Theorem exbtwnz 10465

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 535	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ ((𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)) ∧ (𝑦 ≤ 𝐴 ∧ 𝐴 < (𝑦 + 1)))) → 𝑥 ∈ ℤ)
3	2	zred 9565	. . . . . . . 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 488	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ ((𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)) ∧ (𝑦 ≤ 𝐴 ∧ 𝐴 < (𝑦 + 1)))) → 𝐴 ∈ ℝ)
6		simplrr 536	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ ((𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)) ∧ (𝑦 ≤ 𝐴 ∧ 𝐴 < (𝑦 + 1)))) → 𝑦 ∈ ℤ)
7	6	zred 9565	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ ((𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)) ∧ (𝑦 ≤ 𝐴 ∧ 𝐴 < (𝑦 + 1)))) → 𝑦 ∈ ℝ)
8		1red 8157	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ ((𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)) ∧ (𝑦 ≤ 𝐴 ∧ 𝐴 < (𝑦 + 1)))) → 1 ∈ ℝ)
9	7, 8	readdcld 8172	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ ((𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)) ∧ (𝑦 ≤ 𝐴 ∧ 𝐴 < (𝑦 + 1)))) → (𝑦 + 1) ∈ ℝ)
10		simprll 537	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ ((𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)) ∧ (𝑦 ≤ 𝐴 ∧ 𝐴 < (𝑦 + 1)))) → 𝑥 ≤ 𝐴)
11		simprrr 540	. . . . . . . 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 8267	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ ((𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)) ∧ (𝑦 ≤ 𝐴 ∧ 𝐴 < (𝑦 + 1)))) → 𝑥 < (𝑦 + 1))
13		zleltp1 9498	. . . . . . . 8 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (𝑥 ≤ 𝑦 ↔ 𝑥 < (𝑦 + 1)))
14	2, 6, 13	syl2anc 411	. . . . . . 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 8172	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ ((𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)) ∧ (𝑦 ≤ 𝐴 ∧ 𝐴 < (𝑦 + 1)))) → (𝑥 + 1) ∈ ℝ)
17		simprrl 539	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ ((𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)) ∧ (𝑦 ≤ 𝐴 ∧ 𝐴 < (𝑦 + 1)))) → 𝑦 ≤ 𝐴)
18		simprlr 538	. . . . . . . 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 8267	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ ((𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)) ∧ (𝑦 ≤ 𝐴 ∧ 𝐴 < (𝑦 + 1)))) → 𝑦 < (𝑥 + 1))
20		zleltp1 9498	. . . . . . . 8 ⊢ ((𝑦 ∈ ℤ ∧ 𝑥 ∈ ℤ) → (𝑦 ≤ 𝑥 ↔ 𝑦 < (𝑥 + 1)))
21	6, 2, 20	syl2anc 411	. . . . . . 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 8258	. . . . . 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 950	. . . . 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 2612	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ ℤ ∀𝑦 ∈ ℤ (((𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)) ∧ (𝑦 ≤ 𝐴 ∧ 𝐴 < (𝑦 + 1))) → 𝑥 = 𝑦))
27		breq1 4085	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 ≤ 𝐴 ↔ 𝑦 ≤ 𝐴))
28		oveq1 6007	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 + 1) = (𝑦 + 1))
29	28	breq2d 4094	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐴 < (𝑥 + 1) ↔ 𝐴 < (𝑦 + 1)))
30	27, 29	anbi12d 473	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)) ↔ (𝑦 ≤ 𝐴 ∧ 𝐴 < (𝑦 + 1))))
31	30	rmo4 2996	. . 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 2749	. 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 417	1 ⊢ (𝜑 → ∃!𝑥 ∈ ℤ (𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∈ wcel 2200  ∀wral 2508  ∃wrex 2509  ∃!wreu 2510  ∃*wrmo 2511   class class class wbr 4082  (class class class)co 6000  ℝcr 7994  1c1 7996   + caddc 7998   < clt 8177   ≤ cle 8178  ℤcz 9442

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-cnex 8086  ax-resscn 8087  ax-1cn 8088  ax-1re 8089  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-addcom 8095  ax-addass 8097  ax-distr 8099  ax-i2m1 8100  ax-0lt1 8101  ax-0id 8103  ax-rnegex 8104  ax-cnre 8106  ax-pre-ltirr 8107  ax-pre-ltwlin 8108  ax-pre-lttrn 8109  ax-pre-apti 8110  ax-pre-ltadd 8111

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-br 4083  df-opab 4145  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-iota 5277  df-fun 5319  df-fv 5325  df-riota 5953  df-ov 6003  df-oprab 6004  df-mpo 6005  df-pnf 8179  df-mnf 8180  df-xr 8181  df-ltxr 8182  df-le 8183  df-sub 8315  df-neg 8316  df-inn 9107  df-n0 9366  df-z 9443

This theorem is referenced by:  qbtwnz  10466  apbtwnz  10489