Theorem sge0sn 38335

Description: A sum of a nonnegative extended real is the term. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Hypotheses

Ref	Expression
sge0sn.1	|- ( ph -> A e. V )
sge0sn.2	|- ( ph -> F : { A } --> ( 0 [,] +oo ) )

Assertion

Ref	Expression
sge0sn	|- ( ph -> (Σ^ ` F ) = ( F ` A ) )

Proof of Theorem sge0sn

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

Step	Hyp	Ref	Expression
1		snex 4641	. . . . 5 |- { A } e. _V
2	1	a1i 11	. . . 4 |- ( ( ph /\ ( F ` A ) = +oo ) -> { A } e. _V )
3		sge0sn.2	. . . . 5 |- ( ph -> F : { A } --> ( 0 [,] +oo ) )
4	3	adantr 472	. . . 4 |- ( ( ph /\ ( F ` A ) = +oo ) -> F : { A } --> ( 0 [,] +oo ) )
5		id 22	. . . . . . 7 |- ( ( F ` A ) = +oo -> ( F ` A ) = +oo )
6	5	eqcomd 2477	. . . . . 6 |- ( ( F ` A ) = +oo -> +oo = ( F ` A ) )
7	6	adantl 473	. . . . 5 |- ( ( ph /\ ( F ` A ) = +oo ) -> +oo = ( F ` A ) )
8		ffun 5742	. . . . . . . 8 |- ( F : { A } --> ( 0 [,] +oo ) -> Fun F )
9	3, 8	syl 17	. . . . . . 7 |- ( ph -> Fun F )
10	9	adantr 472	. . . . . 6 |- ( ( ph /\ ( F ` A ) = +oo ) -> Fun F )
11		sge0sn.1	. . . . . . . . 9 |- ( ph -> A e. V )
12		snidg 3986	. . . . . . . . 9 |- ( A e. V -> A e. { A } )
13	11, 12	syl 17	. . . . . . . 8 |- ( ph -> A e. { A } )
14		fdm 5745	. . . . . . . . . 10 |- ( F : { A } --> ( 0 [,] +oo ) -> dom F = { A } )
15	3, 14	syl 17	. . . . . . . . 9 |- ( ph -> dom F = { A } )
16	15	eqcomd 2477	. . . . . . . 8 |- ( ph -> { A } = dom F )
17	13, 16	eleqtrd 2551	. . . . . . 7 |- ( ph -> A e. dom F )
18	17	adantr 472	. . . . . 6 |- ( ( ph /\ ( F ` A ) = +oo ) -> A e. dom F )
19		fvelrn 6030	. . . . . 6 |- ( ( Fun F /\ A e. dom F ) -> ( F ` A ) e. ran F )
20	10, 18, 19	syl2anc 673	. . . . 5 |- ( ( ph /\ ( F ` A ) = +oo ) -> ( F ` A ) e. ran F )
21	7, 20	eqeltrd 2549	. . . 4 |- ( ( ph /\ ( F ` A ) = +oo ) -> +oo e. ran F )
22	2, 4, 21	sge0pnfval 38329	. . 3 |- ( ( ph /\ ( F ` A ) = +oo ) -> (Σ^ ` F ) = +oo )
23		simpr 468	. . 3 |- ( ( ph /\ ( F ` A ) = +oo ) -> ( F ` A ) = +oo )
24	22, 23	eqtr4d 2508	. 2 |- ( ( ph /\ ( F ` A ) = +oo ) -> (Σ^ ` F ) = ( F ` A ) )
25	1	a1i 11	. . . 4 |- ( ( ph /\ -. ( F ` A ) = +oo ) -> { A } e. _V )
26	3	adantr 472	. . . . 5 |- ( ( ph /\ -. ( F ` A ) = +oo ) -> F : { A } --> ( 0 [,] +oo ) )
