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Theorem esumcocn 26465
Description: Lemma for esummulc2 26467 and co. Composing with a continuous function preserves extended sums (Contributed by Thierry Arnoux, 29-Jun-2017.)
Hypotheses
Ref Expression
esumcocn.j  |-  J  =  ( (ordTop `  <_  )t  ( 0 [,] +oo )
)
esumcocn.a  |-  ( ph  ->  A  e.  V )
esumcocn.b  |-  ( (
ph  /\  k  e.  A )  ->  B  e.  ( 0 [,] +oo ) )
esumcocn.1  |-  ( ph  ->  C  e.  ( J  Cn  J ) )
esumcocn.0  |-  ( ph  ->  ( C `  0
)  =  0 )
esumcocn.f  |-  ( (
ph  /\  x  e.  ( 0 [,] +oo )  /\  y  e.  ( 0 [,] +oo )
)  ->  ( C `  ( x +e
y ) )  =  ( ( C `  x ) +e
( C `  y
) ) )
Assertion
Ref Expression
esumcocn  |-  ( ph  ->  ( C ` Σ* k  e.  A B )  = Σ* k  e.  A ( C `  B ) )
Distinct variable groups:    A, k    x, y, k, C    k, V    ph, x, y, k
Allowed substitution hints:    A( x, y)    B( x, y, k)    J( x, y, k)    V( x, y)

Proof of Theorem esumcocn
StepHypRef Expression
1 nfv 1678 . . 3  |-  F/ k
ph
2 nfcv 2577 . . 3  |-  F/_ k A
3 esumcocn.a . . 3  |-  ( ph  ->  A  e.  V )
4 esumcocn.1 . . . . . 6  |-  ( ph  ->  C  e.  ( J  Cn  J ) )
5 xrge0tps 26308 . . . . . . . 8  |-  ( RR*ss  ( 0 [,] +oo ) )  e.  TopSp
6 xrge0base 26079 . . . . . . . . 9  |-  ( 0 [,] +oo )  =  ( Base `  ( RR*ss  ( 0 [,] +oo ) ) )
7 esumcocn.j . . . . . . . . . 10  |-  J  =  ( (ordTop `  <_  )t  ( 0 [,] +oo )
)
8 xrge0topn 26309 . . . . . . . . . 10  |-  ( TopOpen `  ( RR*ss  ( 0 [,] +oo ) ) )  =  ( (ordTop `  <_  )t  ( 0 [,] +oo )
)
97, 8eqtr4i 2464 . . . . . . . . 9  |-  J  =  ( TopOpen `  ( RR*ss  ( 0 [,] +oo ) ) )
106, 9tpsuni 18502 . . . . . . . 8  |-  ( (
RR*ss  ( 0 [,] +oo ) )  e.  TopSp  -> 
( 0 [,] +oo )  =  U. J )
115, 10ax-mp 5 . . . . . . 7  |-  ( 0 [,] +oo )  = 
U. J
1211, 11cnf 18809 . . . . . 6  |-  ( C  e.  ( J  Cn  J )  ->  C : ( 0 [,] +oo ) --> ( 0 [,] +oo ) )
134, 12syl 16 . . . . 5  |-  ( ph  ->  C : ( 0 [,] +oo ) --> ( 0 [,] +oo )
)
1413adantr 462 . . . 4  |-  ( (
ph  /\  k  e.  A )  ->  C : ( 0 [,] +oo ) --> ( 0 [,] +oo ) )
15 esumcocn.b . . . 4  |-  ( (
ph  /\  k  e.  A )  ->  B  e.  ( 0 [,] +oo ) )
1614, 15ffvelrnd 5841 . . 3  |-  ( (
ph  /\  k  e.  A )  ->  ( C `  B )  e.  ( 0 [,] +oo ) )
17 xrge0cmn 17814 . . . . . 6  |-  ( RR*ss  ( 0 [,] +oo ) )  e. CMnd
1817a1i 11 . . . . 5  |-  ( ph  ->  ( RR*ss  ( 0 [,] +oo ) )  e. CMnd )
195a1i 11 . . . . 5  |-  ( ph  ->  ( RR*ss  ( 0 [,] +oo ) )  e.  TopSp )
