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Theorem refsumcn 30802
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 21102 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.
StepHypRef Expression
1 eqid 2460 . . . 4  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
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  (,) )
61tgioo2 21036 . . . . . . . 8  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
75, 6eqtri 2489 . . . . . . 7  |-  K  =  ( ( TopOpen ` fld )t  RR )
87oveq2i 6286 . . . . . 6  |-  ( J  Cn  K )  =  ( J  Cn  (
( TopOpen ` fld )t  RR ) )
94, 8syl6eleq 2558 . . . . 5  |-  ( (
ph  /\  k  e.  A )  ->  (
x  e.  X  |->  B )  e.  ( J  Cn  ( ( TopOpen ` fld )t  RR ) ) )
101cnfldtopon 21018 . . . . . . 7  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
1110a1i 11 . . . . . 6  |-  ( (
ph  /\  k  e.  A )  ->  ( TopOpen
` fld
)  e.  (TopOn `  CC ) )
122adantr 465 . . . . . . . 8  |-  ( (
ph  /\  k  e.  A )  ->  J  e.  (TopOn `  X )
)
13 retopon 20998 . . . . . . . . . 10  |-  ( topGen ` 
ran  (,) )  e.  (TopOn `  RR )
145, 13eqeltri 2544 . . . . . . . . 9  |-  K  e.  (TopOn `  RR )
1514a1i 11 . . . . . . . 8  |-  ( (
ph  /\  k  e.  A )  ->  K  e.  (TopOn `  RR )
)
16 cnf2 19509 . . . . . . . 8  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  RR )  /\  ( x  e.  X  |->  B )  e.  ( J  Cn  K ) )  ->  ( x  e.  X  |->  B ) : X --> RR )
1712, 15, 4, 16syl3anc 1223 . . . . . . 7  |-  ( (
ph  /\  k  e.  A )  ->  (
x  e.  X  |->  B ) : X --> RR )
18 frn 5728 . . . . . . 7  |-  ( ( x  e.  X  |->  B ) : X --> RR  ->  ran  ( x  e.  X  |->  B )  C_  RR )
1917, 18syl 16 . . . . . 6  |-  ( (
ph  /\  k  e.  A )  ->  ran  ( x  e.  X  |->  B )  C_  RR )
20 ax-resscn 9538 . . . . . . 7  |-  RR  C_  CC
2120a1i 11 . . . . . 6  |-  ( (
ph  /\  k  e.  A )  ->  RR  C_  CC )
22 cnrest2 19546 . . . . . 6  |-  ( ( ( TopOpen ` fld )  e.  (TopOn `  CC )  /\  ran  ( x  e.  X  |->  B )  C_  RR  /\  RR  C_  CC )  ->  ( ( x  e.  X  |->  B )  e.  ( J  Cn  ( TopOpen
` fld
) )  <->  ( x  e.  X  |->  B )  e.  ( J  Cn  ( ( TopOpen ` fld )t  RR ) ) ) )
2311, 19, 21, 22syl3anc 1223 . . . . 5  |-  ( (
ph  /\  k  e.  A )  ->  (
( x  e.  X  |->  B )  e.  ( J  Cn  ( TopOpen ` fld )
)  <->  ( x  e.  X  |->  B )  e.  ( J  Cn  (
( TopOpen ` fld )t  RR ) ) ) )
249, 23mpbird 232 . . . 4  |-  ( (
ph  /\  k  e.  A )  ->  (
x  e.  X  |->  B )  e.  ( J  Cn  ( TopOpen ` fld ) ) )
251, 2, 3, 24fsumcnf 30793 . . 3  |-  ( ph  ->  ( x  e.  X  |-> 
sum_ k  e.  A  B )  e.  ( J  Cn  ( TopOpen ` fld )
) )
2610a1i 11 . . . 4  |-  ( ph  ->  ( TopOpen ` fld )  e.  (TopOn `  CC ) )
27 refsumcn.1 . . . . . . . . . . 11  |-  F/ x ph
283adantr 465 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  X )  ->  A  e.  Fin )
29 simpll 753 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  x  e.  X )  /\  k  e.  A )  ->  ph )
30 simpr 461 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  x  e.  X )  /\  k  e.  A )  ->  k  e.  A )
3129, 30jca 532 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  X )  /\  k  e.  A )  ->  ( ph  /\  k  e.  A
) )
32 simplr 754 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  X )  /\  k  e.  A )  ->  x  e.  X )
33 eqid 2460 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  X  |->  B )  =  ( x  e.  X  |->  B )
3433fmpt 6033 . . . . . . . . . . . . . . . 16  |-  ( A. x  e.  X  B  e.  RR  <->  ( x  e.  X  |->  B ) : X --> RR )
