Theorem esumpfinvalf 30138

Description: Same as esumpfinval 30137, 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 30090	. . . 4 |- Σ* k e. A B = U. ( ( RR*s ↾s ( 0 [,] +oo ) ) tsums ( k e. A |-> B ) )
2		xrge0base 29685	. . . . . 6 |- ( 0 [,] +oo ) = ( Base ` ( RR*s ↾s ( 0 [,] +oo ) ) )
3		xrge00 29686	. . . . . 6 |- 0 = ( 0g ` ( RR*s ↾s ( 0 [,] +oo ) ) )
4		xrge0cmn 19788	. . . . . . 7 |- ( RR*s ↾s ( 0 [,] +oo ) ) e. CMnd
5	4	a1i 11	. . . . . 6 |- ( ph -> ( RR*s ↾s ( 0 [,] +oo ) ) e. CMnd )
6		xrge0tps 29988	. . . . . . 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 2764	. . . . . . 7 |- F/_ k ( 0 [,] +oo )
12		icossicc 12260	. . . . . . . 8 |- ( 0 [,) +oo ) C_ ( 0 [,] +oo )
13		esumpfinvalf.b	. . . . . . . 8 |- ( ( ph /\ k e. A ) -> B e. ( 0 [,) +oo ) )
14	12, 13	sseldi 3601	. . . . . . 7 |- ( ( ph /\ k e. A ) -> B e. ( 0 [,] +oo ) )
15		eqid 2622	. . . . . . 7 |- ( k e. A |-> B ) = ( k e. A |-> B )
16	9, 10, 11, 14, 15	fmptdF 29456	. . . . . 6 |- ( ph -> ( k e. A |-> B ) : A --> ( 0 [,] +oo ) )
17		c0ex 10034	. . . . . . . 8 |- 0 e. _V
18	17	a1i 11	. . . . . . 7 |- ( ph -> 0 e. _V )
19	16, 8, 18	fdmfifsupp 8285	. . . . . 6 |- ( ph -> ( k e. A |-> B ) finSupp 0 )
20		xrge0topn 29989	. . . . . . 7 |- ( TopOpen ` ( RR*s ↾s ( 0 [,] +oo ) ) ) = ( (ordTop ` <_ ) ↾t ( 0 [,] +oo ) )
21	20	eqcomi 2631	. . . . . 6 |- ( (ordTop ` <_ ) ↾t ( 0 [,] +oo ) ) = ( TopOpen ` ( RR*s ↾s ( 0 [,] +oo ) ) )
22		xrhaus 29535	. . . . . . . 8 |- (ordTop ` <_ ) e. Haus
23		ovex 6678	. . . . . . . 8 |- ( 0 [,] +oo ) e. _V
24		resthaus 21172	. . . . . . . 8 |- ( ( (ordTop ` <_ ) e. Haus /\ ( 0 [,] +oo ) e. _V ) -> ( (ordTop ` <_ ) ↾t ( 0 [,] +oo ) ) e. Haus )
25	22, 23, 24	mp2an 708	. . . . . . 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 21944	. . . . 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 4446	. . . 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 2668	. . 3 |- ( ph -> Σ* k e. A B = U. { ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. A |-> B ) ) } )
30		ovex 6678	. . . 4 |- ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. A |-> B ) ) e. _V
31	30	unisn 4451	. . 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 2672	. 2 |- ( ph -> Σ* k e. A B = ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. A |-> B ) ) )
33		nfcv 2764	. . . 4 |- F/_ k ( 0 [,) +oo )
34	9, 10, 33, 13, 15	fmptdF 29456	. . 3 |- ( ph -> ( k e. A |-> B ) : A --> ( 0 [,) +oo ) )
35		esumpfinvallem 30136	. . 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 693	. 2 |- ( ph -> (ℂfld gsumg ( k e. A |-> B ) ) = ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. A |-> B ) ) )
37		rge0ssre 12280	. . . . . . . 8 |- ( 0 [,) +oo ) C_ RR
38		ax-resscn 9993	. . . . . . . 8 |- RR C_ CC
39	37, 38	sstri 3612	. . . . . . 7 |- ( 0 [,) +oo ) C_ CC
40	39, 13	sseldi 3601	. . . . . 6 |- ( ( ph /\ k e. A ) -> B e. CC )
41	40	sbt 2419	. . . . 5 |- [ l / k ] ( ( ph /\ k e. A ) -> B e. CC )
42		sbim 2395	. . . . . 6 |- ( [ l / k ] ( ( ph /\ k e. A ) -> B e. CC ) <-> ( [ l / k ] ( ph /\ k e. A ) -> [ l / k ] B e. CC ) )
