Theorem rebtwn2z 10190

Description: A real number can be bounded by integers above and below which are two apart.

The proof starts by finding two integers which are less than and greater than the given real number. 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 weak linearity, and iterating until the range consists of integers which are two apart. (Contributed by Jim Kingdon, 13-Oct-2021.)

Assertion

Ref	Expression
rebtwn2z	⊢ (𝐴 ∈ ℝ → ∃𝑥 ∈ ℤ (𝑥 < 𝐴 ∧ 𝐴 < (𝑥 + 2)))

Distinct variable group:   𝑥,𝐴

Proof of Theorem rebtwn2z

Dummy variables 𝑚 𝑛 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		btwnz 9310	. . 3 ⊢ (𝐴 ∈ ℝ → (∃𝑚 ∈ ℤ 𝑚 < 𝐴 ∧ ∃𝑛 ∈ ℤ 𝐴 < 𝑛))
2		reeanv 2635	. . 3 ⊢ (∃𝑚 ∈ ℤ ∃𝑛 ∈ ℤ (𝑚 < 𝐴 ∧ 𝐴 < 𝑛) ↔ (∃𝑚 ∈ ℤ 𝑚 < 𝐴 ∧ ∃𝑛 ∈ ℤ 𝐴 < 𝑛))
3	1, 2	sylibr 133	. 2 ⊢ (𝐴 ∈ ℝ → ∃𝑚 ∈ ℤ ∃𝑛 ∈ ℤ (𝑚 < 𝐴 ∧ 𝐴 < 𝑛))
4		simpll 519	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑛)) → 𝐴 ∈ ℝ)
5		simplrl 525	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑛)) → 𝑚 ∈ ℤ)
6	5	zred 9313	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑛)) → 𝑚 ∈ ℝ)
7		simplrr 526	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑛)) → 𝑛 ∈ ℤ)
8	7	zred 9313	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑛)) → 𝑛 ∈ ℝ)
9		simprl 521	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑛)) → 𝑚 < 𝐴)
10		simprr 522	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑛)) → 𝐴 < 𝑛)
11	6, 4, 8, 9, 10	lttrd 8024	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑛)) → 𝑚 < 𝑛)
12		znnsub 9242	. . . . . . . 8 ⊢ ((𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑚 < 𝑛 ↔ (𝑛 − 𝑚) ∈ ℕ))
13	12	ad2antlr 481	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑛)) → (𝑚 < 𝑛 ↔ (𝑛 − 𝑚) ∈ ℕ))
14	11, 13	mpbid 146	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑛)) → (𝑛 − 𝑚) ∈ ℕ)
15		elnnuz 9502	. . . . . . . 8 ⊢ ((𝑛 − 𝑚) ∈ ℕ ↔ (𝑛 − 𝑚) ∈ (ℤ≥‘1))
16		eluzp1p1 9491	. . . . . . . 8 ⊢ ((𝑛 − 𝑚) ∈ (ℤ≥‘1) → ((𝑛 − 𝑚) + 1) ∈ (ℤ≥‘(1 + 1)))
17	15, 16	sylbi 120	. . . . . . 7 ⊢ ((𝑛 − 𝑚) ∈ ℕ → ((𝑛 − 𝑚) + 1) ∈ (ℤ≥‘(1 + 1)))
18		df-2 8916	. . . . . . . 8 ⊢ 2 = (1 + 1)
19	18	fveq2i 5489	. . . . . . 7 ⊢ (ℤ≥‘2) = (ℤ≥‘(1 + 1))
20	17, 19	eleqtrrdi 2260	. . . . . 6 ⊢ ((𝑛 − 𝑚) ∈ ℕ → ((𝑛 − 𝑚) + 1) ∈ (ℤ≥‘2))
21	14, 20	syl 14	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑛)) → ((𝑛 − 𝑚) + 1) ∈ (ℤ≥‘2))
22	5	zcnd 9314	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑛)) → 𝑚 ∈ ℂ)
23	7	zcnd 9314	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑛)) → 𝑛 ∈ ℂ)
24	22, 23	pncan3d 8212	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑛)) → (𝑚 + (𝑛 − 𝑚)) = 𝑛)