20 cmnmnd 16285 . . . . . . . 8  |-  ( (
RR*ss  ( 0 [,] +oo ) )  e. CMnd  ->  (
RR*ss  ( 0 [,] +oo ) )  e.  Mnd )
2117, 20ax-mp 5 . . . . . . 7  |-  ( RR*ss  ( 0 [,] +oo ) )  e.  Mnd
2221a1i 11 . . . . . 6  |-  ( ph  ->  ( RR*ss  ( 0 [,] +oo ) )  e.  Mnd )
23 esumcocn.f . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( 0 [,] +oo )  /\  y  e.  ( 0 [,] +oo )
)  ->  ( C `  ( x +e
y ) )  =  ( ( C `  x ) +e
( C `  y
) ) )
24233expib 1185 . . . . . . 7  |-  ( ph  ->  ( ( x  e.  ( 0 [,] +oo )  /\  y  e.  ( 0 [,] +oo )
)  ->  ( C `  ( x +e
y ) )  =  ( ( C `  x ) +e
( C `  y
) ) ) )
2524ralrimivv 2805 . . . . . 6  |-  ( ph  ->  A. x  e.  ( 0 [,] +oo ) A. y  e.  (
0 [,] +oo )
( C `  (
x +e y ) )  =  ( ( C `  x
) +e ( C `  y ) ) )
26 esumcocn.0 . . . . . 6  |-  ( ph  ->  ( C `  0
)  =  0 )
27 xrge0plusg 26081 . . . . . . . 8  |-  +e 
=  ( +g  `  ( RR*ss  ( 0 [,] +oo ) ) )
28 xrge00 26080 . . . . . . . 8  |-  0  =  ( 0g `  ( RR*ss  ( 0 [,] +oo ) ) )
296, 6, 27, 27, 28, 28ismhm 15462 . . . . . . 7  |-  ( C  e.  ( ( RR*ss  ( 0 [,] +oo ) ) MndHom  ( RR*ss  (
0 [,] +oo )
) )  <->  ( (
( RR*ss  ( 0 [,] +oo ) )  e.  Mnd  /\  ( RR*ss  ( 0 [,] +oo ) )  e.  Mnd )  /\  ( C : ( 0 [,] +oo ) --> ( 0 [,] +oo )  /\  A. x  e.  ( 0 [,] +oo ) A. y  e.  (
0 [,] +oo )
( C `  (
x +e y ) )  =  ( ( C `  x
) +e ( C `  y ) )  /\  ( C `
 0 )  =  0 ) ) )
3029biimpri 206 . . . . . 6  |-  ( ( ( ( RR*ss  (
0 [,] +oo )
)  e.  Mnd  /\  ( RR*ss  ( 0 [,] +oo ) )  e.  Mnd )  /\  ( C :
( 0 [,] +oo )
--> ( 0 [,] +oo )  /\  A. x  e.  ( 0 [,] +oo ) A. y  e.  ( 0 [,] +oo )
( C `  (
x +e y ) )  =  ( ( C `  x
) +e ( C `  y ) )  /\  ( C `
 0 )  =  0 ) )  ->  C  e.  ( ( RR*ss  ( 0 [,] +oo ) ) MndHom  ( RR*ss  ( 0 [,] +oo ) ) ) )
3122, 22, 13, 25, 26, 30syl23anc 1220 . . . . 5  |-  ( ph  ->  C  e.  ( (
RR*ss  ( 0 [,] +oo ) ) MndHom  ( RR*ss  ( 0 [,] +oo ) ) ) )
32 eqidd 2442 . . . . . 6  |-  ( ph  ->  ( k  e.  A  |->  B )  =  ( k  e.  A  |->  B ) )
3332, 15fmpt3d 25908 . . . . 5  |-  ( ph  ->  ( k  e.  A  |->  B ) : A --> ( 0 [,] +oo ) )
341, 2, 3, 15esumel 26437 . . . . 5  |-  ( ph  -> Σ* k  e.  A B  e.  ( ( RR*ss  (
0 [,] +oo )
) tsums  ( k  e.  A  |->  B ) ) )
356, 9, 9, 18, 19, 18, 19, 31, 4, 3, 33, 34tsmsmhm 19679 . . . 4  |-  ( ph  ->  ( C ` Σ* k  e.  A B )  e.  ( ( RR*ss  ( 0 [,] +oo ) ) tsums 
( C  o.  (
k  e.  A  |->  B ) ) ) )
3613, 15cofmpt 25916 . . . . 5  |-  ( ph  ->  ( C  o.  (
