Theorem flqeqceilz 10331

Description: A rational number is an integer iff its floor equals its ceiling. (Contributed by Jim Kingdon, 11-Oct-2021.)

Assertion

Ref	Expression
flqeqceilz	⊢ (𝐴 ∈ ℚ → (𝐴 ∈ ℤ ↔ (⌊‘𝐴) = (⌈‘𝐴)))

Proof of Theorem flqeqceilz

Step	Hyp	Ref	Expression
1		flid 10297	. . 3 ⊢ (𝐴 ∈ ℤ → (⌊‘𝐴) = 𝐴)
2		ceilid 10328	. . 3 ⊢ (𝐴 ∈ ℤ → (⌈‘𝐴) = 𝐴)
3	1, 2	eqtr4d 2223	. 2 ⊢ (𝐴 ∈ ℤ → (⌊‘𝐴) = (⌈‘𝐴))
4		flqcl 10286	. . . . . 6 ⊢ (𝐴 ∈ ℚ → (⌊‘𝐴) ∈ ℤ)
5		zq 9639	. . . . . 6 ⊢ ((⌊‘𝐴) ∈ ℤ → (⌊‘𝐴) ∈ ℚ)
6	4, 5	syl 14	. . . . 5 ⊢ (𝐴 ∈ ℚ → (⌊‘𝐴) ∈ ℚ)
7		qdceq 10260	. . . . 5 ⊢ (((⌊‘𝐴) ∈ ℚ ∧ 𝐴 ∈ ℚ) → DECID (⌊‘𝐴) = 𝐴)
8	6, 7	mpancom 422	. . . 4 ⊢ (𝐴 ∈ ℚ → DECID (⌊‘𝐴) = 𝐴)
9		exmiddc 837	. . . 4 ⊢ (DECID (⌊‘𝐴) = 𝐴 → ((⌊‘𝐴) = 𝐴 ∨ ¬ (⌊‘𝐴) = 𝐴))
10	8, 9	syl 14	. . 3 ⊢ (𝐴 ∈ ℚ → ((⌊‘𝐴) = 𝐴 ∨ ¬ (⌊‘𝐴) = 𝐴))
11		eqeq1 2194	. . . . . . 7 ⊢ ((⌊‘𝐴) = 𝐴 → ((⌊‘𝐴) = (⌈‘𝐴) ↔ 𝐴 = (⌈‘𝐴)))
12	11	adantr 276	. . . . . 6 ⊢ (((⌊‘𝐴) = 𝐴 ∧ 𝐴 ∈ ℚ) → ((⌊‘𝐴) = (⌈‘𝐴) ↔ 𝐴 = (⌈‘𝐴)))
13		ceilqidz 10329	. . . . . . . . 9 ⊢ (𝐴 ∈ ℚ → (𝐴 ∈ ℤ ↔ (⌈‘𝐴) = 𝐴))
14		eqcom 2189	. . . . . . . . 9 ⊢ ((⌈‘𝐴) = 𝐴 ↔ 𝐴 = (⌈‘𝐴))
15	13, 14	bitrdi 196	. . . . . . . 8 ⊢ (𝐴 ∈ ℚ → (𝐴 ∈ ℤ ↔ 𝐴 = (⌈‘𝐴)))
16	15	biimprd 158	. . . . . . 7 ⊢ (𝐴 ∈ ℚ → (𝐴 = (⌈‘𝐴) → 𝐴 ∈ ℤ))
17	16	adantl 277	. . . . . 6 ⊢ (((⌊‘𝐴) = 𝐴 ∧ 𝐴 ∈ ℚ) → (𝐴 = (⌈‘𝐴) → 𝐴 ∈ ℤ))
18	12, 17	sylbid 150	. . . . 5 ⊢ (((⌊‘𝐴) = 𝐴 ∧ 𝐴 ∈ ℚ) → ((⌊‘𝐴) = (⌈‘𝐴) → 𝐴 ∈ ℤ))
19	18	ex 115	. . . 4 ⊢ ((⌊‘𝐴) = 𝐴 → (𝐴 ∈ ℚ → ((⌊‘𝐴) = (⌈‘𝐴) → 𝐴 ∈ ℤ)))
20		flqle 10291	. . . . 5 ⊢ (𝐴 ∈ ℚ → (⌊‘𝐴) ≤ 𝐴)
21		df-ne 2358	. . . . . 6 ⊢ ((⌊‘𝐴) ≠ 𝐴 ↔ ¬ (⌊‘𝐴) = 𝐴)
22		necom 2441	. . . . . . 7 ⊢ ((⌊‘𝐴) ≠ 𝐴 ↔ 𝐴 ≠ (⌊‘𝐴))
23		qltlen 9653	. . . . . . . . . . 11 ⊢ (((⌊‘𝐴) ∈ ℚ ∧ 𝐴 ∈ ℚ) → ((⌊‘𝐴) < 𝐴 ↔ ((⌊‘𝐴) ≤ 𝐴 ∧ 𝐴 ≠ (⌊‘𝐴))))
