Theorem rebtwn2z 10344

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 9445	. . 3 ⊢ (𝐴 ∈ ℝ → (∃𝑚 ∈ ℤ 𝑚 < 𝐴 ∧ ∃𝑛 ∈ ℤ 𝐴 < 𝑛))
2		reeanv 2667	. . 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 527	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑛)) → 𝐴 ∈ ℝ)
5		simplrl 535	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑛)) → 𝑚 ∈ ℤ)
6	5	zred 9448	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑛)) → 𝑚 ∈ ℝ)
7		simplrr 536	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑛)) → 𝑛 ∈ ℤ)
8	7	zred 9448	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑛)) → 𝑛 ∈ ℝ)
9		simprl 529	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑛)) → 𝑚 < 𝐴)
10		simprr 531	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑛)) → 𝐴 < 𝑛)
11	6, 4, 8, 9, 10	lttrd 8152	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑛)) → 𝑚 < 𝑛)
12		znnsub 9377	. . . . . . . 8 ⊢ ((𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑚 < 𝑛 ↔ (𝑛 − 𝑚) ∈ ℕ))
13	12	ad2antlr 489	. . . . . . 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 9638	. . . . . . . 8 ⊢ ((𝑛 − 𝑚) ∈ ℕ ↔ (𝑛 − 𝑚) ∈ (ℤ≥‘1))
16		eluzp1p1 9627	. . . . . . . 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 9049	. . . . . . . 8 ⊢ 2 = (1 + 1)
19	18	fveq2i 5561	. . . . . . 7 ⊢ (ℤ≥‘2) = (ℤ≥‘(1 + 1))
20	17, 19	eleqtrrdi 2290	. . . . . 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 9449	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑛)) → 𝑚 ∈ ℂ)
23	7	zcnd 9449	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑛)) → 𝑛 ∈ ℂ)
24	22, 23	pncan3d 8340	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑛)) → (𝑚 + (𝑛 − 𝑚)) = 𝑛)
25	24, 8	eqeltrd 2273	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑛)) → (𝑚 + (𝑛 − 𝑚)) ∈ ℝ)
26	8, 6	resubcld 8407	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑛)) → (𝑛 − 𝑚) ∈ ℝ)
27		1red 8041	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑛)) → 1 ∈ ℝ)
28	26, 27	readdcld 8056	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑛)) → ((𝑛 − 𝑚) + 1) ∈ ℝ)
29	6, 28	readdcld 8056	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑛)) → (𝑚 + ((𝑛 − 𝑚) + 1)) ∈ ℝ)
30	10, 24	breqtrrd 4061	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑛)) → 𝐴 < (𝑚 + (𝑛 − 𝑚)))
31	26	ltp1d 8957	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑛)) → (𝑛 − 𝑚) < ((𝑛 − 𝑚) + 1))
32	26, 28, 6, 31	ltadd2dd 8449	. . . . . . 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 8152	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑛)) → 𝐴 < (𝑚 + ((𝑛 − 𝑚) + 1)))
34		breq1 4036	. . . . . . . 8 ⊢ (𝑦 = 𝑚 → (𝑦 < 𝐴 ↔ 𝑚 < 𝐴))
35		oveq1 5929	. . . . . . . . 9 ⊢ (𝑦 = 𝑚 → (𝑦 + ((𝑛 − 𝑚) + 1)) = (𝑚 + ((𝑛 − 𝑚) + 1)))
36	35	breq2d 4045	. . . . . . . 8 ⊢ (𝑦 = 𝑚 → (𝐴 < (𝑦 + ((𝑛 − 𝑚) + 1)) ↔ 𝐴 < (𝑚 + ((𝑛 − 𝑚) + 1))))
37	34, 36	anbi12d 473	. . . . . . 7 ⊢ (𝑦 = 𝑚 → ((𝑦 < 𝐴 ∧ 𝐴 < (𝑦 + ((𝑛 − 𝑚) + 1))) ↔ (𝑚 < 𝐴 ∧ 𝐴 < (𝑚 + ((𝑛 − 𝑚) + 1)))))
38	37	rspcev 2868	. . . . . 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 1247	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑛)) → ∃𝑦 ∈ ℤ (𝑦 < 𝐴 ∧ 𝐴 < (𝑦 + ((𝑛 − 𝑚) + 1))))
40		rebtwn2zlemshrink 10343	. . . . 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 1249	. . . 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 2622	. 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 104   ↔ wb 105   ∈ wcel 2167  ∃wrex 2476   class class class wbr 4033  ‘cfv 5258  (class class class)co 5922  ℝcr 7878  1c1 7880   + caddc 7882   < clt 8061   − cmin 8197  ℕcn 8990  2c2 9041  ℤcz 9326  ℤ_≥cuz 9601

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-0id 7987  ax-rnegex 7988  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-ltadd 7995  ax-arch 7998

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-inn 8991  df-2 9049  df-n0 9250  df-z 9327  df-uz 9602

This theorem is referenced by:  qbtwnre  10346