Theorem refsumcn 39189

Description: A finite sum of continuous real functions, from a common topological space, is continuous. The class expression for B normally contains free variables k and x to index it. See fsumcn 22673 for the analogous theorem on continuous complex functions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Hypotheses

Ref	Expression
refsumcn.1	|- F/ x ph
refsumcn.2	|- K = ( topGen ` ran (,) )
refsumcn.3	|- ( ph -> J e. (TopOn ` X ) )
refsumcn.4	|- ( ph -> A e. Fin )
refsumcn.5	|- ( ( ph /\ k e. A ) -> ( x e. X |-> B ) e. ( J Cn K ) )

Assertion

Ref	Expression
refsumcn	|- ( ph -> ( x e. X |-> sum_ k e. A B ) e. ( J Cn K ) )

Distinct variable groups:   x, k, A   k, J, x   k, X, x   ph, k

Allowed substitution hints:   ph( x)   B( x, k)   K( x, k)

Proof of Theorem refsumcn

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. . . 4 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
2		refsumcn.3	. . . 4 |- ( ph -> J e. (TopOn ` X ) )
3		refsumcn.4	. . . 4 |- ( ph -> A e. Fin )
4		refsumcn.5	. . . . . 6 |- ( ( ph /\ k e. A ) -> ( x e. X |-> B ) e. ( J Cn K ) )
5		refsumcn.2	. . . . . . . 8 |- K = ( topGen ` ran (,) )
6	1	tgioo2 22606	. . . . . . . 8 |- ( topGen ` ran (,) ) = ( ( TopOpen ` ℂfld ) ↾t RR )
7	5, 6	eqtri 2644	. . . . . . 7 |- K = ( ( TopOpen ` ℂfld ) ↾t RR )
8	7	oveq2i 6661	. . . . . 6 |- ( J Cn K ) = ( J Cn ( ( TopOpen ` ℂfld ) ↾t RR ) )
9	4, 8	syl6eleq 2711	. . . . 5 |- ( ( ph /\ k e. A ) -> ( x e. X |-> B ) e. ( J Cn ( ( TopOpen ` ℂfld ) ↾t RR ) ) )
10	1	cnfldtopon 22586	. . . . . . 7 |- ( TopOpen ` ℂfld ) e. (TopOn ` CC )
11	10	a1i 11	. . . . . 6 |- ( ( ph /\ k e. A ) -> ( TopOpen ` ℂfld ) e. (TopOn ` CC ) )
12	2	adantr 481	. . . . . . . 8 |- ( ( ph /\ k e. A ) -> J e. (TopOn ` X ) )
13		retopon 22567	. . . . . . . . . 10 |- ( topGen ` ran (,) ) e. (TopOn ` RR )
14	5, 13	eqeltri 2697	. . . . . . . . 9 |- K e. (TopOn ` RR )
15	14	a1i 11	. . . . . . . 8 |- ( ( ph /\ k e. A ) -> K e. (TopOn ` RR ) )
16		cnf2 21053	. . . . . . . 8 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` RR ) /\ ( x e. X |-> B ) e. ( J Cn K ) ) -> ( x e. X |-> B ) : X --> RR )
17	12, 15, 4, 16	syl3anc 1326	. . . . . . 7 |- ( ( ph /\ k e. A ) -> ( x e. X |-> B ) : X --> RR )
18		frn 6053	. . . . . . 7 |- ( ( x e. X |-> B ) : X --> RR -> ran ( x e. X |-> B ) C_ RR )
19	17, 18	syl 17	. . . . . 6 |- ( ( ph /\ k e. A ) -> ran ( x e. X |-> B ) C_ RR )
20		ax-resscn 9993	. . . . . . 7 |- RR C_ CC
21	20	a1i 11	. . . . . 6 |- ( ( ph /\ k e. A ) -> RR C_ CC )
22		cnrest2 21090	. . . . . 6 |- ( ( ( TopOpen ` ℂfld ) e. (TopOn ` CC ) /\ ran ( x e. X |-> B ) C_ RR /\ RR C_ CC ) -> ( ( x e. X |-> B ) e. ( J Cn ( TopOpen ` ℂfld ) ) <-> ( x e. X |-> B ) e. ( J Cn ( ( TopOpen ` ℂfld ) ↾t RR ) ) ) )
23	11, 19, 21, 22	syl3anc 1326	. . . . 5 |- ( ( ph /\ k e. A ) -> ( ( x e. X |-> B ) e. ( J Cn ( TopOpen ` ℂfld ) ) <-> ( x e. X |-> B ) e. ( J Cn ( ( TopOpen ` ℂfld ) ↾t RR ) ) ) )
24	9, 23	mpbird 247	. . . 4 |- ( ( ph /\ k e. A ) -> ( x e. X |-> B ) e. ( J Cn ( TopOpen ` ℂfld ) ) )
25	1, 2, 3, 24	fsumcnf 39180	. . 3 |- ( ph -> ( x e. X |-> sum_ k e. A B ) e. ( J Cn ( TopOpen ` ℂfld ) ) )
26	10	a1i 11	. . . 4 |- ( ph -> ( TopOpen ` ℂfld ) e. (TopOn ` CC ) )
