Theorem sge0sup 38243

Description: The arbitrary sum of nonnegative extended reals is the supremum of finite subsums. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Hypotheses

Ref	Expression
sge0sup.x	|- ( ph -> X e. V )
sge0sup.f	|- ( ph -> F : X --> ( 0 [,] +oo ) )

Assertion

Ref	Expression
sge0sup	|- ( ph -> (Σ^ ` F ) = sup ( ran ( x e. ( ~P X i^i Fin ) |-> (Σ^ ` ( F |` x ) ) ) , RR* , < ) )

Distinct variable groups:   x, F   x, X   ph, x

Allowed substitution hint:   V( x)

Proof of Theorem sge0sup

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqidd 2454	. . 3 |- ( ( ph /\ +oo e. ran F ) -> +oo = +oo )
2		sge0sup.x	. . . . 5 |- ( ph -> X e. V )
3	2	adantr 467	. . . 4 |- ( ( ph /\ +oo e. ran F ) -> X e. V )
4		sge0sup.f	. . . . 5 |- ( ph -> F : X --> ( 0 [,] +oo ) )
5	4	adantr 467	. . . 4 |- ( ( ph /\ +oo e. ran F ) -> F : X --> ( 0 [,] +oo ) )
6		simpr 463	. . . 4 |- ( ( ph /\ +oo e. ran F ) -> +oo e. ran F )
7	3, 5, 6	sge0pnfval 38225	. . 3 |- ( ( ph /\ +oo e. ran F ) -> (Σ^ ` F ) = +oo )
8		vex 3050	. . . . . . . . 9 |- x e. _V
9	8	a1i 11	. . . . . . . 8 |- ( ( ph /\ x e. ( ~P X i^i Fin ) ) -> x e. _V )
10	4	adantr 467	. . . . . . . . 9 |- ( ( ph /\ x e. ( ~P X i^i Fin ) ) -> F : X --> ( 0 [,] +oo ) )
11		elinel1 3621	. . . . . . . . . . 11 |- ( x e. ( ~P X i^i Fin ) -> x e. ~P X )
12		elpwi 3962	. . . . . . . . . . 11 |- ( x e. ~P X -> x C_ X )
13	11, 12	syl 17	. . . . . . . . . 10 |- ( x e. ( ~P X i^i Fin ) -> x C_ X )
14	13	adantl 468	. . . . . . . . 9 |- ( ( ph /\ x e. ( ~P X i^i Fin ) ) -> x C_ X )
15	10, 14	fssresd 5755	. . . . . . . 8 |- ( ( ph /\ x e. ( ~P X i^i Fin ) ) -> ( F |` x ) : x --> ( 0 [,] +oo ) )
16	9, 15	sge0xrcl 38237	. . . . . . 7 |- ( ( ph /\ x e. ( ~P X i^i Fin ) ) -> (Σ^ ` ( F |` x ) ) e. RR* )
17	16	adantlr 722	. . . . . 6 |- ( ( ( ph /\ +oo e. ran F ) /\ x e. ( ~P X i^i Fin ) ) -> (Σ^ ` ( F |` x ) ) e. RR* )
18	17	ralrimiva 2804	. . . . 5 |- ( ( ph /\ +oo e. ran F ) -> A. x e. ( ~P X i^i Fin ) (Σ^ ` ( F |` x ) ) e. RR* )
19		eqid 2453	. . . . . 6 |- ( x e. ( ~P X i^i Fin ) |-> (Σ^ ` ( F |` x ) ) ) = ( x e. ( ~P X i^i Fin ) |-> (Σ^ ` ( F |` x ) ) )
20	19	rnmptss 6057	. . . . 5 |- ( A. x e. ( ~P X i^i Fin ) (Σ^ ` ( F |` x ) ) e. RR* -> ran ( x e. ( ~P X i^i Fin ) |-> (Σ^ ` ( F |` x ) ) ) C_ RR* )
21	18, 20	syl 17	. . . 4 |- ( ( ph /\ +oo e. ran F ) -> ran ( x e. ( ~P X i^i Fin ) |-> (Σ^ ` ( F |` x ) ) ) C_ RR* )
22		ffn 5733	. . . . . . . . 9 |- ( F : X --> ( 0 [,] +oo ) -> F Fn X )
23	4, 22	syl 17	. . . . . . . 8 |- ( ph -> F Fn X )
24		fvelrnb 5917	. . . . . . . 8 |- ( F Fn X -> ( +oo e. ran F <-> E. y e. X ( F ` y ) = +oo ) )
25	23, 24	syl 17	. . . . . . 7 |- ( ph -> ( +oo e. ran F <-> E. y e. X ( F ` y ) = +oo ) )
26	25	adantr 467	. . . . . 6 |- ( ( ph /\ +oo e. ran F ) -> ( +oo e. ran F <-> E. y e. X ( F ` y ) = +oo ) )
27	6, 26	mpbid 214	. . . . 5 |- ( ( ph /\ +oo e. ran F ) -> E. y e. X ( F ` y ) = +oo )
28		snelpwi 4648	. . . . . . . . . . . 12 |- ( y e. X -> { y } e. ~P X )
29		snfi 7655	. . . . . . . . . . . . 13 |- { y } e. Fin
30	29	a1i 11	. . . . . . . . . . . 12 |- ( y e. X -> { y } e. Fin )
31	28, 30	elind 3620	. . . . . . . . . . 11 |- ( y e. X -> { y } e. ( ~P X i^i Fin ) )
