. RR /\ ( m e. ZZ /\ n e. ZZ ) ) /\ ( m < A /\ A < n ) ) -> A e. RR )
5		simplrl 505	. . . . . . . . 9 |- ( ( ( A e. RR /\ ( m e. ZZ /\ n e. ZZ ) ) /\ ( m < A /\ A < n ) ) -> m e. ZZ )
6	5	zred 9025	. . . . . . . 8 |- ( ( ( A e. RR /\ ( m e. ZZ /\ n e. ZZ ) ) /\ ( m < A /\ A < n ) ) -> m e. RR )
7		simplrr 506	. . . . . . . . 9 |- ( ( ( A e. RR /\ ( m e. ZZ /\ n e. ZZ ) ) /\ ( m < A /\ A < n ) ) -> n e. ZZ )
8	7	zred 9025	. . . . . . . 8 |- ( ( ( A e. RR /\ ( m e. ZZ /\ n e. ZZ ) ) /\ ( m < A /\ A < n ) ) -> n e. RR )
9		simprl 501	. . . . . . . 8 |- ( ( ( A e. RR /\ ( m e. ZZ /\ n e. ZZ ) ) /\ ( m < A /\ A < n ) ) -> m < A )
10		simprr 502	. . . . . . . 8 |- ( ( ( A e. RR /\ ( m e. ZZ /\ n e. ZZ ) ) /\ ( m < A /\ A < n ) ) -> A < n )
11	6, 4, 8, 9, 10	lttrd 7759	. . . . . . 7 |- ( ( ( A e. RR /\ ( m e. ZZ /\ n e. ZZ ) ) /\ ( m < A /\ A < n ) ) -> m < n )
12		znnsub 8957	. . . . . . . 8 |- ( ( m e. ZZ /\ n e. ZZ ) -> ( m < n <-> ( n - m ) e. NN ) )
13	12	ad2antlr 476	. . . . . . 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 9212	. . . . . . . 8 |- ( ( n - m ) e. NN <-> ( n - m ) e. ( ZZ>= ` 1 ) )
16		eluzp1p1 9201	. . . . . . . 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 8637	. . . . . . . 8 |- 2 = ( 1 + 1 )
19	18	fveq2i 5356	. . . . . . 7 |- ( ZZ>= ` 2 ) = ( ZZ>= ` ( 1 + 1 ) )
20	17, 19	syl6eleqr 2193	. . . . . 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 9026	. . . . . . . . 9 |- ( ( ( A e. RR /\ ( m e. ZZ /\ n e. ZZ ) ) /\ ( m < A /\ A < n ) ) -> m e. CC )
23	7	zcnd 9026	. . . . . . . . 9 |- ( ( ( A e. RR /\ ( m e. ZZ /\ n e. ZZ ) ) /\ ( m < A /\ A < n ) ) -> n e. CC )
24	22, 23	pncan3d 7947	. . . . . . . 8 |- ( ( ( A e. RR /\ ( m e. ZZ /\ n e. ZZ ) ) /\ ( m < A /\ A < n ) ) -> ( m + ( n - m ) ) = n )
25	24, 8	eqeltrd 2176	. . . . . . 7 |- ( ( ( A e. RR /\ ( m e. ZZ /\ n e. ZZ ) ) /\ ( m < A /\ A < n ) ) -> ( m + ( n - m ) ) e. RR )
26	8, 6	resubcld 8010	. . . . . . . . 9 |- ( ( ( A e. RR /\ ( m e. ZZ /\ n e. ZZ ) ) /\ ( m < A /\ A < n ) ) -> ( n - m ) e. RR )
27		1red 7653	. . . . . . . . 9 |- ( ( ( A e. RR /\ ( m e. ZZ /\ n e. ZZ ) ) /\ ( m < A /\ A < n ) ) -> 1 e. RR )
28	26, 27	readdcld 7667	. . . . . . . 8 |- ( ( ( A e. RR /\ ( m e. ZZ /\ n e. ZZ ) ) /\ ( m < A /\ A < n ) ) -> ( ( n - m ) + 1 ) e. RR )
29	6, 28	readdcld 7667	. . . . . . 7 |- ( ( ( A e. RR /\ ( m e. ZZ /\ n e. ZZ ) ) /\ ( m < A /\ A < n ) ) -> ( m + ( ( n - m ) + 1 ) ) e. RR )
30	10, 24	breqtrrd 3901	. . . . . . 7 |- ( ( ( A e. RR /\ ( m e. ZZ /\ n e. ZZ ) ) /\ ( m < A /\ A < n ) ) -> A < ( m + ( n - m ) ) )
31	26	ltp1d 8546	. . . . . . . 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 8051	. . . . . . 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 7759	. . . . . 6 |- ( ( ( A e. RR /\ ( m e. ZZ /\ n e. ZZ ) ) /\ ( m < A /\ A < n ) ) -> A < ( m + ( ( n - m ) + 1 ) ) )
34		breq1 3878	. . . . . . . 8 |- ( y = m -> ( y < A <-> m < A ) )
35		oveq1 5713	. . . . . . . . 9 |- ( y = m -> ( y + ( ( n - m ) + 1 ) ) = ( m + ( ( n - m ) + 1 ) ) )
36	35	breq2d 3887	. . . . . . . 8 |- ( y = m -> ( A < ( y + ( ( n - m ) + 1 ) ) <-> A < ( m + ( ( n - m ) + 1 ) ) ) )
37	34, 36	anbi12d 460	. . . . . . 7 |- ( y = m -> ( ( y < A /\ A < ( y + ( ( n - m ) + 1 ) ) ) <-> ( m < A /\ A < ( m + ( ( n - m ) + 1 ) ) ) ) )
38	37	rspcev 2744	. . . . . 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 1182	. . . . 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 9872	. . . . 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 1184	. . . 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 2516	. 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 1448   E.wrex 2376   class class class wbr 3875   ` cfv 5059  (class class class)co 5706   RRcr 7499   1c1 7501   + caddc 7503   < clt 7672   - cmin 7804   NNcn 8578   2c2 8629   ZZcz 8906   ZZ>=cuz 9176

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-cnex 7586  ax-resscn 7587  ax-1cn 7588  ax-1re 7589  ax-icn 7590  ax-addcl 7591  ax-addrcl 7592  ax-mulcl 7593  ax-addcom 7595  ax-addass 7597  ax-distr 7599  ax-i2m1 7600  ax-0lt1 7601  ax-0id 7603  ax-rnegex 7604  ax-cnre 7606  ax-pre-ltirr 7607  ax-pre-ltwlin 7608  ax-pre-lttrn 7609  ax-pre-ltadd 7611  ax-arch 7614

This theorem depends on definitions:  df-bi 116  df-3or 931  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-nel 2363  df-ral 2380  df-rex 2381  df-reu 2382  df-rab 2384  df-v 2643  df-sbc 2863  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-int 3719  df-br 3876  df-opab 3930  df-mpt 3931  df-id 4153  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-fv 5067  df-riota 5662  df-ov 5709  df-oprab 5710  df-mpo 5711  df-pnf 7674  df-mnf 7675  df-xr 7676  df-ltxr 7677  df-le 7678  df-sub 7806  df-neg 7807  df-inn 8579  df-2 8637  df-n0 8830  df-z 8907  df-uz 9177

This theorem is referenced by:  qbtwnre  9875