Theorem flqeqceilz 10557

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 10521	. . 3 |- ( A e. ZZ -> ( |_ ` A ) = A )
2		ceilid 10554	. . 3 |- ( A e. ZZ -> (⌈ ` A ) = A )
3	1, 2	eqtr4d 2265	. 2 |- ( A e. ZZ -> ( |_ ` A ) = (⌈ ` A ) )
4		flqcl 10510	. . . . . 6 |- ( A e. QQ -> ( |_ ` A ) e. ZZ )
5		zq 9838	. . . . . 6 |- ( ( |_ ` A ) e. ZZ -> ( |_ ` A ) e. QQ )
6	4, 5	syl 14	. . . . 5 |- ( A e. QQ -> ( |_ ` A ) e. QQ )
7		qdceq 10481	. . . . 5 |- ( ( ( |_ ` A ) e. QQ /\ A e. QQ ) -> DECID ( |_ ` A ) = A )
8	6, 7	mpancom 422	. . . 4 |- ( A e. QQ -> DECID ( |_ ` A ) = A )
9		exmiddc 841	. . . 4 |- (DECID ( |_ ` A ) = A -> ( ( |_ ` A ) = A \/ -. ( |_ ` A ) = A ) )
10	8, 9	syl 14	. . 3 |- ( A e. QQ -> ( ( |_ ` A ) = A \/ -. ( |_ ` A ) = A ) )
11		eqeq1 2236	. . . . . . 7 |- ( ( |_ ` A ) = A -> ( ( |_ ` A ) = (⌈ ` A ) <-> A = (⌈ ` A ) ) )
12	11	adantr 276	. . . . . 6 |- ( ( ( |_ ` A ) = A /\ A e. QQ ) -> ( ( |_ ` A ) = (⌈ ` A ) <-> A = (⌈ ` A ) ) )
13		ceilqidz 10555	. . . . . . . . 9 |- ( A e. QQ -> ( A e. ZZ <-> (⌈ ` A ) = A ) )
14		eqcom 2231	. . . . . . . . 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 10515	. . . . 5 |- ( A e. QQ -> ( |_ ` A ) <_ A )
21		df-ne 2401	. . . . . 6 |- ( ( |_ ` A ) =/= A <-> -. ( |_ ` A ) = A )
22		necom 2484	. . . . . . 7 |- ( ( |_ ` A ) =/= A <-> A =/= ( |_ ` A ) )
23		qltlen 9852	. . . . . . . . . . 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 4086	. . . . . . . . . . . . . 14 |- ( ( |_ ` A ) = (⌈ ` A ) -> ( ( |_ ` A ) < A <-> (⌈ ` A ) < A ) )
26	25	adantl 277	. . . . . . . . . . . . 13 |- ( ( A e. QQ /\ ( |_ ` A ) = (⌈ ` A ) ) -> ( ( |_ ` A ) < A <-> (⌈ ` A ) < A ) )
27		ceilqge 10549	. . . . . . . . . . . . . . 15 |- ( A e. QQ -> A <_ (⌈ ` A ) )
28		qre 9837	. . . . . . . . . . . . . . . . 17 |- ( A e. QQ -> A e. RR )
29		ceilqcl 10547	. . . . . . . . . . . . . . . . . 18 |- ( A e. QQ -> (⌈ ` A ) e. ZZ )
30	29	zred 9585	. . . . . . . . . . . . . . . . 17 |- ( A e. QQ -> (⌈ ` A ) e. RR )
31	28, 30	lenltd 8280	. . . . . . . . . . . . . . . 16 |- ( A e. QQ -> ( A <_ (⌈ ` A ) <-> -. (⌈ ` A ) < A ) )
32		pm2.21 620	. . . . . . . . . . . . . . . 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 721	. . 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 713  DECID wdc 839   = wceq 1395   e. wcel 2200   =/= wne 2400   class class class wbr 4083   ` cfv 5321   < clt 8197   <_ cle 8198   ZZcz 9462   QQcq 9831   |_cfl 10505  ⌈cceil 10506

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4259  ax-pr 4294  ax-un 4525  ax-setind 4630  ax-cnex 8106  ax-resscn 8107  ax-1cn 8108  ax-1re 8109  ax-icn 8110  ax-addcl 8111  ax-addrcl 8112  ax-mulcl 8113  ax-mulrcl 8114  ax-addcom 8115  ax-mulcom 8116  ax-addass 8117  ax-mulass 8118  ax-distr 8119  ax-i2m1 8120  ax-0lt1 8121  ax-1rid 8122  ax-0id 8123  ax-rnegex 8124  ax-precex 8125  ax-cnre 8126  ax-pre-ltirr 8127  ax-pre-ltwlin 8128  ax-pre-lttrn 8129  ax-pre-apti 8130  ax-pre-ltadd 8131  ax-pre-mulgt0 8132  ax-pre-mulext 8133  ax-arch 8134

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4385  df-po 4388  df-iso 4389  df-xp 4726  df-rel 4727  df-cnv 4728  df-co 4729  df-dm 4730  df-rn 4731  df-res 4732  df-ima 4733  df-iota 5281  df-fun 5323  df-fn 5324  df-f 5325  df-fv 5329  df-riota 5963  df-ov 6013  df-oprab 6014  df-mpo 6015  df-1st 6295  df-2nd 6296  df-pnf 8199  df-mnf 8200  df-xr 8201  df-ltxr 8202  df-le 8203  df-sub 8335  df-neg 8336  df-reap 8738  df-ap 8745  df-div 8836  df-inn 9127  df-n0 9386  df-z 9463  df-q 9832  df-rp 9867  df-fl 10507  df-ceil 10508

This theorem is referenced by: (None)