Theorem exbtwnz 10685

Description: If a real number is between an integer and its successor, there is a unique greatest integer less than or equal to the real number. (Contributed by Jim Kingdon, 10-May-2022.)

Hypotheses

Ref	Expression
exbtwnz.ex	⊢ (𝜑 → ∃𝑥 ∈ ℤ (𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)))
exbtwnz.a	⊢ (𝜑 → 𝐴 ∈ ℝ)

Assertion

Ref	Expression
exbtwnz	⊢ (𝜑 → ∃!𝑥 ∈ ℤ (𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)))

Distinct variable groups:   𝑥,𝐴   𝜑,𝑥

Proof of Theorem exbtwnz

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		exbtwnz.ex	. 2 ⊢ (𝜑 → ∃𝑥 ∈ ℤ (𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)))
2		simplrl 541	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ ((𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)) ∧ (𝑦 ≤ 𝐴 ∧ 𝐴 < (𝑦 + 1)))) → 𝑥 ∈ ℤ)
3	2	zred 9768	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ ((𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)) ∧ (𝑦 ≤ 𝐴 ∧ 𝐴 < (𝑦 + 1)))) → 𝑥 ∈ ℝ)
4		exbtwnz.a	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℝ)
5	4	ad2antrr 492	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ ((𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)) ∧ (𝑦 ≤ 𝐴 ∧ 𝐴 < (𝑦 + 1)))) → 𝐴 ∈ ℝ)
6		simplrr 542	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ ((𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)) ∧ (𝑦 ≤ 𝐴 ∧ 𝐴 < (𝑦 + 1)))) → 𝑦 ∈ ℤ)
7	6	zred 9768	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ ((𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)) ∧ (𝑦 ≤ 𝐴 ∧ 𝐴 < (𝑦 + 1)))) → 𝑦 ∈ ℝ)
8		1red 8341	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ ((𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)) ∧ (𝑦 ≤ 𝐴 ∧ 𝐴 < (𝑦 + 1)))) → 1 ∈ ℝ)
9	7, 8	readdcld 8355	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ ((𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)) ∧ (𝑦 ≤ 𝐴 ∧ 𝐴 < (𝑦 + 1)))) → (𝑦 + 1) ∈ ℝ)
10		simprll 543	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ ((𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)) ∧ (𝑦 ≤ 𝐴 ∧ 𝐴 < (𝑦 + 1)))) → 𝑥 ≤ 𝐴)
11		simprrr 546	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ ((𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)) ∧ (𝑦 ≤ 𝐴 ∧ 𝐴 < (𝑦 + 1)))) → 𝐴 < (𝑦 + 1))
12	3, 5, 9, 10, 11	lelttrd 8451	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ ((𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)) ∧ (𝑦 ≤ 𝐴 ∧ 𝐴 < (𝑦 + 1)))) → 𝑥 < (𝑦 + 1))
13		zleltp1 9700	. . . . . . . 8 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (𝑥 ≤ 𝑦 ↔ 𝑥 < (𝑦 + 1)))
14	2, 6, 13	syl2anc 415	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ ((𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)) ∧ (𝑦 ≤ 𝐴 ∧ 𝐴 < (𝑦 + 1)))) → (𝑥 ≤ 𝑦 ↔ 𝑥 < (𝑦 + 1)))
15	12, 14	mpbird 167	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ ((𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)) ∧ (𝑦 ≤ 𝐴 ∧ 𝐴 < (𝑦 + 1)))) → 𝑥 ≤ 𝑦)
16	3, 8	readdcld 8355	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ ((𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)) ∧ (𝑦 ≤ 𝐴 ∧ 𝐴 < (𝑦 + 1)))) → (𝑥 + 1) ∈ ℝ)
17		simprrl 545	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ ((𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)) ∧ (𝑦 ≤ 𝐴 ∧ 𝐴 < (𝑦 + 1)))) → 𝑦 ≤ 𝐴)