27		elsni 3985	. . . . . . . . 9 |- ( +oo e. { ( F ` A ) } -> +oo = ( F ` A ) )
28	27	eqcomd 2477	. . . . . . . 8 |- ( +oo e. { ( F ` A ) } -> ( F ` A ) = +oo )
29	28	con3i 142	. . . . . . 7 |- ( -. ( F ` A ) = +oo -> -. +oo e. { ( F ` A ) } )
30	29	adantl 473	. . . . . 6 |- ( ( ph /\ -. ( F ` A ) = +oo ) -> -. +oo e. { ( F ` A ) } )
31	11, 3	rnsnf 37529	. . . . . . . 8 |- ( ph -> ran F = { ( F ` A ) } )
32	31	eqcomd 2477	. . . . . . 7 |- ( ph -> { ( F ` A ) } = ran F )
33	32	adantr 472	. . . . . 6 |- ( ( ph /\ -. ( F ` A ) = +oo ) -> { ( F ` A ) } = ran F )
34	30, 33	neleqtrd 2570	. . . . 5 |- ( ( ph /\ -. ( F ` A ) = +oo ) -> -. +oo e. ran F )
35	26, 34	fge0iccico 38326	. . . 4 |- ( ( ph /\ -. ( F ` A ) = +oo ) -> F : { A } --> ( 0 [,) +oo ) )
36	25, 35	sge0reval 38328	. . 3 |- ( ( ph /\ -. ( F ` A ) = +oo ) -> (Σ^ ` F ) = sup ( ran ( x e. ( ~P { A } i^i Fin ) |-> sum_ y e. x ( F ` y ) ) , RR* , < ) )
37		sum0 13864	. . . . . . . 8 |- sum_ y e. (/) ( F ` y ) = 0
38	37	eqcomi 2480	. . . . . . 7 |- 0 = sum_ y e. (/) ( F ` y )
39	38	a1i 11	. . . . . 6 |- ( ( ph /\ -. ( F ` A ) = +oo ) -> 0 = sum_ y e. (/) ( F ` y ) )
40		nfcvd 2613	. . . . . . . 8 |- ( ( ph /\ -. ( F ` A ) = +oo ) -> F/_ y ( F ` A ) )
41		nfv 1769	. . . . . . . 8 |- F/ y ( ph /\ -. ( F ` A ) = +oo )
42		fveq2 5879	. . . . . . . . 9 |- ( y = A -> ( F ` y ) = ( F ` A ) )
43	42	adantl 473	. . . . . . . 8 |- ( ( ( ph /\ -. ( F ` A ) = +oo ) /\ y = A ) -> ( F ` y ) = ( F ` A ) )
44	11	adantr 472	. . . . . . . 8 |- ( ( ph /\ -. ( F ` A ) = +oo ) -> A e. V )
45		rge0ssre 11766	. . . . . . . . . 10 |- ( 0 [,) +oo ) C_ RR
46		ax-resscn 9614	. . . . . . . . . 10 |- RR C_ CC
47	45, 46	sstri 3427	. . . . . . . . 9 |- ( 0 [,) +oo ) C_ CC
48	44, 12	syl 17	. . . . . . . . . 10 |- ( ( ph /\ -. ( F ` A ) = +oo ) -> A e. { A } )
49	35, 48	ffvelrnd 6038	. . . . . . . . 9 |- ( ( ph /\ -. ( F ` A ) = +oo ) -> ( F ` A ) e. ( 0 [,) +oo ) )
50	47, 49	sseldi 3416	. . . . . . . 8 |- ( ( ph /\ -. ( F ` A ) = +oo ) -> ( F ` A ) e. CC )
51	40, 41, 43, 44, 50	sumsnd 37410	. . . . . . 7 |- ( ( ph /\ -. ( F ` A ) = +oo ) -> sum_ y e. { A } ( F ` y ) = ( F ` A ) )
52	51	eqcomd 2477	. . . . . 6 |- ( ( ph /\ -. ( F ` A ) = +oo ) -> ( F ` A ) = sum_ y e. { A } ( F ` y ) )