32	31	3ad2ant2 1031	. . . . . . . . . 10 |- ( ( ph /\ y e. X /\ ( F ` y ) = +oo ) -> { y } e. ( ~P X i^i Fin ) )
33		simp2 1010	. . . . . . . . . . . 12 |- ( ( ph /\ y e. X /\ ( F ` y ) = +oo ) -> y e. X )
34	4	3ad2ant1 1030	. . . . . . . . . . . . 13 |- ( ( ph /\ y e. X /\ ( F ` y ) = +oo ) -> F : X --> ( 0 [,] +oo ) )
35	33	snssd 4120	. . . . . . . . . . . . 13 |- ( ( ph /\ y e. X /\ ( F ` y ) = +oo ) -> { y } C_ X )
36	34, 35	fssresd 5755	. . . . . . . . . . . 12 |- ( ( ph /\ y e. X /\ ( F ` y ) = +oo ) -> ( F |` { y } ) : { y } --> ( 0 [,] +oo ) )
37	33, 36	sge0sn 38231	. . . . . . . . . . 11 |- ( ( ph /\ y e. X /\ ( F ` y ) = +oo ) -> (Σ^ ` ( F |` { y } ) ) = ( ( F |` { y } ) ` y ) )
38		ssnid 3999	. . . . . . . . . . . . 13 |- y e. { y }
39		fvres 5884	. . . . . . . . . . . . 13 |- ( y e. { y } -> ( ( F |` { y } ) ` y ) = ( F ` y ) )
40	38, 39	ax-mp 5	. . . . . . . . . . . 12 |- ( ( F |` { y } ) ` y ) = ( F ` y )
41	40	a1i 11	. . . . . . . . . . 11 |- ( ( ph /\ y e. X /\ ( F ` y ) = +oo ) -> ( ( F |` { y } ) ` y ) = ( F ` y ) )
42		simp3 1011	. . . . . . . . . . 11 |- ( ( ph /\ y e. X /\ ( F ` y ) = +oo ) -> ( F ` y ) = +oo )
43	37, 41, 42	3eqtrrd 2492	. . . . . . . . . 10 |- ( ( ph /\ y e. X /\ ( F ` y ) = +oo ) -> +oo = (Σ^ ` ( F |` { y } ) ) )
44		reseq2 5103	. . . . . . . . . . . . 13 |- ( x = { y } -> ( F |` x ) = ( F |` { y } ) )
45	44	fveq2d 5874	. . . . . . . . . . . 12 |- ( x = { y } -> (Σ^ ` ( F |` x ) ) = (Σ^ ` ( F |` { y } ) ) )
46	45	eqeq2d 2463	. . . . . . . . . . 11 |- ( x = { y } -> ( +oo = (Σ^ ` ( F |` x ) ) <-> +oo = (Σ^ ` ( F |` { y } ) ) ) )
47	46	rspcev 3152	. . . . . . . . . 10 |- ( ( { y } e. ( ~P X i^i Fin ) /\ +oo = (Σ^ ` ( F |` { y } ) ) ) -> E. x e. ( ~P X i^i Fin ) +oo = (Σ^ ` ( F |` x ) ) )
48	32, 43, 47	syl2anc 667	. . . . . . . . 9 |- ( ( ph /\ y e. X /\ ( F ` y ) = +oo ) -> E. x e. ( ~P X i^i Fin ) +oo = (Σ^ ` ( F |` x ) ) )
49		pnfex 11420	. . . . . . . . . 10 |- +oo e. _V
50	49	a1i 11	. . . . . . . . 9 |- ( ( ph /\ y e. X /\ ( F ` y ) = +oo ) -> +oo e. _V )
51	19, 48, 50	elrnmptd 37463	. . . . . . . 8 |- ( ( ph /\ y e. X /\ ( F ` y ) = +oo ) -> +oo e. ran ( x e. ( ~P X i^i Fin ) |-> (Σ^ ` ( F |` x ) ) ) )
52	51	3exp 1208	. . . . . . 7 |- ( ph -> ( y e. X -> ( ( F ` y ) = +oo -> +oo e. ran ( x e. ( ~P X i^i Fin ) |-> (Σ^ ` ( F |` x ) ) ) ) ) )
53	52	rexlimdv 2879	. . . . . 6 |- ( ph -> ( E. y e. X ( F ` y ) = +oo -> +oo e. ran ( x e. ( ~P X i^i Fin ) |-> (Σ^ ` ( F |` x ) ) ) ) )
54	53	adantr 467	. . . . 5 |- ( ( ph /\ +oo e. ran F ) -> ( E. y e. X ( F ` y ) = +oo -> +oo e. ran ( x e. ( ~P X i^i Fin ) |-> (Σ^ ` ( F |` x ) ) ) ) )
55	27, 54	mpd 15	. . . 4 |- ( ( ph /\ +oo e. ran F ) -> +oo e. ran ( x e. ( ~P X i^i Fin ) |-> (Σ^ ` ( F |` x ) ) ) )
56		supxrpnf 11611	. . . 4 |- ( ( ran ( x e. ( ~P X i^i Fin ) |-> (Σ^ ` ( F |` x ) ) ) C_ RR* /\ +oo e. ran ( x e. ( ~P X i^i Fin ) |-> (Σ^ ` ( F |` x ) ) ) ) -> sup ( ran ( x e. ( ~P X i^i Fin ) |-> (Σ^ ` ( F |` x ) ) ) , RR* , < ) = +oo )
57	21, 55, 56	syl2anc 667	. . 3 |- ( ( ph /\ +oo e. ran F ) -> sup ( ran ( x e. ( ~P X i^i Fin ) |-> (Σ^ ` ( F |` x ) ) ) , RR* , < ) = +oo )
58	1, 7, 57	3eqtr4d 2497	. 2 |- ( ( ph /\ +oo e. ran F ) -> (Σ^ ` F ) = sup ( ran ( x e. ( ~P X i^i Fin ) |-> (Σ^ ` ( F |` x ) ) ) , RR* , < ) )
59	2	adantr 467	. . . 4 |- ( ( ph /\ -. +oo e. ran F ) -> X e. V )
