Theorem esumpfinvalf 28971

Description: Same as esumpfinval 28970, minus distinct variable restrictions. (Contributed by Thierry Arnoux, 28-Aug-2017.) (Proof shortened by AV, 25-Jul-2019.)

Hypotheses

Ref	Expression
esumpfinvalf.1	|- F/_ k A
esumpfinvalf.2	|- F/ k ph
esumpfinvalf.a	|- ( ph -> A e. Fin )
esumpfinvalf.b	|- ( ( ph /\ k e. A ) -> B e. ( 0 [,) +oo ) )

Assertion

Ref	Expression
esumpfinvalf	|- ( ph -> Σ* k e. A B = sum_ k e. A B )

Proof of Theorem esumpfinvalf

Dummy variable l is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-esum 28923	. . . 4 |- Σ* k e. A B = U. ( ( RR*s ↾s ( 0 [,] +oo ) ) tsums ( k e. A |-> B ) )
2		xrge0base 28522	. . . . . 6 |- ( 0 [,] +oo ) = ( Base ` ( RR*s ↾s ( 0 [,] +oo ) ) )
3		xrge00 28523	. . . . . 6 |- 0 = ( 0g ` ( RR*s ↾s ( 0 [,] +oo ) ) )
4		xrge0cmn 19087	. . . . . . 7 |- ( RR*s ↾s ( 0 [,] +oo ) ) e. CMnd
5	4	a1i 11	. . . . . 6 |- ( ph -> ( RR*s ↾s ( 0 [,] +oo ) ) e. CMnd )
6		xrge0tps 28822	. . . . . . 7 |- ( RR*s ↾s ( 0 [,] +oo ) ) e. TopSp
7	6	a1i 11	. . . . . 6 |- ( ph -> ( RR*s ↾s ( 0 [,] +oo ) ) e. TopSp )
8		esumpfinvalf.a	. . . . . 6 |- ( ph -> A e. Fin )
9		esumpfinvalf.2	. . . . . . 7 |- F/ k ph
10		esumpfinvalf.1	. . . . . . 7 |- F/_ k A
11		nfcv 2612	. . . . . . 7 |- F/_ k ( 0 [,] +oo )
12		icossicc 11746	. . . . . . . 8 |- ( 0 [,) +oo ) C_ ( 0 [,] +oo )
13		esumpfinvalf.b	. . . . . . . 8 |- ( ( ph /\ k e. A ) -> B e. ( 0 [,) +oo ) )
14	12, 13	sseldi 3416	. . . . . . 7 |- ( ( ph /\ k e. A ) -> B e. ( 0 [,] +oo ) )
15		eqid 2471	. . . . . . 7 |- ( k e. A |-> B ) = ( k e. A |-> B )
16	9, 10, 11, 14, 15	fmptdF 28331	. . . . . 6 |- ( ph -> ( k e. A |-> B ) : A --> ( 0 [,] +oo ) )
17		c0ex 9655	. . . . . . . 8 |- 0 e. _V
18	17	a1i 11	. . . . . . 7 |- ( ph -> 0 e. _V )
19	16, 8, 18	fdmfifsupp 7911	. . . . . 6 |- ( ph -> ( k e. A |-> B ) finSupp 0 )
20		xrge0topn 28823	. . . . . . 7 |- ( TopOpen ` ( RR*s ↾s ( 0 [,] +oo ) ) ) = ( (ordTop ` <_ ) ↾t ( 0 [,] +oo ) )
21	20	eqcomi 2480	. . . . . 6 |- ( (ordTop ` <_ ) ↾t ( 0 [,] +oo ) ) = ( TopOpen ` ( RR*s ↾s ( 0 [,] +oo ) ) )
22		xrhaus 28430	. . . . . . . 8 |- (ordTop ` <_ ) e. Haus
23		ovex 6336	. . . . . . . 8 |- ( 0 [,] +oo ) e. _V
24		resthaus 20461	. . . . . . . 8 |- ( ( (ordTop ` <_ ) e. Haus /\ ( 0 [,] +oo ) e. _V ) -> ( (ordTop ` <_ ) ↾t ( 0 [,] +oo ) ) e. Haus )
25	22, 23, 24	mp2an 686	. . . . . . 7 |- ( (ordTop ` <_ ) ↾t ( 0 [,] +oo ) ) e. Haus
26	25	a1i 11	. . . . . 6 |- ( ph -> ( (ordTop ` <_ ) ↾t ( 0 [,] +oo ) ) e. Haus )
27	2, 3, 5, 7, 8, 16, 19, 21, 26	haustsmsid 21233	. . . . 5 |- ( ph -> ( ( RR*s ↾s ( 0 [,] +oo ) ) tsums ( k e. A |-> B ) ) = { ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. A |-> B ) ) } )
28	27	unieqd 4200	. . . 4 |- ( ph -> U. ( ( RR*s ↾s ( 0 [,] +oo ) ) tsums ( k e. A |-> B ) ) = U. { ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. A |-> B ) ) } )
29	1, 28	syl5eq 2517	. . 3 |- ( ph -> Σ* k e. A B = U. { ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. A |-> B ) ) } )
30		ovex 6336	. . . 4 |- ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. A |-> B ) ) e. _V
