Theorem flqeqceilz 10581

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 10545	. . 3 ⊢ (𝐴 ∈ ℤ → (⌊‘𝐴) = 𝐴)
2		ceilid 10578	. . 3 ⊢ (𝐴 ∈ ℤ → (⌈‘𝐴) = 𝐴)
3	1, 2	eqtr4d 2267	. 2 ⊢ (𝐴 ∈ ℤ → (⌊‘𝐴) = (⌈‘𝐴))
4		flqcl 10534	. . . . . 6 ⊢ (𝐴 ∈ ℚ → (⌊‘𝐴) ∈ ℤ)
5		zq 9860	. . . . . 6 ⊢ ((⌊‘𝐴) ∈ ℤ → (⌊‘𝐴) ∈ ℚ)
6	4, 5	syl 14	. . . . 5 ⊢ (𝐴 ∈ ℚ → (⌊‘𝐴) ∈ ℚ)
7		qdceq 10505	. . . . 5 ⊢ (((⌊‘𝐴) ∈ ℚ ∧ 𝐴 ∈ ℚ) → DECID (⌊‘𝐴) = 𝐴)
8	6, 7	mpancom 422	. . . 4 ⊢ (𝐴 ∈ ℚ → DECID (⌊‘𝐴) = 𝐴)
9		exmiddc 843	. . . 4 ⊢ (DECID (⌊‘𝐴) = 𝐴 → ((⌊‘𝐴) = 𝐴 ∨ ¬ (⌊‘𝐴) = 𝐴))
10	8, 9	syl 14	. . 3 ⊢ (𝐴 ∈ ℚ → ((⌊‘𝐴) = 𝐴 ∨ ¬ (⌊‘𝐴) = 𝐴))
11		eqeq1 2238	. . . . . . 7 ⊢ ((⌊‘𝐴) = 𝐴 → ((⌊‘𝐴) = (⌈‘𝐴) ↔ 𝐴 = (⌈‘𝐴)))
12	11	adantr 276	. . . . . 6 ⊢ (((⌊‘𝐴) = 𝐴 ∧ 𝐴 ∈ ℚ) → ((⌊‘𝐴) = (⌈‘𝐴) ↔ 𝐴 = (⌈‘𝐴)))
13		ceilqidz 10579	. . . . . . . . 9 ⊢ (𝐴 ∈ ℚ → (𝐴 ∈ ℤ ↔ (⌈‘𝐴) = 𝐴))
14		eqcom 2233	. . . . . . . . 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 10539	. . . . 5 ⊢ (𝐴 ∈ ℚ → (⌊‘𝐴) ≤ 𝐴)
21		df-ne 2403	. . . . . 6 ⊢ ((⌊‘𝐴) ≠ 𝐴 ↔ ¬ (⌊‘𝐴) = 𝐴)
22		necom 2486	. . . . . . 7 ⊢ ((⌊‘𝐴) ≠ 𝐴 ↔ 𝐴 ≠ (⌊‘𝐴))
23		qltlen 9874	. . . . . . . . . . 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 4091	. . . . . . . . . . . . . 14 ⊢ ((⌊‘𝐴) = (⌈‘𝐴) → ((⌊‘𝐴) < 𝐴 ↔ (⌈‘𝐴) < 𝐴))
26	25	adantl 277	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℚ ∧ (⌊‘𝐴) = (⌈‘𝐴)) → ((⌊‘𝐴) < 𝐴 ↔ (⌈‘𝐴) < 𝐴))
27		ceilqge 10573	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ ℚ → 𝐴 ≤ (⌈‘𝐴))
28		qre 9859	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ ℚ → 𝐴 ∈ ℝ)
29		ceilqcl 10571	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ ℚ → (⌈‘𝐴) ∈ ℤ)
30	29	zred 9602	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ ℚ → (⌈‘𝐴) ∈ ℝ)
31	28, 30	lenltd 8297	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ ℚ → (𝐴 ≤ (⌈‘𝐴) ↔ ¬ (⌈‘𝐴) < 𝐴))
32		pm2.21 622	. . . . . . . . . . . . . . . 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 723	. . 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 715  DECID wdc 841   = wceq 1397   ∈ wcel 2202   ≠ wne 2402   class class class wbr 4088  ‘cfv 5326   < clt 8214   ≤ cle 8215  ℤcz 9479  ℚcq 9853  ⌊cfl 10529  ⌈cceil 10530

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-mulrcl 8131  ax-addcom 8132  ax-mulcom 8133  ax-addass 8134  ax-mulass 8135  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-1rid 8139  ax-0id 8140  ax-rnegex 8141  ax-precex 8142  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-apti 8147  ax-pre-ltadd 8148  ax-pre-mulgt0 8149  ax-pre-mulext 8150  ax-arch 8151

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-po 4393  df-iso 4394  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-reap 8755  df-ap 8762  df-div 8853  df-inn 9144  df-n0 9403  df-z 9480  df-q 9854  df-rp 9889  df-fl 10531  df-ceil 10532

This theorem is referenced by: (None)