Theorem flqeqceilz 10389

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 e. QQ -> ( A e. ZZ <-> ( |_ ` A ) = (⌈ ` A ) ) )

Proof of Theorem flqeqceilz

Step	Hyp	Ref	Expression
1		flid 10353	. . 3 |- ( A e. ZZ -> ( |_ ` A ) = A )
2		ceilid 10386	. . 3 |- ( A e. ZZ -> (⌈ ` A ) = A )
3	1, 2	eqtr4d 2229	. 2 |- ( A e. ZZ -> ( |_ ` A ) = (⌈ ` A ) )
4		flqcl 10342	. . . . . 6 |- ( A e. QQ -> ( |_ ` A ) e. ZZ )
5		zq 9691	. . . . . 6 |- ( ( |_ ` A ) e. ZZ -> ( |_ ` A ) e. QQ )
6	4, 5	syl 14	. . . . 5 |- ( A e. QQ -> ( |_ ` A ) e. QQ )
7		qdceq 10314	. . . . 5 |- ( ( ( |_ ` A ) e. QQ /\ A e. QQ ) -> DECID ( |_ ` A ) = A )
8	6, 7	mpancom 422	. . . 4 |- ( A e. QQ -> DECID ( |_ ` A ) = A )
9		exmiddc 837	. . . 4 |- (DECID ( |_ ` A ) = A -> ( ( |_ ` A ) = A \/ -. ( |_ ` A ) = A ) )
10	8, 9	syl 14	. . 3 |- ( A e. QQ -> ( ( |_ ` A ) = A \/ -. ( |_ ` A ) = A ) )
11		eqeq1 2200	. . . . . . 7 |- ( ( |_ ` A ) = A -> ( ( |_ ` A ) = (⌈ ` A ) <-> A = (⌈ ` A ) ) )
12	11	adantr 276	. . . . . 6 |- ( ( ( |_ ` A ) = A /\ A e. QQ ) -> ( ( |_ ` A ) = (⌈ ` A ) <-> A = (⌈ ` A ) ) )
13		ceilqidz 10387	. . . . . . . . 9 |- ( A e. QQ -> ( A e. ZZ <-> (⌈ ` A ) = A ) )
14		eqcom 2195	. . . . . . . . 9 |- ( (⌈ ` A ) = A <-> A = (⌈ ` A ) )
15	13, 14	bitrdi 196	. . . . . . . 8 |- ( A e. QQ -> ( A e. ZZ <-> A = (⌈ ` A ) ) )
16	15	biimprd 158	. . . . . . 7 |- ( A e. QQ -> ( A = (⌈ ` A ) -> A e. ZZ ) )
17	16	adantl 277	. . . . . 6 |- ( ( ( |_ ` A ) = A /\ A e. QQ ) -> ( A = (⌈ ` A ) -> A e. ZZ ) )
18	12, 17	sylbid 150	. . . . 5 |- ( ( ( |_ ` A ) = A /\ A e. QQ ) -> ( ( |_ ` A ) = (⌈ ` A ) -> A e. ZZ ) )
19	18	ex 115	. . . 4 |- ( ( |_ ` A ) = A -> ( A e. QQ -> ( ( |_ ` A ) = (⌈ ` A ) -> A e. ZZ ) ) )
20		flqle 10347	. . . . 5 |- ( A e. QQ -> ( |_ ` A ) <_ A )
21		df-ne 2365	. . . . . 6 |- ( ( |_ ` A ) =/= A <-> -. ( |_ ` A ) = A )
22		necom 2448	. . . . . . 7 |- ( ( |_ ` A ) =/= A <-> A =/= ( |_ ` A ) )
23		qltlen 9705	. . . . . . . . . . 11 |- ( ( ( |_ ` A ) e. QQ /\ A e. QQ ) -> ( ( |_ ` A ) < A <-> ( ( |_ ` A ) <_ A /\ A =/= ( |_ ` A ) ) ) )
24	6, 23	mpancom 422	. . . . . . . . . 10 |- ( A e. QQ -> ( ( |_ ` A ) < A <-> ( ( |_ ` A ) <_ A /\ A =/= ( |_ ` A ) ) ) )
25		breq1 4032	. . . . . . . . . . . . . 14 |- ( ( |_ ` A ) = (⌈ ` A ) -> ( ( |_ ` A ) < A <-> (⌈ ` A ) < A ) )
26	25	adantl 277	. . . . . . . . . . . . 13 |- ( ( A e. QQ /\ ( |_ ` A ) = (⌈ ` A ) ) -> ( ( |_ ` A ) < A <-> (⌈ ` A ) < A ) )
27		ceilqge 10381	. . . . . . . . . . . . . . 15 |- ( A e. QQ -> A <_ (⌈ ` A ) )
28		qre 9690	. . . . . . . . . . . . . . . . 17 |- ( A e. QQ -> A e. RR )
29		ceilqcl 10379	. . . . . . . . . . . . . . . . . 18 |- ( A e. QQ -> (⌈ ` A ) e. ZZ )
30	29	zred 9439	. . . . . . . . . . . . . . . . 17 |- ( A e. QQ -> (⌈ ` A ) e. RR )
31	28, 30	lenltd 8137	. . . . . . . . . . . . . . . 16 |- ( A e. QQ -> ( A <_ (⌈ ` A ) <-> -. (⌈ ` A ) < A ) )
