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Theorem esumcocn 27911
Description: Lemma for esummulc2 27913 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 1683 . . 3  |-  F/ k
ph
2 nfcv 2629 . . 3  |-  F/_ k A
3 esumcocn.a . . 3  |-  ( ph  ->  A  e.  V )
4 esumcocn.1 . . . . . 6  |-  ( ph  ->  C  e.  ( J  Cn  J ) )
5 xrge0tps 27749 . . . . . . . 8  |-  ( RR*ss  ( 0 [,] +oo ) )  e.  TopSp
6 xrge0base 27497 . . . . . . . . 9  |-  ( 0 [,] +oo )  =  ( Base `  ( RR*ss  ( 0 [,] +oo ) ) )
7 esumcocn.j . . . . . . . . . 10  |-  J  =  ( (ordTop `  <_  )t  ( 0 [,] +oo )
)
8 xrge0topn 27750 . . . . . . . . . 10  |-  ( TopOpen `  ( RR*ss  ( 0 [,] +oo ) ) )  =  ( (ordTop `  <_  )t  ( 0 [,] +oo )
)
97, 8eqtr4i 2499 . . . . . . . . 9  |-  J  =  ( TopOpen `  ( RR*ss  ( 0 [,] +oo ) ) )
106, 9tpsuni 19308 . . . . . . . 8  |-  ( (
RR*ss  ( 0 [,] +oo ) )  e.  TopSp  -> 
( 0 [,] +oo )  =  U. J )
115, 10ax-mp 5 . . . . . . 7  |-  ( 0 [,] +oo )  = 
U. J
1211, 11cnf 19615 . . . . . 6  |-  ( C  e.  ( J  Cn  J )  ->  C : ( 0 [,] +oo ) --> ( 0 [,] +oo ) )
134, 12syl 16 . . . . 5  |-  ( ph  ->  C : ( 0 [,] +oo ) --> ( 0 [,] +oo )
)
1413adantr 465 . . . 4  |-  ( (
ph  /\  k  e.  A )  ->  C : ( 0 [,] +oo ) --> ( 0 [,] +oo ) )
15 esumcocn.b . . . 4  |-  ( (
ph  /\  k  e.  A )  ->  B  e.  ( 0 [,] +oo ) )
1614, 15ffvelrnd 6033 . . 3  |-  ( (
ph  /\  k  e.  A )  ->  ( C `  B )  e.  ( 0 [,] +oo ) )
17 xrge0cmn 18330 . . . . . 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 16686 . . . . . . . 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 1199 . . . . . . 7  |-  ( ph  ->  ( ( x  e.  ( 0 [,] +oo )  /\  y  e.  ( 0 [,] +oo )
)  ->  ( C `  ( x +e
y ) )  =  ( ( C `  x ) +e
( C `  y
) ) ) )
2524ralrimivv 2887 . . . . . 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 27499 . . . . . . . 8  |-  +e 
=  ( +g  `  ( RR*ss  ( 0 [,] +oo ) ) )
28 xrge00 27498 . . . . . . . 8  |-  0  =  ( 0g `  ( RR*ss  ( 0 [,] +oo ) ) )
296, 6, 27, 27, 28, 28ismhm 15841 . . . . . . 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 1235 . . . . 5  |-  ( ph  ->  C  e.  ( (
RR*ss  ( 0 [,] +oo ) ) MndHom  ( RR*ss  ( 0 [,] +oo ) ) ) )
32 eqidd 2468 . . . . . 6  |-  ( ph  ->  ( k  e.  A  |->  B )  =  ( k  e.  A  |->  B ) )
3332, 15fmpt3d 27319 . . . . 5  |-  ( ph  ->  ( k  e.  A  |->  B ) : A --> ( 0 [,] +oo ) )
341, 2, 3, 15esumel 27883 . . . . 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 20516 . . . 4  |-  ( ph  ->  ( C ` Σ* k  e.  A B )  e.  ( ( RR*ss  ( 0 [,] +oo ) ) tsums 
( C  o.  (
k  e.  A  |->  B ) ) ) )
3613, 15cofmpt 27327 . . . . 5  |-  ( ph  ->  ( C  o.  (
k  e.  A  |->  B ) )  =  ( k  e.  A  |->  ( C `  B ) ) )
3736oveq2d 6311 . . . 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 2557 . . 3  |-  ( ph  ->  ( C ` Σ* k  e.  A B )  e.  ( ( RR*ss  ( 0 [,] +oo ) ) tsums 
( k  e.  A  |->  ( C `  B
) ) ) )
391, 2, 3, 16, 38esumid 27881 . 2  |-  ( ph  -> Σ* k  e.  A ( C `
 B )  =  ( C ` Σ* k  e.  A B ) )
4039eqcomd 2475 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 973    = wceq 1379    e. wcel 1767   A.wral 2817   U.cuni 4251    |-> cmpt 4511    o. ccom 5009   -->wf 5590   ` cfv 5594  (class class class)co 6295   0cc0 9504   +oocpnf 9637    <_ cle 9641   +ecxad 11328   [,]cicc 11544   ↾s cress 14508   ↾t crest 14693   TopOpenctopn 14694  ordTopcordt 14771   RR*scxrs 14772   Mndcmnd 15793   MndHom cmhm 15837  CMndccmn 16671   TopSpctps 19266    Cn ccn 19593   tsums ctsu 20492  Σ*cesum 27865
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-iin 4334  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-of 6535  df-om 6696  df-1st 6795  df-2nd 6796  df-supp 6914  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-map 7434  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-fsupp 7842  df-fi 7883  df-sup 7913  df-oi 7947  df-card 8332  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-7 10611  df-8 10612  df-9 10613  df-10 10614  df-n0 10808  df-z 10877  df-dec 10989  df-uz 11095  df-q 11195  df-xadd 11331  df-ioo 11545  df-ioc 11546  df-ico 11547  df-icc 11548  df-fz 11685  df-fzo 11805  df-seq 12088  df-hash 12386  df-struct 14509  df-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-ress 14514  df-plusg 14585  df-mulr 14586  df-tset 14591  df-ple 14592  df-ds 14594  df-rest 14695  df-topn 14696  df-0g 14714  df-gsum 14715  df-topgen 14716  df-ordt 14773  df-xrs 14774  df-mre 14858  df-mrc 14859  df-acs 14861  df-ps 15704  df-tsr 15705  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-mhm 15839  df-submnd 15840  df-cntz 16227  df-cmn 16673  df-fbas 18286  df-fg 18287  df-top 19268  df-bases 19270  df-topon 19271  df-topsp 19272  df-ntr 19389  df-nei 19467  df-cn 19596  df-cnp 19597  df-haus 19684  df-fil 20215  df-fm 20307  df-flim 20308  df-flf 20309  df-tsms 20493  df-esum 27866
This theorem is referenced by:  esummulc1  27912
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