Theorem esumsnf 30126

Description: The extended sum of a singleton is the term. (Contributed by Thierry Arnoux, 2-Jan-2017.) (Revised by Thierry Arnoux, 2-May-2020.)

Hypotheses

Ref	Expression
esumsnf.0	|- F/_ k B
esumsnf.1	|- ( ( ph /\ k = M ) -> A = B )
esumsnf.2	|- ( ph -> M e. V )
esumsnf.3	|- ( ph -> B e. ( 0 [,] +oo ) )

Assertion

Ref	Expression
esumsnf	|- ( ph -> Σ* k e. { M } A = B )

Distinct variable groups:   k, M   ph, k

Allowed substitution hints:   A( k)   B( k)   V( k)

Proof of Theorem esumsnf

Dummy variables x l are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-esum 30090	. . 3 |- Σ* k e. { M } A = U. ( ( RR*s ↾s ( 0 [,] +oo ) ) tsums ( k e. { M } |-> A ) )
2	1	a1i 11	. 2 |- ( ph -> Σ* k e. { M } A = U. ( ( RR*s ↾s ( 0 [,] +oo ) ) tsums ( k e. { M } |-> A ) ) )
3		eqid 2622	. . . 4 |- ( RR*s ↾s ( 0 [,] +oo ) ) = ( RR*s ↾s ( 0 [,] +oo ) )
4		snfi 8038	. . . . 5 |- { M } e. Fin
5	4	a1i 11	. . . 4 |- ( ph -> { M } e. Fin )
6		elsni 4194	. . . . . . . . 9 |- ( k e. { M } -> k = M )
7		esumsnf.1	. . . . . . . . 9 |- ( ( ph /\ k = M ) -> A = B )
8	6, 7	sylan2 491	. . . . . . . 8 |- ( ( ph /\ k e. { M } ) -> A = B )
9	8	mpteq2dva 4744	. . . . . . 7 |- ( ph -> ( k e. { M } |-> A ) = ( k e. { M } |-> B ) )
10		esumsnf.2	. . . . . . . 8 |- ( ph -> M e. V )
11		esumsnf.3	. . . . . . . 8 |- ( ph -> B e. ( 0 [,] +oo ) )
12		fmptsn 6433	. . . . . . . . 9 |- ( ( M e. V /\ B e. ( 0 [,] +oo ) ) -> { <. M , B >. } = ( l e. { M } |-> B ) )
13		nfcv 2764	. . . . . . . . . 10 |- F/_ l B
14		esumsnf.0	. . . . . . . . . 10 |- F/_ k B
15		eqidd 2623	. . . . . . . . . 10 |- ( k = l -> B = B )
16	13, 14, 15	cbvmpt 4749	. . . . . . . . 9 |- ( k e. { M } |-> B ) = ( l e. { M } |-> B )
17	12, 16	syl6eqr 2674	. . . . . . . 8 |- ( ( M e. V /\ B e. ( 0 [,] +oo ) ) -> { <. M , B >. } = ( k e. { M } |-> B ) )
18	10, 11, 17	syl2anc 693	. . . . . . 7 |- ( ph -> { <. M , B >. } = ( k e. { M } |-> B ) )
19	9, 18	eqtr4d 2659	. . . . . 6 |- ( ph -> ( k e. { M } |-> A ) = { <. M , B >. } )
20		fsng 6404	. . . . . . 7 |- ( ( M e. V /\ B e. ( 0 [,] +oo ) ) -> ( ( k e. { M } |-> A ) : { M } --> { B } <-> ( k e. { M } |-> A ) = { <. M , B >. } ) )
21	10, 11, 20	syl2anc 693	. . . . . 6 |- ( ph -> ( ( k e. { M } |-> A ) : { M } --> { B } <-> ( k e. { M } |-> A ) = { <. M , B >. } ) )
22	19, 21	mpbird 247	. . . . 5 |- ( ph -> ( k e. { M } |-> A ) : { M } --> { B } )
23	11	snssd 4340	. . . . 5 |- ( ph -> { B } C_ ( 0 [,] +oo ) )
24	22, 23	fssd 6057	. . . 4 |- ( ph -> ( k e. { M } |-> A ) : { M } --> ( 0 [,] +oo ) )
25		xrltso 11974	. . . . . . 7 |- < Or RR*
26	25	a1i 11	. . . . . 6 |- ( ph -> < Or RR* )
27		0xr 10086	. . . . . . 7 |- 0 e. RR*
28	27	a1i 11	. . . . . 6 |- ( ph -> 0 e. RR* )
