Theorem sge0sup 40608

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 2623	. . 3 |- ( ( ph /\ +oo e. ran F ) -> +oo = +oo )
2		sge0sup.x	. . . . 5 |- ( ph -> X e. V )
3	2	adantr 481	. . . 4 |- ( ( ph /\ +oo e. ran F ) -> X e. V )
4		sge0sup.f	. . . . 5 |- ( ph -> F : X --> ( 0 [,] +oo ) )
5	4	adantr 481	. . . 4 |- ( ( ph /\ +oo e. ran F ) -> F : X --> ( 0 [,] +oo ) )
6		simpr 477	. . . 4 |- ( ( ph /\ +oo e. ran F ) -> +oo e. ran F )
7	3, 5, 6	sge0pnfval 40590	. . 3 |- ( ( ph /\ +oo e. ran F ) -> (Σ^ ` F ) = +oo )
8		vex 3203	. . . . . . . . 9 |- x e. _V
9	8	a1i 11	. . . . . . . 8 |- ( ( ph /\ x e. ( ~P X i^i Fin ) ) -> x e. _V )
10	4	adantr 481	. . . . . . . . 9 |- ( ( ph /\ x e. ( ~P X i^i Fin ) ) -> F : X --> ( 0 [,] +oo ) )
11		elinel1 3799	. . . . . . . . . . 11 |- ( x e. ( ~P X i^i Fin ) -> x e. ~P X )
12		elpwi 4168	. . . . . . . . . . 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 482	. . . . . . . . 9 |- ( ( ph /\ x e. ( ~P X i^i Fin ) ) -> x C_ X )
15	10, 14	fssresd 6071	. . . . . . . 8 |- ( ( ph /\ x e. ( ~P X i^i Fin ) ) -> ( F |` x ) : x --> ( 0 [,] +oo ) )
16	9, 15	sge0xrcl 40602	. . . . . . 7 |- ( ( ph /\ x e. ( ~P X i^i Fin ) ) -> (Σ^ ` ( F |` x ) ) e. RR* )
17	16	adantlr 751	. . . . . 6 |- ( ( ( ph /\ +oo e. ran F ) /\ x e. ( ~P X i^i Fin ) ) -> (Σ^ ` ( F |` x ) ) e. RR* )
18	17	ralrimiva 2966	. . . . 5 |- ( ( ph /\ +oo e. ran F ) -> A. x e. ( ~P X i^i Fin ) (Σ^ ` ( F |` x ) ) e. RR* )
19		eqid 2622	. . . . . 6 |- ( x e. ( ~P X i^i Fin ) |-> (Σ^ ` ( F |` x ) ) ) = ( x e. ( ~P X i^i Fin ) |-> (Σ^ ` ( F |` x ) ) )
20	19	rnmptss 6392	. . . . 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 6045	. . . . . . . . 9 |- ( F : X --> ( 0 [,] +oo ) -> F Fn X )
23	4, 22	syl 17	. . . . . . . 8 |- ( ph -> F Fn X )
24		fvelrnb 6243	. . . . . . . 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 481	. . . . . 6 |- ( ( ph /\ +oo e. ran F ) -> ( +oo e. ran F <-> E. y e. X ( F ` y ) = +oo ) )
27	6, 26	mpbid 222	. . . . 5 |- ( ( ph /\ +oo e. ran F ) -> E. y e. X ( F ` y ) = +oo )
28		snelpwi 4912	. . . . . . . . . . . 12 |- ( y e. X -> { y } e. ~P X )
29		snfi 8038	. . . . . . . . . . . . 13 |- { y } e. Fin
30	29	a1i 11	. . . . . . . . . . . 12 |- ( y e. X -> { y } e. Fin )
31	28, 30	elind 3798	. . . . . . . . . . 11 |- ( y e. X -> { y } e. ( ~P X i^i Fin ) )
32	31	3ad2ant2 1083	. . . . . . . . . 10 |- ( ( ph /\ y e. X /\ ( F ` y ) = +oo ) -> { y } e. ( ~P X i^i Fin ) )
33		simp2 1062	. . . . . . . . . . . 12 |- ( ( ph /\ y e. X /\ ( F ` y ) = +oo ) -> y e. X )
34	4	3ad2ant1 1082	. . . . . . . . . . . . 13 |- ( ( ph /\ y e. X /\ ( F ` y ) = +oo ) -> F : X --> ( 0 [,] +oo ) )
35	33	snssd 4340	. . . . . . . . . . . . 13 |- ( ( ph /\ y e. X /\ ( F ` y ) = +oo ) -> { y } C_ X )
36	34, 35	fssresd 6071	. . . . . . . . . . . 12 |- ( ( ph /\ y e. X /\ ( F ` y ) = +oo ) -> ( F |` { y } ) : { y } --> ( 0 [,] +oo ) )
37	33, 36	sge0sn 40596	. . . . . . . . . . 11 |- ( ( ph /\ y e. X /\ ( F ` y ) = +oo ) -> (Σ^ ` ( F |` { y } ) ) = ( ( F |` { y } ) ` y ) )
38		vsnid 4209	. . . . . . . . . . . . 13 |- y e. { y }
39		fvres 6207	. . . . . . . . . . . . 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 1063	. . . . . . . . . . 11 |- ( ( ph /\ y e. X /\ ( F ` y ) = +oo ) -> ( F ` y ) = +oo )
43	37, 41, 42	3eqtrrd 2661	. . . . . . . . . 10 |- ( ( ph /\ y e. X /\ ( F ` y ) = +oo ) -> +oo = (Σ^ ` ( F |` { y } ) ) )
44		reseq2 5391	. . . . . . . . . . . . 13 |- ( x = { y } -> ( F |` x ) = ( F |` { y } ) )
45	44	fveq2d 6195	. . . . . . . . . . . 12 |- ( x = { y } -> (Σ^ ` ( F |` x ) ) = (Σ^ ` ( F |` { y } ) ) )
46	45	eqeq2d 2632	. . . . . . . . . . 11 |- ( x = { y } -> ( +oo = (Σ^ ` ( F |` x ) ) <-> +oo = (Σ^ ` ( F |` { y } ) ) ) )
