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Theorem refsumcn 27701
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 18374 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 2283 . . . 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 18309 . . . . . . . 8  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
75, 6eqtri 2303 . . . . . . 7  |-  K  =  ( ( TopOpen ` fld )t  RR )
87oveq2i 5869 . . . . . 6  |-  ( J  Cn  K )  =  ( J  Cn  (
( TopOpen ` fld )t  RR ) )
94, 8syl6eleq 2373 . . . . 5  |-  ( (
ph  /\  k  e.  A )  ->  (
x  e.  X  |->  B )  e.  ( J  Cn  ( ( TopOpen ` fld )t  RR ) ) )
101cnfldtopon 18292 . . . . . . . 8  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
1110a1i 10 . . . . . . 7  |-  ( (
ph  /\  k  e.  A )  ->  ( TopOpen
` fld
)  e.  (TopOn `  CC ) )
122adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  A )  ->  J  e.  (TopOn `  X )
)
13 retopon 18272 . . . . . . . . . . . 12  |-  ( topGen ` 
ran  (,) )  e.  (TopOn `  RR )
145, 13eqeltri 2353 . . . . . . . . . . 11  |-  K  e.  (TopOn `  RR )
1514a1i 10 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  A )  ->  K  e.  (TopOn `  RR )
)
1612, 15, 43jca 1132 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  A )  ->  ( J  e.  (TopOn `  X
)  /\  K  e.  (TopOn `  RR )  /\  ( x  e.  X  |->  B )  e.  ( J  Cn  K ) ) )
17 cnf2 16979 . . . . . . . . 9  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  RR )  /\  ( x  e.  X  |->  B )  e.  ( J  Cn  K ) )  ->  ( x  e.  X  |->  B ) : X --> RR )
1816, 17syl 15 . . . . . . . 8  |-  ( (
ph  /\  k  e.  A )  ->  (
x  e.  X  |->  B ) : X --> RR )
19 frn 5395 . . . . . . . 8  |-  ( ( x  e.  X  |->  B ) : X --> RR  ->  ran  ( x  e.  X  |->  B )  C_  RR )
2018, 19syl 15 . . . . . . 7  |-  ( (
ph  /\  k  e.  A )  ->  ran  ( x  e.  X  |->  B )  C_  RR )
21 ax-resscn 8794 . . . . . . . 8  |-  RR  C_  CC
2221a1i 10 . . . . . . 7  |-  ( (
ph  /\  k  e.  A )  ->  RR  C_  CC )
2311, 20, 223jca 1132 . . . . . 6  |-  ( (
ph  /\  k  e.  A )  ->  (
( TopOpen ` fld )  e.  (TopOn `  CC )  /\  ran  ( x  e.  X  |->  B )  C_  RR  /\  RR  C_  CC )
)
24 cnrest2 17014 . . . . . 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 ) ) ) )
2523, 24syl 15 . . . . 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 ) ) ) )
269, 25mpbird 223 . . . 4  |-  ( (
ph  /\  k  e.  A )  ->  (
x  e.  X  |->  B )  e.  ( J  Cn  ( TopOpen ` fld ) ) )
271, 2, 3, 26fsumcnf 27692 . . 3  |-  ( ph  ->  ( x  e.  X  |-> 
sum_ k  e.  A  B )  e.  ( J  Cn  ( TopOpen ` fld )
) )
2810a1i 10 . . . . 5  |-  ( ph  ->  ( TopOpen ` fld )  e.  (TopOn `  CC ) )
29 refsumcn.1 . . . . . . . . . . . 12  |-  F/ x ph
303adantr 451 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  X )  ->  A  e.  Fin )
31 simpll 730 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  x  e.  X )  /\  k  e.  A )  ->  ph )
32 simpr 447 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  x  e.  X )  /\  k  e.  A )  ->  k  e.  A )
3331, 32jca 518 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  x  e.  X )  /\  k  e.  A )  ->  ( ph  /\  k  e.  A
) )
34 simplr 731 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  x  e.  X )  /\  k  e.  A )  ->  x  e.  X )