60	4	adantr 467	. . . . 5 |- ( ( ph /\ -. +oo e. ran F ) -> F : X --> ( 0 [,] +oo ) )
61		simpr 463	. . . . 5 |- ( ( ph /\ -. +oo e. ran F ) -> -. +oo e. ran F )
62	60, 61	fge0iccico 38222	. . . 4 |- ( ( ph /\ -. +oo e. ran F ) -> F : X --> ( 0 [,) +oo ) )
63	59, 62	sge0reval 38224	. . 3 |- ( ( ph /\ -. +oo e. ran F ) -> (Σ^ ` F ) = sup ( ran ( x e. ( ~P X i^i Fin ) |-> sum_ y e. x ( F ` y ) ) , RR* , < ) )
64		elinel2 3622	. . . . . . . . 9 |- ( x e. ( ~P X i^i Fin ) -> x e. Fin )
65	64	adantl 468	. . . . . . . 8 |- ( ( ( ph /\ -. +oo e. ran F ) /\ x e. ( ~P X i^i Fin ) ) -> x e. Fin )
66	15	adantlr 722	. . . . . . . . 9 |- ( ( ( ph /\ -. +oo e. ran F ) /\ x e. ( ~P X i^i Fin ) ) -> ( F |` x ) : x --> ( 0 [,] +oo ) )
67		nelrnres 37472	. . . . . . . . . 10 |- ( -. +oo e. ran F -> -. +oo e. ran ( F |` x ) )
68	67	ad2antlr 734	. . . . . . . . 9 |- ( ( ( ph /\ -. +oo e. ran F ) /\ x e. ( ~P X i^i Fin ) ) -> -. +oo e. ran ( F |` x ) )
69	66, 68	fge0iccico 38222	. . . . . . . 8 |- ( ( ( ph /\ -. +oo e. ran F ) /\ x e. ( ~P X i^i Fin ) ) -> ( F |` x ) : x --> ( 0 [,) +oo ) )
70	65, 69	sge0fsum 38239	. . . . . . 7 |- ( ( ( ph /\ -. +oo e. ran F ) /\ x e. ( ~P X i^i Fin ) ) -> (Σ^ ` ( F |` x ) ) = sum_ y e. x ( ( F |` x ) ` y ) )
71		simpr 463	. . . . . . . . . 10 |- ( ( x e. ( ~P X i^i Fin ) /\ y e. x ) -> y e. x )
72		fvres 5884	. . . . . . . . . 10 |- ( y e. x -> ( ( F |` x ) ` y ) = ( F ` y ) )
73	71, 72	syl 17	. . . . . . . . 9 |- ( ( x e. ( ~P X i^i Fin ) /\ y e. x ) -> ( ( F |` x ) ` y ) = ( F ` y ) )
74	73	sumeq2dv 13781	. . . . . . . 8 |- ( x e. ( ~P X i^i Fin ) -> sum_ y e. x ( ( F |` x ) ` y ) = sum_ y e. x ( F ` y ) )
75	74	adantl 468	. . . . . . 7 |- ( ( ( ph /\ -. +oo e. ran F ) /\ x e. ( ~P X i^i Fin ) ) -> sum_ y e. x ( ( F |` x ) ` y ) = sum_ y e. x ( F ` y ) )
76	70, 75	eqtrd 2487	. . . . . 6 |- ( ( ( ph /\ -. +oo e. ran F ) /\ x e. ( ~P X i^i Fin ) ) -> (Σ^ ` ( F |` x ) ) = sum_ y e. x ( F ` y ) )
77	76	mpteq2dva 4492	. . . . 5 |- ( ( ph /\ -. +oo e. ran F ) -> ( x e. ( ~P X i^i Fin ) |-> (Σ^ ` ( F |` x ) ) ) = ( x e. ( ~P X i^i Fin ) |-> sum_ y e. x ( F ` y ) ) )
78	77	rneqd 5065	. . . 4 |- ( ( ph /\ -. +oo e. ran F ) -> ran ( x e. ( ~P X i^i Fin ) |-> (Σ^ ` ( F |` x ) ) ) = ran ( x e. ( ~P X i^i Fin ) |-> sum_ y e. x ( F ` y ) ) )
79	78	supeq1d 7965	. . 3 |- ( ( ph /\ -. +oo e. ran F ) -> sup ( ran ( x e. ( ~P X i^i Fin ) |-> (Σ^ ` ( F |` x ) ) ) , RR* , < ) = sup ( ran ( x e. ( ~P X i^i Fin ) |-> sum_ y e. x ( F ` y ) ) , RR* , < ) )
80	63, 79	eqtr4d 2490	. 2 |- ( ( ph /\ -. +oo e. ran F ) -> (Σ^ ` F ) = sup ( ran ( x e. ( ~P X i^i Fin ) |-> (Σ^ ` ( F |` x ) ) ) , RR* , < ) )
81	58, 80	pm2.61dan 801	1 |- ( ph -> (Σ^ ` F ) = sup ( ran ( x e. ( ~P X i^i Fin ) |-> (Σ^ ` ( F |` x ) ) ) , RR* , < ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 188   /\ wa 371   /\ w3a 986   = wceq 1446   e. wcel 1889   A.wral 2739   E.wrex 2740   _Vcvv 3047   i^i cin 3405   C_ wss 3406   ~Pcpw 3953   {csn 3970   |-> cmpt 4464   ran crn 4838   |` cres 4839   Fn wfn 5580   -->wf 5581   ` cfv 5585  (class class class)co 6295   Fincfn 7574   supcsup 7959   0cc0 9544   +oocpnf 9677   RR*cxr 9679   < clt 9680   [,]cicc 11645   sum_csu 13764  Σ^{^}csumge0 38214