3517, 34sylibr 212 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  k  e.  A )  ->  A. x  e.  X  B  e.  RR )
36 rsp 2823 . . . . . . . . . . . . . . 15  |-  ( A. x  e.  X  B  e.  RR  ->  ( x  e.  X  ->  B  e.  RR ) )
3735, 36syl 16 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  k  e.  A )  ->  (
x  e.  X  ->  B  e.  RR )
)
3831, 32, 37sylc 60 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  X )  /\  k  e.  A )  ->  B  e.  RR )
3928, 38fsumrecl 13505 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  X )  ->  sum_ k  e.  A  B  e.  RR )
4039ex 434 . . . . . . . . . . 11  |-  ( ph  ->  ( x  e.  X  -> 
sum_ k  e.  A  B  e.  RR )
)
4127, 40ralrimi 2857 . . . . . . . . . 10  |-  ( ph  ->  A. x  e.  X  sum_ k  e.  A  B  e.  RR )
42 eqid 2460 . . . . . . . . . . 11  |-  ( x  e.  X  |->  sum_ k  e.  A  B )  =  ( x  e.  X  |->  sum_ k  e.  A  B )
4342fnmpt 5698 . . . . . . . . . 10  |-  ( A. x  e.  X  sum_ k  e.  A  B  e.  RR  ->  ( x  e.  X  |->  sum_ k  e.  A  B )  Fn  X )
4441, 43syl 16 . . . . . . . . 9  |-  ( ph  ->  ( x  e.  X  |-> 
sum_ k  e.  A  B )  Fn  X
)
45 nfcv 2622 . . . . . . . . . 10  |-  F/_ x X
46 nfcv 2622 . . . . . . . . . 10  |-  F/_ x
y
47 nfmpt1 4529 . . . . . . . . . 10  |-  F/_ x
( x  e.  X  |-> 
sum_ k  e.  A  B )
4845, 46, 47fvelrnbf 30790 . . . . . . . . 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 ) )
4944, 48syl 16 . . . . . . . 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 ) )
5049biimpa 484 . . . . . . 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 )
5147nfrn 5236 . . . . . . . . . 10  |-  F/_ x ran  ( x  e.  X  |-> 
sum_ k  e.  A  B )
5251nfcri 2615 . . . . . . . . 9  |-  F/ x  y  e.  ran  ( x  e.  X  |->  sum_ k  e.  A  B )
5327, 52nfan 1870 . . . . . . . 8  |-  F/ x
( ph  /\  y  e.  ran  ( x  e.  X  |->  sum_ k  e.  A  B ) )
54 nfcv 2622 . . . . . . . . 9  |-  F/_ x RR
5554nfcri 2615 . . . . . . . 8  |-  F/ x  y  e.  RR
56 simpr 461 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  X )  ->  x  e.  X )
5756, 39jca 532 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  X )  ->  (
x  e.  X  /\  sum_ k  e.  A  B  e.  RR ) )
5842fvmpt2 5948 . . . . . . . . . . . . . 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 )
5957, 58syl 16 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  X )  ->  (
( x  e.  X  |-> 
sum_ k  e.  A  B ) `  x
)  =  sum_ k  e.  A  B )
60593adant3 1011 . . . . . . . . . . . 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 993 . . . . . . . . . . . 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 )
6260, 61eqtr3d 2503 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  X  /\  ( ( x  e.  X  |->  sum_ k  e.  A  B ) `  x )  =  y )  ->  sum_ k  e.  A  B  =  y )
63393adant3 1011 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  X  /\  ( ( x  e.  X  |->  sum_ k  e.  A  B ) `  x )  =  y )  ->  sum_ k  e.  A  B  e.  RR )
6462, 63eqeltrrd 2549 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  X  /\  ( ( x  e.  X  |->  sum_ k  e.  A  B ) `  x )  =  y )  ->  y  e.  RR )
65643adant1r 1216 . . . . . . . . 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 )
66653exp 1190 . . . . . . . 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 ) ) )
6753, 55, 66rexlimd 2940 . . . . . . 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 ) )
6850, 67mpd 15 . . . . . 6  |-  ( (
ph  /\  y  e.  ran  ( x  e.  X  |-> 
sum_ k  e.  A  B ) )  -> 
y  e.  RR )
