Theorem flqeqceilz 10091

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 10057	. . 3 |- ( A e. ZZ -> ( |_ ` A ) = A )
2		ceilid 10088	. . 3 |- ( A e. ZZ -> (⌈ ` A ) = A )
3	1, 2	eqtr4d 2175	. 2 |- ( A e. ZZ -> ( |_ ` A ) = (⌈ ` A ) )
4		flqcl 10046	. . . . . 6 |- ( A e. QQ -> ( |_ ` A ) e. ZZ )
5		zq 9418	. . . . . 6 |- ( ( |_ ` A ) e. ZZ -> ( |_ ` A ) e. QQ )
6	4, 5	syl 14	. . . . 5 |- ( A e. QQ -> ( |_ ` A ) e. QQ )
7		qdceq 10024	. . . . 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 2146	. . . . . . 7 |- ( ( |_ ` A ) = A -> ( ( |_ ` A ) = (⌈ ` A ) <-> A = (⌈ ` A ) ) )
12	11	adantr 274	. . . . . 6 |- ( ( ( |_ ` A ) = A /\ A e. QQ ) -> ( ( |_ ` A ) = (⌈ ` A ) <-> A = (⌈ ` A ) ) )
13		ceilqidz 10089	. . . . . . . . 9 |- ( A e. QQ -> ( A e. ZZ <-> (⌈ ` A ) = A ) )
14		eqcom 2141	. . . . . . . . 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 10051	. . . . 5 |- ( A e. QQ -> ( |_ ` A ) <_ A )
21		df-ne 2309	. . . . . 6 |- ( ( |_ ` A ) =/= A <-> -. ( |_ ` A ) = A )
22		necom 2392	. . . . . . 7 |- ( ( |_ ` A ) =/= A <-> A =/= ( |_ ` A ) )
23		qltlen 9432	. . . . . . . . . . 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 3932	. . . . . . . . . . . . . 14 |- ( ( |_ ` A ) = (⌈ ` A ) -> ( ( |_ ` A ) < A <-> (⌈ ` A ) < A ) )
26	25	adantl 275	. . . . . . . . . . . . 13 |- ( ( A e. QQ /\ ( |_ ` A ) = (⌈ ` A ) ) -> ( ( |_ ` A ) < A <-> (⌈ ` A ) < A ) )
27		ceilqge 10083	. . . . . . . . . . . . . . 15 |- ( A e. QQ -> A <_ (⌈ ` A ) )
28		qre 9417	. . . . . . . . . . . . . . . . 17 |- ( A e. QQ -> A e. RR )
29		ceilqcl 10081	. . . . . . . . . . . . . . . . . 18 |- ( A e. QQ -> (⌈ ` A ) e. ZZ )
30	29	zred 9173	. . . . . . . . . . . . . . . . 17 |- ( A e. QQ -> (⌈ ` A ) e. RR )
31	28, 30	lenltd 7880	. . . . . . . . . . . . . . . 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 2308   class class class wbr 3929   ` cfv 5123   < clt 7800   <_ cle 7801   ZZcz 9054   QQcq 9411   |_cfl 10041  ⌈cceil 10042

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 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7711  ax-resscn 7712  ax-1cn 7713  ax-1re 7714  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-mulrcl 7719  ax-addcom 7720  ax-mulcom 7721  ax-addass 7722  ax-mulass 7723  ax-distr 7724  ax-i2m1 7725  ax-0lt1 7726  ax-1rid 7727  ax-0id 7728  ax-rnegex 7729  ax-precex 7730  ax-cnre 7731  ax-pre-ltirr 7732  ax-pre-ltwlin 7733  ax-pre-lttrn 7734  ax-pre-apti 7735  ax-pre-ltadd 7736  ax-pre-mulgt0 7737  ax-pre-mulext 7738  ax-arch 7739

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 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-po 4218  df-iso 4219  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806  df-sub 7935  df-neg 7936  df-reap 8337  df-ap 8344  df-div 8433  df-inn 8721  df-n0 8978  df-z 9055  df-q 9412  df-rp 9442  df-fl 10043  df-ceil 10044

This theorem is referenced by: (None)