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-inf2 8151  ax-cnex 9600  ax-resscn 9601  ax-1cn 9602  ax-icn 9603  ax-addcl 9604  ax-addrcl 9605  ax-mulcl 9606  ax-mulrcl 9607  ax-mulcom 9608  ax-addass 9609  ax-mulass 9610  ax-distr 9611  ax-i2m1 9612  ax-1ne0 9613  ax-1rid 9614  ax-rnegex 9615  ax-rrecex 9616  ax-cnre 9617  ax-pre-lttri 9618  ax-pre-lttrn 9619  ax-pre-ltadd 9620  ax-pre-mulgt0 9621  ax-pre-sup 9622

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-fal 1452  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-reu 2746  df-rmo 2747  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-int 4238  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-se 4797  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-isom 5594  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-oadd 7191  df-er 7368  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-sup 7961  df-oi 8030  df-card 8378  df-pnf 9682  df-mnf 9683  df-xr 9684  df-ltxr 9685  df-le 9686  df-sub 9867  df-neg 9868  df-div 10277  df-nn 10617  df-2 10675  df-3 10676  df-n0 10877  df-z 10945  df-uz 11167  df-rp 11310  df-ico 11648  df-icc 11649  df-fz 11792  df-fzo 11923  df-seq 12221  df-exp 12280  df-hash 12523  df-cj 13174  df-re 13175  df-im 13176  df-sqrt 13310  df-abs 13311  df-clim 13564  df-sum 13765  df-sumge0 38215

This theorem is referenced by:  sge0gerp  38247  sge0pnffigt  38248  sge0lefi  38250