Theorem exbtwnz 10200

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 530	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ ((𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)) ∧ (𝑦 ≤ 𝐴 ∧ 𝐴 < (𝑦 + 1)))) → 𝑥 ∈ ℤ)
3	2	zred 9327	. . . . . . . 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 485	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ ((𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)) ∧ (𝑦 ≤ 𝐴 ∧ 𝐴 < (𝑦 + 1)))) → 𝐴 ∈ ℝ)
6		simplrr 531	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ ((𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)) ∧ (𝑦 ≤ 𝐴 ∧ 𝐴 < (𝑦 + 1)))) → 𝑦 ∈ ℤ)
7	6	zred 9327	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ ((𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)) ∧ (𝑦 ≤ 𝐴 ∧ 𝐴 < (𝑦 + 1)))) → 𝑦 ∈ ℝ)
8		1red 7928	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ ((𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)) ∧ (𝑦 ≤ 𝐴 ∧ 𝐴 < (𝑦 + 1)))) → 1 ∈ ℝ)
9	7, 8	readdcld 7942	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ ((𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)) ∧ (𝑦 ≤ 𝐴 ∧ 𝐴 < (𝑦 + 1)))) → (𝑦 + 1) ∈ ℝ)
10		simprll 532	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ ((𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)) ∧ (𝑦 ≤ 𝐴 ∧ 𝐴 < (𝑦 + 1)))) → 𝑥 ≤ 𝐴)
11		simprrr 535	. . . . . . . 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 8037	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ ((𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)) ∧ (𝑦 ≤ 𝐴 ∧ 𝐴 < (𝑦 + 1)))) → 𝑥 < (𝑦 + 1))
13		zleltp1 9260	. . . . . . . 8 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (𝑥 ≤ 𝑦 ↔ 𝑥 < (𝑦 + 1)))
14	2, 6, 13	syl2anc 409	. . . . . . 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 166	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ ((𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)) ∧ (𝑦 ≤ 𝐴 ∧ 𝐴 < (𝑦 + 1)))) → 𝑥 ≤ 𝑦)
16	3, 8	readdcld 7942	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ ((𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)) ∧ (𝑦 ≤ 𝐴 ∧ 𝐴 < (𝑦 + 1)))) → (𝑥 + 1) ∈ ℝ)
17		simprrl 534	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ ((𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)) ∧ (𝑦 ≤ 𝐴 ∧ 𝐴 < (𝑦 + 1)))) → 𝑦 ≤ 𝐴)
18		simprlr 533	. . . . . . . 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 8037	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ ((𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)) ∧ (𝑦 ≤ 𝐴 ∧ 𝐴 < (𝑦 + 1)))) → 𝑦 < (𝑥 + 1))
20		zleltp1 9260	. . . . . . . 8 ⊢ ((𝑦 ∈ ℤ ∧ 𝑥 ∈ ℤ) → (𝑦 ≤ 𝑥 ↔ 𝑦 < (𝑥 + 1)))
21	6, 2, 20	syl2anc 409	. . . . . . 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 166	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ ((𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)) ∧ (𝑦 ≤ 𝐴 ∧ 𝐴 < (𝑦 + 1)))) → 𝑦 ≤ 𝑥)
23	3, 7	letri3d 8028	. . . . . 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 939	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ ((𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)) ∧ (𝑦 ≤ 𝐴 ∧ 𝐴 < (𝑦 + 1)))) → 𝑥 = 𝑦)
25	24	ex 114	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → (((𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)) ∧ (𝑦 ≤ 𝐴 ∧ 𝐴 < (𝑦 + 1))) → 𝑥 = 𝑦))
26	25	ralrimivva 2552	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ ℤ ∀𝑦 ∈ ℤ (((𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)) ∧ (𝑦 ≤ 𝐴 ∧ 𝐴 < (𝑦 + 1))) → 𝑥 = 𝑦))
27		breq1 3990	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 ≤ 𝐴 ↔ 𝑦 ≤ 𝐴))
28		oveq1 5858	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 + 1) = (𝑦 + 1))
29	28	breq2d 3999	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐴 < (𝑥 + 1) ↔ 𝐴 < (𝑦 + 1)))
30	27, 29	anbi12d 470	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)) ↔ (𝑦 ≤ 𝐴 ∧ 𝐴 < (𝑦 + 1))))
31	30	rmo4 2923	. . 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 133	. 2 ⊢ (𝜑 → ∃*𝑥 ∈ ℤ (𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)))
33		reu5 2682	. 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 415	1 ⊢ (𝜑 → ∃!𝑥 ∈ ℤ (𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∈ wcel 2141  ∀wral 2448  ∃wrex 2449  ∃!wreu 2450  ∃*wrmo 2451   class class class wbr 3987  (class class class)co 5851  ℝcr 7766  1c1 7768   + caddc 7770   < clt 7947   ≤ cle 7948  ℤcz 9205

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-cnex 7858  ax-resscn 7859  ax-1cn 7860  ax-1re 7861  ax-icn 7862  ax-addcl 7863  ax-addrcl 7864  ax-mulcl 7865  ax-addcom 7867  ax-addass 7869  ax-distr 7871  ax-i2m1 7872  ax-0lt1 7873  ax-0id 7875  ax-rnegex 7876  ax-cnre 7878  ax-pre-ltirr 7879  ax-pre-ltwlin 7880  ax-pre-lttrn 7881  ax-pre-apti 7882  ax-pre-ltadd 7883

This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-br 3988  df-opab 4049  df-id 4276  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-iota 5158  df-fun 5198  df-fv 5204  df-riota 5807  df-ov 5854  df-oprab 5855  df-mpo 5856  df-pnf 7949  df-mnf 7950  df-xr 7951  df-ltxr 7952  df-le 7953  df-sub 8085  df-neg 8086  df-inn 8872  df-n0 9129  df-z 9206

This theorem is referenced by:  qbtwnz  10201  apbtwnz  10223