Theorem rebtwn2z 10255

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 9372	. . 3 ⊢ (𝐴 ∈ ℝ → (∃𝑚 ∈ ℤ 𝑚 < 𝐴 ∧ ∃𝑛 ∈ ℤ 𝐴 < 𝑛))
2		reeanv 2647	. . 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 9375	. . . . . . . 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 9375	. . . . . . . 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 8083	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑛)) → 𝑚 < 𝑛)
12		znnsub 9304	. . . . . . . 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 9564	. . . . . . . 8 ⊢ ((𝑛 − 𝑚) ∈ ℕ ↔ (𝑛 − 𝑚) ∈ (ℤ≥‘1))
16		eluzp1p1 9553	. . . . . . . 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 8978	. . . . . . . 8 ⊢ 2 = (1 + 1)
19	18	fveq2i 5519	. . . . . . 7 ⊢ (ℤ≥‘2) = (ℤ≥‘(1 + 1))
20	17, 19	eleqtrrdi 2271	. . . . . 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 9376	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑛)) → 𝑚 ∈ ℂ)
23	7	zcnd 9376	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑛)) → 𝑛 ∈ ℂ)
24	22, 23	pncan3d 8271	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑛)) → (𝑚 + (𝑛 − 𝑚)) = 𝑛)
25	24, 8	eqeltrd 2254	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑛)) → (𝑚 + (𝑛 − 𝑚)) ∈ ℝ)
26	8, 6	resubcld 8338	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑛)) → (𝑛 − 𝑚) ∈ ℝ)
27		1red 7972	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑛)) → 1 ∈ ℝ)
28	26, 27	readdcld 7987	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑛)) → ((𝑛 − 𝑚) + 1) ∈ ℝ)
29	6, 28	readdcld 7987	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑛)) → (𝑚 + ((𝑛 − 𝑚) + 1)) ∈ ℝ)
30	10, 24	breqtrrd 4032	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑛)) → 𝐴 < (𝑚 + (𝑛 − 𝑚)))
31	26	ltp1d 8887	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑛)) → (𝑛 − 𝑚) < ((𝑛 − 𝑚) + 1))
32	26, 28, 6, 31	ltadd2dd 8379	. . . . . . 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 8083	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑛)) → 𝐴 < (𝑚 + ((𝑛 − 𝑚) + 1)))
34		breq1 4007	. . . . . . . 8 ⊢ (𝑦 = 𝑚 → (𝑦 < 𝐴 ↔ 𝑚 < 𝐴))
35		oveq1 5882	. . . . . . . . 9 ⊢ (𝑦 = 𝑚 → (𝑦 + ((𝑛 − 𝑚) + 1)) = (𝑚 + ((𝑛 − 𝑚) + 1)))
36	35	breq2d 4016	. . . . . . . 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 2842	. . . . . 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 1236	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑛)) → ∃𝑦 ∈ ℤ (𝑦 < 𝐴 ∧ 𝐴 < (𝑦 + ((𝑛 − 𝑚) + 1))))
40		rebtwn2zlemshrink 10254	. . . . 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 1238	. . . 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 2602	. 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 2148  ∃wrex 2456   class class class wbr 4004  ‘cfv 5217  (class class class)co 5875  ℝcr 7810  1c1 7812   + caddc 7814   < clt 7992   − cmin 8128  ℕcn 8919  2c2 8970  ℤcz 9253  ℤ_≥cuz 9528

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-cnex 7902  ax-resscn 7903  ax-1cn 7904  ax-1re 7905  ax-icn 7906  ax-addcl 7907  ax-addrcl 7908  ax-mulcl 7909  ax-addcom 7911  ax-addass 7913  ax-distr 7915  ax-i2m1 7916  ax-0lt1 7917  ax-0id 7919  ax-rnegex 7920  ax-cnre 7922  ax-pre-ltirr 7923  ax-pre-ltwlin 7924  ax-pre-lttrn 7925  ax-pre-ltadd 7927  ax-arch 7930

This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-fv 5225  df-riota 5831  df-ov 5878  df-oprab 5879  df-mpo 5880  df-pnf 7994  df-mnf 7995  df-xr 7996  df-ltxr 7997  df-le 7998  df-sub 8130  df-neg 8131  df-inn 8920  df-2 8978  df-n0 9177  df-z 9254  df-uz 9529

This theorem is referenced by:  qbtwnre  10257