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Theorem refsumcn 27804
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 18390 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 2296 . . . 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 18325 . . . . . . . 8  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
75, 6eqtri 2316 . . . . . . 7  |-  K  =  ( ( TopOpen ` fld )t  RR )
87oveq2i 5885 . . . . . 6  |-  ( J  Cn  K )  =  ( J  Cn  (
( TopOpen ` fld )t  RR ) )
94, 8syl6eleq 2386 . . . . 5  |-  ( (
ph  /\  k  e.  A )  ->  (
x  e.  X  |->  B )  e.  ( J  Cn  ( ( TopOpen ` fld )t  RR ) ) )
101cnfldtopon 18308 . . . . . . . 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 18288 . . . . . . . . . . . 12  |-  ( topGen ` 
ran  (,) )  e.  (TopOn `  RR )
145, 13eqeltri 2366 . . . . . . . . . . 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 16995 . . . . . . . . 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 5411 . . . . . . . 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 8810 . . . . . . . 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 17030 . . . . . 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 27795 . . 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 2296 . . . . . . . . . . . . . . . . . 18  |-  ( x  e.  X  |->  B )  =  ( x  e.  X  |->  B )
3635fmpt 5697 . . . . . . . . . . . . . . . . 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 2616 . . . . . . . . . . . . . . . 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 12223 . . . . . . . . . . . . 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 2637 . . . . . . . . . . 11  |-  ( ph  ->  A. x  e.  X  sum_ k  e.  A  B  e.  RR )
44 eqid 2296 . . . . . . . . . . . 12  |-  ( x  e.  X  |->  sum_ k  e.  A  B )  =  ( x  e.  X  |->  sum_ k  e.  A  B )
4544fnmpt 5386 . . . . . . . . . . 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 2432 . . . . . . . . . . 11  |-  F/_ x X
48 nfcv 2432 . . . . . . . . . . 11  |-  F/_ x
y
49 nfmpt1 4125 . . . . . . . . . . 11  |-  F/_ x
( x  e.  X  |-> 
sum_ k  e.  A  B )
5047, 48, 49fvelrnbf 27792 . . . . . . . . . 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 4937 . . . . . . . . . . 11  |-  F/_ x ran  ( x  e.  X  |-> 
sum_ k  e.  A  B )
5448, 53nfel 2440 . . . . . . . . . 10  |-  F/ x  y  e.  ran  ( x  e.  X  |->  sum_ k  e.  A  B )
5529, 54nfan 1783 . . . . . . . . 9  |-  F/ x
( ph  /\  y  e.  ran  ( x  e.  X  |->  sum_ k  e.  A  B ) )
56 nfcv 2432 . . . . . . . . . 10  |-  F/_ x RR
5748, 56nfel 2440 . . . . . . . . 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 5624 . . . . . . . . . . . . . . 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 2330 . . . . . . . . . . . 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 2371 . . . . . . . . . . 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 2677 . . . . . . . 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 3198 . . . . 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 17030 . . . 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 2387 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 1534    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557    C_ wss 3165    e. cmpt 4093   ran crn 4706    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874   Fincfn 6879   CCcc 8751   RRcr 8752   (,)cioo 10672   sum_csu 12174   ↾t crest 13341   TopOpenctopn 13342   topGenctg 13358  ℂfldccnfld 16393  TopOnctopon 16648    Cn ccn 16970
This theorem is referenced by:  refsum2cnlem1  27811
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-fi 7181  df-sup 7210  df-oi 7241  df-card 7588  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-ioo 10676  df-icc 10679  df-fz 10799  df-fzo 10887  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-clim 11978  df-sum 12175  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-starv 13239  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-hom 13248  df-cco 13249  df-rest 13343  df-topn 13344  df-topgen 13360  df-pt 13361  df-prds 13364  df-xrs 13419  df-0g 13420  df-gsum 13421  df-qtop 13426  df-imas 13427  df-xps 13429  df-mre 13504  df-mrc 13505  df-acs 13507  df-mnd 14383  df-submnd 14432  df-mulg 14508  df-cntz 14809  df-cmn 15107  df-xmet 16389  df-met 16390  df-bl 16391  df-mopn 16392  df-cnfld 16394  df-top 16652  df-bases 16654  df-topon 16655  df-topsp 16656  df-cn 16973  df-cnp 16974  df-tx 17273  df-hmeo 17462  df-xms 17901  df-ms 17902  df-tms 17903
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