Theorem rebtwn2z 10326

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 9439	. . 3 |- ( A e. RR -> ( E. m e. ZZ m < A /\ E. n e. ZZ A < n ) )
2		reeanv 2664	. . 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 134	. 2 |- ( A e. RR -> E. m e. ZZ E. n e. ZZ ( m < A /\ A < n ) )
4		simpll 527	. . . . 5 |- ( ( ( A e. RR /\ ( m e. ZZ /\ n e. ZZ ) ) /\ ( m < A /\ A < n ) ) -> A e. RR )
5		simplrl 535	. . . . . . . . 9 |- ( ( ( A e. RR /\ ( m e. ZZ /\ n e. ZZ ) ) /\ ( m < A /\ A < n ) ) -> m e. ZZ )
6	5	zred 9442	. . . . . . . 8 |- ( ( ( A e. RR /\ ( m e. ZZ /\ n e. ZZ ) ) /\ ( m < A /\ A < n ) ) -> m e. RR )
7		simplrr 536	. . . . . . . . 9 |- ( ( ( A e. RR /\ ( m e. ZZ /\ n e. ZZ ) ) /\ ( m < A /\ A < n ) ) -> n e. ZZ )
8	7	zred 9442	. . . . . . . 8 |- ( ( ( A e. RR /\ ( m e. ZZ /\ n e. ZZ ) ) /\ ( m < A /\ A < n ) ) -> n e. RR )
9		simprl 529	. . . . . . . 8 |- ( ( ( A e. RR /\ ( m e. ZZ /\ n e. ZZ ) ) /\ ( m < A /\ A < n ) ) -> m < A )
10		simprr 531	. . . . . . . 8 |- ( ( ( A e. RR /\ ( m e. ZZ /\ n e. ZZ ) ) /\ ( m < A /\ A < n ) ) -> A < n )
11	6, 4, 8, 9, 10	lttrd 8147	. . . . . . 7 |- ( ( ( A e. RR /\ ( m e. ZZ /\ n e. ZZ ) ) /\ ( m < A /\ A < n ) ) -> m < n )
12		znnsub 9371	. . . . . . . 8 |- ( ( m e. ZZ /\ n e. ZZ ) -> ( m < n <-> ( n - m ) e. NN ) )
13	12	ad2antlr 489	. . . . . . 7 |- ( ( ( A e. RR /\ ( m e. ZZ /\ n e. ZZ ) ) /\ ( m < A /\ A < n ) ) -> ( m < n <-> ( n - m ) e. NN ) )
14	11, 13	mpbid 147	. . . . . 6 |- ( ( ( A e. RR /\ ( m e. ZZ /\ n e. ZZ ) ) /\ ( m < A /\ A < n ) ) -> ( n - m ) e. NN )
15		elnnuz 9632	. . . . . . . 8 |- ( ( n - m ) e. NN <-> ( n - m ) e. ( ZZ>= ` 1 ) )
16		eluzp1p1 9621	. . . . . . . 8 |- ( ( n - m ) e. ( ZZ>= ` 1 ) -> ( ( n - m ) + 1 ) e. ( ZZ>= ` ( 1 + 1 ) ) )
17	15, 16	sylbi 121	. . . . . . 7 |- ( ( n - m ) e. NN -> ( ( n - m ) + 1 ) e. ( ZZ>= ` ( 1 + 1 ) ) )
18		df-2 9043	. . . . . . . 8 |- 2 = ( 1 + 1 )
19	18	fveq2i 5558	. . . . . . 7 |- ( ZZ>= ` 2 ) = ( ZZ>= ` ( 1 + 1 ) )
20	17, 19	eleqtrrdi 2287	. . . . . 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 9443	. . . . . . . . 9 |- ( ( ( A e. RR /\ ( m e. ZZ /\ n e. ZZ ) ) /\ ( m < A /\ A < n ) ) -> m e. CC )
23	7	zcnd 9443	. . . . . . . . 9 |- ( ( ( A e. RR /\ ( m e. ZZ /\ n e. ZZ ) ) /\ ( m < A /\ A < n ) ) -> n e. CC )
24	22, 23	pncan3d 8335	. . . . . . . 8 |- ( ( ( A e. RR /\ ( m e. ZZ /\ n e. ZZ ) ) /\ ( m < A /\ A < n ) ) -> ( m + ( n - m ) ) = n )
25	24, 8	eqeltrd 2270	. . . . . . 7 |- ( ( ( A e. RR /\ ( m e. ZZ /\ n e. ZZ ) ) /\ ( m < A /\ A < n ) ) -> ( m + ( n - m ) ) e. RR )