29		elxrge0 12281	. . . . . . . 8 |- ( B e. ( 0 [,] +oo ) <-> ( B e. RR* /\ 0 <_ B ) )
30	11, 29	sylib 208	. . . . . . 7 |- ( ph -> ( B e. RR* /\ 0 <_ B ) )
31	30	simpld 475	. . . . . 6 |- ( ph -> B e. RR* )
32		suppr 8377	. . . . . 6 |- ( ( < Or RR* /\ 0 e. RR* /\ B e. RR* ) -> sup ( { 0 , B } , RR* , < ) = if ( B < 0 , 0 , B ) )
33	26, 28, 31, 32	syl3anc 1326	. . . . 5 |- ( ph -> sup ( { 0 , B } , RR* , < ) = if ( B < 0 , 0 , B ) )
34		0fin 8188	. . . . . . . . . . 11 |- (/) e. Fin
35	34	a1i 11	. . . . . . . . . 10 |- ( ph -> (/) e. Fin )
36		reseq2 5391	. . . . . . . . . . . . . 14 |- ( x = (/) -> ( ( k e. { M } |-> A ) |` x ) = ( ( k e. { M } |-> A ) |` (/) ) )
37		res0 5400	. . . . . . . . . . . . . 14 |- ( ( k e. { M } |-> A ) |` (/) ) = (/)
38	36, 37	syl6eq 2672	. . . . . . . . . . . . 13 |- ( x = (/) -> ( ( k e. { M } |-> A ) |` x ) = (/) )
39	38	oveq2d 6666	. . . . . . . . . . . 12 |- ( x = (/) -> ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( ( k e. { M } |-> A ) |` x ) ) = ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg (/) ) )
40		xrge00 29686	. . . . . . . . . . . . 13 |- 0 = ( 0g ` ( RR*s ↾s ( 0 [,] +oo ) ) )
41	40	gsum0 17278	. . . . . . . . . . . 12 |- ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg (/) ) = 0
42	39, 41	syl6eq 2672	. . . . . . . . . . 11 |- ( x = (/) -> ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( ( k e. { M } |-> A ) |` x ) ) = 0 )
43	42	adantl 482	. . . . . . . . . 10 |- ( ( ph /\ x = (/) ) -> ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( ( k e. { M } |-> A ) |` x ) ) = 0 )
44		reseq2 5391	. . . . . . . . . . . . 13 |- ( x = { M } -> ( ( k e. { M } |-> A ) |` x ) = ( ( k e. { M } |-> A ) |` { M } ) )
45		ssid 3624	. . . . . . . . . . . . . 14 |- { M } C_ { M }
46		resmpt 5449	. . . . . . . . . . . . . 14 |- ( { M } C_ { M } -> ( ( k e. { M } |-> A ) |` { M } ) = ( k e. { M } |-> A ) )
47	45, 46	ax-mp 5	. . . . . . . . . . . . 13 |- ( ( k e. { M } |-> A ) |` { M } ) = ( k e. { M } |-> A )
48	44, 47	syl6eq 2672	. . . . . . . . . . . 12 |- ( x = { M } -> ( ( k e. { M } |-> A ) |` x ) = ( k e. { M } |-> A ) )
49	48	oveq2d 6666	. . . . . . . . . . 11 |- ( x = { M } -> ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( ( k e. { M } |-> A ) |` x ) ) = ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. { M } |-> A ) ) )
50		xrge0base 29685	. . . . . . . . . . . 12 |- ( 0 [,] +oo ) = ( Base ` ( RR*s ↾s ( 0 [,] +oo ) ) )
51		xrge0cmn 19788	. . . . . . . . . . . . . 14 |- ( RR*s ↾s ( 0 [,] +oo ) ) e. CMnd
52		cmnmnd 18208	. . . . . . . . . . . . . 14 |- ( ( RR*s ↾s ( 0 [,] +oo ) ) e. CMnd -> ( RR*s ↾s ( 0 [,] +oo ) ) e. Mnd )
53	51, 52	ax-mp 5	. . . . . . . . . . . . 13 |- ( RR*s ↾s ( 0 [,] +oo ) ) e. Mnd