31	30	unisn 4205	. . 3 |- U. { ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. A |-> B ) ) } = ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. A |-> B ) )
32	29, 31	syl6eq 2521	. 2 |- ( ph -> Σ* k e. A B = ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. A |-> B ) ) )
33		nfcv 2612	. . . 4 |- F/_ k ( 0 [,) +oo )
34	9, 10, 33, 13, 15	fmptdF 28331	. . 3 |- ( ph -> ( k e. A |-> B ) : A --> ( 0 [,) +oo ) )
35		esumpfinvallem 28969	. . 3 |- ( ( A e. Fin /\ ( k e. A |-> B ) : A --> ( 0 [,) +oo ) ) -> (ℂfld gsumg ( k e. A |-> B ) ) = ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. A |-> B ) ) )
36	8, 34, 35	syl2anc 673	. 2 |- ( ph -> (ℂfld gsumg ( k e. A |-> B ) ) = ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. A |-> B ) ) )
37		rge0ssre 11766	. . . . . . . 8 |- ( 0 [,) +oo ) C_ RR
38		ax-resscn 9614	. . . . . . . 8 |- RR C_ CC
39	37, 38	sstri 3427	. . . . . . 7 |- ( 0 [,) +oo ) C_ CC
40	39, 13	sseldi 3416	. . . . . 6 |- ( ( ph /\ k e. A ) -> B e. CC )
41	40	sbt 2268	. . . . 5 |- [ l / k ] ( ( ph /\ k e. A ) -> B e. CC )
42		sbim 2244	. . . . . 6 |- ( [ l / k ] ( ( ph /\ k e. A ) -> B e. CC ) <-> ( [ l / k ] ( ph /\ k e. A ) -> [ l / k ] B e. CC ) )
43		sban 2248	. . . . . . . 8 |- ( [ l / k ] ( ph /\ k e. A ) <-> ( [ l / k ] ph /\ [ l / k ] k e. A ) )
44	9	sbf 2229	. . . . . . . . 9 |- ( [ l / k ] ph <-> ph )
45	10	clelsb3f 28195	. . . . . . . . 9 |- ( [ l / k ] k e. A <-> l e. A )
46	44, 45	anbi12i 711	. . . . . . . 8 |- ( ( [ l / k ] ph /\ [ l / k ] k e. A ) <-> ( ph /\ l e. A ) )
47	43, 46	bitri 257	. . . . . . 7 |- ( [ l / k ] ( ph /\ k e. A ) <-> ( ph /\ l e. A ) )
48		sbsbc 3259	. . . . . . . 8 |- ( [ l / k ] B e. CC <-> [. l / k ]. B e. CC )
49		vex 3034	. . . . . . . . 9 |- l e. _V
50		sbcel1g 3780	. . . . . . . . 9 |- ( l e. _V -> ( [. l / k ]. B e. CC <-> [_ l / k ]_ B e. CC ) )
51	49, 50	ax-mp 5	. . . . . . . 8 |- ( [. l / k ]. B e. CC <-> [_ l / k ]_ B e. CC )
52	48, 51	bitri 257	. . . . . . 7 |- ( [ l / k ] B e. CC <-> [_ l / k ]_ B e. CC )
53	47, 52	imbi12i 333	. . . . . 6 |- ( ( [ l / k ] ( ph /\ k e. A ) -> [ l / k ] B e. CC ) <-> ( ( ph /\ l e. A ) -> [_ l / k ]_ B e. CC ) )
54	42, 53	bitri 257	. . . . 5 |- ( [ l / k ] ( ( ph /\ k e. A ) -> B e. CC ) <-> ( ( ph /\ l e. A ) -> [_ l / k ]_ B e. CC ) )
55	41, 54	mpbi 213	. . . 4 |- ( ( ph /\ l e. A ) -> [_ l / k ]_ B e. CC )
56	8, 55	gsumfsum 19111	. . 3 |- ( ph -> (ℂfld gsumg ( l e. A |-> [_ l / k ]_ B ) ) = sum_ l e. A [_ l / k ]_ B )
57		nfcv 2612	. . . . 5 |- F/_ l A
58		nfcv 2612	. . . . 5 |- F/_ l B
59		nfcsb1v 3365	. . . . 5 |- F/_ k [_ l / k ]_ B
60		csbeq1a 3358	. . . . 5 |- ( k = l -> B = [_ l / k ]_ B )
61	10, 57, 58, 59, 60	cbvmptf 4486	. . . 4 |- ( k e. A |-> B ) = ( l e. A |-> [_ l / k ]_ B )
62	61	oveq2i 6319	. . 3 |- (ℂfld gsumg ( k e. A |-> B ) ) = (ℂfld gsumg ( l e. A |-> [_ l / k ]_ B ) )
63	60, 57, 10, 58, 59	cbvsum 13838	. . 3 |- sum_ k e. A B = sum_ l e. A [_ l / k ]_ B
64	56, 62, 63	3eqtr4g 2530	. 2 |- ( ph -> (ℂfld gsumg ( k e. A |-> B ) ) = sum_ k e. A B )
65	32, 36, 64	3eqtr2d 2511	1 |- ( ph -> Σ* k e. A B = sum_ k e. A B )

