Theorem exbtwnz 10634

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 537	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ ((𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)) ∧ (𝑦 ≤ 𝐴 ∧ 𝐴 < (𝑦 + 1)))) → 𝑥 ∈ ℤ)
3	2	zred 9718	. . . . . . . 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 538	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ ((𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)) ∧ (𝑦 ≤ 𝐴 ∧ 𝐴 < (𝑦 + 1)))) → 𝑦 ∈ ℤ)
7	6	zred 9718	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ ((𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)) ∧ (𝑦 ≤ 𝐴 ∧ 𝐴 < (𝑦 + 1)))) → 𝑦 ∈ ℝ)
8		1red 8305	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ ((𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)) ∧ (𝑦 ≤ 𝐴 ∧ 𝐴 < (𝑦 + 1)))) → 1 ∈ ℝ)
9	7, 8	readdcld 8319	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ ((𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)) ∧ (𝑦 ≤ 𝐴 ∧ 𝐴 < (𝑦 + 1)))) → (𝑦 + 1) ∈ ℝ)
10		simprll 539	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ ((𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)) ∧ (𝑦 ≤ 𝐴 ∧ 𝐴 < (𝑦 + 1)))) → 𝑥 ≤ 𝐴)
11		simprrr 542	. . . . . . . 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 8414	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ ((𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)) ∧ (𝑦 ≤ 𝐴 ∧ 𝐴 < (𝑦 + 1)))) → 𝑥 < (𝑦 + 1))
13		zleltp1 9650	. . . . . . . 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 8319	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ ((𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)) ∧ (𝑦 ≤ 𝐴 ∧ 𝐴 < (𝑦 + 1)))) → (𝑥 + 1) ∈ ℝ)
17		simprrl 541	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ ((𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)) ∧ (𝑦 ≤ 𝐴 ∧ 𝐴 < (𝑦 + 1)))) → 𝑦 ≤ 𝐴)
18		simprlr 540	. . . . . . . 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 8414	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ ((𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)) ∧ (𝑦 ≤ 𝐴 ∧ 𝐴 < (𝑦 + 1)))) → 𝑦 < (𝑥 + 1))
20		zleltp1 9650	. . . . . . . 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 8405	. . . . . 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 953	. . . . 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 2626	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ ℤ ∀𝑦 ∈ ℤ (((𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)) ∧ (𝑦 ≤ 𝐴 ∧ 𝐴 < (𝑦 + 1))) → 𝑥 = 𝑦))
27		breq1 4117	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 ≤ 𝐴 ↔ 𝑦 ≤ 𝐴))
28		oveq1 6065	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 + 1) = (𝑦 + 1))
29	28	breq2d 4126	. . . . 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 3013	. . 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 2764	. 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 2205  ∀wral 2522  ∃wrex 2523  ∃!wreu 2524  ∃*wrmo 2525   class class class wbr 4114  (class class class)co 6058  ℝcr 8142  1c1 8144   + caddc 8146   < clt 8324   ≤ cle 8325  ℤcz 9594

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-br 4115  df-opab 4177  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-iota 5317  df-fun 5359  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-inn 9255  df-n0 9514  df-z 9595

This theorem is referenced by:  qbtwnz  10635  apbtwnz  10658