Theorem rebtwn2z 10689

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 9765	. . 3 ⊢ (𝐴 ∈ ℝ → (∃𝑚 ∈ ℤ 𝑚 < 𝐴 ∧ ∃𝑛 ∈ ℤ 𝐴 < 𝑛))
2		reeanv 2721	. . 3 ⊢ (∃𝑚 ∈ ℤ ∃𝑛 ∈ ℤ (𝑚 < 𝐴 ∧ 𝐴 < 𝑛) ↔ (∃𝑚 ∈ ℤ 𝑚 < 𝐴 ∧ ∃𝑛 ∈ ℤ 𝐴 < 𝑛))
3	1, 2	sylibr 134	. 2 ⊢ (𝐴 ∈ ℝ → ∃𝑚 ∈ ℤ ∃𝑛 ∈ ℤ (𝑚 < 𝐴 ∧ 𝐴 < 𝑛))
4		simpll 531	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑛)) → 𝐴 ∈ ℝ)
5		simplrl 541	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑛)) → 𝑚 ∈ ℤ)
6	5	zred 9768	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑛)) → 𝑚 ∈ ℝ)
7		simplrr 542	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑛)) → 𝑛 ∈ ℤ)
8	7	zred 9768	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑛)) → 𝑛 ∈ ℝ)
9		simprl 535	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑛)) → 𝑚 < 𝐴)
10		simprr 537	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑛)) → 𝐴 < 𝑛)
11	6, 4, 8, 9, 10	lttrd 8452	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑛)) → 𝑚 < 𝑛)
12		znnsub 9696	. . . . . . . 8 ⊢ ((𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑚 < 𝑛 ↔ (𝑛 − 𝑚) ∈ ℕ))
13	12	ad2antlr 493	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑛)) → (𝑚 < 𝑛 ↔ (𝑛 − 𝑚) ∈ ℕ))
14	11, 13	mpbid 147	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑛)) → (𝑛 − 𝑚) ∈ ℕ)
15		elnnuz 9959	. . . . . . . 8 ⊢ ((𝑛 − 𝑚) ∈ ℕ ↔ (𝑛 − 𝑚) ∈ (ℤ≥‘1))
16		eluzp1p1 9948	. . . . . . . 8 ⊢ ((𝑛 − 𝑚) ∈ (ℤ≥‘1) → ((𝑛 − 𝑚) + 1) ∈ (ℤ≥‘(1 + 1)))
17	15, 16	sylbi 121	. . . . . . 7 ⊢ ((𝑛 − 𝑚) ∈ ℕ → ((𝑛 − 𝑚) + 1) ∈ (ℤ≥‘(1 + 1)))
18		df-2 9363	. . . . . . . 8 ⊢ 2 = (1 + 1)
19	18	fveq2i 5698	. . . . . . 7 ⊢ (ℤ≥‘2) = (ℤ≥‘(1 + 1))
20	17, 19	eleqtrrdi 2332	. . . . . 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 9769	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑛)) → 𝑚 ∈ ℂ)
23	7	zcnd 9769	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑛)) → 𝑛 ∈ ℂ)
24	22, 23	pncan3d 8640	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑛)) → (𝑚 + (𝑛 − 𝑚)) = 𝑛)
25	24, 8	eqeltrd 2315	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑛)) → (𝑚 + (𝑛 − 𝑚)) ∈ ℝ)
26	8, 6	resubcld 8708	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑛)) → (𝑛 − 𝑚) ∈ ℝ)
27		1red 8341	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑛)) → 1 ∈ ℝ)
28	26, 27	readdcld 8355	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑛)) → ((𝑛 − 𝑚) + 1) ∈ ℝ)
29	6, 28	readdcld 8355	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑛)) → (𝑚 + ((𝑛 − 𝑚) + 1)) ∈ ℝ)
30	10, 24	breqtrrd 4158	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑛)) → 𝐴 < (𝑚 + (𝑛 − 𝑚)))
31	26	ltp1d 9260	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑛)) → (𝑛 − 𝑚) < ((𝑛 − 𝑚) + 1))
32	26, 28, 6, 31	ltadd2dd 8750	. . . . . . 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 8452	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑛)) → 𝐴 < (𝑚 + ((𝑛 − 𝑚) + 1)))
34		breq1 4133	. . . . . . . 8 ⊢ (𝑦 = 𝑚 → (𝑦 < 𝐴 ↔ 𝑚 < 𝐴))
35		oveq1 6092	. . . . . . . . 9 ⊢ (𝑦 = 𝑚 → (𝑦 + ((𝑛 − 𝑚) + 1)) = (𝑚 + ((𝑛 − 𝑚) + 1)))
36	35	breq2d 4142	. . . . . . . 8 ⊢ (𝑦 = 𝑚 → (𝐴 < (𝑦 + ((𝑛 − 𝑚) + 1)) ↔ 𝐴 < (𝑚 + ((𝑛 − 𝑚) + 1))))
37	34, 36	anbi12d 477	. . . . . . 7 ⊢ (𝑦 = 𝑚 → ((𝑦 < 𝐴 ∧ 𝐴 < (𝑦 + ((𝑛 − 𝑚) + 1))) ↔ (𝑚 < 𝐴 ∧ 𝐴 < (𝑚 + ((𝑛 − 𝑚) + 1)))))
38	37	rspcev 2929	. . . . . 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 1276	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑛)) → ∃𝑦 ∈ ℤ (𝑦 < 𝐴 ∧ 𝐴 < (𝑦 + ((𝑛 − 𝑚) + 1))))
40		rebtwn2zlemshrink 10688	. . . . 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 1278	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑛)) → ∃𝑥 ∈ ℤ (𝑥 < 𝐴 ∧ 𝐴 < (𝑥 + 2)))
42	41	ex 115	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → ((𝑚 < 𝐴 ∧ 𝐴 < 𝑛) → ∃𝑥 ∈ ℤ (𝑥 < 𝐴 ∧ 𝐴 < (𝑥 + 2))))
43	42	rexlimdvva 2676	. 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)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ↔ wb 105   ∈ wcel 2209  ∃wrex 2529   class class class wbr 4130  ‘cfv 5377  (class class class)co 6085  ℝcr 8178  1c1 8180   + caddc 8182   < clt 8360   − cmin 8497  ℕcn 9304  2c2 9355  ℤcz 9644  ℤ_≥cuz 9921

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 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-0id 8287  ax-rnegex 8288  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-ltadd 8295  ax-arch 8298

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-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-inn 9305  df-2 9363  df-n0 9564  df-z 9645  df-uz 9922

This theorem is used by:  qbtwnre  10691