26	8, 6	resubcld 8402	. . . . . . . . 9 |- ( ( ( A e. RR /\ ( m e. ZZ /\ n e. ZZ ) ) /\ ( m < A /\ A < n ) ) -> ( n - m ) e. RR )
27		1red 8036	. . . . . . . . 9 |- ( ( ( A e. RR /\ ( m e. ZZ /\ n e. ZZ ) ) /\ ( m < A /\ A < n ) ) -> 1 e. RR )
28	26, 27	readdcld 8051	. . . . . . . 8 |- ( ( ( A e. RR /\ ( m e. ZZ /\ n e. ZZ ) ) /\ ( m < A /\ A < n ) ) -> ( ( n - m ) + 1 ) e. RR )
29	6, 28	readdcld 8051	. . . . . . 7 |- ( ( ( A e. RR /\ ( m e. ZZ /\ n e. ZZ ) ) /\ ( m < A /\ A < n ) ) -> ( m + ( ( n - m ) + 1 ) ) e. RR )
30	10, 24	breqtrrd 4058	. . . . . . 7 |- ( ( ( A e. RR /\ ( m e. ZZ /\ n e. ZZ ) ) /\ ( m < A /\ A < n ) ) -> A < ( m + ( n - m ) ) )
31	26	ltp1d 8951	. . . . . . . 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 8443	. . . . . . 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 8147	. . . . . 6 |- ( ( ( A e. RR /\ ( m e. ZZ /\ n e. ZZ ) ) /\ ( m < A /\ A < n ) ) -> A < ( m + ( ( n - m ) + 1 ) ) )
34		breq1 4033	. . . . . . . 8 |- ( y = m -> ( y < A <-> m < A ) )
35		oveq1 5926	. . . . . . . . 9 |- ( y = m -> ( y + ( ( n - m ) + 1 ) ) = ( m + ( ( n - m ) + 1 ) ) )
36	35	breq2d 4042	. . . . . . . 8 |- ( y = m -> ( A < ( y + ( ( n - m ) + 1 ) ) <-> A < ( m + ( ( n - m ) + 1 ) ) ) )
37	34, 36	anbi12d 473	. . . . . . 7 |- ( y = m -> ( ( y < A /\ A < ( y + ( ( n - m ) + 1 ) ) ) <-> ( m < A /\ A < ( m + ( ( n - m ) + 1 ) ) ) ) )
38	37	rspcev 2865	. . . . . 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 1247	. . . . 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 10325	. . . . 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 1249	. . . 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 115	. . 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 2619	. 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 104   <-> wb 105   e. wcel 2164   E.wrex 2473   class class class wbr 4030   ` cfv 5255  (class class class)co 5919   RRcr 7873   1c1 7875   + caddc 7877   < clt 8056   - cmin 8192   NNcn 8984   2c2 9035   ZZcz 9320   ZZ>=cuz 9595

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-addcom 7974  ax-addass 7976  ax-distr 7978  ax-i2m1 7979  ax-0lt1 7980  ax-0id 7982  ax-rnegex 7983  ax-cnre 7985  ax-pre-ltirr 7986  ax-pre-ltwlin 7987  ax-pre-lttrn 7988  ax-pre-ltadd 7990  ax-arch 7993

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2987  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-pnf 8058  df-mnf 8059  df-xr 8060  df-ltxr 8061  df-le 8062  df-sub 8194  df-neg 8195  df-inn 8985  df-2 9043  df-n0 9244  df-z 9321  df-uz 9596

This theorem is referenced by:  qbtwnre  10328