Theorem flqeqceilz 10738

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 10702	. . 3 ⊢ (𝐴 ∈ ℤ → (⌊‘𝐴) = 𝐴)
2		ceilid 10735	. . 3 ⊢ (𝐴 ∈ ℤ → (⌈‘𝐴) = 𝐴)
3	1, 2	eqtr4d 2274	. 2 ⊢ (𝐴 ∈ ℤ → (⌊‘𝐴) = (⌈‘𝐴))
4		flqcl 10691	. . . . . 6 ⊢ (𝐴 ∈ ℚ → (⌊‘𝐴) ∈ ℤ)
5		zq 10009	. . . . . 6 ⊢ ((⌊‘𝐴) ∈ ℤ → (⌊‘𝐴) ∈ ℚ)
6	4, 5	syl 14	. . . . 5 ⊢ (𝐴 ∈ ℚ → (⌊‘𝐴) ∈ ℚ)
7		qdceq 10662	. . . . 5 ⊢ (((⌊‘𝐴) ∈ ℚ ∧ 𝐴 ∈ ℚ) → DECID (⌊‘𝐴) = 𝐴)
8	6, 7	mpancom 426	. . . 4 ⊢ (𝐴 ∈ ℚ → DECID (⌊‘𝐴) = 𝐴)
9		exmiddc 848	. . . 4 ⊢ (DECID (⌊‘𝐴) = 𝐴 → ((⌊‘𝐴) = 𝐴 ∨ ¬ (⌊‘𝐴) = 𝐴))
10	8, 9	syl 14	. . 3 ⊢ (𝐴 ∈ ℚ → ((⌊‘𝐴) = 𝐴 ∨ ¬ (⌊‘𝐴) = 𝐴))
11		eqeq1 2245	. . . . . . 7 ⊢ ((⌊‘𝐴) = 𝐴 → ((⌊‘𝐴) = (⌈‘𝐴) ↔ 𝐴 = (⌈‘𝐴)))
12	11	adantr 276	. . . . . 6 ⊢ (((⌊‘𝐴) = 𝐴 ∧ 𝐴 ∈ ℚ) → ((⌊‘𝐴) = (⌈‘𝐴) ↔ 𝐴 = (⌈‘𝐴)))
13		ceilqidz 10736	. . . . . . . . 9 ⊢ (𝐴 ∈ ℚ → (𝐴 ∈ ℤ ↔ (⌈‘𝐴) = 𝐴))
14		eqcom 2240	. . . . . . . . 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 10696	. . . . 5 ⊢ (𝐴 ∈ ℚ → (⌊‘𝐴) ≤ 𝐴)
21		df-ne 2421	. . . . . 6 ⊢ ((⌊‘𝐴) ≠ 𝐴 ↔ ¬ (⌊‘𝐴) = 𝐴)
22		necom 2504	. . . . . . 7 ⊢ ((⌊‘𝐴) ≠ 𝐴 ↔ 𝐴 ≠ (⌊‘𝐴))
23		qltlen 10023	. . . . . . . . . . 11 ⊢ (((⌊‘𝐴) ∈ ℚ ∧ 𝐴 ∈ ℚ) → ((⌊‘𝐴) < 𝐴 ↔ ((⌊‘𝐴) ≤ 𝐴 ∧ 𝐴 ≠ (⌊‘𝐴))))
24	6, 23	mpancom 426	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℚ → ((⌊‘𝐴) < 𝐴 ↔ ((⌊‘𝐴) ≤ 𝐴 ∧ 𝐴 ≠ (⌊‘𝐴))))
25		breq1 4131	. . . . . . . . . . . . . 14 ⊢ ((⌊‘𝐴) = (⌈‘𝐴) → ((⌊‘𝐴) < 𝐴 ↔ (⌈‘𝐴) < 𝐴))
26	25	adantl 277	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℚ ∧ (⌊‘𝐴) = (⌈‘𝐴)) → ((⌊‘𝐴) < 𝐴 ↔ (⌈‘𝐴) < 𝐴))
27		ceilqge 10730	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ ℚ → 𝐴 ≤ (⌈‘𝐴))
28		qre 10008	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ ℚ → 𝐴 ∈ ℝ)
29		ceilqcl 10728	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ ℚ → (⌈‘𝐴) ∈ ℤ)
30	29	zred 9751	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ ℚ → (⌈‘𝐴) ∈ ℝ)
31	28, 30	lenltd 8438	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ ℚ → (𝐴 ≤ (⌈‘𝐴) ↔ ¬ (⌈‘𝐴) < 𝐴))
32		pm2.21 626	. . . . . . . . . . . . . . . 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 728	. . 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 720  DECID wdc 846   = wceq 1402   ∈ wcel 2209   ≠ wne 2420   class class class wbr 4128  ‘cfv 5375   < clt 8354   ≤ cle 8355  ℤcz 9627  ℚcq 10002  ⌊cfl 10686  ⌈cceil 10687

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-cnex 8264  ax-resscn 8265  ax-1cn 8266  ax-1re 8267  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-mulrcl 8272  ax-addcom 8273  ax-mulcom 8274  ax-addass 8275  ax-mulass 8276  ax-distr 8277  ax-i2m1 8278  ax-0lt1 8279  ax-1rid 8280  ax-0id 8281  ax-rnegex 8282  ax-precex 8283  ax-cnre 8284  ax-pre-ltirr 8285  ax-pre-ltwlin 8286  ax-pre-lttrn 8287  ax-pre-apti 8288  ax-pre-ltadd 8289  ax-pre-mulgt0 8290  ax-pre-mulext 8291  ax-arch 8292

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-po 4439  df-iso 4440  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-fv 5383  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-pnf 8356  df-mnf 8357  df-xr 8358  df-ltxr 8359  df-le 8360  df-sub 8493  df-neg 8494  df-reap 8897  df-ap 8904  df-div 8997  df-inn 9288  df-n0 9547  df-z 9628  df-q 10003  df-rp 10038  df-fl 10688  df-ceil 10689

This theorem is referenced by: (None)