Theorem esumpfinvalf 26669

Description: Same as esumpfinval 26668, 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 26628	. . . 4 |- Σ* k e. A B = U. ( ( RR*s ↾s ( 0 [,] +oo ) ) tsums ( k e. A |-> B ) )
2		xrge0base 26290	. . . . . 6 |- ( 0 [,] +oo ) = ( Base ` ( RR*s ↾s ( 0 [,] +oo ) ) )
3		xrge00 26291	. . . . . 6 |- 0 = ( 0g ` ( RR*s ↾s ( 0 [,] +oo ) ) )
4		xrge0cmn 17979	. . . . . . 7 |- ( RR*s ↾s ( 0 [,] +oo ) ) e. CMnd
5	4	a1i 11	. . . . . 6 |- ( ph -> ( RR*s ↾s ( 0 [,] +oo ) ) e. CMnd )
6		xrge0tps 26516	. . . . . . 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 2616	. . . . . . 7 |- F/_ k ( 0 [,] +oo )
12		icossicc 11492	. . . . . . . 8 |- ( 0 [,) +oo ) C_ ( 0 [,] +oo )
13		esumpfinvalf.b	. . . . . . . 8 |- ( ( ph /\ k e. A ) -> B e. ( 0 [,) +oo ) )
14	12, 13	sseldi 3461	. . . . . . 7 |- ( ( ph /\ k e. A ) -> B e. ( 0 [,] +oo ) )
15		eqid 2454	. . . . . . 7 |- ( k e. A |-> B ) = ( k e. A |-> B )
16	9, 10, 11, 14, 15	fmptdF 26122	. . . . . 6 |- ( ph -> ( k e. A |-> B ) : A --> ( 0 [,] +oo ) )
17		c0ex 9490	. . . . . . . 8 |- 0 e. _V
18	17	a1i 11	. . . . . . 7 |- ( ph -> 0 e. _V )
19	16, 8, 18	fdmfifsupp 7740	. . . . . 6 |- ( ph -> ( k e. A |-> B ) finSupp 0 )
20		xrge0topn 26517	. . . . . . 7 |- ( TopOpen ` ( RR*s ↾s ( 0 [,] +oo ) ) ) = ( (ordTop ` <_ ) ↾t ( 0 [,] +oo ) )
21	20	eqcomi 2467	. . . . . 6 |- ( (ordTop ` <_ ) ↾t ( 0 [,] +oo ) ) = ( TopOpen ` ( RR*s ↾s ( 0 [,] +oo ) ) )
22		xrhaus 26207	. . . . . . . 8 |- (ordTop ` <_ ) e. Haus
23		ovex 6224	. . . . . . . 8 |- ( 0 [,] +oo ) e. _V
24		resthaus 19103	. . . . . . . 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 19842	. . . . 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 4208	. . . 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 2507	. . 3 |- ( ph -> Σ* k e. A B = U. { ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. A |-> B ) ) } )
30		ovex 6224	. . . 4 |- ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. A |-> B ) ) e. _V
31	30	unisn 4213	. . 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 2511	. 2 |- ( ph -> Σ* k e. A B = ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. A |-> B ) ) )
33		nfcv 2616	. . . 4 |- F/_ k ( 0 [,) +oo )
34	9, 10, 33, 13, 15	fmptdF 26122	. . 3 |- ( ph -> ( k e. A |-> B ) : A --> ( 0 [,) +oo ) )
35		esumpfinvallem 26667	. . 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 11509	. . . . . . . 8 |- ( 0 [,) +oo ) C_ RR
38		ax-resscn 9449	. . . . . . . 8 |- RR C_ CC
39	37, 38	sstri 3472	. . . . . . 7 |- ( 0 [,) +oo ) C_ CC
40	39, 13	sseldi 3461	. . . . . 6 |- ( ( ph /\ k e. A ) -> B e. CC )
41	40	sbt 2127	. . . . 5 |- [ l / k ] ( ( ph /\ k e. A ) -> B e. CC )
42		sbim 2097	. . . . . 6 |- ( [ l / k ] ( ( ph /\ k e. A ) -> B e. CC ) <-> ( [ l / k ] ( ph /\ k e. A ) -> [ l / k ] B e. CC ) )
43		sban 2101	. . . . . . . 8 |- ( [ l / k ] ( ph /\ k e. A ) <-> ( [ l / k ] ph /\ [ l / k ] k e. A ) )
44	9	sbf 2081	. . . . . . . . 9 |- ( [ l / k ] ph <-> ph )
45	10	clelsb3f 26015	. . . . . . . . 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 3296	. . . . . . . 8 |- ( [ l / k ] B e. CC <-> [. l / k ]. B e. CC )
49		vex 3079	. . . . . . . . 9 |- l e. _V
50		sbcel1g 3788	. . . . . . . . 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 18003	. . 3 |- ( ph -> (ℂfld gsumg ( l e. A |-> [_ l / k ]_ B ) ) = sum_ l e. A [_ l / k ]_ B )
57		nfcv 2616	. . . . 5 |- F/_ l A
58		nfcv 2616	. . . . 5 |- F/_ l B
59		nfcsb1v 3410	. . . . 5 |- F/_ k [_ l / k ]_ B
60		csbeq1a 3403	. . . . 5 |- ( k = l -> B = [_ l / k ]_ B )
61	10, 57, 58, 59, 60	cbvmptf 26121	. . . 4 |- ( k e. A |-> B ) = ( l e. A |-> [_ l / k ]_ B )
62	61	oveq2i 6210	. . 3 |- (ℂfld gsumg ( k e. A |-> B ) ) = (ℂfld gsumg ( l e. A |-> [_ l / k ]_ B ) )
63	60, 57, 10, 58, 59	cbvsum 13289	. . 3 |- sum_ k e. A B = sum_ l e. A [_ l / k ]_ B
64	56, 62, 63	3eqtr4g 2520	. 2 |- ( ph -> (ℂfld gsumg ( k e. A |-> B ) ) = sum_ k e. A B )
65	32, 36, 64	3eqtr2d 2501	1 |- ( ph -> Σ* k e. A B = sum_ k e. A B )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1370   F/wnf 1590   [wsb 1702   e. wcel 1758   F/_wnfc 2602   _Vcvv 3076   [.wsbc 3292   [_csb 3394   {csn 3984   U.cuni 4198   |-> cmpt 4457   -->wf 5521   ` cfv 5525  (class class class)co 6199   Fincfn 7419   CCcc 9390   RRcr 9391   0cc0 9392   +oocpnf 9525   <_ cle 9529   [,)cico 11412   [,]cicc 11413   sum_csu 13280   ↾_s cress 14292   ↾_t crest 14477   TopOpenctopn 14478   gsum_g cgsu 14497  ordTopcordt 14555   RR*scxrs 14556  CMndccmn 16397  ℂ_fldccnfld 17942   TopSpctps 18632   Hauscha 19043   tsums ctsu 19827  Σ^*cesum 26627