25	24, 8	eqeltrd 2243	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑛)) → (𝑚 + (𝑛 − 𝑚)) ∈ ℝ)
26	8, 6	resubcld 8279	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑛)) → (𝑛 − 𝑚) ∈ ℝ)
27		1red 7914	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑛)) → 1 ∈ ℝ)
28	26, 27	readdcld 7928	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑛)) → ((𝑛 − 𝑚) + 1) ∈ ℝ)
29	6, 28	readdcld 7928	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑛)) → (𝑚 + ((𝑛 − 𝑚) + 1)) ∈ ℝ)
30	10, 24	breqtrrd 4010	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑛)) → 𝐴 < (𝑚 + (𝑛 − 𝑚)))
31	26	ltp1d 8825	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑛)) → (𝑛 − 𝑚) < ((𝑛 − 𝑚) + 1))
32	26, 28, 6, 31	ltadd2dd 8320	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑛)) → (𝑚 + (𝑛 − 𝑚)) < (𝑚 + ((𝑛 − 𝑚) + 1)))
33	4, 25, 29, 30, 32	lttrd 8024	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑛)) → 𝐴 < (𝑚 + ((𝑛 − 𝑚) + 1)))
34		breq1 3985	. . . . . . . 8 ⊢ (𝑦 = 𝑚 → (𝑦 < 𝐴 ↔ 𝑚 < 𝐴))
35		oveq1 5849	. . . . . . . . 9 ⊢ (𝑦 = 𝑚 → (𝑦 + ((𝑛 − 𝑚) + 1)) = (𝑚 + ((𝑛 − 𝑚) + 1)))
36	35	breq2d 3994	. . . . . . . 8 ⊢ (𝑦 = 𝑚 → (𝐴 < (𝑦 + ((𝑛 − 𝑚) + 1)) ↔ 𝐴 < (𝑚 + ((𝑛 − 𝑚) + 1))))
37	34, 36	anbi12d 465	. . . . . . 7 ⊢ (𝑦 = 𝑚 → ((𝑦 < 𝐴 ∧ 𝐴 < (𝑦 + ((𝑛 − 𝑚) + 1))) ↔ (𝑚 < 𝐴 ∧ 𝐴 < (𝑚 + ((𝑛 − 𝑚) + 1)))))
38	37	rspcev 2830	. . . . . 6 ⊢ ((𝑚 ∈ ℤ ∧ (𝑚 < 𝐴 ∧ 𝐴 < (𝑚 + ((𝑛 − 𝑚) + 1)))) → ∃𝑦 ∈ ℤ (𝑦 < 𝐴 ∧ 𝐴 < (𝑦 + ((𝑛 − 𝑚) + 1))))
39	5, 9, 33, 38	syl12anc 1226	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑛)) → ∃𝑦 ∈ ℤ (𝑦 < 𝐴 ∧ 𝐴 < (𝑦 + ((𝑛 − 𝑚) + 1))))
40		rebtwn2zlemshrink 10189	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑛 − 𝑚) + 1) ∈ (ℤ≥‘2) ∧ ∃𝑦 ∈ ℤ (𝑦 < 𝐴 ∧ 𝐴 < (𝑦 + ((𝑛 − 𝑚) + 1)))) → ∃𝑥 ∈ ℤ (𝑥 < 𝐴 ∧ 𝐴 < (𝑥 + 2)))
41	4, 21, 39, 40	syl3anc 1228	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑛)) → ∃𝑥 ∈ ℤ (𝑥 < 𝐴 ∧ 𝐴 < (𝑥 + 2)))
42	41	ex 114	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → ((𝑚 < 𝐴 ∧ 𝐴 < 𝑛) → ∃𝑥 ∈ ℤ (𝑥 < 𝐴 ∧ 𝐴 < (𝑥 + 2))))
43	42	rexlimdvva 2591	. 2 ⊢ (𝐴 ∈ ℝ → (∃𝑚 ∈ ℤ ∃𝑛 ∈ ℤ (𝑚 < 𝐴 ∧ 𝐴 < 𝑛) → ∃𝑥 ∈ ℤ (𝑥 < 𝐴 ∧ 𝐴 < (𝑥 + 2))))
44	3, 43	mpd 13	1 ⊢ (𝐴 ∈ ℝ → ∃𝑥 ∈ ℤ (𝑥 < 𝐴 ∧ 𝐴 < (𝑥 + 2)))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∈ wcel 2136  ∃wrex 2445   class class class wbr 3982  ‘cfv 5188  (class class class)co 5842  ℝcr 7752  1c1 7754   + caddc 7756   < clt 7933   − cmin 8069  ℕcn 8857  2c2 8908  ℤcz 9191  ℤ_≥cuz 9466

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-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  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-2 8916  df-n0 9115  df-z 9192  df-uz 9467

This theorem is referenced by:  qbtwnre  10192