27		refsumcn.1	. . . . . . . . . . 11 |- F/ x ph
28	3	adantr 481	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. X ) -> A e. Fin )
29		simpll 790	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ x e. X ) /\ k e. A ) -> ph )
30		simpr 477	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ x e. X ) /\ k e. A ) -> k e. A )
31	29, 30	jca 554	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x e. X ) /\ k e. A ) -> ( ph /\ k e. A ) )
32		simplr 792	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x e. X ) /\ k e. A ) -> x e. X )
33		eqid 2622	. . . . . . . . . . . . . . . . 17 |- ( x e. X |-> B ) = ( x e. X |-> B )
34	33	fmpt 6381	. . . . . . . . . . . . . . . 16 |- ( A. x e. X B e. RR <-> ( x e. X |-> B ) : X --> RR )
35	17, 34	sylibr 224	. . . . . . . . . . . . . . 15 |- ( ( ph /\ k e. A ) -> A. x e. X B e. RR )
36		rsp 2929	. . . . . . . . . . . . . . 15 |- ( A. x e. X B e. RR -> ( x e. X -> B e. RR ) )
37	35, 36	syl 17	. . . . . . . . . . . . . 14 |- ( ( ph /\ k e. A ) -> ( x e. X -> B e. RR ) )
38	31, 32, 37	sylc 65	. . . . . . . . . . . . 13 |- ( ( ( ph /\ x e. X ) /\ k e. A ) -> B e. RR )
39	28, 38	fsumrecl 14465	. . . . . . . . . . . 12 |- ( ( ph /\ x e. X ) -> sum_ k e. A B e. RR )
40	39	ex 450	. . . . . . . . . . 11 |- ( ph -> ( x e. X -> sum_ k e. A B e. RR ) )
41	27, 40	ralrimi 2957	. . . . . . . . . 10 |- ( ph -> A. x e. X sum_ k e. A B e. RR )
42		eqid 2622	. . . . . . . . . . 11 |- ( x e. X |-> sum_ k e. A B ) = ( x e. X |-> sum_ k e. A B )
43	42	fnmpt 6020	. . . . . . . . . 10 |- ( A. x e. X sum_ k e. A B e. RR -> ( x e. X |-> sum_ k e. A B ) Fn X )
44	41, 43	syl 17	. . . . . . . . 9 |- ( ph -> ( x e. X |-> sum_ k e. A B ) Fn X )
45		nfcv 2764	. . . . . . . . . 10 |- F/_ x X
46		nfcv 2764	. . . . . . . . . 10 |- F/_ x y
47		nfmpt1 4747	. . . . . . . . . 10 |- F/_ x ( x e. X |-> sum_ k e. A B )
48	45, 46, 47	fvelrnbf 39177	. . . . . . . . 9 |- ( ( x e. X |-> sum_ k e. A B ) Fn X -> ( y e. ran ( x e. X |-> sum_ k e. A B ) <-> E. x e. X ( ( x e. X |-> sum_ k e. A B ) ` x ) = y ) )
49	44, 48	syl 17	. . . . . . . 8 |- ( ph -> ( y e. ran ( x e. X |-> sum_ k e. A B ) <-> E. x e. X ( ( x e. X |-> sum_ k e. A B ) ` x ) = y ) )
50	49	biimpa 501	. . . . . . 7 |- ( ( ph /\ y e. ran ( x e. X |-> sum_ k e. A B ) ) -> E. x e. X ( ( x e. X |-> sum_ k e. A B ) ` x ) = y )
51	47	nfrn 5368	. . . . . . . . . 10 |- F/_ x ran ( x e. X |-> sum_ k e. A B )
52	51	nfcri 2758	. . . . . . . . 9 |- F/ x y e. ran ( x e. X |-> sum_ k e. A B )
53	27, 52	nfan 1828	. . . . . . . 8 |- F/ x ( ph /\ y e. ran ( x e. X |-> sum_ k e. A B ) )
54		nfcv 2764	. . . . . . . . 9 |- F/_ x RR
55	54	nfcri 2758	. . . . . . . 8 |- F/ x y e. RR
56		simpr 477	. . . . . . . . . . . . . . 15 |- ( ( ph /\ x e. X ) -> x e. X )
57	56, 39	jca 554	. . . . . . . . . . . . . 14 |- ( ( ph /\ x e. X ) -> ( x e. X /\ sum_ k e. A B e. RR ) )
58	42	fvmpt2 6291	. . . . . . . . . . . . . 14 |- ( ( x e. X /\ sum_ k e. A B e. RR ) -> ( ( x e. X |-> sum_ k e. A B ) ` x ) = sum_ k e. A B )
59	57, 58	syl 17	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. X ) -> ( ( x e. X |-> sum_ k e. A B ) ` x ) = sum_ k e. A B )
60	59	3adant3 1081	. . . . . . . . . . . 12 |- ( ( ph /\ x e. X /\ ( ( x e. X |-> sum_ k e. A B ) ` x ) = y ) -> ( ( x e. X |-> sum_ k e. A B ) ` x ) = sum_ k e. A B )
