Theorem flqeqceilz 10527

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 10491	. . 3 ⊢ (𝐴 ∈ ℤ → (⌊‘𝐴) = 𝐴)
2		ceilid 10524	. . 3 ⊢ (𝐴 ∈ ℤ → (⌈‘𝐴) = 𝐴)
3	1, 2	eqtr4d 2265	. 2 ⊢ (𝐴 ∈ ℤ → (⌊‘𝐴) = (⌈‘𝐴))
4		flqcl 10480	. . . . . 6 ⊢ (𝐴 ∈ ℚ → (⌊‘𝐴) ∈ ℤ)
5		zq 9809	. . . . . 6 ⊢ ((⌊‘𝐴) ∈ ℤ → (⌊‘𝐴) ∈ ℚ)
6	4, 5	syl 14	. . . . 5 ⊢ (𝐴 ∈ ℚ → (⌊‘𝐴) ∈ ℚ)
7		qdceq 10451	. . . . 5 ⊢ (((⌊‘𝐴) ∈ ℚ ∧ 𝐴 ∈ ℚ) → DECID (⌊‘𝐴) = 𝐴)
8	6, 7	mpancom 422	. . . 4 ⊢ (𝐴 ∈ ℚ → DECID (⌊‘𝐴) = 𝐴)
9		exmiddc 841	. . . 4 ⊢ (DECID (⌊‘𝐴) = 𝐴 → ((⌊‘𝐴) = 𝐴 ∨ ¬ (⌊‘𝐴) = 𝐴))
10	8, 9	syl 14	. . 3 ⊢ (𝐴 ∈ ℚ → ((⌊‘𝐴) = 𝐴 ∨ ¬ (⌊‘𝐴) = 𝐴))
11		eqeq1 2236	. . . . . . 7 ⊢ ((⌊‘𝐴) = 𝐴 → ((⌊‘𝐴) = (⌈‘𝐴) ↔ 𝐴 = (⌈‘𝐴)))
12	11	adantr 276	. . . . . 6 ⊢ (((⌊‘𝐴) = 𝐴 ∧ 𝐴 ∈ ℚ) → ((⌊‘𝐴) = (⌈‘𝐴) ↔ 𝐴 = (⌈‘𝐴)))
13		ceilqidz 10525	. . . . . . . . 9 ⊢ (𝐴 ∈ ℚ → (𝐴 ∈ ℤ ↔ (⌈‘𝐴) = 𝐴))
14		eqcom 2231	. . . . . . . . 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 10485	. . . . 5 ⊢ (𝐴 ∈ ℚ → (⌊‘𝐴) ≤ 𝐴)
21		df-ne 2401	. . . . . 6 ⊢ ((⌊‘𝐴) ≠ 𝐴 ↔ ¬ (⌊‘𝐴) = 𝐴)
22		necom 2484	. . . . . . 7 ⊢ ((⌊‘𝐴) ≠ 𝐴 ↔ 𝐴 ≠ (⌊‘𝐴))
23		qltlen 9823	. . . . . . . . . . 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 4085	. . . . . . . . . . . . . 14 ⊢ ((⌊‘𝐴) = (⌈‘𝐴) → ((⌊‘𝐴) < 𝐴 ↔ (⌈‘𝐴) < 𝐴))
26	25	adantl 277	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℚ ∧ (⌊‘𝐴) = (⌈‘𝐴)) → ((⌊‘𝐴) < 𝐴 ↔ (⌈‘𝐴) < 𝐴))
27		ceilqge 10519	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ ℚ → 𝐴 ≤ (⌈‘𝐴))
28		qre 9808	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ ℚ → 𝐴 ∈ ℝ)
29		ceilqcl 10517	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ ℚ → (⌈‘𝐴) ∈ ℤ)
30	29	zred 9557	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ ℚ → (⌈‘𝐴) ∈ ℝ)
31	28, 30	lenltd 8252	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ ℚ → (𝐴 ≤ (⌈‘𝐴) ↔ ¬ (⌈‘𝐴) < 𝐴))
32		pm2.21 620	. . . . . . . . . . . . . . . 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 721	. . 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 713  DECID wdc 839   = wceq 1395   ∈ wcel 2200   ≠ wne 2400   class class class wbr 4082  ‘cfv 5314   < clt 8169   ≤ cle 8170  ℤcz 9434  ℚcq 9802  ⌊cfl 10475  ⌈cceil 10476

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 4521  ax-setind 4626  ax-cnex 8078  ax-resscn 8079  ax-1cn 8080  ax-1re 8081  ax-icn 8082  ax-addcl 8083  ax-addrcl 8084  ax-mulcl 8085  ax-mulrcl 8086  ax-addcom 8087  ax-mulcom 8088  ax-addass 8089  ax-mulass 8090  ax-distr 8091  ax-i2m1 8092  ax-0lt1 8093  ax-1rid 8094  ax-0id 8095  ax-rnegex 8096  ax-precex 8097  ax-cnre 8098  ax-pre-ltirr 8099  ax-pre-ltwlin 8100  ax-pre-lttrn 8101  ax-pre-apti 8102  ax-pre-ltadd 8103  ax-pre-mulgt0 8104  ax-pre-mulext 8105  ax-arch 8106

This theorem depends on definitions:  df-bi 117  df-dc 840  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-csb 3125  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-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4381  df-po 4384  df-iso 4385  df-xp 4722  df-rel 4723  df-cnv 4724  df-co 4725  df-dm 4726  df-rn 4727  df-res 4728  df-ima 4729  df-iota 5274  df-fun 5316  df-fn 5317  df-f 5318  df-fv 5322  df-riota 5947  df-ov 5997  df-oprab 5998  df-mpo 5999  df-1st 6276  df-2nd 6277  df-pnf 8171  df-mnf 8172  df-xr 8173  df-ltxr 8174  df-le 8175  df-sub 8307  df-neg 8308  df-reap 8710  df-ap 8717  df-div 8808  df-inn 9099  df-n0 9358  df-z 9435  df-q 9803  df-rp 9838  df-fl 10477  df-ceil 10478

This theorem is referenced by: (None)