18		simprlr 544	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ ((𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)) ∧ (𝑦 ≤ 𝐴 ∧ 𝐴 < (𝑦 + 1)))) → 𝐴 < (𝑥 + 1))
19	7, 5, 16, 17, 18	lelttrd 8451	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ ((𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)) ∧ (𝑦 ≤ 𝐴 ∧ 𝐴 < (𝑦 + 1)))) → 𝑦 < (𝑥 + 1))
20		zleltp1 9700	. . . . . . . 8 ⊢ ((𝑦 ∈ ℤ ∧ 𝑥 ∈ ℤ) → (𝑦 ≤ 𝑥 ↔ 𝑦 < (𝑥 + 1)))
21	6, 2, 20	syl2anc 415	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ ((𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)) ∧ (𝑦 ≤ 𝐴 ∧ 𝐴 < (𝑦 + 1)))) → (𝑦 ≤ 𝑥 ↔ 𝑦 < (𝑥 + 1)))
22	19, 21	mpbird 167	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ ((𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)) ∧ (𝑦 ≤ 𝐴 ∧ 𝐴 < (𝑦 + 1)))) → 𝑦 ≤ 𝑥)
23	3, 7	letri3d 8441	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ ((𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)) ∧ (𝑦 ≤ 𝐴 ∧ 𝐴 < (𝑦 + 1)))) → (𝑥 = 𝑦 ↔ (𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥)))
24	15, 22, 23	mpbir2and 957	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ ((𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)) ∧ (𝑦 ≤ 𝐴 ∧ 𝐴 < (𝑦 + 1)))) → 𝑥 = 𝑦)
25	24	ex 115	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → (((𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)) ∧ (𝑦 ≤ 𝐴 ∧ 𝐴 < (𝑦 + 1))) → 𝑥 = 𝑦))
26	25	ralrimivva 2632	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ ℤ ∀𝑦 ∈ ℤ (((𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)) ∧ (𝑦 ≤ 𝐴 ∧ 𝐴 < (𝑦 + 1))) → 𝑥 = 𝑦))
27		breq1 4133	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 ≤ 𝐴 ↔ 𝑦 ≤ 𝐴))
28		oveq1 6092	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 + 1) = (𝑦 + 1))
29	28	breq2d 4142	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐴 < (𝑥 + 1) ↔ 𝐴 < (𝑦 + 1)))
30	27, 29	anbi12d 477	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)) ↔ (𝑦 ≤ 𝐴 ∧ 𝐴 < (𝑦 + 1))))
31	30	rmo4 3019	. . 3 ⊢ (∃*𝑥 ∈ ℤ (𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)) ↔ ∀𝑥 ∈ ℤ ∀𝑦 ∈ ℤ (((𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)) ∧ (𝑦 ≤ 𝐴 ∧ 𝐴 < (𝑦 + 1))) → 𝑥 = 𝑦))
32	26, 31	sylibr 134	. 2 ⊢ (𝜑 → ∃*𝑥 ∈ ℤ (𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)))
33		reu5 2770	. 2 ⊢ (∃!𝑥 ∈ ℤ (𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)) ↔ (∃𝑥 ∈ ℤ (𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)) ∧ ∃*𝑥 ∈ ℤ (𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1))))
34	1, 32, 33	sylanbrc 421	1 ⊢ (𝜑 → ∃!𝑥 ∈ ℤ (𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ↔ wb 105   ∈ wcel 2209  ∀wral 2528  ∃wrex 2529  ∃!wreu 2530  ∃*wrmo 2531   class class class wbr 4130  (class class class)co 6085  ℝcr 8178  1c1 8180   + caddc 8182   < clt 8360   ≤ cle 8361  ℤcz 9644

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-apti 8294  ax-pre-ltadd 8295

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-rmo 2536  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-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-iota 5337  df-fun 5379  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-n0 9564  df-z 9645

This theorem is used by:  qbtwnz  10686  apbtwnz  10709