Theorem exbtwnzlemex 10185

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	⊢ (𝜑 → 𝐴 ∈ ℝ)
exbtwnzlemex.tri	⊢ ((𝜑 ∧ 𝑛 ∈ ℤ) → (𝑛 ≤ 𝐴 ∨ 𝐴 < 𝑛))

Assertion

Ref	Expression
exbtwnzlemex	⊢ (𝜑 → ∃𝑥 ∈ ℤ (𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)))

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

Allowed substitution hint:   𝜑(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 ⊢ (𝜑 → 𝐴 ∈ ℝ)
2		btwnz 9310	. . . 4 ⊢ (𝐴 ∈ ℝ → (∃𝑚 ∈ ℤ 𝑚 < 𝐴 ∧ ∃𝑗 ∈ ℤ 𝐴 < 𝑗))
3	1, 2	syl 14	. . 3 ⊢ (𝜑 → (∃𝑚 ∈ ℤ 𝑚 < 𝐴 ∧ ∃𝑗 ∈ ℤ 𝐴 < 𝑗))
4		reeanv 2635	. . 3 ⊢ (∃𝑚 ∈ ℤ ∃𝑗 ∈ ℤ (𝑚 < 𝐴 ∧ 𝐴 < 𝑗) ↔ (∃𝑚 ∈ ℤ 𝑚 < 𝐴 ∧ ∃𝑗 ∈ ℤ 𝐴 < 𝑗))
5	3, 4	sylibr 133	. 2 ⊢ (𝜑 → ∃𝑚 ∈ ℤ ∃𝑗 ∈ ℤ (𝑚 < 𝐴 ∧ 𝐴 < 𝑗))
6		simplrl 525	. . . . . 6 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑗)) → 𝑚 ∈ ℤ)
7	6	zred 9313	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑗)) → 𝑚 ∈ ℝ)
8	1	ad2antrr 480	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑗)) → 𝐴 ∈ ℝ)
9		simprl 521	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑗)) → 𝑚 < 𝐴)
10	7, 8, 9	ltled 8017	. . . . . 6 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑗)) → 𝑚 ≤ 𝐴)
11		simprr 522	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑗)) → 𝐴 < 𝑗)
12	6	zcnd 9314	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑗)) → 𝑚 ∈ ℂ)
13		simplrr 526	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑗)) → 𝑗 ∈ ℤ)
14	13	zcnd 9314	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑗)) → 𝑗 ∈ ℂ)
15	12, 14	pncan3d 8212	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑗)) → (𝑚 + (𝑗 − 𝑚)) = 𝑗)
16	11, 15	breqtrrd 4010	. . . . . 6 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑗)) → 𝐴 < (𝑚 + (𝑗 − 𝑚)))
17		breq1 3985	. . . . . . . 8 ⊢ (𝑦 = 𝑚 → (𝑦 ≤ 𝐴 ↔ 𝑚 ≤ 𝐴))
18		oveq1 5849	. . . . . . . . 9 ⊢ (𝑦 = 𝑚 → (𝑦 + (𝑗 − 𝑚)) = (𝑚 + (𝑗 − 𝑚)))
19	18	breq2d 3994	. . . . . . . 8 ⊢ (𝑦 = 𝑚 → (𝐴 < (𝑦 + (𝑗 − 𝑚)) ↔ 𝐴 < (𝑚 + (𝑗 − 𝑚))))
20	17, 19	anbi12d 465	. . . . . . 7 ⊢ (𝑦 = 𝑚 → ((𝑦 ≤ 𝐴 ∧ 𝐴 < (𝑦 + (𝑗 − 𝑚))) ↔ (𝑚 ≤ 𝐴 ∧ 𝐴 < (𝑚 + (𝑗 − 𝑚)))))
21	20	rspcev 2830	. . . . . 6 ⊢ ((𝑚 ∈ ℤ ∧ (𝑚 ≤ 𝐴 ∧ 𝐴 < (𝑚 + (𝑗 − 𝑚)))) → ∃𝑦 ∈ ℤ (𝑦 ≤ 𝐴 ∧ 𝐴 < (𝑦 + (𝑗 − 𝑚))))
22	6, 10, 16, 21	syl12anc 1226	. . . . 5 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑗)) → ∃𝑦 ∈ ℤ (𝑦 ≤ 𝐴 ∧ 𝐴 < (𝑦 + (𝑗 − 𝑚))))
23	13	zred 9313	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑗)) → 𝑗 ∈ ℝ)
24	7, 8, 23, 9, 11	lttrd 8024	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑗)) → 𝑚 < 𝑗)
25		znnsub 9242	. . . . . . . 8 ⊢ ((𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ) → (𝑚 < 𝑗 ↔ (𝑗 − 𝑚) ∈ ℕ))