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4510  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481  ax-inf2 7957  ax-cnex 9448  ax-resscn 9449  ax-1cn 9450  ax-icn 9451  ax-addcl 9452  ax-addrcl 9453  ax-mulcl 9454  ax-mulrcl 9455  ax-mulcom 9456  ax-addass 9457  ax-mulass 9458  ax-distr 9459  ax-i2m1 9460  ax-1ne0 9461  ax-1rid 9462  ax-rnegex 9463  ax-rrecex 9464  ax-cnre 9465  ax-pre-lttri 9466  ax-pre-lttrn 9467  ax-pre-ltadd 9468  ax-pre-mulgt0 9469  ax-pre-sup 9470  ax-addf 9471  ax-mulf 9472

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-nel 2650  df-ral 2803  df-rex 2804  df-reu 2805  df-rmo 2806  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-pss 3451  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-tp 3989  df-op 3991  df-uni 4199  df-int 4236  df-iun 4280  df-iin 4281  df-br 4400  df-opab 4458  df-mpt 4459  df-tr 4493  df-eprel 4739  df-id 4743  df-po 4748  df-so 4749  df-fr 4786  df-se 4787  df-we 4788  df-ord 4829  df-on 4830  df-lim 4831  df-suc 4832  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-isom 5534  df-riota 6160  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-om 6586  df-1st 6686  df-2nd 6687  df-supp 6800  df-recs 6941  df-rdg 6975  df-1o 7029  df-oadd 7033  df-er 7210  df-map 7325  df-en 7420  df-dom 7421  df-sdom 7422  df-fin 7423  df-fsupp 7731  df-fi 7771  df-sup 7801  df-oi 7834  df-card 8219  df-pnf 9530  df-mnf 9531  df-xr 9532  df-ltxr 9533  df-le 9534  df-sub 9707  df-neg 9708  df-div 10104  df-nn 10433  df-2 10490  df-3 10491  df-4 10492  df-5 10493  df-6 10494  df-7 10495  df-8 10496  df-9 10497  df-10 10498  df-n0 10690  df-z 10757  df-dec 10866  df-uz 10972  df-rp 11102  df-xadd 11200  df-ioo 11414  df-ico 11416  df-icc 11417  df-fz 11554  df-fzo 11665  df-seq 11923  df-exp 11982  df-hash 12220  df-cj 12705  df-re 12706  df-im 12707  df-sqr 12841  df-abs 12842  df-clim 13083  df-sum 13281  df-struct 14293  df-ndx 14294  df-slot 14295  df-base 14296  df-sets 14297  df-ress 14298  df-plusg 14369  df-mulr 14370  df-starv 14371  df-tset 14375  df-ple 14376  df-ds 14378  df-unif 14379  df-rest 14479  df-topn 14480  df-0g 14498  df-gsum 14499  df-topgen 14500  df-ordt 14557  df-xrs 14558  df-ps 15488  df-tsr 15489  df-mnd 15533  df-submnd 15583  df-grp 15663  df-minusg 15664  df-cntz 15953  df-cmn 16399  df-abl 16400  df-mgp 16713  df-ur 16725  df-rng 16769  df-cring 16770  df-fbas 17938  df-fg 17939  df-cnfld 17943  df-top 18634  df-bases 18636  df-topon 18637  df-topsp 18638  df-cld 18754  df-ntr 18755  df-cls 18756  df-nei 18833  df-cn 18962  df-haus 19050  df-fil 19550  df-fm 19642  df-flim 19643  df-flf 19644  df-tsms 19828  df-esum 26628

This theorem is referenced by:  volfiniune  26789