Theorem exbtwnzlemex 10175

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 9301	. . . 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 2633	. . 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 9304	. . . . . . 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 8008	. . . . . 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 9305	. . . . . . . 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 9305	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑗)) → 𝑗 ∈ ℂ)
15	12, 14	pncan3d 8203	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑗)) → (𝑚 + (𝑗 − 𝑚)) = 𝑗)
16	11, 15	breqtrrd 4004	. . . . . 6 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑗)) → 𝐴 < (𝑚 + (𝑗 − 𝑚)))
17		breq1 3979	. . . . . . . 8 ⊢ (𝑦 = 𝑚 → (𝑦 ≤ 𝐴 ↔ 𝑚 ≤ 𝐴))
18		oveq1 5843	. . . . . . . . 9 ⊢ (𝑦 = 𝑚 → (𝑦 + (𝑗 − 𝑚)) = (𝑚 + (𝑗 − 𝑚)))
19	18	breq2d 3988	. . . . . . . 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 2825	. . . . . 6 ⊢ ((𝑚 ∈ ℤ ∧ (𝑚 ≤ 𝐴 ∧ 𝐴 < (𝑚 + (𝑗 − 𝑚)))) → ∃𝑦 ∈ ℤ (𝑦 ≤ 𝐴 ∧ 𝐴 < (𝑦 + (𝑗 − 𝑚))))
22	6, 10, 16, 21	syl12anc 1225	. . . . 5 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑗)) → ∃𝑦 ∈ ℤ (𝑦 ≤ 𝐴 ∧ 𝐴 < (𝑦 + (𝑗 − 𝑚))))
23	13	zred 9304	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑗)) → 𝑗 ∈ ℝ)
24	7, 8, 23, 9, 11	lttrd 8015	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑗)) → 𝑚 < 𝑗)
25		znnsub 9233	. . . . . . . 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 2537	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑛 ∈ ℤ (𝑛 ≤ 𝐴 ∨ 𝐴 < 𝑛))
30		breq1 3979	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑎 → (𝑛 ≤ 𝐴 ↔ 𝑎 ≤ 𝐴))
31		breq2 3980	. . . . . . . . . . 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 2689	. . . . . . . . 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 2552	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑗)) ∧ 𝑎 ∈ ℤ) → (𝑎 ≤ 𝐴 ∨ 𝐴 < 𝑎))
37	27, 8, 36	exbtwnzlemshrink 10174	. . . . 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 2589	. 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 2135  ∀wral 2442  ∃wrex 2443   class class class wbr 3976  (class class class)co 5836  ℝcr 7743  1c1 7745   + caddc 7747   < clt 7924   ≤ cle 7925   − cmin 8060  ℕcn 8848  ℤcz 9182

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 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-sep 4094  ax-pow 4147  ax-pr 4181  ax-un 4405  ax-setind 4508  ax-cnex 7835  ax-resscn 7836  ax-1cn 7837  ax-1re 7838  ax-icn 7839  ax-addcl 7840  ax-addrcl 7841  ax-mulcl 7842  ax-addcom 7844  ax-addass 7846  ax-distr 7848  ax-i2m1 7849  ax-0lt1 7850  ax-0id 7852  ax-rnegex 7853  ax-cnre 7855  ax-pre-ltirr 7856  ax-pre-ltwlin 7857  ax-pre-lttrn 7858  ax-pre-ltadd 7860  ax-arch 7863

This theorem depends on definitions:  df-bi 116  df-3or 968  df-3an 969  df-tru 1345  df-fal 1348  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ne 2335  df-nel 2430  df-ral 2447  df-rex 2448  df-reu 2449  df-rab 2451  df-v 2723  df-sbc 2947  df-dif 3113  df-un 3115  df-in 3117  df-ss 3124  df-pw 3555  df-sn 3576  df-pr 3577  df-op 3579  df-uni 3784  df-int 3819  df-br 3977  df-opab 4038  df-id 4265  df-xp 4604  df-rel 4605  df-cnv 4606  df-co 4607  df-dm 4608  df-iota 5147  df-fun 5184  df-fv 5190  df-riota 5792  df-ov 5839  df-oprab 5840  df-mpo 5841  df-pnf 7926  df-mnf 7927  df-xr 7928  df-ltxr 7929  df-le 7930  df-sub 8062  df-neg 8063  df-inn 8849  df-n0 9106  df-z 9183

This theorem is referenced by:  qbtwnz  10177  apbtwnz  10199