24	6, 23	mpancom 422	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℚ → ((⌊‘𝐴) < 𝐴 ↔ ((⌊‘𝐴) ≤ 𝐴 ∧ 𝐴 ≠ (⌊‘𝐴))))
25		breq1 4018	. . . . . . . . . . . . . 14 ⊢ ((⌊‘𝐴) = (⌈‘𝐴) → ((⌊‘𝐴) < 𝐴 ↔ (⌈‘𝐴) < 𝐴))
26	25	adantl 277	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℚ ∧ (⌊‘𝐴) = (⌈‘𝐴)) → ((⌊‘𝐴) < 𝐴 ↔ (⌈‘𝐴) < 𝐴))
27		ceilqge 10323	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ ℚ → 𝐴 ≤ (⌈‘𝐴))
28		qre 9638	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ ℚ → 𝐴 ∈ ℝ)
29		ceilqcl 10321	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ ℚ → (⌈‘𝐴) ∈ ℤ)
30	29	zred 9388	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ ℚ → (⌈‘𝐴) ∈ ℝ)
31	28, 30	lenltd 8088	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ ℚ → (𝐴 ≤ (⌈‘𝐴) ↔ ¬ (⌈‘𝐴) < 𝐴))
32		pm2.21 618	. . . . . . . . . . . . . . . 16 ⊢ (¬ (⌈‘𝐴) < 𝐴 → ((⌈‘𝐴) < 𝐴 → 𝐴 ∈ ℤ))
33	31, 32	biimtrdi 163	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ ℚ → (𝐴 ≤ (⌈‘𝐴) → ((⌈‘𝐴) < 𝐴 → 𝐴 ∈ ℤ)))
34	27, 33	mpd 13	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℚ → ((⌈‘𝐴) < 𝐴 → 𝐴 ∈ ℤ))
35	34	adantr 276	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℚ ∧ (⌊‘𝐴) = (⌈‘𝐴)) → ((⌈‘𝐴) < 𝐴 → 𝐴 ∈ ℤ))
36	26, 35	sylbid 150	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℚ ∧ (⌊‘𝐴) = (⌈‘𝐴)) → ((⌊‘𝐴) < 𝐴 → 𝐴 ∈ ℤ))
37	36	ex 115	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℚ → ((⌊‘𝐴) = (⌈‘𝐴) → ((⌊‘𝐴) < 𝐴 → 𝐴 ∈ ℤ)))
38	37	com23 78	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℚ → ((⌊‘𝐴) < 𝐴 → ((⌊‘𝐴) = (⌈‘𝐴) → 𝐴 ∈ ℤ)))
39	24, 38	sylbird 170	. . . . . . . . 9 ⊢ (𝐴 ∈ ℚ → (((⌊‘𝐴) ≤ 𝐴 ∧ 𝐴 ≠ (⌊‘𝐴)) → ((⌊‘𝐴) = (⌈‘𝐴) → 𝐴 ∈ ℤ)))
40	39	expd 258	. . . . . . . 8 ⊢ (𝐴 ∈ ℚ → ((⌊‘𝐴) ≤ 𝐴 → (𝐴 ≠ (⌊‘𝐴) → ((⌊‘𝐴) = (⌈‘𝐴) → 𝐴 ∈ ℤ))))
41	40	com3r 79	. . . . . . 7 ⊢ (𝐴 ≠ (⌊‘𝐴) → (𝐴 ∈ ℚ → ((⌊‘𝐴) ≤ 𝐴 → ((⌊‘𝐴) = (⌈‘𝐴) → 𝐴 ∈ ℤ))))