32		pm2.21 618	. . . . . . . . . . . . . . . 16 |- ( -. (⌈ ` A ) < A -> ( (⌈ ` A ) < A -> A e. ZZ ) )
33	31, 32	biimtrdi 163	. . . . . . . . . . . . . . 15 |- ( A e. QQ -> ( A <_ (⌈ ` A ) -> ( (⌈ ` A ) < A -> A e. ZZ ) ) )
34	27, 33	mpd 13	. . . . . . . . . . . . . 14 |- ( A e. QQ -> ( (⌈ ` A ) < A -> A e. ZZ ) )
35	34	adantr 276	. . . . . . . . . . . . 13 |- ( ( A e. QQ /\ ( |_ ` A ) = (⌈ ` A ) ) -> ( (⌈ ` A ) < A -> A e. ZZ ) )
36	26, 35	sylbid 150	. . . . . . . . . . . 12 |- ( ( A e. QQ /\ ( |_ ` A ) = (⌈ ` A ) ) -> ( ( |_ ` A ) < A -> A e. ZZ ) )
37	36	ex 115	. . . . . . . . . . 11 |- ( A e. QQ -> ( ( |_ ` A ) = (⌈ ` A ) -> ( ( |_ ` A ) < A -> A e. ZZ ) ) )
38	37	com23 78	. . . . . . . . . 10 |- ( A e. QQ -> ( ( |_ ` A ) < A -> ( ( |_ ` A ) = (⌈ ` A ) -> A e. ZZ ) ) )
39	24, 38	sylbird 170	. . . . . . . . 9 |- ( A e. QQ -> ( ( ( |_ ` A ) <_ A /\ A =/= ( |_ ` A ) ) -> ( ( |_ ` A ) = (⌈ ` A ) -> A e. ZZ ) ) )
40	39	expd 258	. . . . . . . 8 |- ( A e. QQ -> ( ( |_ ` A ) <_ A -> ( A =/= ( |_ ` A ) -> ( ( |_ ` A ) = (⌈ ` A ) -> A e. ZZ ) ) ) )
41	40	com3r 79	. . . . . . 7 |- ( A =/= ( |_ ` A ) -> ( A e. QQ -> ( ( |_ ` A ) <_ A -> ( ( |_ ` A ) = (⌈ ` A ) -> A e. ZZ ) ) ) )
42	22, 41	sylbi 121	. . . . . 6 |- ( ( |_ ` A ) =/= A -> ( A e. QQ -> ( ( |_ ` A ) <_ A -> ( ( |_ ` A ) = (⌈ ` A ) -> A e. ZZ ) ) ) )
43	21, 42	sylbir 135	. . . . 5 |- ( -. ( |_ ` A ) = A -> ( A e. QQ -> ( ( |_ ` A ) <_ A -> ( ( |_ ` A ) = (⌈ ` A ) -> A e. ZZ ) ) ) )
44	20, 43	mpdi 43	. . . 4 |- ( -. ( |_ ` A ) = A -> ( A e. QQ -> ( ( |_ ` A ) = (⌈ ` A ) -> A e. ZZ ) ) )
45	19, 44	jaoi 717	. . 3 |- ( ( ( |_ ` A ) = A \/ -. ( |_ ` A ) = A ) -> ( A e. QQ -> ( ( |_ ` A ) = (⌈ ` A ) -> A e. ZZ ) ) )
46	10, 45	mpcom 36	. 2 |- ( A e. QQ -> ( ( |_ ` A ) = (⌈ ` A ) -> A e. ZZ ) )
47	3, 46	impbid2 143	1 |- ( A e. QQ -> ( A e. ZZ <-> ( |_ ` A ) = (⌈ ` A ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 709  DECID wdc 835   = wceq 1364   e. wcel 2164   =/= wne 2364   class class class wbr 4029   ` cfv 5254   < clt 8054   <_ cle 8055   ZZcz 9317   QQcq 9684   |_cfl 10337  ⌈cceil 10338

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-mulrcl 7971  ax-addcom 7972  ax-mulcom 7973  ax-addass 7974  ax-mulass 7975  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-1rid 7979  ax-0id 7980  ax-rnegex 7981  ax-precex 7982  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-apti 7987  ax-pre-ltadd 7988  ax-pre-mulgt0 7989  ax-pre-mulext 7990  ax-arch 7991

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-po 4327  df-iso 4328  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-reap 8594  df-ap 8601  df-div 8692  df-inn 8983  df-n0 9241  df-z 9318  df-q 9685  df-rp 9720  df-fl 10339  df-ceil 10340

This theorem is referenced by: (None)