Theorem esumpfinvalf 27722

Description: Same as esumpfinval 27721, 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 27681	. . . 4 |- Σ* k e. A B = U. ( ( RR*s ↾s ( 0 [,] +oo ) ) tsums ( k e. A |-> B ) )
2		xrge0base 27335	. . . . . 6 |- ( 0 [,] +oo ) = ( Base ` ( RR*s ↾s ( 0 [,] +oo ) ) )
3		xrge00 27336	. . . . . 6 |- 0 = ( 0g ` ( RR*s ↾s ( 0 [,] +oo ) ) )
4		xrge0cmn 18228	. . . . . . 7 |- ( RR*s ↾s ( 0 [,] +oo ) ) e. CMnd
5	4	a1i 11	. . . . . 6 |- ( ph -> ( RR*s ↾s ( 0 [,] +oo ) ) e. CMnd )
6		xrge0tps 27560	. . . . . . 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 2629	. . . . . . 7 |- F/_ k ( 0 [,] +oo )
12		icossicc 11607	. . . . . . . 8 |- ( 0 [,) +oo ) C_ ( 0 [,] +oo )
13		esumpfinvalf.b	. . . . . . . 8 |- ( ( ph /\ k e. A ) -> B e. ( 0 [,) +oo ) )
14	12, 13	sseldi 3502	. . . . . . 7 |- ( ( ph /\ k e. A ) -> B e. ( 0 [,] +oo ) )
15		eqid 2467	. . . . . . 7 |- ( k e. A |-> B ) = ( k e. A |-> B )
16	9, 10, 11, 14, 15	fmptdF 27167	. . . . . 6 |- ( ph -> ( k e. A |-> B ) : A --> ( 0 [,] +oo ) )
17		c0ex 9586	. . . . . . . 8 |- 0 e. _V
18	17	a1i 11	. . . . . . 7 |- ( ph -> 0 e. _V )
19	16, 8, 18	fdmfifsupp 7835	. . . . . 6 |- ( ph -> ( k e. A |-> B ) finSupp 0 )
20		xrge0topn 27561	. . . . . . 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 27252	. . . . . . . 8 |- (ordTop ` <_ ) e. Haus
23		ovex 6307	. . . . . . . 8 |- ( 0 [,] +oo ) e. _V
24		resthaus 19635	. . . . . . . 8 |- ( ( (ordTop ` <_ ) e. Haus /\ ( 0 [,] +oo ) e. _V ) -> ( (ordTop ` <_ ) ↾t ( 0 [,] +oo ) ) e. Haus )
25	22, 23, 24	mp2an 672	. . . . . . 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 20374	. . . . 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 4255	. . . 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 2520	. . 3 |- ( ph -> Σ* k e. A B = U. { ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. A |-> B ) ) } )
30		ovex 6307	. . . 4 |- ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. A |-> B ) ) e. _V
31	30	unisn 4260	. . 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 2524	. 2 |- ( ph -> Σ* k e. A B = ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. A |-> B ) ) )
33		nfcv 2629	. . . 4 |- F/_ k ( 0 [,) +oo )
34	9, 10, 33, 13, 15	fmptdF 27167	. . 3 |- ( ph -> ( k e. A |-> B ) : A --> ( 0 [,) +oo ) )
35		esumpfinvallem 27720	. . 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 661	. 2 |- ( ph -> (ℂfld gsumg ( k e. A |-> B ) ) = ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. A |-> B ) ) )
37		rge0ssre 11624	. . . . . . . 8 |- ( 0 [,) +oo ) C_ RR
38		ax-resscn 9545	. . . . . . . 8 |- RR C_ CC
39	37, 38	sstri 3513	. . . . . . 7 |- ( 0 [,) +oo ) C_ CC
40	39, 13	sseldi 3502	. . . . . 6 |- ( ( ph /\ k e. A ) -> B e. CC )
41	40	sbt 2140	. . . . 5 |- [ l / k ] ( ( ph /\ k e. A ) -> B e. CC )
42		sbim 2110	. . . . . 6 |- ( [ l / k ] ( ( ph /\ k e. A ) -> B e. CC ) <-> ( [ l / k ] ( ph /\ k e. A ) -> [ l / k ] B e. CC ) )
43		sban 2114	. . . . . . . 8 |- ( [ l / k ] ( ph /\ k e. A ) <-> ( [ l / k ] ph /\ [ l / k ] k e. A ) )
44	9	sbf 2094	. . . . . . . . 9 |- ( [ l / k ] ph <-> ph )
45	10	clelsb3f 27055	. . . . . . . . 9 |- ( [ l / k ] k e. A <-> l e. A )
46	44, 45	anbi12i 697	. . . . . . . 8 |- ( ( [ l / k ] ph /\ [ l / k ] k e. A ) <-> ( ph /\ l e. A ) )
47	43, 46	bitri 249	. . . . . . 7 |- ( [ l / k ] ( ph /\ k e. A ) <-> ( ph /\ l e. A ) )
48		sbsbc 3335	. . . . . . . 8 |- ( [ l / k ] B e. CC <-> [. l / k ]. B e. CC )
49		vex 3116	. . . . . . . . 9 |- l e. _V
50		sbcel1g 3829	. . . . . . . . 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 249	. . . . . . 7 |- ( [ l / k ] B e. CC <-> [_ l / k ]_ B e. CC )
53	47, 52	imbi12i 326	. . . . . 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 249	. . . . 5 |- ( [ l / k ] ( ( ph /\ k e. A ) -> B e. CC ) <-> ( ( ph /\ l e. A ) -> [_ l / k ]_ B e. CC ) )
55	41, 54	mpbi 208	. . . 4 |- ( ( ph /\ l e. A ) -> [_ l / k ]_ B e. CC )
56	8, 55	gsumfsum 18252	. . 3 |- ( ph -> (ℂfld gsumg ( l e. A |-> [_ l / k ]_ B ) ) = sum_ l e. A [_ l / k ]_ B )
57		nfcv 2629	. . . . 5 |- F/_ l A
58		nfcv 2629	. . . . 5 |- F/_ l B
59		nfcsb1v 3451	. . . . 5 |- F/_ k [_ l / k ]_ B
60		csbeq1a 3444	. . . . 5 |- ( k = l -> B = [_ l / k ]_ B )
61	10, 57, 58, 59, 60	cbvmptf 27166	. . . 4 |- ( k e. A |-> B ) = ( l e. A |-> [_ l / k ]_ B )
62	61	oveq2i 6293	. . 3 |- (ℂfld gsumg ( k e. A |-> B ) ) = (ℂfld gsumg ( l e. A |-> [_ l / k ]_ B ) )
63	60, 57, 10, 58, 59	cbvsum 13476	. . 3 |- sum_ k e. A B = sum_ l e. A [_ l / k ]_ B
64	56, 62, 63	3eqtr4g 2533	. 2 |- ( ph -> (ℂfld gsumg ( k e. A |-> B ) ) = sum_ k e. A B )
65	32, 36, 64	3eqtr2d 2514	1 |- ( ph -> Σ* k e. A B = sum_ k e. A B )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1379   F/wnf 1599   [wsb 1711   e. wcel 1767   F/_wnfc 2615   _Vcvv 3113   [.wsbc 3331   [_csb 3435   {csn 4027   U.cuni 4245   |-> cmpt 4505   -->wf 5582   ` cfv 5586  (class class class)co 6282   Fincfn 7513   CCcc 9486   RRcr 9487   0cc0 9488   +oocpnf 9621   <_ cle 9625   [,)cico 11527   [,]cicc 11528   sum_csu 13467   ↾_s cress 14487   ↾_t crest 14672   TopOpenctopn 14673   gsum_g cgsu 14692  ordTopcordt 14750   RR*scxrs 14751  CMndccmn 16594  ℂ_fldccnfld 18191   TopSpctps 19164   Hauscha 19575   tsums ctsu 20359  Σ^*cesum 27680

