Theorem sge0sup 38347

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 2472	. . 3 |- ( ( ph /\ +oo e. ran F ) -> +oo = +oo )
2		sge0sup.x	. . . . 5 |- ( ph -> X e. V )
3	2	adantr 472	. . . 4 |- ( ( ph /\ +oo e. ran F ) -> X e. V )
4		sge0sup.f	. . . . 5 |- ( ph -> F : X --> ( 0 [,] +oo ) )
5	4	adantr 472	. . . 4 |- ( ( ph /\ +oo e. ran F ) -> F : X --> ( 0 [,] +oo ) )
6		simpr 468	. . . 4 |- ( ( ph /\ +oo e. ran F ) -> +oo e. ran F )
7	3, 5, 6	sge0pnfval 38329	. . 3 |- ( ( ph /\ +oo e. ran F ) -> (Σ^ ` F ) = +oo )
8		vex 3034	. . . . . . . . 9 |- x e. _V
9	8	a1i 11	. . . . . . . 8 |- ( ( ph /\ x e. ( ~P X i^i Fin ) ) -> x e. _V )
10	4	adantr 472	. . . . . . . . 9 |- ( ( ph /\ x e. ( ~P X i^i Fin ) ) -> F : X --> ( 0 [,] +oo ) )
11		elinel1 3610	. . . . . . . . . . 11 |- ( x e. ( ~P X i^i Fin ) -> x e. ~P X )
12		elpwi 3951	. . . . . . . . . . 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 473	. . . . . . . . 9 |- ( ( ph /\ x e. ( ~P X i^i Fin ) ) -> x C_ X )
15	10, 14	fssresd 5762	. . . . . . . 8 |- ( ( ph /\ x e. ( ~P X i^i Fin ) ) -> ( F |` x ) : x --> ( 0 [,] +oo ) )
16	9, 15	sge0xrcl 38341	. . . . . . 7 |- ( ( ph /\ x e. ( ~P X i^i Fin ) ) -> (Σ^ ` ( F |` x ) ) e. RR* )
17	16	adantlr 729	. . . . . 6 |- ( ( ( ph /\ +oo e. ran F ) /\ x e. ( ~P X i^i Fin ) ) -> (Σ^ ` ( F |` x ) ) e. RR* )
18	17	ralrimiva 2809	. . . . 5 |- ( ( ph /\ +oo e. ran F ) -> A. x e. ( ~P X i^i Fin ) (Σ^ ` ( F |` x ) ) e. RR* )
19		eqid 2471	. . . . . 6 |- ( x e. ( ~P X i^i Fin ) |-> (Σ^ ` ( F |` x ) ) ) = ( x e. ( ~P X i^i Fin ) |-> (Σ^ ` ( F |` x ) ) )
20	19	rnmptss 6068	. . . . 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 5739	. . . . . . . . 9 |- ( F : X --> ( 0 [,] +oo ) -> F Fn X )
23	4, 22	syl 17	. . . . . . . 8 |- ( ph -> F Fn X )
24		fvelrnb 5926	. . . . . . . 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 472	. . . . . 6 |- ( ( ph /\ +oo e. ran F ) -> ( +oo e. ran F <-> E. y e. X ( F ` y ) = +oo ) )
27	6, 26	mpbid 215	. . . . 5 |- ( ( ph /\ +oo e. ran F ) -> E. y e. X ( F ` y ) = +oo )
28		snelpwi 4645	. . . . . . . . . . . 12 |- ( y e. X -> { y } e. ~P X )
29		snfi 7668	. . . . . . . . . . . . 13 |- { y } e. Fin
30	29	a1i 11	. . . . . . . . . . . 12 |- ( y e. X -> { y } e. Fin )
31	28, 30	elind 3609	. . . . . . . . . . 11 |- ( y e. X -> { y } e. ( ~P X i^i Fin ) )
32	31	3ad2ant2 1052	. . . . . . . . . 10 |- ( ( ph /\ y e. X /\ ( F ` y ) = +oo ) -> { y } e. ( ~P X i^i Fin ) )
33		simp2 1031	. . . . . . . . . . . 12 |- ( ( ph /\ y e. X /\ ( F ` y ) = +oo ) -> y e. X )
34	4	3ad2ant1 1051	. . . . . . . . . . . . 13 |- ( ( ph /\ y e. X /\ ( F ` y ) = +oo ) -> F : X --> ( 0 [,] +oo ) )
35	33	snssd 4108	. . . . . . . . . . . . 13 |- ( ( ph /\ y e. X /\ ( F ` y ) = +oo ) -> { y } C_ X )
36	34, 35	fssresd 5762	. . . . . . . . . . . 12 |- ( ( ph /\ y e. X /\ ( F ` y ) = +oo ) -> ( F |` { y } ) : { y } --> ( 0 [,] +oo ) )
37	33, 36	sge0sn 38335	. . . . . . . . . . 11 |- ( ( ph /\ y e. X /\ ( F ` y ) = +oo ) -> (Σ^ ` ( F |` { y } ) ) = ( ( F |` { y } ) ` y ) )
38		ssnid 3989	. . . . . . . . . . . . 13 |- y e. { y }
39		fvres 5893	. . . . . . . . . . . . 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 1032	. . . . . . . . . . 11 |- ( ( ph /\ y e. X /\ ( F ` y ) = +oo ) -> ( F ` y ) = +oo )
43	37, 41, 42	3eqtrrd 2510	. . . . . . . . . 10 |- ( ( ph /\ y e. X /\ ( F ` y ) = +oo ) -> +oo = (Σ^ ` ( F |` { y } ) ) )
44		reseq2 5106	. . . . . . . . . . . . 13 |- ( x = { y } -> ( F |` x ) = ( F |` { y } ) )
45	44	fveq2d 5883	. . . . . . . . . . . 12 |- ( x = { y } -> (Σ^ ` ( F |` x ) ) = (Σ^ ` ( F |` { y } ) ) )
46	45	eqeq2d 2481	. . . . . . . . . . 11 |- ( x = { y } -> ( +oo = (Σ^ ` ( F |` x ) ) <-> +oo = (Σ^ ` ( F |` { y } ) ) ) )
47	46	rspcev 3136	. . . . . . . . . 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 673	. . . . . . . . 9 |- ( ( ph /\ y e. X /\ ( F ` y ) = +oo ) -> E. x e. ( ~P X i^i Fin ) +oo = (Σ^ ` ( F |` x ) ) )