54	53	a1i 11	. . . . . . . . . . . 12 |- ( ph -> ( RR*s ↾s ( 0 [,] +oo ) ) e. Mnd )
55		nfv 1843	. . . . . . . . . . . 12 |- F/ k ph
56	50, 54, 10, 11, 7, 55, 14	gsumsnfd 18351	. . . . . . . . . . 11 |- ( ph -> ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. { M } |-> A ) ) = B )
57	49, 56	sylan9eqr 2678	. . . . . . . . . 10 |- ( ( ph /\ x = { M } ) -> ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( ( k e. { M } |-> A ) |` x ) ) = B )
58	35, 5, 28, 11, 43, 57	fmptpr 6438	. . . . . . . . 9 |- ( ph -> { <. (/) , 0 >. , <. { M } , B >. } = ( x e. { (/) , { M } } |-> ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( ( k e. { M } |-> A ) |` x ) ) ) )
59		pwsn 4428	. . . . . . . . . . . . 13 |- ~P { M } = { (/) , { M } }
60		prssi 4353	. . . . . . . . . . . . . 14 |- ( ( (/) e. Fin /\ { M } e. Fin ) -> { (/) , { M } } C_ Fin )
61	34, 4, 60	mp2an 708	. . . . . . . . . . . . 13 |- { (/) , { M } } C_ Fin
62	59, 61	eqsstri 3635	. . . . . . . . . . . 12 |- ~P { M } C_ Fin
63		df-ss 3588	. . . . . . . . . . . 12 |- ( ~P { M } C_ Fin <-> ( ~P { M } i^i Fin ) = ~P { M } )
64	62, 63	mpbi 220	. . . . . . . . . . 11 |- ( ~P { M } i^i Fin ) = ~P { M }
65	64, 59	eqtri 2644	. . . . . . . . . 10 |- ( ~P { M } i^i Fin ) = { (/) , { M } }
66		eqid 2622	. . . . . . . . . 10 |- ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( ( k e. { M } |-> A ) |` x ) ) = ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( ( k e. { M } |-> A ) |` x ) )
67	65, 66	mpteq12i 4742	. . . . . . . . 9 |- ( x e. ( ~P { M } i^i Fin ) |-> ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( ( k e. { M } |-> A ) |` x ) ) ) = ( x e. { (/) , { M } } |-> ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( ( k e. { M } |-> A ) |` x ) ) )
68	58, 67	syl6eqr 2674	. . . . . . . 8 |- ( ph -> { <. (/) , 0 >. , <. { M } , B >. } = ( x e. ( ~P { M } i^i Fin ) |-> ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( ( k e. { M } |-> A ) |` x ) ) ) )
69	68	rneqd 5353	. . . . . . 7 |- ( ph -> ran { <. (/) , 0 >. , <. { M } , B >. } = ran ( x e. ( ~P { M } i^i Fin ) |-> ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( ( k e. { M } |-> A ) |` x ) ) ) )
70		rnpropg 5615	. . . . . . . 8 |- ( ( (/) e. Fin /\ { M } e. Fin ) -> ran { <. (/) , 0 >. , <. { M } , B >. } = { 0 , B } )
71	35, 5, 70	syl2anc 693	. . . . . . 7 |- ( ph -> ran { <. (/) , 0 >. , <. { M } , B >. } = { 0 , B } )
72	69, 71	eqtr3d 2658	. . . . . 6 |- ( ph -> ran ( x e. ( ~P { M } i^i Fin ) |-> ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( ( k e. { M } |-> A ) |` x ) ) ) = { 0 , B } )
73	72	supeq1d 8352	. . . . 5 |- ( ph -> sup ( ran ( x e. ( ~P { M } i^i Fin ) |-> ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( ( k e. { M } |-> A ) |` x ) ) ) , RR* , < ) = sup ( { 0 , B } , RR* , < ) )