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 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566  ax-addf 9567  ax-mulf 9568

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 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-supp 6899  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-map 7419  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-fsupp 7826  df-fi 7867  df-sup 7897  df-oi 7931  df-card 8316  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10973  df-uz 11079  df-rp 11217  df-xadd 11315  df-ioo 11529  df-ico 11531  df-icc 11532  df-fz 11669  df-fzo 11789  df-seq 12072  df-exp 12131  df-hash 12370  df-cj 12891  df-re 12892  df-im 12893  df-sqrt 13027  df-abs 13028  df-clim 13270  df-sum 13468  df-struct 14488  df-ndx 14489  df-slot 14490  df-base 14491  df-sets 14492  df-ress 14493  df-plusg 14564  df-mulr 14565  df-starv 14566  df-tset 14570  df-ple 14571  df-ds 14573  df-unif 14574  df-rest 14674  df-topn 14675  df-0g 14693  df-gsum 14694  df-topgen 14695  df-ordt 14752  df-xrs 14753  df-ps 15683  df-tsr 15684  df-mnd 15728  df-submnd 15778  df-grp 15858  df-minusg 15859  df-cntz 16150  df-cmn 16596  df-abl 16597  df-mgp 16932  df-ur 16944  df-rng 16988  df-cring 16989  df-fbas 18187  df-fg 18188  df-cnfld 18192  df-top 19166  df-bases 19168  df-topon 19169  df-topsp 19170  df-cld 19286  df-ntr 19287  df-cls 19288  df-nei 19365  df-cn 19494  df-haus 19582  df-fil 20082  df-fm 20174  df-flim 20175  df-flf 20176  df-tsms 20360  df-esum 27681

This theorem is referenced by:  volfiniune  27842