49		pnfex 11436	. . . . . . . . . 10 |- +oo e. _V
50	49	a1i 11	. . . . . . . . 9 |- ( ( ph /\ y e. X /\ ( F ` y ) = +oo ) -> +oo e. _V )
51	19, 48, 50	elrnmptd 37524	. . . . . . . 8 |- ( ( ph /\ y e. X /\ ( F ` y ) = +oo ) -> +oo e. ran ( x e. ( ~P X i^i Fin ) |-> (Σ^ ` ( F |` x ) ) ) )
52	51	3exp 1230	. . . . . . 7 |- ( ph -> ( y e. X -> ( ( F ` y ) = +oo -> +oo e. ran ( x e. ( ~P X i^i Fin ) |-> (Σ^ ` ( F |` x ) ) ) ) ) )
53	52	rexlimdv 2870	. . . . . 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 472	. . . . 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 11629	. . . 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 673	. . 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 2515	. 2 |- ( ( ph /\ +oo e. ran F ) -> (Σ^ ` F ) = sup ( ran ( x e. ( ~P X i^i Fin ) |-> (Σ^ ` ( F |` x ) ) ) , RR* , < ) )
59	2	adantr 472	. . . 4 |- ( ( ph /\ -. +oo e. ran F ) -> X e. V )
60	4	adantr 472	. . . . 5 |- ( ( ph /\ -. +oo e. ran F ) -> F : X --> ( 0 [,] +oo ) )
61		simpr 468	. . . . 5 |- ( ( ph /\ -. +oo e. ran F ) -> -. +oo e. ran F )
62	60, 61	fge0iccico 38326	. . . 4 |- ( ( ph /\ -. +oo e. ran F ) -> F : X --> ( 0 [,) +oo ) )
63	59, 62	sge0reval 38328	. . 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 3611	. . . . . . . . 9 |- ( x e. ( ~P X i^i Fin ) -> x e. Fin )
65	64	adantl 473	. . . . . . . 8 |- ( ( ( ph /\ -. +oo e. ran F ) /\ x e. ( ~P X i^i Fin ) ) -> x e. Fin )
66	15	adantlr 729	. . . . . . . . 9 |- ( ( ( ph /\ -. +oo e. ran F ) /\ x e. ( ~P X i^i Fin ) ) -> ( F |` x ) : x --> ( 0 [,] +oo ) )
67		nelrnres 37533	. . . . . . . . . 10 |- ( -. +oo e. ran F -> -. +oo e. ran ( F |` x ) )
68	67	ad2antlr 741	. . . . . . . . 9 |- ( ( ( ph /\ -. +oo e. ran F ) /\ x e. ( ~P X i^i Fin ) ) -> -. +oo e. ran ( F |` x ) )
69	66, 68	fge0iccico 38326	. . . . . . . 8 |- ( ( ( ph /\ -. +oo e. ran F ) /\ x e. ( ~P X i^i Fin ) ) -> ( F |` x ) : x --> ( 0 [,) +oo ) )
70	65, 69	sge0fsum 38343	. . . . . . 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 468	. . . . . . . . . 10 |- ( ( x e. ( ~P X i^i Fin ) /\ y e. x ) -> y e. x )
72		fvres 5893	. . . . . . . . . 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 13846	. . . . . . . 8 |- ( x e. ( ~P X i^i Fin ) -> sum_ y e. x ( ( F |` x ) ` y ) = sum_ y e. x ( F ` y ) )
75	74	adantl 473	. . . . . . 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 2505	. . . . . 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 4482	. . . . 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 5068	. . . 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 7978	. . 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 2508	. 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 808	1 |- ( ph -> (Σ^ ` F ) = sup ( ran ( x e. ( ~P X i^i Fin ) |-> (Σ^ ` ( F |` x ) ) ) , RR* , < ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 189   /\ wa 376   /\ w3a 1007   = wceq 1452   e. wcel 1904   A.wral 2756   E.wrex 2757   _Vcvv 3031   i^i cin 3389   C_ wss 3390   ~Pcpw 3942   {csn 3959   |-> cmpt 4454   ran crn 4840   |` cres 4841   Fn wfn 5584   -->wf 5585   ` cfv 5589  (class class class)co 6308   Fincfn 7587   supcsup 7972   0cc0 9557   +oocpnf 9690   RR*cxr 9692   < clt 9693   [,]cicc 11663   sum_csu 13829  Σ^{^}csumge0 38318

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-sup 7974  df-oi 8043  df-card 8391  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-n0 10894  df-z 10962  df-uz 11183  df-rp 11326  df-ico 11666  df-icc 11667  df-fz 11811  df-fzo 11943  df-seq 12252  df-exp 12311  df-hash 12554  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-clim 13629  df-sum 13830  df-sumge0 38319

This theorem is referenced by:  sge0gerp  38351  sge0pnffigt  38352  sge0lefi  38354