Theorem dnicn 32482

Description: The "distance to nearest integer" function is continuous. (Contributed by Asger C. Ipsen, 4-Apr-2021.)

Hypothesis

Ref	Expression
dnicn.1	|- T = ( x e. RR |-> ( abs ` ( ( |_ ` ( x + ( 1 / 2 ) ) ) - x ) ) )

Assertion

Ref	Expression
dnicn	|- T e. ( RR -cn-> RR )

Proof of Theorem dnicn

Dummy variables d e y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dnicn.1	. . 3 |- T = ( x e. RR |-> ( abs ` ( ( |_ ` ( x + ( 1 / 2 ) ) ) - x ) ) )
2	1	dnif 32464	. 2 |- T : RR --> RR
3		simpr 477	. . . 4 |- ( ( y e. RR /\ e e. RR+ ) -> e e. RR+ )
4		simplr 792	. . . . . . . . . . 11 |- ( ( ( ( y e. RR /\ e e. RR+ ) /\ z e. RR ) /\ ( abs ` ( z - y ) ) < e ) -> z e. RR )
5	1, 4	dnicld2 32463	. . . . . . . . . 10 |- ( ( ( ( y e. RR /\ e e. RR+ ) /\ z e. RR ) /\ ( abs ` ( z - y ) ) < e ) -> ( T ` z ) e. RR )
6		simplll 798	. . . . . . . . . . 11 |- ( ( ( ( y e. RR /\ e e. RR+ ) /\ z e. RR ) /\ ( abs ` ( z - y ) ) < e ) -> y e. RR )
7	1, 6	dnicld2 32463	. . . . . . . . . 10 |- ( ( ( ( y e. RR /\ e e. RR+ ) /\ z e. RR ) /\ ( abs ` ( z - y ) ) < e ) -> ( T ` y ) e. RR )
8	5, 7	resubcld 10458	. . . . . . . . 9 |- ( ( ( ( y e. RR /\ e e. RR+ ) /\ z e. RR ) /\ ( abs ` ( z - y ) ) < e ) -> ( ( T ` z ) - ( T ` y ) ) e. RR )
9	8	recnd 10068	. . . . . . . 8 |- ( ( ( ( y e. RR /\ e e. RR+ ) /\ z e. RR ) /\ ( abs ` ( z - y ) ) < e ) -> ( ( T ` z ) - ( T ` y ) ) e. CC )
10	9	abscld 14175	. . . . . . 7 |- ( ( ( ( y e. RR /\ e e. RR+ ) /\ z e. RR ) /\ ( abs ` ( z - y ) ) < e ) -> ( abs ` ( ( T ` z ) - ( T ` y ) ) ) e. RR )
11	4, 6	resubcld 10458	. . . . . . . . 9 |- ( ( ( ( y e. RR /\ e e. RR+ ) /\ z e. RR ) /\ ( abs ` ( z - y ) ) < e ) -> ( z - y ) e. RR )
12	11	recnd 10068	. . . . . . . 8 |- ( ( ( ( y e. RR /\ e e. RR+ ) /\ z e. RR ) /\ ( abs ` ( z - y ) ) < e ) -> ( z - y ) e. CC )
13	12	abscld 14175	. . . . . . 7 |- ( ( ( ( y e. RR /\ e e. RR+ ) /\ z e. RR ) /\ ( abs ` ( z - y ) ) < e ) -> ( abs ` ( z - y ) ) e. RR )
14	3	ad2antrr 762	. . . . . . . 8 |- ( ( ( ( y e. RR /\ e e. RR+ ) /\ z e. RR ) /\ ( abs ` ( z - y ) ) < e ) -> e e. RR+ )
15	14	rpred 11872	. . . . . . 7 |- ( ( ( ( y e. RR /\ e e. RR+ ) /\ z e. RR ) /\ ( abs ` ( z - y ) ) < e ) -> e e. RR )
16	1, 6, 4	dnibnd 32481	. . . . . . 7 |- ( ( ( ( y e. RR /\ e e. RR+ ) /\ z e. RR ) /\ ( abs ` ( z - y ) ) < e ) -> ( abs ` ( ( T ` z ) - ( T ` y ) ) ) <_ ( abs ` ( z - y ) ) )
17		simpr 477	. . . . . . 7 |- ( ( ( ( y e. RR /\ e e. RR+ ) /\ z e. RR ) /\ ( abs ` ( z - y ) ) < e ) -> ( abs ` ( z - y ) ) < e )
