Theorem exbtwnzlemex 10695

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 9770	. . . 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 2721	. . 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 541	. . . . . 6 |- ( ( ( ph /\ ( m e. ZZ /\ j e. ZZ ) ) /\ ( m < A /\ A < j ) ) -> m e. ZZ )
7	6	zred 9773	. . . . . . 7 |- ( ( ( ph /\ ( m e. ZZ /\ j e. ZZ ) ) /\ ( m < A /\ A < j ) ) -> m e. RR )
8	1	ad2antrr 492	. . . . . . 7 |- ( ( ( ph /\ ( m e. ZZ /\ j e. ZZ ) ) /\ ( m < A /\ A < j ) ) -> A e. RR )
9		simprl 535	. . . . . . 7 |- ( ( ( ph /\ ( m e. ZZ /\ j e. ZZ ) ) /\ ( m < A /\ A < j ) ) -> m < A )
10	7, 8, 9	ltled 8447	. . . . . 6 |- ( ( ( ph /\ ( m e. ZZ /\ j e. ZZ ) ) /\ ( m < A /\ A < j ) ) -> m <_ A )
11		simprr 537	. . . . . . 7 |- ( ( ( ph /\ ( m e. ZZ /\ j e. ZZ ) ) /\ ( m < A /\ A < j ) ) -> A < j )
12	6	zcnd 9774	. . . . . . . 8 |- ( ( ( ph /\ ( m e. ZZ /\ j e. ZZ ) ) /\ ( m < A /\ A < j ) ) -> m e. CC )
13		simplrr 542	. . . . . . . . 9 |- ( ( ( ph /\ ( m e. ZZ /\ j e. ZZ ) ) /\ ( m < A /\ A < j ) ) -> j e. ZZ )
14	13	zcnd 9774	. . . . . . . 8 |- ( ( ( ph /\ ( m e. ZZ /\ j e. ZZ ) ) /\ ( m < A /\ A < j ) ) -> j e. CC )
15	12, 14	pncan3d 8642	. . . . . . 7 |- ( ( ( ph /\ ( m e. ZZ /\ j e. ZZ ) ) /\ ( m < A /\ A < j ) ) -> ( m + ( j - m ) ) = j )
16	11, 15	breqtrrd 4158	. . . . . 6 |- ( ( ( ph /\ ( m e. ZZ /\ j e. ZZ ) ) /\ ( m < A /\ A < j ) ) -> A < ( m + ( j - m ) ) )
17		breq1 4133	. . . . . . . 8 |- ( y = m -> ( y <_ A <-> m <_ A ) )
18		oveq1 6092	. . . . . . . . 9 |- ( y = m -> ( y + ( j - m ) ) = ( m + ( j - m ) ) )
19	18	breq2d 4142	. . . . . . . 8 |- ( y = m -> ( A < ( y + ( j - m ) ) <-> A < ( m + ( j - m ) ) ) )
20	17, 19	anbi12d 477	. . . . . . 7 |- ( y = m -> ( ( y <_ A /\ A < ( y + ( j - m ) ) ) <-> ( m <_ A /\ A < ( m + ( j - m ) ) ) ) )
21	20	rspcev 2929	. . . . . 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 1276	. . . . 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 9773	. . . . . . . 8 |- ( ( ( ph /\ ( m e. ZZ /\ j e. ZZ ) ) /\ ( m < A /\ A < j ) ) -> j e. RR )
24	7, 8, 23, 9, 11	lttrd 8454	. . . . . . 7 |- ( ( ( ph /\ ( m e. ZZ /\ j e. ZZ ) ) /\ ( m < A /\ A < j ) ) -> m < j )
25		znnsub 9701	. . . . . . . 8 |- ( ( m e. ZZ /\ j e. ZZ ) -> ( m < j <-> ( j - m ) e. NN ) )
26	25	ad2antlr 493	. . . . . . 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 2623	. . . . . . . . 9 |- ( ph -> A. n e. ZZ ( n <_ A \/ A < n ) )
30		breq1 4133	. . . . . . . . . . 11 |- ( n = a -> ( n <_ A <-> a <_ A ) )
31		breq2 4134	. . . . . . . . . . 11 |- ( n = a -> ( A < n <-> A < a ) )
32	30, 31	orbi12d 805	. . . . . . . . . 10 |- ( n = a -> ( ( n <_ A \/ A < n ) <-> ( a <_ A \/ A < a ) ) )
33	32	cbvralv 2786	. . . . . . . . 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 492	. . . . . . 7 |- ( ( ( ph /\ ( m e. ZZ /\ j e. ZZ ) ) /\ ( m < A /\ A < j ) ) -> A. a e. ZZ ( a <_ A \/ A < a ) )
36	35	r19.21bi 2638	. . . . . 6 |- ( ( ( ( ph /\ ( m e. ZZ /\ j e. ZZ ) ) /\ ( m < A /\ A < j ) ) /\ a e. ZZ ) -> ( a <_ A \/ A < a ) )
37	27, 8, 36	exbtwnzlemshrink 10694	. . . . 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 425	. . . 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 2676	. 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 ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 720   e. wcel 2209   A.wral 2528   E.wrex 2529   class class class wbr 4130  (class class class)co 6085   RRcr 8179   1c1 8181   + caddc 8183   < clt 8361   <_ cle 8362   - cmin 8499   NNcn 9307   ZZcz 9649

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8271  ax-resscn 8272  ax-1cn 8273  ax-1re 8274  ax-icn 8275  ax-addcl 8276  ax-addrcl 8277  ax-mulcl 8278  ax-addcom 8280  ax-addass 8282  ax-distr 8284  ax-i2m1 8285  ax-0lt1 8286  ax-0id 8288  ax-rnegex 8289  ax-cnre 8291  ax-pre-ltirr 8292  ax-pre-ltwlin 8293  ax-pre-lttrn 8294  ax-pre-ltadd 8296  ax-arch 8299

This proof depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4131  df-opab 4193  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-iota 5337  df-fun 5379  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-pnf 8363  df-mnf 8364  df-xr 8365  df-ltxr 8366  df-le 8367  df-sub 8501  df-neg 8502  df-inn 9308  df-n0 9569  df-z 9650

This theorem is used by:  qbtwnz  10697  apbtwnz  10720