Theorem reperflem 22621

Description: A subset of the real numbers that is closed under addition with real numbers is perfect. (Contributed by Mario Carneiro, 26-Dec-2016.)

Hypotheses

Ref	Expression
recld2.1	|- J = ( TopOpen ` ℂfld )
reperflem.2	|- ( ( u e. S /\ v e. RR ) -> ( u + v ) e. S )
reperflem.3	|- S C_ CC

Assertion

Ref	Expression
reperflem	|- ( J ↾t S ) e. Perf

Distinct variable groups:   u, J   v, u, S

Allowed substitution hint:   J( v)

Proof of Theorem reperflem

Dummy variables n r are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnxmet 22576	. . . . . . 7 |- ( abs o. - ) e. ( *Met ` CC )
2		reperflem.3	. . . . . . . 8 |- S C_ CC
3	2	sseli 3599	. . . . . . 7 |- ( u e. S -> u e. CC )
4		recld2.1	. . . . . . . . 9 |- J = ( TopOpen ` ℂfld )
5	4	cnfldtopn 22585	. . . . . . . 8 |- J = ( MetOpen ` ( abs o. - ) )
6	5	neibl 22306	. . . . . . 7 |- ( ( ( abs o. - ) e. ( *Met ` CC ) /\ u e. CC ) -> ( n e. ( ( nei ` J ) ` { u } ) <-> ( n C_ CC /\ E. r e. RR+ ( u ( ball ` ( abs o. - ) ) r ) C_ n ) ) )
7	1, 3, 6	sylancr 695	. . . . . 6 |- ( u e. S -> ( n e. ( ( nei ` J ) ` { u } ) <-> ( n C_ CC /\ E. r e. RR+ ( u ( ball ` ( abs o. - ) ) r ) C_ n ) ) )
8		reperflem.2	. . . . . . . . . . . . . . . . 17 |- ( ( u e. S /\ v e. RR ) -> ( u + v ) e. S )
9	8	ralrimiva 2966	. . . . . . . . . . . . . . . 16 |- ( u e. S -> A. v e. RR ( u + v ) e. S )
10		rpre 11839	. . . . . . . . . . . . . . . . 17 |- ( r e. RR+ -> r e. RR )
11	10	rehalfcld 11279	. . . . . . . . . . . . . . . 16 |- ( r e. RR+ -> ( r / 2 ) e. RR )
12		oveq2 6658	. . . . . . . . . . . . . . . . . 18 |- ( v = ( r / 2 ) -> ( u + v ) = ( u + ( r / 2 ) ) )
13	12	eleq1d 2686	. . . . . . . . . . . . . . . . 17 |- ( v = ( r / 2 ) -> ( ( u + v ) e. S <-> ( u + ( r / 2 ) ) e. S ) )
14	13	rspccva 3308	. . . . . . . . . . . . . . . 16 |- ( ( A. v e. RR ( u + v ) e. S /\ ( r / 2 ) e. RR ) -> ( u + ( r / 2 ) ) e. S )
15	9, 11, 14	syl2an 494	. . . . . . . . . . . . . . 15 |- ( ( u e. S /\ r e. RR+ ) -> ( u + ( r / 2 ) ) e. S )
16	2, 15	sseldi 3601	. . . . . . . . . . . . . 14 |- ( ( u e. S /\ r e. RR+ ) -> ( u + ( r / 2 ) ) e. CC )
17	3	adantr 481	. . . . . . . . . . . . . 14 |- ( ( u e. S /\ r e. RR+ ) -> u e. CC )
18		eqid 2622	. . . . . . . . . . . . . . 15 |- ( abs o. - ) = ( abs o. - )
19	18	cnmetdval 22574	. . . . . . . . . . . . . 14 |- ( ( ( u + ( r / 2 ) ) e. CC /\ u e. CC ) -> ( ( u + ( r / 2 ) ) ( abs o. - ) u ) = ( abs ` ( ( u + ( r / 2 ) ) - u ) ) )
20	16, 17, 19	syl2anc 693	. . . . . . . . . . . . 13 |- ( ( u e. S /\ r e. RR+ ) -> ( ( u + ( r / 2 ) ) ( abs o. - ) u ) = ( abs ` ( ( u + ( r / 2 ) ) - u ) ) )
21		simpr 477	. . . . . . . . . . . . . . . . 17 |- ( ( u e. S /\ r e. RR+ ) -> r e. RR+ )
22	21	rphalfcld 11884	. . . . . . . . . . . . . . . 16 |- ( ( u e. S /\ r e. RR+ ) -> ( r / 2 ) e. RR+ )
23	22	rpcnd 11874	. . . . . . . . . . . . . . 15 |- ( ( u e. S /\ r e. RR+ ) -> ( r / 2 ) e. CC )
24	17, 23	pncan2d 10394	. . . . . . . . . . . . . 14 |- ( ( u e. S /\ r e. RR+ ) -> ( ( u + ( r / 2 ) ) - u ) = ( r / 2 ) )
25	24	fveq2d 6195	. . . . . . . . . . . . 13 |- ( ( u e. S /\ r e. RR+ ) -> ( abs ` ( ( u + ( r / 2 ) ) - u ) ) = ( abs ` ( r / 2 ) ) )