26	25	ad2antlr 481	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑗)) → (𝑚 < 𝑗 ↔ (𝑗 − 𝑚) ∈ ℕ))
27	24, 26	mpbid 146	. . . . . 6 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑗)) → (𝑗 − 𝑚) ∈ ℕ)
28		exbtwnzlemex.tri	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℤ) → (𝑛 ≤ 𝐴 ∨ 𝐴 < 𝑛))
29	28	ralrimiva 2539	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑛 ∈ ℤ (𝑛 ≤ 𝐴 ∨ 𝐴 < 𝑛))
30		breq1 3985	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑎 → (𝑛 ≤ 𝐴 ↔ 𝑎 ≤ 𝐴))
31		breq2 3986	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑎 → (𝐴 < 𝑛 ↔ 𝐴 < 𝑎))
32	30, 31	orbi12d 783	. . . . . . . . . 10 ⊢ (𝑛 = 𝑎 → ((𝑛 ≤ 𝐴 ∨ 𝐴 < 𝑛) ↔ (𝑎 ≤ 𝐴 ∨ 𝐴 < 𝑎)))
33	32	cbvralv 2692	. . . . . . . . 9 ⊢ (∀𝑛 ∈ ℤ (𝑛 ≤ 𝐴 ∨ 𝐴 < 𝑛) ↔ ∀𝑎 ∈ ℤ (𝑎 ≤ 𝐴 ∨ 𝐴 < 𝑎))
34	29, 33	sylib 121	. . . . . . . 8 ⊢ (𝜑 → ∀𝑎 ∈ ℤ (𝑎 ≤ 𝐴 ∨ 𝐴 < 𝑎))
35	34	ad2antrr 480	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑗)) → ∀𝑎 ∈ ℤ (𝑎 ≤ 𝐴 ∨ 𝐴 < 𝑎))
36	35	r19.21bi 2554	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑗)) ∧ 𝑎 ∈ ℤ) → (𝑎 ≤ 𝐴 ∨ 𝐴 < 𝑎))
37	27, 8, 36	exbtwnzlemshrink 10184	. . . . 5 ⊢ ((((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑗)) ∧ ∃𝑦 ∈ ℤ (𝑦 ≤ 𝐴 ∧ 𝐴 < (𝑦 + (𝑗 − 𝑚)))) → ∃𝑥 ∈ ℤ (𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)))
38	22, 37	mpdan 418	. . . 4 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑗)) → ∃𝑥 ∈ ℤ (𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)))
39	38	ex 114	. . 3 ⊢ ((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → ((𝑚 < 𝐴 ∧ 𝐴 < 𝑗) → ∃𝑥 ∈ ℤ (𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1))))
40	39	rexlimdvva 2591	. 2 ⊢ (𝜑 → (∃𝑚 ∈ ℤ ∃𝑗 ∈ ℤ (𝑚 < 𝐴 ∧ 𝐴 < 𝑗) → ∃𝑥 ∈ ℤ (𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1))))
41	5, 40	mpd 13	1 ⊢ (𝜑 → ∃𝑥 ∈ ℤ (𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 698   ∈ wcel 2136  ∀wral 2444  ∃wrex 2445   class class class wbr 3982  (class class class)co 5842  ℝcr 7752  1c1 7754   + caddc 7756   < clt 7933   ≤ cle 7934   − cmin 8069  ℕcn 8857  ℤcz 9191

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-addcom 7853  ax-addass 7855  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-0id 7861  ax-rnegex 7862  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-ltadd 7869  ax-arch 7872

This theorem depends on definitions:  df-bi 116  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-br 3983  df-opab 4044  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-iota 5153  df-fun 5190  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-inn 8858  df-n0 9115  df-z 9192

This theorem is referenced by:  qbtwnz  10187  apbtwnz  10209