Theorem flqeqceilz 10626

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 10590	. . 3 |- ( A e. ZZ -> ( |_ ` A ) = A )
2		ceilid 10623	. . 3 |- ( A e. ZZ -> (⌈ ` A ) = A )
3	1, 2	eqtr4d 2267	. 2 |- ( A e. ZZ -> ( |_ ` A ) = (⌈ ` A ) )
4		flqcl 10579	. . . . . 6 |- ( A e. QQ -> ( |_ ` A ) e. ZZ )
5		zq 9904	. . . . . 6 |- ( ( |_ ` A ) e. ZZ -> ( |_ ` A ) e. QQ )
6	4, 5	syl 14	. . . . 5 |- ( A e. QQ -> ( |_ ` A ) e. QQ )
7		qdceq 10550	. . . . 5 |- ( ( ( |_ ` A ) e. QQ /\ A e. QQ ) -> DECID ( |_ ` A ) = A )
8	6, 7	mpancom 422	. . . 4 |- ( A e. QQ -> DECID ( |_ ` A ) = A )
9		exmiddc 844	. . . 4 |- (DECID ( |_ ` A ) = A -> ( ( |_ ` A ) = A \/ -. ( |_ ` A ) = A ) )
10	8, 9	syl 14	. . 3 |- ( A e. QQ -> ( ( |_ ` A ) = A \/ -. ( |_ ` A ) = A ) )
11		eqeq1 2238	. . . . . . 7 |- ( ( |_ ` A ) = A -> ( ( |_ ` A ) = (⌈ ` A ) <-> A = (⌈ ` A ) ) )
12	11	adantr 276	. . . . . 6 |- ( ( ( |_ ` A ) = A /\ A e. QQ ) -> ( ( |_ ` A ) = (⌈ ` A ) <-> A = (⌈ ` A ) ) )
13		ceilqidz 10624	. . . . . . . . 9 |- ( A e. QQ -> ( A e. ZZ <-> (⌈ ` A ) = A ) )
14		eqcom 2233	. . . . . . . . 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 10584	. . . . 5 |- ( A e. QQ -> ( |_ ` A ) <_ A )
21		df-ne 2404	. . . . . 6 |- ( ( |_ ` A ) =/= A <-> -. ( |_ ` A ) = A )
22		necom 2487	. . . . . . 7 |- ( ( |_ ` A ) =/= A <-> A =/= ( |_ ` A ) )
23		qltlen 9918	. . . . . . . . . . 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 4096	. . . . . . . . . . . . . 14 |- ( ( |_ ` A ) = (⌈ ` A ) -> ( ( |_ ` A ) < A <-> (⌈ ` A ) < A ) )
26	25	adantl 277	. . . . . . . . . . . . 13 |- ( ( A e. QQ /\ ( |_ ` A ) = (⌈ ` A ) ) -> ( ( |_ ` A ) < A <-> (⌈ ` A ) < A ) )
27		ceilqge 10618	. . . . . . . . . . . . . . 15 |- ( A e. QQ -> A <_ (⌈ ` A ) )
28		qre 9903	. . . . . . . . . . . . . . . . 17 |- ( A e. QQ -> A e. RR )
29		ceilqcl 10616	. . . . . . . . . . . . . . . . . 18 |- ( A e. QQ -> (⌈ ` A ) e. ZZ )
30	29	zred 9646	. . . . . . . . . . . . . . . . 17 |- ( A e. QQ -> (⌈ ` A ) e. RR )
31	28, 30	lenltd 8339	. . . . . . . . . . . . . . . 16 |- ( A e. QQ -> ( A <_ (⌈ ` A ) <-> -. (⌈ ` A ) < A ) )
32		pm2.21 622	. . . . . . . . . . . . . . . 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 724	. . 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 716  DECID wdc 842   = wceq 1398   e. wcel 2202   =/= wne 2403   class class class wbr 4093   ` cfv 5333   < clt 8256   <_ cle 8257   ZZcz 9523   QQcq 9897   |_cfl 10574  ⌈cceil 10575

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-mulrcl 8174  ax-addcom 8175  ax-mulcom 8176  ax-addass 8177  ax-mulass 8178  ax-distr 8179  ax-i2m1 8180  ax-0lt1 8181  ax-1rid 8182  ax-0id 8183  ax-rnegex 8184  ax-precex 8185  ax-cnre 8186  ax-pre-ltirr 8187  ax-pre-ltwlin 8188  ax-pre-lttrn 8189  ax-pre-apti 8190  ax-pre-ltadd 8191  ax-pre-mulgt0 8192  ax-pre-mulext 8193  ax-arch 8194

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-po 4399  df-iso 4400  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-pnf 8258  df-mnf 8259  df-xr 8260  df-ltxr 8261  df-le 8262  df-sub 8394  df-neg 8395  df-reap 8797  df-ap 8804  df-div 8895  df-inn 9186  df-n0 9445  df-z 9524  df-q 9898  df-rp 9933  df-fl 10576  df-ceil 10577

This theorem is referenced by: (None)