Theorem exbtwnzlemex 10499

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 9589	. . . 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 2701	. . 3 ⊢ (∃𝑚 ∈ ℤ ∃𝑗 ∈ ℤ (𝑚 < 𝐴 ∧ 𝐴 < 𝑗) ↔ (∃𝑚 ∈ ℤ 𝑚 < 𝐴 ∧ ∃𝑗 ∈ ℤ 𝐴 < 𝑗))
5	3, 4	sylibr 134	. 2 ⊢ (𝜑 → ∃𝑚 ∈ ℤ ∃𝑗 ∈ ℤ (𝑚 < 𝐴 ∧ 𝐴 < 𝑗))
6		simplrl 535	. . . . . 6 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑗)) → 𝑚 ∈ ℤ)
7	6	zred 9592	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑗)) → 𝑚 ∈ ℝ)
8	1	ad2antrr 488	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑗)) → 𝐴 ∈ ℝ)
9		simprl 529	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑗)) → 𝑚 < 𝐴)
10	7, 8, 9	ltled 8288	. . . . . 6 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑗)) → 𝑚 ≤ 𝐴)
11		simprr 531	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑗)) → 𝐴 < 𝑗)
12	6	zcnd 9593	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑗)) → 𝑚 ∈ ℂ)
13		simplrr 536	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑗)) → 𝑗 ∈ ℤ)
14	13	zcnd 9593	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑗)) → 𝑗 ∈ ℂ)
15	12, 14	pncan3d 8483	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑗)) → (𝑚 + (𝑗 − 𝑚)) = 𝑗)
16	11, 15	breqtrrd 4114	. . . . . 6 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑗)) → 𝐴 < (𝑚 + (𝑗 − 𝑚)))
17		breq1 4089	. . . . . . . 8 ⊢ (𝑦 = 𝑚 → (𝑦 ≤ 𝐴 ↔ 𝑚 ≤ 𝐴))
18		oveq1 6020	. . . . . . . . 9 ⊢ (𝑦 = 𝑚 → (𝑦 + (𝑗 − 𝑚)) = (𝑚 + (𝑗 − 𝑚)))
19	18	breq2d 4098	. . . . . . . 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 2908	. . . . . 6 ⊢ ((𝑚 ∈ ℤ ∧ (𝑚 ≤ 𝐴 ∧ 𝐴 < (𝑚 + (𝑗 − 𝑚)))) → ∃𝑦 ∈ ℤ (𝑦 ≤ 𝐴 ∧ 𝐴 < (𝑦 + (𝑗 − 𝑚))))
22	6, 10, 16, 21	syl12anc 1269	. . . . 5 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑗)) → ∃𝑦 ∈ ℤ (𝑦 ≤ 𝐴 ∧ 𝐴 < (𝑦 + (𝑗 − 𝑚))))
23	13	zred 9592	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑗)) → 𝑗 ∈ ℝ)
24	7, 8, 23, 9, 11	lttrd 8295	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑗)) → 𝑚 < 𝑗)
25		znnsub 9521	. . . . . . . 8 ⊢ ((𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ) → (𝑚 < 𝑗 ↔ (𝑗 − 𝑚) ∈ ℕ))
26	25	ad2antlr 489	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑗)) → (𝑚 < 𝑗 ↔ (𝑗 − 𝑚) ∈ ℕ))
27	24, 26	mpbid 147	. . . . . 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 2603	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑛 ∈ ℤ (𝑛 ≤ 𝐴 ∨ 𝐴 < 𝑛))
30		breq1 4089	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑎 → (𝑛 ≤ 𝐴 ↔ 𝑎 ≤ 𝐴))
31		breq2 4090	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑎 → (𝐴 < 𝑛 ↔ 𝐴 < 𝑎))
32	30, 31	orbi12d 798	. . . . . . . . . 10 ⊢ (𝑛 = 𝑎 → ((𝑛 ≤ 𝐴 ∨ 𝐴 < 𝑛) ↔ (𝑎 ≤ 𝐴 ∨ 𝐴 < 𝑎)))
33	32	cbvralv 2765	. . . . . . . . 9 ⊢ (∀𝑛 ∈ ℤ (𝑛 ≤ 𝐴 ∨ 𝐴 < 𝑛) ↔ ∀𝑎 ∈ ℤ (𝑎 ≤ 𝐴 ∨ 𝐴 < 𝑎))
34	29, 33	sylib 122	. . . . . . . 8 ⊢ (𝜑 → ∀𝑎 ∈ ℤ (𝑎 ≤ 𝐴 ∨ 𝐴 < 𝑎))
35	34	ad2antrr 488	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑗)) → ∀𝑎 ∈ ℤ (𝑎 ≤ 𝐴 ∨ 𝐴 < 𝑎))
36	35	r19.21bi 2618	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑗)) ∧ 𝑎 ∈ ℤ) → (𝑎 ≤ 𝐴 ∨ 𝐴 < 𝑎))
37	27, 8, 36	exbtwnzlemshrink 10498	. . . . 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 421	. . . 4 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑗)) → ∃𝑥 ∈ ℤ (𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)))
39	38	ex 115	. . 3 ⊢ ((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → ((𝑚 < 𝐴 ∧ 𝐴 < 𝑗) → ∃𝑥 ∈ ℤ (𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1))))
40	39	rexlimdvva 2656	. 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 104   ↔ wb 105   ∨ wo 713   ∈ wcel 2200  ∀wral 2508  ∃wrex 2509   class class class wbr 4086  (class class class)co 6013  ℝcr 8021  1c1 8023   + caddc 8025   < clt 8204   ≤ cle 8205   − cmin 8340  ℕcn 9133  ℤcz 9469

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-addcom 8122  ax-addass 8124  ax-distr 8126  ax-i2m1 8127  ax-0lt1 8128  ax-0id 8130  ax-rnegex 8131  ax-cnre 8133  ax-pre-ltirr 8134  ax-pre-ltwlin 8135  ax-pre-lttrn 8136  ax-pre-ltadd 8138  ax-arch 8141

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-br 4087  df-opab 4149  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-iota 5284  df-fun 5326  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-pnf 8206  df-mnf 8207  df-xr 8208  df-ltxr 8209  df-le 8210  df-sub 8342  df-neg 8343  df-inn 9134  df-n0 9393  df-z 9470

This theorem is referenced by:  qbtwnz  10501  apbtwnz  10524