42	22, 41	sylbi 121	. . . . . 6 ⊢ ((⌊‘𝐴) ≠ 𝐴 → (𝐴 ∈ ℚ → ((⌊‘𝐴) ≤ 𝐴 → ((⌊‘𝐴) = (⌈‘𝐴) → 𝐴 ∈ ℤ))))
43	21, 42	sylbir 135	. . . . 5 ⊢ (¬ (⌊‘𝐴) = 𝐴 → (𝐴 ∈ ℚ → ((⌊‘𝐴) ≤ 𝐴 → ((⌊‘𝐴) = (⌈‘𝐴) → 𝐴 ∈ ℤ))))
44	20, 43	mpdi 43	. . . 4 ⊢ (¬ (⌊‘𝐴) = 𝐴 → (𝐴 ∈ ℚ → ((⌊‘𝐴) = (⌈‘𝐴) → 𝐴 ∈ ℤ)))
45	19, 44	jaoi 717	. . 3 ⊢ (((⌊‘𝐴) = 𝐴 ∨ ¬ (⌊‘𝐴) = 𝐴) → (𝐴 ∈ ℚ → ((⌊‘𝐴) = (⌈‘𝐴) → 𝐴 ∈ ℤ)))
46	10, 45	mpcom 36	. 2 ⊢ (𝐴 ∈ ℚ → ((⌊‘𝐴) = (⌈‘𝐴) → 𝐴 ∈ ℤ))
47	3, 46	impbid2 143	1 ⊢ (𝐴 ∈ ℚ → (𝐴 ∈ ℤ ↔ (⌊‘𝐴) = (⌈‘𝐴)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 709  DECID wdc 835   = wceq 1363   ∈ wcel 2158   ≠ wne 2357   class class class wbr 4015  ‘cfv 5228   < clt 8005   ≤ cle 8006  ℤcz 9266  ℚcq 9632  ⌊cfl 10281  ⌈cceil 10282

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-cnex 7915  ax-resscn 7916  ax-1cn 7917  ax-1re 7918  ax-icn 7919  ax-addcl 7920  ax-addrcl 7921  ax-mulcl 7922  ax-mulrcl 7923  ax-addcom 7924  ax-mulcom 7925  ax-addass 7926  ax-mulass 7927  ax-distr 7928  ax-i2m1 7929  ax-0lt1 7930  ax-1rid 7931  ax-0id 7932  ax-rnegex 7933  ax-precex 7934  ax-cnre 7935  ax-pre-ltirr 7936  ax-pre-ltwlin 7937  ax-pre-lttrn 7938  ax-pre-apti 7939  ax-pre-ltadd 7940  ax-pre-mulgt0 7941  ax-pre-mulext 7942  ax-arch 7943

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rmo 2473  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-po 4308  df-iso 4309  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-fv 5236  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6154  df-2nd 6155  df-pnf 8007  df-mnf 8008  df-xr 8009  df-ltxr 8010  df-le 8011  df-sub 8143  df-neg 8144  df-reap 8545  df-ap 8552  df-div 8643  df-inn 8933  df-n0 9190  df-z 9267  df-q 9633  df-rp 9667  df-fl 10283  df-ceil 10284

This theorem is referenced by: (None)