26	22	rpred 11872	. . . . . . . . . . . . . 14 |- ( ( u e. S /\ r e. RR+ ) -> ( r / 2 ) e. RR )
27	22	rpge0d 11876	. . . . . . . . . . . . . 14 |- ( ( u e. S /\ r e. RR+ ) -> 0 <_ ( r / 2 ) )
28	26, 27	absidd 14161	. . . . . . . . . . . . 13 |- ( ( u e. S /\ r e. RR+ ) -> ( abs ` ( r / 2 ) ) = ( r / 2 ) )
29	20, 25, 28	3eqtrd 2660	. . . . . . . . . . . 12 |- ( ( u e. S /\ r e. RR+ ) -> ( ( u + ( r / 2 ) ) ( abs o. - ) u ) = ( r / 2 ) )
30		rphalflt 11860	. . . . . . . . . . . . 13 |- ( r e. RR+ -> ( r / 2 ) < r )
31	30	adantl 482	. . . . . . . . . . . 12 |- ( ( u e. S /\ r e. RR+ ) -> ( r / 2 ) < r )
32	29, 31	eqbrtrd 4675	. . . . . . . . . . 11 |- ( ( u e. S /\ r e. RR+ ) -> ( ( u + ( r / 2 ) ) ( abs o. - ) u ) < r )
33	1	a1i 11	. . . . . . . . . . . 12 |- ( ( u e. S /\ r e. RR+ ) -> ( abs o. - ) e. ( *Met ` CC ) )
34		rpxr 11840	. . . . . . . . . . . . 13 |- ( r e. RR+ -> r e. RR* )
35	34	adantl 482	. . . . . . . . . . . 12 |- ( ( u e. S /\ r e. RR+ ) -> r e. RR* )
36		elbl3 22197	. . . . . . . . . . . 12 |- ( ( ( ( abs o. - ) e. ( *Met ` CC ) /\ r e. RR* ) /\ ( u e. CC /\ ( u + ( r / 2 ) ) e. CC ) ) -> ( ( u + ( r / 2 ) ) e. ( u ( ball ` ( abs o. - ) ) r ) <-> ( ( u + ( r / 2 ) ) ( abs o. - ) u ) < r ) )
37	33, 35, 17, 16, 36	syl22anc 1327	. . . . . . . . . . 11 |- ( ( u e. S /\ r e. RR+ ) -> ( ( u + ( r / 2 ) ) e. ( u ( ball ` ( abs o. - ) ) r ) <-> ( ( u + ( r / 2 ) ) ( abs o. - ) u ) < r ) )
38	32, 37	mpbird 247	. . . . . . . . . 10 |- ( ( u e. S /\ r e. RR+ ) -> ( u + ( r / 2 ) ) e. ( u ( ball ` ( abs o. - ) ) r ) )
39	22	rpne0d 11877	. . . . . . . . . . . . 13 |- ( ( u e. S /\ r e. RR+ ) -> ( r / 2 ) =/= 0 )
40	24, 39	eqnetrd 2861	. . . . . . . . . . . 12 |- ( ( u e. S /\ r e. RR+ ) -> ( ( u + ( r / 2 ) ) - u ) =/= 0 )
41	16, 17, 40	subne0ad 10403	. . . . . . . . . . 11 |- ( ( u e. S /\ r e. RR+ ) -> ( u + ( r / 2 ) ) =/= u )
42		eldifsn 4317	. . . . . . . . . . 11 |- ( ( u + ( r / 2 ) ) e. ( S \ { u } ) <-> ( ( u + ( r / 2 ) ) e. S /\ ( u + ( r / 2 ) ) =/= u ) )
43	15, 41, 42	sylanbrc 698	. . . . . . . . . 10 |- ( ( u e. S /\ r e. RR+ ) -> ( u + ( r / 2 ) ) e. ( S \ { u } ) )
44		inelcm 4032	. . . . . . . . . 10 |- ( ( ( u + ( r / 2 ) ) e. ( u ( ball ` ( abs o. - ) ) r ) /\ ( u + ( r / 2 ) ) e. ( S \ { u } ) ) -> ( ( u ( ball ` ( abs o. - ) ) r ) i^i ( S \ { u } ) ) =/= (/) )
45	38, 43, 44	syl2anc 693	. . . . . . . . 9 |- ( ( u e. S /\ r e. RR+ ) -> ( ( u ( ball ` ( abs o. - ) ) r ) i^i ( S \ { u } ) ) =/= (/) )
46		ssrin 3838	. . . . . . . . . 10 |- ( ( u ( ball ` ( abs o. - ) ) r ) C_ n -> ( ( u ( ball ` ( abs o. - ) ) r ) i^i ( S \ { u } ) ) C_ ( n i^i ( S \ { u } ) ) )
47		ssn0 3976	. . . . . . . . . . 11 |- ( ( ( ( u ( ball ` ( abs o. - ) ) r ) i^i ( S \ { u } ) ) C_ ( n i^i ( S \ { u } ) ) /\ ( ( u ( ball ` ( abs o. - ) ) r ) i^i ( S \ { u } ) ) =/= (/) ) -> ( n i^i ( S \ { u } ) ) =/= (/) )
48	47	ex 450	. . . . . . . . . 10 |- ( ( ( u ( ball ` ( abs o. - ) ) r ) i^i ( S \ { u } ) ) C_ ( n i^i ( S \ { u } ) ) -> ( ( ( u ( ball ` ( abs o. - ) ) r ) i^i ( S \ { u } ) ) =/= (/) -> ( n i^i ( S \ { u } ) ) =/= (/) ) )
