Theorem exbtwnzlemex 10509

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 9599	. . . 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 2703	. . 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 134	. 2 |- ( ph -> E. m e. ZZ E. j e. ZZ ( m < A /\ A < j ) )
6		simplrl 537	. . . . . 6 |- ( ( ( ph /\ ( m e. ZZ /\ j e. ZZ ) ) /\ ( m < A /\ A < j ) ) -> m e. ZZ )
7	6	zred 9602	. . . . . . 7 |- ( ( ( ph /\ ( m e. ZZ /\ j e. ZZ ) ) /\ ( m < A /\ A < j ) ) -> m e. RR )
8	1	ad2antrr 488	. . . . . . 7 |- ( ( ( ph /\ ( m e. ZZ /\ j e. ZZ ) ) /\ ( m < A /\ A < j ) ) -> A e. RR )
9		simprl 531	. . . . . . 7 |- ( ( ( ph /\ ( m e. ZZ /\ j e. ZZ ) ) /\ ( m < A /\ A < j ) ) -> m < A )
10	7, 8, 9	ltled 8298	. . . . . 6 |- ( ( ( ph /\ ( m e. ZZ /\ j e. ZZ ) ) /\ ( m < A /\ A < j ) ) -> m <_ A )
11		simprr 533	. . . . . . 7 |- ( ( ( ph /\ ( m e. ZZ /\ j e. ZZ ) ) /\ ( m < A /\ A < j ) ) -> A < j )
12	6	zcnd 9603	. . . . . . . 8 |- ( ( ( ph /\ ( m e. ZZ /\ j e. ZZ ) ) /\ ( m < A /\ A < j ) ) -> m e. CC )
13		simplrr 538	. . . . . . . . 9 |- ( ( ( ph /\ ( m e. ZZ /\ j e. ZZ ) ) /\ ( m < A /\ A < j ) ) -> j e. ZZ )
14	13	zcnd 9603	. . . . . . . 8 |- ( ( ( ph /\ ( m e. ZZ /\ j e. ZZ ) ) /\ ( m < A /\ A < j ) ) -> j e. CC )
15	12, 14	pncan3d 8493	. . . . . . 7 |- ( ( ( ph /\ ( m e. ZZ /\ j e. ZZ ) ) /\ ( m < A /\ A < j ) ) -> ( m + ( j - m ) ) = j )
16	11, 15	breqtrrd 4116	. . . . . 6 |- ( ( ( ph /\ ( m e. ZZ /\ j e. ZZ ) ) /\ ( m < A /\ A < j ) ) -> A < ( m + ( j - m ) ) )
17		breq1 4091	. . . . . . . 8 |- ( y = m -> ( y <_ A <-> m <_ A ) )
18		oveq1 6025	. . . . . . . . 9 |- ( y = m -> ( y + ( j - m ) ) = ( m + ( j - m ) ) )
19	18	breq2d 4100	. . . . . . . 8 |- ( y = m -> ( A < ( y + ( j - m ) ) <-> A < ( m + ( j - m ) ) ) )
20	17, 19	anbi12d 473	. . . . . . 7 |- ( y = m -> ( ( y <_ A /\ A < ( y + ( j - m ) ) ) <-> ( m <_ A /\ A < ( m + ( j - m ) ) ) ) )
21	20	rspcev 2910	. . . . . 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 1271	. . . . 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 9602	. . . . . . . 8 |- ( ( ( ph /\ ( m e. ZZ /\ j e. ZZ ) ) /\ ( m < A /\ A < j ) ) -> j e. RR )
24	7, 8, 23, 9, 11	lttrd 8305	. . . . . . 7 |- ( ( ( ph /\ ( m e. ZZ /\ j e. ZZ ) ) /\ ( m < A /\ A < j ) ) -> m < j )
25		znnsub 9531	. . . . . . . 8 |- ( ( m e. ZZ /\ j e. ZZ ) -> ( m < j <-> ( j - m ) e. NN ) )
26	25	ad2antlr 489	. . . . . . 7 |- ( ( ( ph /\ ( m e. ZZ /\ j e. ZZ ) ) /\ ( m < A /\ A < j ) ) -> ( m < j <-> ( j - m ) e. NN ) )
27	24, 26	mpbid 147	. . . . . 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 2605	. . . . . . . . 9 |- ( ph -> A. n e. ZZ ( n <_ A \/ A < n ) )
30		breq1 4091	. . . . . . . . . . 11 |- ( n = a -> ( n <_ A <-> a <_ A ) )
31		breq2 4092	. . . . . . . . . . 11 |- ( n = a -> ( A < n <-> A < a ) )
32	30, 31	orbi12d 800	. . . . . . . . . 10 |- ( n = a -> ( ( n <_ A \/ A < n ) <-> ( a <_ A \/ A < a ) ) )
33	32	cbvralv 2767	. . . . . . . . 9 |- ( A. n e. ZZ ( n <_ A \/ A < n ) <-> A. a e. ZZ ( a <_ A \/ A < a ) )
34	29, 33	sylib 122	. . . . . . . 8 |- ( ph -> A. a e. ZZ ( a <_ A \/ A < a ) )
35	34	ad2antrr 488	. . . . . . 7 |- ( ( ( ph /\ ( m e. ZZ /\ j e. ZZ ) ) /\ ( m < A /\ A < j ) ) -> A. a e. ZZ ( a <_ A \/ A < a ) )
36	35	r19.21bi 2620	. . . . . 6 |- ( ( ( ( ph /\ ( m e. ZZ /\ j e. ZZ ) ) /\ ( m < A /\ A < j ) ) /\ a e. ZZ ) -> ( a <_ A \/ A < a ) )
37	27, 8, 36	exbtwnzlemshrink 10508	. . . . 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 421	. . . 4 |- ( ( ( ph /\ ( m e. ZZ /\ j e. ZZ ) ) /\ ( m < A /\ A < j ) ) -> E. x e. ZZ ( x <_ A /\ A < ( x + 1 ) ) )
39	38	ex 115	. . 3 |- ( ( ph /\ ( m e. ZZ /\ j e. ZZ ) ) -> ( ( m < A /\ A < j ) -> E. x e. ZZ ( x <_ A /\ A < ( x + 1 ) ) ) )
40	39	rexlimdvva 2658	. 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 104   <-> wb 105   \/ wo 715   e. wcel 2202   A.wral 2510   E.wrex 2511   class class class wbr 4088  (class class class)co 6018   RRcr 8031   1c1 8033   + caddc 8035   < clt 8214   <_ cle 8215   - cmin 8350   NNcn 9143   ZZcz 9479

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-addcom 8132  ax-addass 8134  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-0id 8140  ax-rnegex 8141  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-ltadd 8148  ax-arch 8151

This theorem depends on definitions:  df-bi 117  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-iota 5286  df-fun 5328  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-inn 9144  df-n0 9403  df-z 9480

This theorem is referenced by:  qbtwnz  10511  apbtwnz  10534