43		sban 2399	. . . . . . . 8 |- ( [ l / k ] ( ph /\ k e. A ) <-> ( [ l / k ] ph /\ [ l / k ] k e. A ) )
44	9	sbf 2380	. . . . . . . . 9 |- ( [ l / k ] ph <-> ph )
45	10	clelsb3f 2768	. . . . . . . . 9 |- ( [ l / k ] k e. A <-> l e. A )
46	44, 45	anbi12i 733	. . . . . . . 8 |- ( ( [ l / k ] ph /\ [ l / k ] k e. A ) <-> ( ph /\ l e. A ) )
47	43, 46	bitri 264	. . . . . . 7 |- ( [ l / k ] ( ph /\ k e. A ) <-> ( ph /\ l e. A ) )
48		sbsbc 3439	. . . . . . . 8 |- ( [ l / k ] B e. CC <-> [. l / k ]. B e. CC )
49		vex 3203	. . . . . . . . 9 |- l e. _V
50		sbcel1g 3987	. . . . . . . . 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 264	. . . . . . 7 |- ( [ l / k ] B e. CC <-> [_ l / k ]_ B e. CC )
53	47, 52	imbi12i 340	. . . . . 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 264	. . . . 5 |- ( [ l / k ] ( ( ph /\ k e. A ) -> B e. CC ) <-> ( ( ph /\ l e. A ) -> [_ l / k ]_ B e. CC ) )
55	41, 54	mpbi 220	. . . 4 |- ( ( ph /\ l e. A ) -> [_ l / k ]_ B e. CC )
56	8, 55	gsumfsum 19813	. . 3 |- ( ph -> (ℂfld gsumg ( l e. A |-> [_ l / k ]_ B ) ) = sum_ l e. A [_ l / k ]_ B )
57		nfcv 2764	. . . . 5 |- F/_ l A
58		nfcv 2764	. . . . 5 |- F/_ l B
59		nfcsb1v 3549	. . . . 5 |- F/_ k [_ l / k ]_ B
60		csbeq1a 3542	. . . . 5 |- ( k = l -> B = [_ l / k ]_ B )
61	10, 57, 58, 59, 60	cbvmptf 4748	. . . 4 |- ( k e. A |-> B ) = ( l e. A |-> [_ l / k ]_ B )
62	61	oveq2i 6661	. . 3 |- (ℂfld gsumg ( k e. A |-> B ) ) = (ℂfld gsumg ( l e. A |-> [_ l / k ]_ B ) )
63	60, 57, 10, 58, 59	cbvsum 14425	. . 3 |- sum_ k e. A B = sum_ l e. A [_ l / k ]_ B
64	56, 62, 63	3eqtr4g 2681	. 2 |- ( ph -> (ℂfld gsumg ( k e. A |-> B ) ) = sum_ k e. A B )
65	32, 36, 64	3eqtr2d 2662	1 |- ( ph -> Σ* k e. A B = sum_ k e. A B )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   F/wnf 1708   [wsb 1880   e. wcel 1990   F/_wnfc 2751   _Vcvv 3200   [.wsbc 3435   [_csb 3533   {csn 4177   U.cuni 4436   |-> cmpt 4729   -->wf 5884   ` cfv 5888  (class class class)co 6650   Fincfn 7955   CCcc 9934   RRcr 9935   0cc0 9936   +oocpnf 10071   <_ cle 10075   [,)cico 12177   [,]cicc 12178   sum_csu 14416   ↾_s cress 15858   ↾_t crest 16081   TopOpenctopn 16082   gsum_g cgsu 16101  ordTopcordt 16159   RR*scxrs 16160  CMndccmn 18193  ℂ_fldccnfld 19746   TopSpctps 20736   Hauscha 21112   tsums ctsu 21929  Σ^*cesum 30089

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-rp 11833  df-xadd 11947  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-sum 14417  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-ordt 16161  df-xrs 16162  df-ps 17200  df-tsr 17201  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-grp 17425  df-minusg 17426  df-cntz 17750  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-ring 18549  df-cring 18550  df-fbas 19743  df-fg 19744  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-cn 21031  df-haus 21119  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-tsms 21930  df-esum 30090

This theorem is referenced by:  volfiniune  30293