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Theorem esumcocn 28748
Description: Lemma for esummulc2 28750 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 1754 . . 3  |-  F/ k
ph
2 nfcv 2591 . . 3  |-  F/_ k A
3 esumcocn.a . . 3  |-  ( ph  ->  A  e.  V )
4 esumcocn.1 . . . . . 6  |-  ( ph  ->  C  e.  ( J  Cn  J ) )
5 xrge0tps 28595 . . . . . . . 8  |-  ( RR*ss  ( 0 [,] +oo ) )  e.  TopSp
6 xrge0base 28292 . . . . . . . . 9  |-  ( 0 [,] +oo )  =  ( Base `  ( RR*ss  ( 0 [,] +oo ) ) )
7 esumcocn.j . . . . . . . . . 10  |-  J  =  ( (ordTop `  <_  )t  ( 0 [,] +oo )
)
8 xrge0topn 28596 . . . . . . . . . 10  |-  ( TopOpen `  ( RR*ss  ( 0 [,] +oo ) ) )  =  ( (ordTop `  <_  )t  ( 0 [,] +oo )
)
97, 8eqtr4i 2461 . . . . . . . . 9  |-  J  =  ( TopOpen `  ( RR*ss  ( 0 [,] +oo ) ) )
106, 9tpsuni 19888 . . . . . . . 8  |-  ( (
RR*ss  ( 0 [,] +oo ) )  e.  TopSp  -> 
( 0 [,] +oo )  =  U. J )
115, 10ax-mp 5 . . . . . . 7  |-  ( 0 [,] +oo )  = 
U. J
1211, 11cnf 20197 . . . . . 6  |-  ( C  e.  ( J  Cn  J )  ->  C : ( 0 [,] +oo ) --> ( 0 [,] +oo ) )
134, 12syl 17 . . . . 5  |-  ( ph  ->  C : ( 0 [,] +oo ) --> ( 0 [,] +oo )
)
1413adantr 466 . . . 4  |-  ( (
ph  /\  k  e.  A )  ->  C : ( 0 [,] +oo ) --> ( 0 [,] +oo ) )
15 esumcocn.b . . . 4  |-  ( (
ph  /\  k  e.  A )  ->  B  e.  ( 0 [,] +oo ) )
1614, 15ffvelrnd 6038 . . 3  |-  ( (
ph  /\  k  e.  A )  ->  ( C `  B )  e.  ( 0 [,] +oo ) )
17 xrge0cmn 18949 . . . . . 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 17384 . . . . . . . 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 1208 . . . . . . 7  |-  ( ph  ->  ( ( x  e.  ( 0 [,] +oo )  /\  y  e.  ( 0 [,] +oo )
)  ->  ( C `  ( x +e
y ) )  =  ( ( C `  x ) +e
( C `  y
) ) ) )
2524ralrimivv 2852 . . . . . 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 28294 . . . . . . . 8  |-  +e 
=  ( +g  `  ( RR*ss  ( 0 [,] +oo ) ) )
28 xrge00 28293 . . . . . . . 8  |-  0  =  ( 0g `  ( RR*ss  ( 0 [,] +oo ) ) )
296, 6, 27, 27, 28, 28ismhm 16539 . . . . . . 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 209 . . . . . 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 1271 . . . . 5  |-  ( ph  ->  C  e.  ( (
RR*ss  ( 0 [,] +oo ) ) MndHom  ( RR*ss  ( 0 [,] +oo ) ) ) )
32 eqidd 2430 . . . . . 6  |-  ( ph  ->  ( k  e.  A  |->  B )  =  ( k  e.  A  |->  B ) )
3332, 15fmpt3d 6062 . . . . 5  |-  ( ph  ->  ( k  e.  A  |->  B ) : A --> ( 0 [,] +oo ) )
341, 2, 3, 15esumel 28715 . . . . 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 21095 . . . 4  |-  ( ph  ->  ( C ` Σ* k  e.  A B )  e.  ( ( RR*ss  ( 0 [,] +oo ) ) tsums 
( C  o.  (
k  e.  A  |->  B ) ) ) )
3613, 15cofmpt 28114 . . . . 5  |-  ( ph  ->  ( C  o.  (
k  e.  A  |->  B ) )  =  ( k  e.  A  |->  ( C `  B ) ) )
3736oveq2d 6321 . . . 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 2519 . . 3  |-  ( ph  ->  ( C ` Σ* k  e.  A B )  e.  ( ( RR*ss  ( 0 [,] +oo ) ) tsums 
( k  e.  A  |->  ( C `  B
) ) ) )
391, 2, 3, 16, 38esumid 28712 . 2  |-  ( ph  -> Σ* k  e.  A ( C `
 B )  =  ( C ` Σ* k  e.  A B ) )
4039eqcomd 2437 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 370    /\ w3a 982    = wceq 1437    e. wcel 1870   A.wral 2782   U.cuni 4222    |-> cmpt 4484    o. ccom 4858   -->wf 5597   ` cfv 5601  (class class class)co 6305   0cc0 9538   +oocpnf 9671    <_ cle 9675   +ecxad 11407   [,]cicc 11638   ↾s cress 15085   ↾t crest 15282   TopOpenctopn 15283  ordTopcordt 15360   RR*scxrs 15361   Mndcmnd 16490   MndHom cmhm 16535  CMndccmn 17369   TopSpctps 19854    Cn ccn 20175   tsums ctsu 21075  Σ*cesum 28695
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-pre-sup 9616
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-iin 4305  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-se 4814  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-of 6545  df-om 6707  df-1st 6807  df-2nd 6808  df-supp 6926  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-oadd 7194  df-er 7371  df-map 7482  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-fsupp 7890  df-fi 7931  df-sup 7962  df-oi 8025  df-card 8372  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-q 11265  df-xadd 11410  df-ioo 11639  df-ioc 11640  df-ico 11641  df-icc 11642  df-fz 11783  df-fzo 11914  df-seq 12211  df-hash 12513  df-struct 15086  df-ndx 15087  df-slot 15088  df-base 15089  df-sets 15090  df-ress 15091  df-plusg 15166  df-mulr 15167  df-tset 15172  df-ple 15173  df-ds 15175  df-rest 15284  df-topn 15285  df-0g 15303  df-gsum 15304  df-topgen 15305  df-ordt 15362  df-xrs 15363  df-mre 15447  df-mrc 15448  df-acs 15450  df-ps 16401  df-tsr 16402  df-mgm 16443  df-sgrp 16482  df-mnd 16492  df-mhm 16537  df-submnd 16538  df-cntz 16926  df-cmn 17371  df-fbas 18906  df-fg 18907  df-top 19856  df-bases 19857  df-topon 19858  df-topsp 19859  df-ntr 19970  df-nei 20049  df-cn 20178  df-cnp 20179  df-haus 20266  df-fil 20796  df-fm 20888  df-flim 20889  df-flf 20890  df-tsms 21076  df-esum 28696
This theorem is referenced by:  esummulc1  28749
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