Theorem flqeqceilz 10084

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 10050	. . 3 |- ( A e. ZZ -> ( |_ ` A ) = A )
2		ceilid 10081	. . 3 |- ( A e. ZZ -> (⌈ ` A ) = A )
3	1, 2	eqtr4d 2173	. 2 |- ( A e. ZZ -> ( |_ ` A ) = (⌈ ` A ) )
4		flqcl 10039	. . . . . 6 |- ( A e. QQ -> ( |_ ` A ) e. ZZ )
5		zq 9411	. . . . . 6 |- ( ( |_ ` A ) e. ZZ -> ( |_ ` A ) e. QQ )
6	4, 5	syl 14	. . . . 5 |- ( A e. QQ -> ( |_ ` A ) e. QQ )
7		qdceq 10017	. . . . 5 |- ( ( ( |_ ` A ) e. QQ /\ A e. QQ ) -> DECID ( |_ ` A ) = A )
8	6, 7	mpancom 418	. . . 4 |- ( A e. QQ -> DECID ( |_ ` A ) = A )
9		exmiddc 821	. . . 4 |- (DECID ( |_ ` A ) = A -> ( ( |_ ` A ) = A \/ -. ( |_ ` A ) = A ) )
10	8, 9	syl 14	. . 3 |- ( A e. QQ -> ( ( |_ ` A ) = A \/ -. ( |_ ` A ) = A ) )
11		eqeq1 2144	. . . . . . 7 |- ( ( |_ ` A ) = A -> ( ( |_ ` A ) = (⌈ ` A ) <-> A = (⌈ ` A ) ) )
12	11	adantr 274	. . . . . 6 |- ( ( ( |_ ` A ) = A /\ A e. QQ ) -> ( ( |_ ` A ) = (⌈ ` A ) <-> A = (⌈ ` A ) ) )
13		ceilqidz 10082	. . . . . . . . 9 |- ( A e. QQ -> ( A e. ZZ <-> (⌈ ` A ) = A ) )
14		eqcom 2139	. . . . . . . . 9 |- ( (⌈ ` A ) = A <-> A = (⌈ ` A ) )
15	13, 14	syl6bb 195	. . . . . . . 8 |- ( A e. QQ -> ( A e. ZZ <-> A = (⌈ ` A ) ) )
16	15	biimprd 157	. . . . . . 7 |- ( A e. QQ -> ( A = (⌈ ` A ) -> A e. ZZ ) )
17	16	adantl 275	. . . . . 6 |- ( ( ( |_ ` A ) = A /\ A e. QQ ) -> ( A = (⌈ ` A ) -> A e. ZZ ) )
18	12, 17	sylbid 149	. . . . 5 |- ( ( ( |_ ` A ) = A /\ A e. QQ ) -> ( ( |_ ` A ) = (⌈ ` A ) -> A e. ZZ ) )
19	18	ex 114	. . . 4 |- ( ( |_ ` A ) = A -> ( A e. QQ -> ( ( |_ ` A ) = (⌈ ` A ) -> A e. ZZ ) ) )
20		flqle 10044	. . . . 5 |- ( A e. QQ -> ( |_ ` A ) <_ A )
21		df-ne 2307	. . . . . 6 |- ( ( |_ ` A ) =/= A <-> -. ( |_ ` A ) = A )
22		necom 2390	. . . . . . 7 |- ( ( |_ ` A ) =/= A <-> A =/= ( |_ ` A ) )
23		qltlen 9425	. . . . . . . . . . 11 |- ( ( ( |_ ` A ) e. QQ /\ A e. QQ ) -> ( ( |_ ` A ) < A <-> ( ( |_ ` A ) <_ A /\ A =/= ( |_ ` A ) ) ) )
24	6, 23	mpancom 418	. . . . . . . . . 10 |- ( A e. QQ -> ( ( |_ ` A ) < A <-> ( ( |_ ` A ) <_ A /\ A =/= ( |_ ` A ) ) ) )
25		breq1 3927	. . . . . . . . . . . . . 14 |- ( ( |_ ` A ) = (⌈ ` A ) -> ( ( |_ ` A ) < A <-> (⌈ ` A ) < A ) )
26	25	adantl 275	. . . . . . . . . . . . 13 |- ( ( A e. QQ /\ ( |_ ` A ) = (⌈ ` A ) ) -> ( ( |_ ` A ) < A <-> (⌈ ` A ) < A ) )
27		ceilqge 10076	. . . . . . . . . . . . . . 15 |- ( A e. QQ -> A <_ (⌈ ` A ) )
28		qre 9410	. . . . . . . . . . . . . . . . 17 |- ( A e. QQ -> A e. RR )
29		ceilqcl 10074	. . . . . . . . . . . . . . . . . 18 |- ( A e. QQ -> (⌈ ` A ) e. ZZ )
30	29	zred 9166	. . . . . . . . . . . . . . . . 17 |- ( A e. QQ -> (⌈ ` A ) e. RR )
31	28, 30	lenltd 7873	. . . . . . . . . . . . . . . 16 |- ( A e. QQ -> ( A <_ (⌈ ` A ) <-> -. (⌈ ` A ) < A ) )
32		pm2.21 606	. . . . . . . . . . . . . . . 16 |- ( -. (⌈ ` A ) < A -> ( (⌈ ` A ) < A -> A e. ZZ ) )
33	31, 32	syl6bi 162	. . . . . . . . . . . . . . 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 274	. . . . . . . . . . . . 13 |- ( ( A e. QQ /\ ( |_ ` A ) = (⌈ ` A ) ) -> ( (⌈ ` A ) < A -> A e. ZZ ) )
36	26, 35	sylbid 149	. . . . . . . . . . . 12 |- ( ( A e. QQ /\ ( |_ ` A ) = (⌈ ` A ) ) -> ( ( |_ ` A ) < A -> A e. ZZ ) )
37	36	ex 114	. . . . . . . . . . 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 169	. . . . . . . . 9 |- ( A e. QQ -> ( ( ( |_ ` A ) <_ A /\ A =/= ( |_ ` A ) ) -> ( ( |_ ` A ) = (⌈ ` A ) -> A e. ZZ ) ) )
40	39	expd 256	. . . . . . . 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 120	. . . . . 6 |- ( ( |_ ` A ) =/= A -> ( A e. QQ -> ( ( |_ ` A ) <_ A -> ( ( |_ ` A ) = (⌈ ` A ) -> A e. ZZ ) ) ) )
43	21, 42	sylbir 134	. . . . 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 705	. . 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 142	1 |- ( A e. QQ -> ( A e. ZZ <-> ( |_ ` A ) = (⌈ ` A ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 697  DECID wdc 819   = wceq 1331   e. wcel 1480   =/= wne 2306   class class class wbr 3924   ` cfv 5118   < clt 7793   <_ cle 7794   ZZcz 9047   QQcq 9404   |_cfl 10034  ⌈cceil 10035

