Theorem rebtwn2z 10638

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 9715	. . 3 |- ( A e. RR -> ( E. m e. ZZ m < A /\ E. n e. ZZ A < n ) )
2		reeanv 2715	. . 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 537	. . . . . . . . 9 |- ( ( ( A e. RR /\ ( m e. ZZ /\ n e. ZZ ) ) /\ ( m < A /\ A < n ) ) -> m e. ZZ )
6	5	zred 9718	. . . . . . . 8 |- ( ( ( A e. RR /\ ( m e. ZZ /\ n e. ZZ ) ) /\ ( m < A /\ A < n ) ) -> m e. RR )
7		simplrr 538	. . . . . . . . 9 |- ( ( ( A e. RR /\ ( m e. ZZ /\ n e. ZZ ) ) /\ ( m < A /\ A < n ) ) -> n e. ZZ )
8	7	zred 9718	. . . . . . . 8 |- ( ( ( A e. RR /\ ( m e. ZZ /\ n e. ZZ ) ) /\ ( m < A /\ A < n ) ) -> n e. RR )
9		simprl 531	. . . . . . . 8 |- ( ( ( A e. RR /\ ( m e. ZZ /\ n e. ZZ ) ) /\ ( m < A /\ A < n ) ) -> m < A )
10		simprr 533	. . . . . . . 8 |- ( ( ( A e. RR /\ ( m e. ZZ /\ n e. ZZ ) ) /\ ( m < A /\ A < n ) ) -> A < n )
11	6, 4, 8, 9, 10	lttrd 8415	. . . . . . 7 |- ( ( ( A e. RR /\ ( m e. ZZ /\ n e. ZZ ) ) /\ ( m < A /\ A < n ) ) -> m < n )
12		znnsub 9646	. . . . . . . 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 9909	. . . . . . . 8 |- ( ( n - m ) e. NN <-> ( n - m ) e. ( ZZ>= ` 1 ) )
16		eluzp1p1 9898	. . . . . . . 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 9313	. . . . . . . 8 |- 2 = ( 1 + 1 )
19	18	fveq2i 5678	. . . . . . 7 |- ( ZZ>= ` 2 ) = ( ZZ>= ` ( 1 + 1 ) )
20	17, 19	eleqtrrdi 2328	. . . . . 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 9719	. . . . . . . . 9 |- ( ( ( A e. RR /\ ( m e. ZZ /\ n e. ZZ ) ) /\ ( m < A /\ A < n ) ) -> m e. CC )
23	7	zcnd 9719	. . . . . . . . 9 |- ( ( ( A e. RR /\ ( m e. ZZ /\ n e. ZZ ) ) /\ ( m < A /\ A < n ) ) -> n e. CC )
24	22, 23	pncan3d 8603	. . . . . . . 8 |- ( ( ( A e. RR /\ ( m e. ZZ /\ n e. ZZ ) ) /\ ( m < A /\ A < n ) ) -> ( m + ( n - m ) ) = n )
25	24, 8	eqeltrd 2311	. . . . . . 7 |- ( ( ( A e. RR /\ ( m e. ZZ /\ n e. ZZ ) ) /\ ( m < A /\ A < n ) ) -> ( m + ( n - m ) ) e. RR )
26	8, 6	resubcld 8671	. . . . . . . . 9 |- ( ( ( A e. RR /\ ( m e. ZZ /\ n e. ZZ ) ) /\ ( m < A /\ A < n ) ) -> ( n - m ) e. RR )
27		1red 8305	. . . . . . . . 9 |- ( ( ( A e. RR /\ ( m e. ZZ /\ n e. ZZ ) ) /\ ( m < A /\ A < n ) ) -> 1 e. RR )
28	26, 27	readdcld 8319	. . . . . . . 8 |- ( ( ( A e. RR /\ ( m e. ZZ /\ n e. ZZ ) ) /\ ( m < A /\ A < n ) ) -> ( ( n - m ) + 1 ) e. RR )
29	6, 28	readdcld 8319	. . . . . . 7 |- ( ( ( A e. RR /\ ( m e. ZZ /\ n e. ZZ ) ) /\ ( m < A /\ A < n ) ) -> ( m + ( ( n - m ) + 1 ) ) e. RR )
30	10, 24	breqtrrd 4142	. . . . . . 7 |- ( ( ( A e. RR /\ ( m e. ZZ /\ n e. ZZ ) ) /\ ( m < A /\ A < n ) ) -> A < ( m + ( n - m ) ) )
31	26	ltp1d 9221	. . . . . . . 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 8713	. . . . . . 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 8415	. . . . . 6 |- ( ( ( A e. RR /\ ( m e. ZZ /\ n e. ZZ ) ) /\ ( m < A /\ A < n ) ) -> A < ( m + ( ( n - m ) + 1 ) ) )
34		breq1 4117	. . . . . . . 8 |- ( y = m -> ( y < A <-> m < A ) )
35		oveq1 6065	. . . . . . . . 9 |- ( y = m -> ( y + ( ( n - m ) + 1 ) ) = ( m + ( ( n - m ) + 1 ) ) )
36	35	breq2d 4126	. . . . . . . 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 2923	. . . . . 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 1272	. . . . 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 10637	. . . . 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 1274	. . . 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 2670	. 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 2205   E.wrex 2523   class class class wbr 4114   ` cfv 5357  (class class class)co 6058   RRcr 8142   1c1 8144   + caddc 8146   < clt 8324   - cmin 8460   NNcn 9254   2c2 9305   ZZcz 9594   ZZ>=cuz 9871

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 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-ltadd 8259  ax-arch 8262

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 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-inn 9255  df-2 9313  df-n0 9514  df-z 9595  df-uz 9872

This theorem is referenced by:  qbtwnre  10640