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 376   = wceq 1452   F/wnf 1675   [wsb 1805   e. wcel 1904   F/_wnfc 2599   _Vcvv 3031   [.wsbc 3255   [_csb 3349   {csn 3959   U.cuni 4190   |-> cmpt 4454   -->wf 5585   ` cfv 5589  (class class class)co 6308   Fincfn 7587   CCcc 9555   RRcr 9556   0cc0 9557   +oocpnf 9690   <_ cle 9694   [,)cico 11662   [,]cicc 11663   sum_csu 13829   ↾_s cress 15200   ↾_t crest 15397   TopOpenctopn 15398   gsum_g cgsu 15417  ordTopcordt 15475   RR*scxrs 15476  CMndccmn 17508  ℂ_fldccnfld 19047   TopSpctps 19996   Hauscha 20401   tsums ctsu 21218  Σ^*cesum 28922

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635  ax-addf 9636  ax-mulf 9637

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-supp 6934  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-map 7492  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fsupp 7902  df-fi 7943  df-sup 7974  df-oi 8043  df-card 8391  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-rp 11326  df-xadd 11433  df-ico 11666  df-icc 11667  df-fz 11811  df-fzo 11943  df-seq 12252  df-exp 12311  df-hash 12554  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-clim 13629  df-sum 13830  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-starv 15283  df-tset 15287  df-ple 15288  df-ds 15290  df-unif 15291  df-rest 15399  df-topn 15400  df-0g 15418  df-gsum 15419  df-topgen 15420  df-ordt 15477  df-xrs 15478  df-ps 16524  df-tsr 16525  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-submnd 16661  df-grp 16751  df-minusg 16752  df-cntz 17049  df-cmn 17510  df-abl 17511  df-mgp 17802  df-ur 17814  df-ring 17860  df-cring 17861  df-fbas 19044  df-fg 19045  df-cnfld 19048  df-top 19998  df-bases 19999  df-topon 20000  df-topsp 20001  df-cld 20111  df-ntr 20112  df-cls 20113  df-nei 20191  df-cn 20320  df-haus 20408  df-fil 20939  df-fm 21031  df-flim 21032  df-flf 21033  df-tsms 21219  df-esum 28923

This theorem is referenced by:  volfiniune  29126