47	46	rspcev 3309	. . . . . . . . . 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 693	. . . . . . . . 9 |- ( ( ph /\ y e. X /\ ( F ` y ) = +oo ) -> E. x e. ( ~P X i^i Fin ) +oo = (Σ^ ` ( F |` x ) ) )
49		pnfex 10093	. . . . . . . . . 10 |- +oo e. _V
50	49	a1i 11	. . . . . . . . 9 |- ( ( ph /\ y e. X /\ ( F ` y ) = +oo ) -> +oo e. _V )
51	19, 48, 50	elrnmptd 39366	. . . . . . . 8 |- ( ( ph /\ y e. X /\ ( F ` y ) = +oo ) -> +oo e. ran ( x e. ( ~P X i^i Fin ) |-> (Σ^ ` ( F |` x ) ) ) )
52	51	3exp 1264	. . . . . . 7 |- ( ph -> ( y e. X -> ( ( F ` y ) = +oo -> +oo e. ran ( x e. ( ~P X i^i Fin ) |-> (Σ^ ` ( F |` x ) ) ) ) ) )
53	52	rexlimdv 3030	. . . . . 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 481	. . . . 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 12148	. . . 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 693	. . 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 2666	. 2 |- ( ( ph /\ +oo e. ran F ) -> (Σ^ ` F ) = sup ( ran ( x e. ( ~P X i^i Fin ) |-> (Σ^ ` ( F |` x ) ) ) , RR* , < ) )
59	2	adantr 481	. . . 4 |- ( ( ph /\ -. +oo e. ran F ) -> X e. V )
60	4	adantr 481	. . . . 5 |- ( ( ph /\ -. +oo e. ran F ) -> F : X --> ( 0 [,] +oo ) )
61		simpr 477	. . . . 5 |- ( ( ph /\ -. +oo e. ran F ) -> -. +oo e. ran F )
62	60, 61	fge0iccico 40587	. . . 4 |- ( ( ph /\ -. +oo e. ran F ) -> F : X --> ( 0 [,) +oo ) )
63	59, 62	sge0reval 40589	. . 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 3800	. . . . . . . . 9 |- ( x e. ( ~P X i^i Fin ) -> x e. Fin )
65	64	adantl 482	. . . . . . . 8 |- ( ( ( ph /\ -. +oo e. ran F ) /\ x e. ( ~P X i^i Fin ) ) -> x e. Fin )
66	15	adantlr 751	. . . . . . . . 9 |- ( ( ( ph /\ -. +oo e. ran F ) /\ x e. ( ~P X i^i Fin ) ) -> ( F |` x ) : x --> ( 0 [,] +oo ) )
67		nelrnres 39374	. . . . . . . . . 10 |- ( -. +oo e. ran F -> -. +oo e. ran ( F |` x ) )
68	67	ad2antlr 763	. . . . . . . . 9 |- ( ( ( ph /\ -. +oo e. ran F ) /\ x e. ( ~P X i^i Fin ) ) -> -. +oo e. ran ( F |` x ) )
69	66, 68	fge0iccico 40587	. . . . . . . 8 |- ( ( ( ph /\ -. +oo e. ran F ) /\ x e. ( ~P X i^i Fin ) ) -> ( F |` x ) : x --> ( 0 [,) +oo ) )
70	65, 69	sge0fsum 40604	. . . . . . 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 477	. . . . . . . . . 10 |- ( ( x e. ( ~P X i^i Fin ) /\ y e. x ) -> y e. x )
72		fvres 6207	. . . . . . . . . 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 14433	. . . . . . . 8 |- ( x e. ( ~P X i^i Fin ) -> sum_ y e. x ( ( F |` x ) ` y ) = sum_ y e. x ( F ` y ) )
75	74	adantl 482	. . . . . . 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 2656	. . . . . 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 4744	. . . . 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 5353	. . . 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 8352	. . 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 2659	. 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 832	1 |- ( ph -> (Σ^ ` F ) = sup ( ran ( x e. ( ~P X i^i Fin ) |-> (Σ^ ` ( F |` x ) ) ) , RR* , < ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   _Vcvv 3200   i^i cin 3573   C_ wss 3574   ~Pcpw 4158   {csn 4177   |-> cmpt 4729   ran crn 5115   |` cres 5116   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   Fincfn 7955   supcsup 8346   0cc0 9936   +oocpnf 10071   RR*cxr 10073   < clt 10074   [,]cicc 12178   sum_csu 14416  Σ^{^}csumge0 40579

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-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-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-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  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-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  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-sumge0 40580

This theorem is referenced by:  sge0gerp  40612  sge0pnffigt  40613  sge0lefi  40615