Theorem exbtwnzlemex 10390

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 9491	. . . 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 2675	. . 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 9494	. . . . . . 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 8190	. . . . . 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 9495	. . . . . . . 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 9495	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑗)) → 𝑗 ∈ ℂ)
15	12, 14	pncan3d 8385	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑗)) → (𝑚 + (𝑗 − 𝑚)) = 𝑗)
16	11, 15	breqtrrd 4071	. . . . . 6 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑗)) → 𝐴 < (𝑚 + (𝑗 − 𝑚)))
17		breq1 4046	. . . . . . . 8 ⊢ (𝑦 = 𝑚 → (𝑦 ≤ 𝐴 ↔ 𝑚 ≤ 𝐴))
18		oveq1 5950	. . . . . . . . 9 ⊢ (𝑦 = 𝑚 → (𝑦 + (𝑗 − 𝑚)) = (𝑚 + (𝑗 − 𝑚)))
19	18	breq2d 4055	. . . . . . . 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 2876	. . . . . 6 ⊢ ((𝑚 ∈ ℤ ∧ (𝑚 ≤ 𝐴 ∧ 𝐴 < (𝑚 + (𝑗 − 𝑚)))) → ∃𝑦 ∈ ℤ (𝑦 ≤ 𝐴 ∧ 𝐴 < (𝑦 + (𝑗 − 𝑚))))
22	6, 10, 16, 21	syl12anc 1247	. . . . 5 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑗)) → ∃𝑦 ∈ ℤ (𝑦 ≤ 𝐴 ∧ 𝐴 < (𝑦 + (𝑗 − 𝑚))))
23	13	zred 9494	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑗)) → 𝑗 ∈ ℝ)
24	7, 8, 23, 9, 11	lttrd 8197	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑗)) → 𝑚 < 𝑗)
25		znnsub 9423	. . . . . . . 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 2578	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑛 ∈ ℤ (𝑛 ≤ 𝐴 ∨ 𝐴 < 𝑛))
30		breq1 4046	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑎 → (𝑛 ≤ 𝐴 ↔ 𝑎 ≤ 𝐴))
31		breq2 4047	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑎 → (𝐴 < 𝑛 ↔ 𝐴 < 𝑎))
32	30, 31	orbi12d 794	. . . . . . . . . 10 ⊢ (𝑛 = 𝑎 → ((𝑛 ≤ 𝐴 ∨ 𝐴 < 𝑛) ↔ (𝑎 ≤ 𝐴 ∨ 𝐴 < 𝑎)))
33	32	cbvralv 2737	. . . . . . . . 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 2593	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴 ∧ 𝐴 < 𝑗)) ∧ 𝑎 ∈ ℤ) → (𝑎 ≤ 𝐴 ∨ 𝐴 < 𝑎))
37	27, 8, 36	exbtwnzlemshrink 10389	. . . . 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 2630	. 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 709   ∈ wcel 2175  ∀wral 2483  ∃wrex 2484   class class class wbr 4043  (class class class)co 5943  ℝcr 7923  1c1 7925   + caddc 7927   < clt 8106   ≤ cle 8107   − cmin 8242  ℕcn 9035  ℤcz 9371

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584  ax-cnex 8015  ax-resscn 8016  ax-1cn 8017  ax-1re 8018  ax-icn 8019  ax-addcl 8020  ax-addrcl 8021  ax-mulcl 8022  ax-addcom 8024  ax-addass 8026  ax-distr 8028  ax-i2m1 8029  ax-0lt1 8030  ax-0id 8032  ax-rnegex 8033  ax-cnre 8035  ax-pre-ltirr 8036  ax-pre-ltwlin 8037  ax-pre-lttrn 8038  ax-pre-ltadd 8040  ax-arch 8043

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-nel 2471  df-ral 2488  df-rex 2489  df-reu 2490  df-rab 2492  df-v 2773  df-sbc 2998  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-br 4044  df-opab 4105  df-id 4339  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-iota 5231  df-fun 5272  df-fv 5278  df-riota 5898  df-ov 5946  df-oprab 5947  df-mpo 5948  df-pnf 8108  df-mnf 8109  df-xr 8110  df-ltxr 8111  df-le 8112  df-sub 8244  df-neg 8245  df-inn 9036  df-n0 9295  df-z 9372

This theorem is referenced by:  qbtwnz  10392  apbtwnz  10415