Theorem exbtwnzlemex 9661

Description: Existence of an integer so that a given real number is between the integer and its successor. The real number must satisfy the n <_ A \/ A < n hypothesis. For example either a rational number or a number which is irrational (in the sense of being apart from any rational number) will meet this condition.

The proof starts by finding two integers which are less than and greater than A. 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 the n <_ A \/ A < n hypothesis, and iterating until the range consists of two consecutive integers. (Contributed by Jim Kingdon, 8-Oct-2021.)

Hypotheses

Ref	Expression
exbtwnzlemex.a	|- ( ph -> A e. RR )
exbtwnzlemex.tri	|- ( ( ph /\ n e. ZZ ) -> ( n <_ A \/ A < n ) )

Assertion

Ref	Expression
exbtwnzlemex	|- ( ph -> E. x e. ZZ ( x <_ A /\ A < ( x + 1 ) ) )

Distinct variable groups:   A, n   x, A   ph, n

Allowed substitution hint:   ph( x)

Proof of Theorem exbtwnzlemex

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

Step	Hyp	Ref	Expression
1		exbtwnzlemex.a	. . . 4 |- ( ph -> A e. RR )
2		btwnz 8865	. . . 4 |- ( A e. RR -> ( E. m e. ZZ m < A /\ E. j e. ZZ A < j ) )
3	1, 2	syl 14	. . 3 |- ( ph -> ( E. m e. ZZ m < A /\ E. j e. ZZ A < j ) )
4		reeanv 2536	. . 3 |- ( E. m e. ZZ E. j e. ZZ ( m < A /\ A < j ) <-> ( E. m e. ZZ m < A /\ E. j e. ZZ A < j ) )
5	3, 4	sylibr 132	. 2 |- ( ph -> E. m e. ZZ E. j e. ZZ ( m < A /\ A < j ) )
6		simplrl 502	. . . . . 6 |- ( ( ( ph /\ ( m e. ZZ /\ j e. ZZ ) ) /\ ( m < A /\ A < j ) ) -> m e. ZZ )
7	6	zred 8868	. . . . . . 7 |- ( ( ( ph /\ ( m e. ZZ /\ j e. ZZ ) ) /\ ( m < A /\ A < j ) ) -> m e. RR )
8	1	ad2antrr 472	. . . . . . 7 |- ( ( ( ph /\ ( m e. ZZ /\ j e. ZZ ) ) /\ ( m < A /\ A < j ) ) -> A e. RR )
9		simprl 498	. . . . . . 7 |- ( ( ( ph /\ ( m e. ZZ /\ j e. ZZ ) ) /\ ( m < A /\ A < j ) ) -> m < A )
10	7, 8, 9	ltled 7602	. . . . . 6 |- ( ( ( ph /\ ( m e. ZZ /\ j e. ZZ ) ) /\ ( m < A /\ A < j ) ) -> m <_ A )
11		simprr 499	. . . . . . 7 |- ( ( ( ph /\ ( m e. ZZ /\ j e. ZZ ) ) /\ ( m < A /\ A < j ) ) -> A < j )
12	6	zcnd 8869	. . . . . . . 8 |- ( ( ( ph /\ ( m e. ZZ /\ j e. ZZ ) ) /\ ( m < A /\ A < j ) ) -> m e. CC )
13		simplrr 503	. . . . . . . . 9 |- ( ( ( ph /\ ( m e. ZZ /\ j e. ZZ ) ) /\ ( m < A /\ A < j ) ) -> j e. ZZ )
14	13	zcnd 8869	. . . . . . . 8 |- ( ( ( ph /\ ( m e. ZZ /\ j e. ZZ ) ) /\ ( m < A /\ A < j ) ) -> j e. CC )
15	12, 14	pncan3d 7796	. . . . . . 7 |- ( ( ( ph /\ ( m e. ZZ /\ j e. ZZ ) ) /\ ( m < A /\ A < j ) ) -> ( m + ( j - m ) ) = j )
16	11, 15	breqtrrd 3871	. . . . . 6 |- ( ( ( ph /\ ( m e. ZZ /\ j e. ZZ ) ) /\ ( m < A /\ A < j ) ) -> A < ( m + ( j - m ) ) )
17		breq1 3848	. . . . . . . 8 |- ( y = m -> ( y <_ A <-> m <_ A ) )
18		oveq1 5659	. . . . . . . . 9 |- ( y = m -> ( y + ( j - m ) ) = ( m + ( j - m ) ) )
19	18	breq2d 3857	. . . . . . . 8 |- ( y = m -> ( A < ( y + ( j - m ) ) <-> A < ( m + ( j - m ) ) ) )
20	17, 19	anbi12d 457	. . . . . . 7 |- ( y = m -> ( ( y <_ A /\ A < ( y + ( j - m ) ) ) <-> ( m <_ A /\ A < ( m + ( j - m ) ) ) ) )
21	20	rspcev 2722	. . . . . 6 |- ( ( m e. ZZ /\ ( m <_ A /\ A < ( m + ( j - m ) ) ) ) -> E. y e. ZZ ( y <_ A /\ A < ( y + ( j - m ) ) ) )
22	6, 10, 16, 21	syl12anc 1172	. . . . 5 |- ( ( ( ph /\ ( m e. ZZ /\ j e. ZZ ) ) /\ ( m < A /\ A < j ) ) -> E. y e. ZZ ( y <_ A /\ A < ( y + ( j - m ) ) ) )
