Theorem esumpfinvalf 28305

Description: Same as esumpfinval 28304, 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 28257	. . . 4 |- Σ* k e. A B = U. ( ( RR*s ↾s ( 0 [,] +oo ) ) tsums ( k e. A |-> B ) )
2		xrge0base 27907	. . . . . 6 |- ( 0 [,] +oo ) = ( Base ` ( RR*s ↾s ( 0 [,] +oo ) ) )
3		xrge00 27908	. . . . . 6 |- 0 = ( 0g ` ( RR*s ↾s ( 0 [,] +oo ) ) )
4		xrge0cmn 18655	. . . . . . 7 |- ( RR*s ↾s ( 0 [,] +oo ) ) e. CMnd
5	4	a1i 11	. . . . . 6 |- ( ph -> ( RR*s ↾s ( 0 [,] +oo ) ) e. CMnd )
6		xrge0tps 28159	. . . . . . 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 11614	. . . . . . . 8 |- ( 0 [,) +oo ) C_ ( 0 [,] +oo )
13		esumpfinvalf.b	. . . . . . . 8 |- ( ( ph /\ k e. A ) -> B e. ( 0 [,) +oo ) )
14	12, 13	sseldi 3487	. . . . . . 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 27716	. . . . . 6 |- ( ph -> ( k e. A |-> B ) : A --> ( 0 [,] +oo ) )
17		c0ex 9579	. . . . . . . 8 |- 0 e. _V
18	17	a1i 11	. . . . . . 7 |- ( ph -> 0 e. _V )
19	16, 8, 18	fdmfifsupp 7831	. . . . . 6 |- ( ph -> ( k e. A |-> B ) finSupp 0 )
20		xrge0topn 28160	. . . . . . 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 27818	. . . . . . . 8 |- (ordTop ` <_ ) e. Haus
23		ovex 6298	. . . . . . . 8 |- ( 0 [,] +oo ) e. _V
24		resthaus 20036	. . . . . . . 8 |- ( ( (ordTop ` <_ ) e. Haus /\ ( 0 [,] +oo ) e. _V ) -> ( (ordTop ` <_ ) ↾t ( 0 [,] +oo ) ) e. Haus )
25	22, 23, 24	mp2an 670	. . . . . . 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 20805	. . . . 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 4245	. . . 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 6298	. . . 4 |- ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. A |-> B ) ) e. _V
31	30	unisn 4250	. . 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 27716	. . 3 |- ( ph -> ( k e. A |-> B ) : A --> ( 0 [,) +oo ) )
35		esumpfinvallem 28303	. . 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 659	. 2 |- ( ph -> (ℂfld gsumg ( k e. A |-> B ) ) = ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. A |-> B ) ) )
37		rge0ssre 11631	. . . . . . . 8 |- ( 0 [,) +oo ) C_ RR
38		ax-resscn 9538	. . . . . . . 8 |- RR C_ CC
39	37, 38	sstri 3498	. . . . . . 7 |- ( 0 [,) +oo ) C_ CC
40	39, 13	sseldi 3487	. . . . . 6 |- ( ( ph /\ k e. A ) -> B e. CC )
41	40	sbt 2164	. . . . 5 |- [ l / k ] ( ( ph /\ k e. A ) -> B e. CC )
42		sbim 2138	. . . . . 6 |- ( [ l / k ] ( ( ph /\ k e. A ) -> B e. CC ) <-> ( [ l / k ] ( ph /\ k e. A ) -> [ l / k ] B e. CC ) )
43		sban 2142	. . . . . . . 8 |- ( [ l / k ] ( ph /\ k e. A ) <-> ( [ l / k ] ph /\ [ l / k ] k e. A ) )
44	9	sbf 2123	. . . . . . . . 9 |- ( [ l / k ] ph <-> ph )
45	10	clelsb3f 27577	. . . . . . . . 9 |- ( [ l / k ] k e. A <-> l e. A )
46	44, 45	anbi12i 695	. . . . . . . 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 3328	. . . . . . . 8 |- ( [ l / k ] B e. CC <-> [. l / k ]. B e. CC )
49		vex 3109	. . . . . . . . 9 |- l e. _V
50		sbcel1g 3826	. . . . . . . . 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 324	. . . . . 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 18679	. . 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 3436	. . . . 5 |- F/_ k [_ l / k ]_ B
60		csbeq1a 3429	. . . . 5 |- ( k = l -> B = [_ l / k ]_ B )
61	10, 57, 58, 59, 60	cbvmptf 27715	. . . 4 |- ( k e. A |-> B ) = ( l e. A |-> [_ l / k ]_ B )
62	61	oveq2i 6281	. . 3 |- (ℂfld gsumg ( k e. A |-> B ) ) = (ℂfld gsumg ( l e. A |-> [_ l / k ]_ B ) )
63	60, 57, 10, 58, 59	cbvsum 13599	. . 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 367   = wceq 1398   F/wnf 1621   [wsb 1744   e. wcel 1823   F/_wnfc 2602   _Vcvv 3106   [.wsbc 3324   [_csb 3420   {csn 4016   U.cuni 4235   |-> cmpt 4497   -->wf 5566   ` cfv 5570  (class class class)co 6270   Fincfn 7509   CCcc 9479   RRcr 9480   0cc0 9481   +oocpnf 9614   <_ cle 9618   [,)cico 11534   [,]cicc 11535   sum_csu 13590   ↾_s cress 14717   ↾_t crest 14910   TopOpenctopn 14911   gsum_g cgsu 14930  ordTopcordt 14988   RR*scxrs 14989  CMndccmn 16997  ℂ_fldccnfld 18615   TopSpctps 19564   Hauscha 19976   tsums ctsu 20790  Σ^*cesum 28256

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559  ax-addf 9560  ax-mulf 9561

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-iin 4318  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-supp 6892  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-fsupp 7822  df-fi 7863  df-sup 7893  df-oi 7927  df-card 8311  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  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 10977  df-uz 11083  df-rp 11222  df-xadd 11322  df-ico 11538  df-icc 11539  df-fz 11676  df-fzo 11800  df-seq 12090  df-exp 12149  df-hash 12388  df-cj 13014  df-re 13015  df-im 13016  df-sqrt 13150  df-abs 13151  df-clim 13393  df-sum 13591  df-struct 14718  df-ndx 14719  df-slot 14720  df-base 14721  df-sets 14722  df-ress 14723  df-plusg 14797  df-mulr 14798  df-starv 14799  df-tset 14803  df-ple 14804  df-ds 14806  df-unif 14807  df-rest 14912  df-topn 14913  df-0g 14931  df-gsum 14932  df-topgen 14933  df-ordt 14990  df-xrs 14991  df-ps 16029  df-tsr 16030  df-mgm 16071  df-sgrp 16110  df-mnd 16120  df-submnd 16166  df-grp 16256  df-minusg 16257  df-cntz 16554  df-cmn 16999  df-abl 17000  df-mgp 17337  df-ur 17349  df-ring 17395  df-cring 17396  df-fbas 18611  df-fg 18612  df-cnfld 18616  df-top 19566  df-bases 19568  df-topon 19569  df-topsp 19570  df-cld 19687  df-ntr 19688  df-cls 19689  df-nei 19766  df-cn 19895  df-haus 19983  df-fil 20513  df-fm 20605  df-flim 20606  df-flf 20607  df-tsms 20791  df-esum 28257

This theorem is referenced by:  volfiniune  28439