6968ex 434 . . . . 5  |-  ( ph  ->  ( y  e.  ran  ( x  e.  X  |-> 
sum_ k  e.  A  B )  ->  y  e.  RR ) )
7069ssrdv 3503 . . . 4  |-  ( ph  ->  ran  ( x  e.  X  |->  sum_ k  e.  A  B )  C_  RR )
7120a1i 11 . . . 4  |-  ( ph  ->  RR  C_  CC )
72 cnrest2 19546 . . . 4  |-  ( ( ( TopOpen ` fld )  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 ` fld )
)  <->  ( x  e.  X  |->  sum_ k  e.  A  B )  e.  ( J  Cn  ( (
TopOpen ` fld )t  RR ) ) ) )
7326, 70, 71, 72syl3anc 1223 . . 3  |-  ( ph  ->  ( ( x  e.  X  |->  sum_ k  e.  A  B )  e.  ( J  Cn  ( TopOpen ` fld )
)  <->  ( x  e.  X  |->  sum_ k  e.  A  B )  e.  ( J  Cn  ( (
TopOpen ` fld )t  RR ) ) ) )
7425, 73mpbid 210 . 2  |-  ( ph  ->  ( x  e.  X  |-> 
sum_ k  e.  A  B )  e.  ( J  Cn  ( (
TopOpen ` fld )t  RR ) ) )
7574, 8syl6eleqr 2559 1  |-  ( ph  ->  ( x  e.  X  |-> 
sum_ k  e.  A  B )  e.  ( J  Cn  K ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 968    = wceq 1374   F/wnf 1594    e. wcel 1762   A.wral 2807   E.wrex 2808    C_ wss 3469    |-> cmpt 4498   ran crn 4993    Fn wfn 5574   -->wf 5575   ` cfv 5579  (class class class)co 6275   Fincfn 7506   CCcc 9479   RRcr 9480   (,)cioo 11518   sum_csu 13457   ↾t crest 14665   TopOpenctopn 14666   topGenctg 14682  ℂfldccnfld 18184  TopOnctopon 19155    Cn ccn 19484
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-inf2 8047  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559  ax-addf 9560
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-fal 1380  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-iin 4321  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-se 4832  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-isom 5588  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-of 6515  df-om 6672  df-1st 6774  df-2nd 6775  df-supp 6892  df-recs 7032  df-rdg 7066  df-1o 7120  df-2o 7121  df-oadd 7124  df-er 7301  df-map 7412  df-ixp 7460  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-fsupp 7819  df-fi 7860  df-sup 7890  df-oi 7924  df-card 8309  df-cda 8537  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196  df-nn 10526  df-2 10583  df-3 10584  df-4 10585  df-5 10586  df-6 10587  df-7 10588  df-8 10589  df-9 10590  df-10 10591  df-n0 10785  df-z 10854  df-dec 10966  df-uz 11072  df-q 11172  df-rp 11210  df-xneg 11307  df-xadd 11308  df-xmul 11309  df-ioo 11522  df-icc 11525  df-fz 11662  df-fzo 11782  df-seq 12064  df-exp 12123  df-hash 12361  df-cj 12882  df-re 12883  df-im 12884  df-sqr 13018  df-abs 13019  df-clim 13260  df-sum 13458  df-struct 14481  df-ndx 14482  df-slot 14483  df-base 14484  df-sets 14485  df-ress 14486  df-plusg 14557  df-mulr 14558  df-starv 14559  df-sca 14560  df-vsca 14561  df-ip 14562  df-tset 14563  df-ple 14564  df-ds 14566  df-unif 14567  df-hom 14568  df-cco 14569  df-rest 14667  df-topn 14668  df-0g 14686  df-gsum 14687  df-topgen 14688  df-pt 14689  df-prds 14692  df-xrs 14746  df-qtop 14751  df-imas 14752  df-xps 14754  df-mre 14830  df-mrc 14831  df-acs 14833  df-mnd 15721  df-submnd 15771  df-mulg 15854  df-cntz 16143  df-cmn 16589  df-psmet 18175  df-xmet 18176  df-met 18177  df-bl 18178  df-mopn 18179  df-cnfld 18185  df-top 19159  df-bases 19161  df-topon 19162  df-topsp 19163  df-cn 19487  df-cnp 19488  df-tx 19791  df-hmeo 19984  df-xms 20551  df-ms 20552  df-tms 20553
This theorem is referenced by:  refsum2cnlem1  30809
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