k  e.  A  |->  B ) )  =  ( k  e.  A  |->  ( C `  B ) ) )
3736oveq2d 6106 . . . 4  |-  ( ph  ->  ( ( RR*ss  (
0 [,] +oo )
) tsums  ( C  o.  ( k  e.  A  |->  B ) ) )  =  ( ( RR*ss  ( 0 [,] +oo ) ) tsums  ( k  e.  A  |->  ( C `
 B ) ) ) )
3835, 37eleqtrd 2517 . . 3  |-  ( ph  ->  ( C ` Σ* k  e.  A B )  e.  ( ( RR*ss  ( 0 [,] +oo ) ) tsums 
( k  e.  A  |->  ( C `  B
) ) ) )
391, 2, 3, 16, 38esumid 26435 . 2  |-  ( ph  -> Σ* k  e.  A ( C `
 B )  =  ( C ` Σ* k  e.  A B ) )
4039eqcomd 2446 1  |-  ( ph  ->  ( C ` Σ* k  e.  A B )  = Σ* k  e.  A ( C `  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761   A.wral 2713   U.cuni 4088    e. cmpt 4347    o. ccom 4840   -->wf 5411   ` cfv 5415  (class class class)co 6090   0cc0 9278   +oocpnf 9411    <_ cle 9415   +ecxad 11083   [,]cicc 11299   ↾s cress 14171   ↾t crest 14355   TopOpenctopn 14356  ordTopcordt 14433   RR*scxrs 14434   Mndcmnd 15405   MndHom cmhm 15458  CMndccmn 16270   TopSpctps 18460    Cn ccn 18787   tsums ctsu 19655  Σ*cesum 26419
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 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-iin 4171  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-om 6476  df-1st 6576  df-2nd 6577  df-supp 6690  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-map 7212  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-fsupp 7617  df-fi 7657  df-sup 7687  df-oi 7720  df-card 8105  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-10 10384  df-n0 10576  df-z 10643  df-dec 10752  df-uz 10858  df-q 10950  df-xadd 11086  df-ioo 11300  df-ioc 11301  df-ico 11302  df-icc 11303  df-fz 11434  df-fzo 11545  df-seq 11803  df-hash 12100  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177  df-plusg 14247  df-mulr 14248  df-tset 14253  df-ple 14254  df-ds 14256  df-rest 14357  df-topn 14358  df-0g 14376  df-gsum 14377  df-topgen 14378  df-ordt 14435  df-xrs 14436  df-mre 14520  df-mrc 14521  df-acs 14523  df-ps 15366  df-tsr 15367  df-mnd 15411  df-mhm 15460  df-submnd 15461  df-cntz 15828  df-cmn 16272  df-fbas 17773  df-fg 17774  df-top 18462  df-bases 18464  df-topon 18465  df-topsp 18466  df-ntr 18583  df-nei 18661  df-cn 18790  df-cnp 18791  df-haus 18878  df-fil 19378  df-fm 19470  df-flim 19471  df-flf 19472  df-tsms 19656  df-esum 26420
This theorem is referenced by:  esummulc1  26466
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