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-cnex 7704  ax-resscn 7705  ax-1cn 7706  ax-1re 7707  ax-icn 7708  ax-addcl 7709  ax-addrcl 7710  ax-mulcl 7711  ax-mulrcl 7712  ax-addcom 7713  ax-mulcom 7714  ax-addass 7715  ax-mulass 7716  ax-distr 7717  ax-i2m1 7718  ax-0lt1 7719  ax-1rid 7720  ax-0id 7721  ax-rnegex 7722  ax-precex 7723  ax-cnre 7724  ax-pre-ltirr 7725  ax-pre-ltwlin 7726  ax-pre-lttrn 7727  ax-pre-apti 7728  ax-pre-ltadd 7729  ax-pre-mulgt0 7730  ax-pre-mulext 7731  ax-arch 7732

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-nel 2402  df-ral 2419  df-rex 2420  df-reu 2421  df-rmo 2422  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-po 4213  df-iso 4214  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-fv 5126  df-riota 5723  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-pnf 7795  df-mnf 7796  df-xr 7797  df-ltxr 7798  df-le 7799  df-sub 7928  df-neg 7929  df-reap 8330  df-ap 8337  df-div 8426  df-inn 8714  df-n0 8971  df-z 9048  df-q 9405  df-rp 9435  df-fl 10036  df-ceil 10037

This theorem is referenced by: (None)