35 eqid 2283 . . . . . . . . . . . . . . . . . 18  |-  ( x  e.  X  |->  B )  =  ( x  e.  X  |->  B )
3635fmpt 5681 . . . . . . . . . . . . . . . . 17  |-  ( A. x  e.  X  B  e.  RR  <->  ( x  e.  X  |->  B ) : X --> RR )
3718, 36sylibr 203 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  k  e.  A )  ->  A. x  e.  X  B  e.  RR )
38 rsp 2603 . . . . . . . . . . . . . . . 16  |-  ( A. x  e.  X  B  e.  RR  ->  ( x  e.  X  ->  B  e.  RR ) )
3937, 38syl 15 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  k  e.  A )  ->  (
x  e.  X  ->  B  e.  RR )
)
4033, 34, 39sylc 56 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  X )  /\  k  e.  A )  ->  B  e.  RR )
4130, 40fsumrecl 12207 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  X )  ->  sum_ k  e.  A  B  e.  RR )
4241ex 423 . . . . . . . . . . . 12  |-  ( ph  ->  ( x  e.  X  -> 
sum_ k  e.  A  B  e.  RR )
)
4329, 42ralrimi 2624 . . . . . . . . . . 11  |-  ( ph  ->  A. x  e.  X  sum_ k  e.  A  B  e.  RR )
44 eqid 2283 . . . . . . . . . . . 12  |-  ( x  e.  X  |->  sum_ k  e.  A  B )  =  ( x  e.  X  |->  sum_ k  e.  A  B )
4544fnmpt 5370 . . . . . . . . . . 11  |-  ( A. x  e.  X  sum_ k  e.  A  B  e.  RR  ->  ( x  e.  X  |->  sum_ k  e.  A  B )  Fn  X )
4643, 45syl 15 . . . . . . . . . 10  |-  ( ph  ->  ( x  e.  X  |-> 
sum_ k  e.  A  B )  Fn  X
)
47 nfcv 2419 . . . . . . . . . . 11  |-  F/_ x X
48 nfcv 2419 . . . . . . . . . . 11  |-  F/_ x
y
49 nfmpt1 4109 . . . . . . . . . . 11  |-  F/_ x
( x  e.  X  |-> 
sum_ k  e.  A  B )
5047, 48, 49fvelrnbf 27689 . . . . . . . . . 10  |-  ( ( 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 ) )
5146, 50syl 15 . . . . . . . . 9  |-  ( 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 ) )
5251biimpa 470 . . . . . . . 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 )
5349nfrn 4921 . . . . . . . . . . 11  |-  F/_ x ran  ( x  e.  X  |-> 
sum_ k  e.  A  B )
5448, 53nfel 2427 . . . . . . . . . 10  |-  F/ x  y  e.  ran  ( x  e.  X  |->  sum_ k  e.  A  B )
5529, 54nfan 1771 . . . . . . . . 9  |-  F/ x
( ph  /\  y  e.  ran  ( x  e.  X  |->  sum_ k  e.  A  B ) )
56 nfcv 2419 . . . . . . . . . 10  |-  F/_ x RR
5748, 56nfel 2427 . . . . . . . . 9  |-  F/ x  y  e.  RR
58 simp1l 979 . . . . . . . . . . . 12  |-  ( ( ( 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 )  ->  ph )
59 simp2 956 . . . . . . . . . . . 12  |-  ( ( ( 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 )  ->  x  e.  X
)
60 simp3 957 . . . . . . . . . . . 12  |-  ( ( ( 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 )  ->  ( ( x  e.  X  |->  sum_ k  e.  A  B ) `  x )  =  y )
6158, 59, 603jca 1132 . . . . . . . . . . 11  |-  ( ( ( 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 )  ->  ( ph  /\  x  e.  X  /\  ( ( x  e.  X  |->  sum_ k  e.  A  B ) `  x
)  =  y ) )
62 simpr 447 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  X )  ->  x  e.  X )
6362, 41jca 518 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  X )  ->  (
x  e.  X  /\  sum_ k  e.  A  B  e.  RR ) )
6444fvmpt2 5608 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  X  /\  sum_ k  e.  A  B  e.  RR )  ->  (
( x  e.  X  |-> 
sum_ k  e.  A  B ) `  x
)  =  sum_ k  e.  A  B )
6563, 64syl 15 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  X )  ->  (
