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Theorem refsumcn 36785
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 21666 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 2402 . . . 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 21600 . . . . . . . 8  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
75, 6eqtri 2431 . . . . . . 7  |-  K  =  ( ( TopOpen ` fld )t  RR )
87oveq2i 6289 . . . . . 6  |-  ( J  Cn  K )  =  ( J  Cn  (
( TopOpen ` fld )t  RR ) )
94, 8syl6eleq 2500 . . . . 5  |-  ( (
ph  /\  k  e.  A )  ->  (
x  e.  X  |->  B )  e.  ( J  Cn  ( ( TopOpen ` fld )t  RR ) ) )
101cnfldtopon 21582 . . . . . . 7  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
1110a1i 11 . . . . . 6  |-  ( (
ph  /\  k  e.  A )  ->  ( TopOpen
` fld
)  e.  (TopOn `  CC ) )
122adantr 463 . . . . . . . 8  |-  ( (
ph  /\  k  e.  A )  ->  J  e.  (TopOn `  X )
)
13 retopon 21562 . . . . . . . . . 10  |-  ( topGen ` 
ran  (,) )  e.  (TopOn `  RR )
145, 13eqeltri 2486 . . . . . . . . 9  |-  K  e.  (TopOn `  RR )
1514a1i 11 . . . . . . . 8  |-  ( (
ph  /\  k  e.  A )  ->  K  e.  (TopOn `  RR )
)
16 cnf2 20043 . . . . . . . 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 1230 . . . . . . 7  |-  ( (
ph  /\  k  e.  A )  ->  (
x  e.  X  |->  B ) : X --> RR )
18 frn 5720 . . . . . . 7  |-  ( ( x  e.  X  |->  B ) : X --> RR  ->  ran  ( x  e.  X  |->  B )  C_  RR )
1917, 18syl 17 . . . . . 6  |-  ( (
ph  /\  k  e.  A )  ->  ran  ( x  e.  X  |->  B )  C_  RR )
20 ax-resscn 9579 . . . . . . 7  |-  RR  C_  CC
2120a1i 11 . . . . . 6  |-  ( (
ph  /\  k  e.  A )  ->  RR  C_  CC )
22 cnrest2 20080 . . . . . 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 1230 . . . . 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 36776 . . 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 463 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  X )  ->  A  e.  Fin )
29 simpll 752 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  x  e.  X )  /\  k  e.  A )  ->  ph )
30 simpr 459 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  x  e.  X )  /\  k  e.  A )  ->  k  e.  A )
3129, 30jca 530 . . . . . . . . . . . . . 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 2402 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  X  |->  B )  =  ( x  e.  X  |->  B )
3433fmpt 6030 . . . . . . . . . . . . . . . 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 2770 . . . . . . . . . . . . . . 15  |-  ( A. x  e.  X  B  e.  RR  ->  ( x  e.  X  ->  B  e.  RR ) )
3735, 36syl 17 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  k  e.  A )  ->  (
x  e.  X  ->  B  e.  RR )
)
3831, 32, 37sylc 59 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  X )  /\  k  e.  A )  ->  B  e.  RR )
3928, 38fsumrecl 13705 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  X )  ->  sum_ k  e.  A  B  e.  RR )
4039ex 432 . . . . . . . . . . 11  |-  ( ph  ->  ( x  e.  X  -> 
sum_ k  e.  A  B  e.  RR )
)
4127, 40ralrimi 2804 . . . . . . . . . 10  |-  ( ph  ->  A. x  e.  X  sum_ k  e.  A  B  e.  RR )
42 eqid 2402 . . . . . . . . . . 11  |-  ( x  e.  X  |->  sum_ k  e.  A  B )  =  ( x  e.  X  |->  sum_ k  e.  A  B )
4342fnmpt 5690 . . . . . . . . . 10  |-  ( A. x  e.  X  sum_ k  e.  A  B  e.  RR  ->  ( x  e.  X  |->  sum_ k  e.  A  B )  Fn  X )
4441, 43syl 17 . . . . . . . . 9  |-  ( ph  ->  ( x  e.  X  |-> 
sum_ k  e.  A  B )  Fn  X
)
45 nfcv 2564 . . . . . . . . . 10  |-  F/_ x X
46 nfcv 2564 . . . . . . . . . 10  |-  F/_ x
y
47 nfmpt1 4484 . . . . . . . . . 10  |-  F/_ x
( x  e.  X  |-> 
sum_ k  e.  A  B )
4845, 46, 47fvelrnbf 36773 . . . . . . . . 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 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 ) )
5049biimpa 482 . . . . . . 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 5066 . . . . . . . . . 10  |-  F/_ x ran  ( x  e.  X  |-> 
sum_ k  e.  A  B )
5251nfcri 2557 . . . . . . . . 9  |-  F/ x  y  e.  ran  ( x  e.  X  |->  sum_ k  e.  A  B )
5327, 52nfan 1956 . . . . . . . 8  |-  F/ x
( ph  /\  y  e.  ran  ( x  e.  X  |->  sum_ k  e.  A  B ) )
54 nfcv 2564 . . . . . . . . 9  |-  F/_ x RR
5554nfcri 2557 . . . . . . . 8  |-  F/ x  y  e.  RR
56 simpr 459 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  X )  ->  x  e.  X )