18	10, 13, 15, 16, 17	lelttrd 10195	. . . . . 6 |- ( ( ( ( y e. RR /\ e e. RR+ ) /\ z e. RR ) /\ ( abs ` ( z - y ) ) < e ) -> ( abs ` ( ( T ` z ) - ( T ` y ) ) ) < e )
19	18	ex 450	. . . . 5 |- ( ( ( y e. RR /\ e e. RR+ ) /\ z e. RR ) -> ( ( abs ` ( z - y ) ) < e -> ( abs ` ( ( T ` z ) - ( T ` y ) ) ) < e ) )
20	19	ralrimiva 2966	. . . 4 |- ( ( y e. RR /\ e e. RR+ ) -> A. z e. RR ( ( abs ` ( z - y ) ) < e -> ( abs ` ( ( T ` z ) - ( T ` y ) ) ) < e ) )
21		breq2 4657	. . . . . . 7 |- ( d = e -> ( ( abs ` ( z - y ) ) < d <-> ( abs ` ( z - y ) ) < e ) )
22	21	imbi1d 331	. . . . . 6 |- ( d = e -> ( ( ( abs ` ( z - y ) ) < d -> ( abs ` ( ( T ` z ) - ( T ` y ) ) ) < e ) <-> ( ( abs ` ( z - y ) ) < e -> ( abs ` ( ( T ` z ) - ( T ` y ) ) ) < e ) ) )
23	22	ralbidv 2986	. . . . 5 |- ( d = e -> ( A. z e. RR ( ( abs ` ( z - y ) ) < d -> ( abs ` ( ( T ` z ) - ( T ` y ) ) ) < e ) <-> A. z e. RR ( ( abs ` ( z - y ) ) < e -> ( abs ` ( ( T ` z ) - ( T ` y ) ) ) < e ) ) )
24	23	rspcev 3309	. . . 4 |- ( ( e e. RR+ /\ A. z e. RR ( ( abs ` ( z - y ) ) < e -> ( abs ` ( ( T ` z ) - ( T ` y ) ) ) < e ) ) -> E. d e. RR+ A. z e. RR ( ( abs ` ( z - y ) ) < d -> ( abs ` ( ( T ` z ) - ( T ` y ) ) ) < e ) )
25	3, 20, 24	syl2anc 693	. . 3 |- ( ( y e. RR /\ e e. RR+ ) -> E. d e. RR+ A. z e. RR ( ( abs ` ( z - y ) ) < d -> ( abs ` ( ( T ` z ) - ( T ` y ) ) ) < e ) )
26	25	rgen2 2975	. 2 |- A. y e. RR A. e e. RR+ E. d e. RR+ A. z e. RR ( ( abs ` ( z - y ) ) < d -> ( abs ` ( ( T ` z ) - ( T ` y ) ) ) < e )
27		ax-resscn 9993	. . 3 |- RR C_ CC
28		elcncf2 22693	. . 3 |- ( ( RR C_ CC /\ RR C_ CC ) -> ( T e. ( RR -cn-> RR ) <-> ( T : RR --> RR /\ A. y e. RR A. e e. RR+ E. d e. RR+ A. z e. RR ( ( abs ` ( z - y ) ) < d -> ( abs ` ( ( T ` z ) - ( T ` y ) ) ) < e ) ) ) )
29	27, 27, 28	mp2an 708	. 2 |- ( T e. ( RR -cn-> RR ) <-> ( T : RR --> RR /\ A. y e. RR A. e e. RR+ E. d e. RR+ A. z e. RR ( ( abs ` ( z - y ) ) < d -> ( abs ` ( ( T ` z ) - ( T ` y ) ) ) < e ) ) )
30	2, 26, 29	mpbir2an 955	1 |- T e. ( RR -cn-> RR )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   C_ wss 3574   class class class wbr 4653   |-> cmpt 4729   -->wf 5884   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   1c1 9937   + caddc 9939   < clt 10074   - cmin 10266   / cdiv 10684   2c2 11070   RR+crp 11832   |_cfl 12591   abscabs 13974   -cn->ccncf 22679

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fl 12593  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-cncf 22681

This theorem is referenced by:  knoppcnlem10  32492