( x  e.  X  |-> 
sum_ k  e.  A  B ) `  x
)  =  sum_ k  e.  A  B )
66653adant3 975 . . . . . . . . . . . . 13  |-  ( (
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 )
67 simp3 957 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  X  /\  ( ( x  e.  X  |->  sum_ k  e.  A  B ) `  x )  =  y )  ->  ( (
x  e.  X  |->  sum_ k  e.  A  B
) `  x )  =  y )
6866, 67eqtr3d 2317 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  X  /\  ( ( x  e.  X  |->  sum_ k  e.  A  B ) `  x )  =  y )  ->  sum_ k  e.  A  B  =  y )
69413adant3 975 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  X  /\  ( ( x  e.  X  |->  sum_ k  e.  A  B ) `  x )  =  y )  ->  sum_ k  e.  A  B  e.  RR )
7068, 69eqeltrrd 2358 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  X  /\  ( ( x  e.  X  |->  sum_ k  e.  A  B ) `  x )  =  y )  ->  y  e.  RR )
7161, 70syl 15 . . . . . . . . . 10  |-  ( ( ( 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 )
72713exp 1150 . . . . . . . . 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 ) ) )
7355, 57, 72rexlimd 2664 . . . . . . . 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  ->  y  e.  RR ) )
7452, 73mpd 14 . . . . . . 7  |-  ( (
ph  /\  y  e.  ran  ( x  e.  X  |-> 
sum_ k  e.  A  B ) )  -> 
y  e.  RR )
7574ex 423 . . . . . 6  |-  ( ph  ->  ( y  e.  ran  ( x  e.  X  |-> 
sum_ k  e.  A  B )  ->  y  e.  RR ) )
7675ssrdv 3185 . . . . 5  |-  ( ph  ->  ran  ( x  e.  X  |->  sum_ k  e.  A  B )  C_  RR )
7721a1i 10 . . . . 5  |-  ( ph  ->  RR  C_  CC )
7828, 76, 773jca 1132 . . . 4  |-  ( ph  ->  ( ( TopOpen ` fld )  e.  (TopOn `  CC )  /\  ran  ( x  e.  X  |-> 
sum_ k  e.  A  B )  C_  RR  /\  RR  C_  CC )
)
79 cnrest2 17014 . . . 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 ) ) ) )
8078, 79syl 15 . . 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 ) ) ) )
8127, 80mpbid 201 . 2  |-  ( ph  ->  ( x  e.  X  |-> 
sum_ k  e.  A  B )  e.  ( J  Cn  ( (
TopOpen ` fld )t  RR ) ) )
8281, 8syl6eleqr 2374 1  |-  ( ph  ->  ( x  e.  X  |-> 
sum_ k  e.  A  B )  e.  ( J  Cn  K ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934   F/wnf 1531    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544    C_ wss 3152    e. cmpt 4077   ran crn 4690    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   Fincfn 6863   CCcc 8735   RRcr 8736   (,)cioo 10656   sum_csu 12158   ↾t crest 13325   TopOpenctopn 13326   topGenctg 13342  ℂfldccnfld 16377  TopOnctopon 16632    Cn ccn 16954
This theorem is referenced by:  refsum2cnlem1  27708
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-oi 7225  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ioo 10660  df-icc 10663  df-fz 10783  df-fzo 10871  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-sum 12159  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-rest 13327  df-topn 13328  df-topgen 13344  df-pt 13345  df-prds 13348  df-xrs 13403  df-0g 13404  df-gsum 13405  df-qtop 13410  df-imas 13411  df-xps 13413  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-mulg 14492  df-cntz 14793  df-cmn 15091  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cn 16957  df-cnp 16958  df-tx 17257  df-hmeo 17446  df-xms 17885  df-ms 17886  df-tms 17887
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