5756, 39jca 530 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  X )  ->  (
x  e.  X  /\  sum_ k  e.  A  B  e.  RR ) )
5842fvmpt2 5941 . . . . . . . . . . . . . 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 17 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  X )  ->  (
( x  e.  X  |-> 
sum_ k  e.  A  B ) `  x
)  =  sum_ k  e.  A  B )
60593adant3 1017 . . . . . . . . . . . 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 999 . . . . . . . . . . . 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 2445 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  X  /\  ( ( x  e.  X  |->  sum_ k  e.  A  B ) `  x )  =  y )  ->  sum_ k  e.  A  B  =  y )
63393adant3 1017 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  X  /\  ( ( x  e.  X  |->  sum_ k  e.  A  B ) `  x )  =  y )  ->  sum_ k  e.  A  B  e.  RR )
6462, 63eqeltrrd 2491 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  X  /\  ( ( x  e.  X  |->  sum_ k  e.  A  B ) `  x )  =  y )  ->  y  e.  RR )
65643adant1r 1223 . . . . . . . . 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 1196 . . . . . . . 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 2888 . . . . . . 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 432 . . . . 5  |-  ( ph  ->  ( y  e.  ran  ( x  e.  X  |-> 
sum_ k  e.  A  B )  ->  y  e.  RR ) )
7069ssrdv 3448 . . . 4  |-  ( ph  ->  ran  ( x  e.  X  |->  sum_ k  e.  A  B )  C_  RR )
7120a1i 11 . . . 4  |-  ( ph  ->  RR  C_  CC )
72 cnrest2 20080 . . . 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 1230 . . 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 2501 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 367    /\ w3a 974    = wceq 1405   F/wnf 1637    e. wcel 1842   A.wral 2754   E.wrex 2755    C_ wss 3414    |-> cmpt 4453   ran crn 4824    Fn wfn 5564   -->wf 5565   ` cfv 5569  (class class class)co 6278   Fincfn 7554   CCcc 9520   RRcr 9521   (,)cioo 11582   sum_csu 13657   ↾t crest 15035   TopOpenctopn 15036   topGenctg 15052  ℂfldccnfld 18740  TopOnctopon 19687    Cn ccn 20018
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-inf2 8091  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599  ax-pre-sup 9600  ax-addf 9601
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-iin 4274  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-se 4783  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-isom 5578  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-of 6521  df-om 6684  df-1st 6784  df-2nd 6785  df-supp 6903  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-2o 7168  df-oadd 7171  df-er 7348  df-map 7459  df-ixp 7508  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-fsupp 7864  df-fi 7905  df-sup 7935  df-oi 7969  df-card 8352  df-cda 8580  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-div 10248  df-nn 10577  df-2 10635  df-3 10636  df-4 10637  df-5 10638  df-6 10639  df-7 10640  df-8 10641  df-9 10642  df-10 10643  df-n0 10837  df-z 10906  df-dec 11020  df-uz 11128  df-q 11228  df-rp 11266  df-xneg 11371  df-xadd 11372  df-xmul 11373  df-ioo 11586  df-icc 11589  df-fz 11727  df-fzo 11855  df-seq 12152  df-exp 12211  df-hash 12453  df-cj 13081  df-re 13082  df-im 13083  df-sqrt 13217  df-abs 13218  df-clim 13460  df-sum 13658  df-struct 14843  df-ndx 14844  df-slot 14845  df-base 14846  df-sets 14847  df-ress 14848  df-plusg 14922  df-mulr 14923  df-starv 14924  df-sca 14925  df-vsca 14926  df-ip 14927  df-tset 14928  df-ple 14929  df-ds 14931  df-unif 14932  df-hom 14933  df-cco 14934  df-rest 15037  df-topn 15038  df-0g 15056  df-gsum 15057  df-topgen 15058  df-pt 15059  df-prds 15062  df-xrs 15116  df-qtop 15121  df-imas 15122  df-xps 15124  df-mre 15200  df-mrc 15201  df-acs 15203  df-mgm 16196  df-sgrp 16235  df-mnd 16245  df-submnd 16291  df-mulg 16384  df-cntz 16679  df-cmn 17124  df-psmet 18731  df-xmet 18732  df-met 18733  df-bl 18734  df-mopn 18735  df-cnfld 18741  df-top 19691  df-bases 19693  df-topon 19694  df-topsp 19695  df-cn 20021  df-cnp 20022  df-tx 20355  df-hmeo 20548  df-xms 21115  df-ms 21116  df-tms 21117
This theorem is referenced by:  refsum2cnlem1  36792
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