49	46, 48	syl 17	. . . . . . . . 9 |- ( ( u ( ball ` ( abs o. - ) ) r ) C_ n -> ( ( ( u ( ball ` ( abs o. - ) ) r ) i^i ( S \ { u } ) ) =/= (/) -> ( n i^i ( S \ { u } ) ) =/= (/) ) )
50	45, 49	syl5com 31	. . . . . . . 8 |- ( ( u e. S /\ r e. RR+ ) -> ( ( u ( ball ` ( abs o. - ) ) r ) C_ n -> ( n i^i ( S \ { u } ) ) =/= (/) ) )
51	50	rexlimdva 3031	. . . . . . 7 |- ( u e. S -> ( E. r e. RR+ ( u ( ball ` ( abs o. - ) ) r ) C_ n -> ( n i^i ( S \ { u } ) ) =/= (/) ) )
52	51	adantld 483	. . . . . 6 |- ( u e. S -> ( ( n C_ CC /\ E. r e. RR+ ( u ( ball ` ( abs o. - ) ) r ) C_ n ) -> ( n i^i ( S \ { u } ) ) =/= (/) ) )
53	7, 52	sylbid 230	. . . . 5 |- ( u e. S -> ( n e. ( ( nei ` J ) ` { u } ) -> ( n i^i ( S \ { u } ) ) =/= (/) ) )
54	53	ralrimiv 2965	. . . 4 |- ( u e. S -> A. n e. ( ( nei ` J ) ` { u } ) ( n i^i ( S \ { u } ) ) =/= (/) )
55	4	cnfldtop 22587	. . . . . 6 |- J e. Top
56	4	cnfldtopon 22586	. . . . . . . 8 |- J e. (TopOn ` CC )
57	56	toponunii 20721	. . . . . . 7 |- CC = U. J
58	57	islp2 20949	. . . . . 6 |- ( ( J e. Top /\ S C_ CC /\ u e. CC ) -> ( u e. ( ( limPt ` J ) ` S ) <-> A. n e. ( ( nei ` J ) ` { u } ) ( n i^i ( S \ { u } ) ) =/= (/) ) )
59	55, 2, 58	mp3an12 1414	. . . . 5 |- ( u e. CC -> ( u e. ( ( limPt ` J ) ` S ) <-> A. n e. ( ( nei ` J ) ` { u } ) ( n i^i ( S \ { u } ) ) =/= (/) ) )
60	3, 59	syl 17	. . . 4 |- ( u e. S -> ( u e. ( ( limPt ` J ) ` S ) <-> A. n e. ( ( nei ` J ) ` { u } ) ( n i^i ( S \ { u } ) ) =/= (/) ) )
61	54, 60	mpbird 247	. . 3 |- ( u e. S -> u e. ( ( limPt ` J ) ` S ) )
62	61	ssriv 3607	. 2 |- S C_ ( ( limPt ` J ) ` S )
63		eqid 2622	. . . 4 |- ( J ↾t S ) = ( J ↾t S )
64	57, 63	restperf 20988	. . 3 |- ( ( J e. Top /\ S C_ CC ) -> ( ( J ↾t S ) e. Perf <-> S C_ ( ( limPt ` J ) ` S ) ) )
65	55, 2, 64	mp2an 708	. 2 |- ( ( J ↾t S ) e. Perf <-> S C_ ( ( limPt ` J ) ` S ) )
66	62, 65	mpbir 221	1 |- ( J ↾t S ) e. Perf

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   \ cdif 3571   i^i cin 3573   C_ wss 3574   (/)c0 3915   {csn 4177   class class class wbr 4653   o. ccom 5118   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   + caddc 9939   RR*cxr 10073   < clt 10074   - cmin 10266   / cdiv 10684   2c2 11070   RR+crp 11832   abscabs 13974   ↾_t crest 16081   TopOpenctopn 16082   *Metcxmt 19731   ballcbl 19733  ℂ_fldccnfld 19746   Topctop 20698   neicnei 20901   limPtclp 20938  Perfcperf 20939

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-rep 4771  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-int 4476  df-iun 4522  df-iin 4523  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-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fi 8317  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-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-fz 12327  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-plusg 15954  df-mulr 15955  df-starv 15956  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-rest 16083  df-topn 16084  df-topgen 16104  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-lp 20940  df-perf 20941  df-xms 22125  df-ms 22126

This theorem is referenced by:  reperf  22622  cnperf  22623