61		simp3 1063	. . . . . . . . . . . 12 |- ( ( ph /\ x e. X /\ ( ( x e. X |-> sum_ k e. A B ) ` x ) = y ) -> ( ( x e. X |-> sum_ k e. A B ) ` x ) = y )
62	60, 61	eqtr3d 2658	. . . . . . . . . . 11 |- ( ( ph /\ x e. X /\ ( ( x e. X |-> sum_ k e. A B ) ` x ) = y ) -> sum_ k e. A B = y )
63	39	3adant3 1081	. . . . . . . . . . 11 |- ( ( ph /\ x e. X /\ ( ( x e. X |-> sum_ k e. A B ) ` x ) = y ) -> sum_ k e. A B e. RR )
64	62, 63	eqeltrrd 2702	. . . . . . . . . 10 |- ( ( ph /\ x e. X /\ ( ( x e. X |-> sum_ k e. A B ) ` x ) = y ) -> y e. RR )
65	64	3adant1r 1319	. . . . . . . . 9 |- ( ( ( ph /\ y e. ran ( x e. X |-> sum_ k e. A B ) ) /\ x e. X /\ ( ( x e. X |-> sum_ k e. A B ) ` x ) = y ) -> y e. RR )
66	65	3exp 1264	. . . . . . . 8 |- ( ( ph /\ y e. ran ( x e. X |-> sum_ k e. A B ) ) -> ( x e. X -> ( ( ( x e. X |-> sum_ k e. A B ) ` x ) = y -> y e. RR ) ) )
67	53, 55, 66	rexlimd 3026	. . . . . . 7 |- ( ( ph /\ y e. ran ( x e. X |-> sum_ k e. A B ) ) -> ( E. x e. X ( ( x e. X |-> sum_ k e. A B ) ` x ) = y -> y e. RR ) )
68	50, 67	mpd 15	. . . . . 6 |- ( ( ph /\ y e. ran ( x e. X |-> sum_ k e. A B ) ) -> y e. RR )
69	68	ex 450	. . . . 5 |- ( ph -> ( y e. ran ( x e. X |-> sum_ k e. A B ) -> y e. RR ) )
70	69	ssrdv 3609	. . . 4 |- ( ph -> ran ( x e. X |-> sum_ k e. A B ) C_ RR )
71	20	a1i 11	. . . 4 |- ( ph -> RR C_ CC )
72		cnrest2 21090	. . . 4 |- ( ( ( TopOpen ` ℂfld ) e. (TopOn ` CC ) /\ ran ( x e. X |-> sum_ k e. A B ) C_ RR /\ RR C_ CC ) -> ( ( x e. X |-> sum_ k e. A B ) e. ( J Cn ( TopOpen ` ℂfld ) ) <-> ( x e. X |-> sum_ k e. A B ) e. ( J Cn ( ( TopOpen ` ℂfld ) ↾t RR ) ) ) )
73	26, 70, 71, 72	syl3anc 1326	. . 3 |- ( ph -> ( ( x e. X |-> sum_ k e. A B ) e. ( J Cn ( TopOpen ` ℂfld ) ) <-> ( x e. X |-> sum_ k e. A B ) e. ( J Cn ( ( TopOpen ` ℂfld ) ↾t RR ) ) ) )
74	25, 73	mpbid 222	. 2 |- ( ph -> ( x e. X |-> sum_ k e. A B ) e. ( J Cn ( ( TopOpen ` ℂfld ) ↾t RR ) ) )
75	74, 8	syl6eleqr 2712	1 |- ( ph -> ( x e. X |-> sum_ k e. A B ) e. ( J Cn K ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   F/wnf 1708   e. wcel 1990   A.wral 2912   E.wrex 2913   C_ wss 3574   |-> cmpt 4729   ran crn 5115   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   Fincfn 7955   CCcc 9934   RRcr 9935   (,)cioo 12175   sum_csu 14416   ↾_t crest 16081   TopOpenctopn 16082   topGenctg 16098  ℂ_fldccnfld 19746  TopOnctopon 20715   Cn ccn 21028

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-inf2 8538  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  ax-addf 10015

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  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-se 5074  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-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  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-ioo 12179  df-icc 12182  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-sum 14417  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-cmn 18195  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-cn 21031  df-cnp 21032  df-tx 21365  df-hmeo 21558  df-xms 22125  df-ms 22126  df-tms 22127

This theorem is referenced by:  refsum2cnlem1  39196