74	30	simprd 479	. . . . . . . . 9 |- ( ph -> 0 <_ B )
75		xrlenlt 10103	. . . . . . . . . 10 |- ( ( 0 e. RR* /\ B e. RR* ) -> ( 0 <_ B <-> -. B < 0 ) )
76	28, 31, 75	syl2anc 693	. . . . . . . . 9 |- ( ph -> ( 0 <_ B <-> -. B < 0 ) )
77	74, 76	mpbid 222	. . . . . . . 8 |- ( ph -> -. B < 0 )
78		eqidd 2623	. . . . . . . 8 |- ( ph -> B = B )
79	77, 78	jca 554	. . . . . . 7 |- ( ph -> ( -. B < 0 /\ B = B ) )
80	79	olcd 408	. . . . . 6 |- ( ph -> ( ( B < 0 /\ B = 0 ) \/ ( -. B < 0 /\ B = B ) ) )
81		eqif 4126	. . . . . 6 |- ( B = if ( B < 0 , 0 , B ) <-> ( ( B < 0 /\ B = 0 ) \/ ( -. B < 0 /\ B = B ) ) )
82	80, 81	sylibr 224	. . . . 5 |- ( ph -> B = if ( B < 0 , 0 , B ) )
83	33, 73, 82	3eqtr4rd 2667	. . . 4 |- ( ph -> B = sup ( ran ( x e. ( ~P { M } i^i Fin ) |-> ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( ( k e. { M } |-> A ) |` x ) ) ) , RR* , < ) )
84	3, 5, 24, 83	xrge0tsmsd 29785	. . 3 |- ( ph -> ( ( RR*s ↾s ( 0 [,] +oo ) ) tsums ( k e. { M } |-> A ) ) = { B } )
85	84	unieqd 4446	. 2 |- ( ph -> U. ( ( RR*s ↾s ( 0 [,] +oo ) ) tsums ( k e. { M } |-> A ) ) = U. { B } )
86		unisng 4452	. . 3 |- ( B e. ( 0 [,] +oo ) -> U. { B } = B )
87	11, 86	syl 17	. 2 |- ( ph -> U. { B } = B )
88	2, 85, 87	3eqtrd 2660	1 |- ( ph -> Σ* k e. { M } A = B )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   F/_wnfc 2751   i^i cin 3573   C_ wss 3574   (/)c0 3915   ifcif 4086   ~Pcpw 4158   {csn 4177   {cpr 4179   <.cop 4183   U.cuni 4436   class class class wbr 4653   |-> cmpt 4729   Or wor 5034   ran crn 5115   |` cres 5116   -->wf 5884  (class class class)co 6650   Fincfn 7955   supcsup 8346   0cc0 9936   +oocpnf 10071   RR*cxr 10073   < clt 10074   <_ cle 10075   [,]cicc 12178   ↾_s cress 15858   gsum_g cgsu 16101   RR*scxrs 16160   Mndcmnd 17294  CMndccmn 18193   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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  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-of 6897  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-inf 8349  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-q 11789  df-xadd 11947  df-ioo 12179  df-ioc 12180  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-seq 12802  df-hash 13118  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-tset 15960  df-ple 15961  df-ds 15964  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-ordt 16161  df-xrs 16162  df-mre 16246  df-mrc 16247  df-acs 16249  df-ps 17200  df-tsr 17201  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-cmn 18195  df-fbas 19743  df-fg 19744  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-ntr 20824  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:  esumsn  30127  esum2dlem  30154