23	13	zred 8868	. . . . . . . 8 |- ( ( ( ph /\ ( m e. ZZ /\ j e. ZZ ) ) /\ ( m < A /\ A < j ) ) -> j e. RR )
24	7, 8, 23, 9, 11	lttrd 7609	. . . . . . 7 |- ( ( ( ph /\ ( m e. ZZ /\ j e. ZZ ) ) /\ ( m < A /\ A < j ) ) -> m < j )
25		znnsub 8801	. . . . . . . 8 |- ( ( m e. ZZ /\ j e. ZZ ) -> ( m < j <-> ( j - m ) e. NN ) )
26	25	ad2antlr 473	. . . . . . 7 |- ( ( ( ph /\ ( m e. ZZ /\ j e. ZZ ) ) /\ ( m < A /\ A < j ) ) -> ( m < j <-> ( j - m ) e. NN ) )
27	24, 26	mpbid 145	. . . . . 6 |- ( ( ( ph /\ ( m e. ZZ /\ j e. ZZ ) ) /\ ( m < A /\ A < j ) ) -> ( j - m ) e. NN )
28		exbtwnzlemex.tri	. . . . . . . . . 10 |- ( ( ph /\ n e. ZZ ) -> ( n <_ A \/ A < n ) )
29	28	ralrimiva 2446	. . . . . . . . 9 |- ( ph -> A. n e. ZZ ( n <_ A \/ A < n ) )
30		breq1 3848	. . . . . . . . . . 11 |- ( n = a -> ( n <_ A <-> a <_ A ) )
31		breq2 3849	. . . . . . . . . . 11 |- ( n = a -> ( A < n <-> A < a ) )
32	30, 31	orbi12d 742	. . . . . . . . . 10 |- ( n = a -> ( ( n <_ A \/ A < n ) <-> ( a <_ A \/ A < a ) ) )
33	32	cbvralv 2590	. . . . . . . . 9 |- ( A. n e. ZZ ( n <_ A \/ A < n ) <-> A. a e. ZZ ( a <_ A \/ A < a ) )
34	29, 33	sylib 120	. . . . . . . 8 |- ( ph -> A. a e. ZZ ( a <_ A \/ A < a ) )
35	34	ad2antrr 472	. . . . . . 7 |- ( ( ( ph /\ ( m e. ZZ /\ j e. ZZ ) ) /\ ( m < A /\ A < j ) ) -> A. a e. ZZ ( a <_ A \/ A < a ) )
36	35	r19.21bi 2461	. . . . . 6 |- ( ( ( ( ph /\ ( m e. ZZ /\ j e. ZZ ) ) /\ ( m < A /\ A < j ) ) /\ a e. ZZ ) -> ( a <_ A \/ A < a ) )
37	27, 8, 36	exbtwnzlemshrink 9660	. . . . 5 |- ( ( ( ( ph /\ ( m e. ZZ /\ j e. ZZ ) ) /\ ( m < A /\ A < j ) ) /\ E. y e. ZZ ( y <_ A /\ A < ( y + ( j - m ) ) ) ) -> E. x e. ZZ ( x <_ A /\ A < ( x + 1 ) ) )
38	22, 37	mpdan 412	. . . 4 |- ( ( ( ph /\ ( m e. ZZ /\ j e. ZZ ) ) /\ ( m < A /\ A < j ) ) -> E. x e. ZZ ( x <_ A /\ A < ( x + 1 ) ) )
39	38	ex 113	. . 3 |- ( ( ph /\ ( m e. ZZ /\ j e. ZZ ) ) -> ( ( m < A /\ A < j ) -> E. x e. ZZ ( x <_ A /\ A < ( x + 1 ) ) ) )
40	39	rexlimdvva 2496	. 2 |- ( ph -> ( E. m e. ZZ E. j e. ZZ ( m < A /\ A < j ) -> E. x e. ZZ ( x <_ A /\ A < ( x + 1 ) ) ) )
41	5, 40	mpd 13	1 |- ( ph -> E. x e. ZZ ( x <_ A /\ A < ( x + 1 ) ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   \/ wo 664   e. wcel 1438   A.wral 2359   E.wrex 2360   class class class wbr 3845  (class class class)co 5652   RRcr 7349   1c1 7351   + caddc 7353   < clt 7522   <_ cle 7523   - cmin 7653   NNcn 8422   ZZcz 8750

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-cnex 7436  ax-resscn 7437  ax-1cn 7438  ax-1re 7439  ax-icn 7440  ax-addcl 7441  ax-addrcl 7442  ax-mulcl 7443  ax-addcom 7445  ax-addass 7447  ax-distr 7449  ax-i2m1 7450  ax-0lt1 7451  ax-0id 7453  ax-rnegex 7454  ax-cnre 7456  ax-pre-ltirr 7457  ax-pre-ltwlin 7458  ax-pre-lttrn 7459  ax-pre-ltadd 7461  ax-arch 7464

This theorem depends on definitions:  df-bi 115  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-br 3846  df-opab 3900  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-iota 4980  df-fun 5017  df-fv 5023  df-riota 5608  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-pnf 7524  df-mnf 7525  df-xr 7526  df-ltxr 7527  df-le 7528  df-sub 7655  df-neg 7656  df-inn 8423  df-n0 8674  df-z 8751

This theorem is referenced by:  qbtwnz  9663  apbtwnz  9681