53	39, 52	preq12d 4050	. . . . 5 |- ( ( ph /\ -. ( F ` A ) = +oo ) -> { 0 , ( F ` A ) } = { sum_ y e. (/) ( F ` y ) , sum_ y e. { A } ( F ` y ) } )
54	53	supeq1d 7978	. . . 4 |- ( ( ph /\ -. ( F ` A ) = +oo ) -> sup ( { 0 , ( F ` A ) } , RR* , < ) = sup ( { sum_ y e. (/) ( F ` y ) , sum_ y e. { A } ( F ` y ) } , RR* , < ) )
55		xrltso 11463	. . . . . . . 8 |- < Or RR*
56	55	a1i 11	. . . . . . 7 |- ( ph -> < Or RR* )
57		0xr 9705	. . . . . . . 8 |- 0 e. RR*
58	57	a1i 11	. . . . . . 7 |- ( ph -> 0 e. RR* )
59		iccssxr 11742	. . . . . . . 8 |- ( 0 [,] +oo ) C_ RR*
60	3, 13	ffvelrnd 6038	. . . . . . . 8 |- ( ph -> ( F ` A ) e. ( 0 [,] +oo ) )
61	59, 60	sseldi 3416	. . . . . . 7 |- ( ph -> ( F ` A ) e. RR* )
62		suppr 8005	. . . . . . 7 |- ( ( < Or RR* /\ 0 e. RR* /\ ( F ` A ) e. RR* ) -> sup ( { 0 , ( F ` A ) } , RR* , < ) = if ( ( F ` A ) < 0 , 0 , ( F ` A ) ) )
63	56, 58, 61, 62	syl3anc 1292	. . . . . 6 |- ( ph -> sup ( { 0 , ( F ` A ) } , RR* , < ) = if ( ( F ` A ) < 0 , 0 , ( F ` A ) ) )
64		pnfxr 11435	. . . . . . . . . . 11 |- +oo e. RR*
65	64	a1i 11	. . . . . . . . . 10 |- ( ph -> +oo e. RR* )
66	58, 65, 60	3jca 1210	. . . . . . . . 9 |- ( ph -> ( 0 e. RR* /\ +oo e. RR* /\ ( F ` A ) e. ( 0 [,] +oo ) ) )
67		iccgelb 11716	. . . . . . . . 9 |- ( ( 0 e. RR* /\ +oo e. RR* /\ ( F ` A ) e. ( 0 [,] +oo ) ) -> 0 <_ ( F ` A ) )
68	66, 67	syl 17	. . . . . . . 8 |- ( ph -> 0 <_ ( F ` A ) )
69	58, 61	xrlenltd 9718	. . . . . . . 8 |- ( ph -> ( 0 <_ ( F ` A ) <-> -. ( F ` A ) < 0 ) )
70	68, 69	mpbid 215	. . . . . . 7 |- ( ph -> -. ( F ` A ) < 0 )
71	70	iffalsed 3883	. . . . . 6 |- ( ph -> if ( ( F ` A ) < 0 , 0 , ( F ` A ) ) = ( F ` A ) )
72	63, 71	eqtr2d 2506	. . . . 5 |- ( ph -> ( F ` A ) = sup ( { 0 , ( F ` A ) } , RR* , < ) )
73	72	adantr 472	. . . 4 |- ( ( ph /\ -. ( F ` A ) = +oo ) -> ( F ` A ) = sup ( { 0 , ( F ` A ) } , RR* , < ) )
74		pwsn 4184	. . . . . . . . . . . 12 |- ~P { A } = { (/) , { A } }
75	74	ineq1i 3621	. . . . . . . . . . 11 |- ( ~P { A } i^i Fin ) = ( { (/) , { A } } i^i Fin )
76		0fin 7817	. . . . . . . . . . . . 13 |- (/) e. Fin
77		snfi 7668	. . . . . . . . . . . . 13 |- { A } e. Fin
78		prssi 4119	. . . . . . . . . . . . 13 |- ( ( (/) e. Fin /\ { A } e. Fin ) -> { (/) , { A } } C_ Fin )
79	76, 77, 78	mp2an 686	. . . . . . . . . . . 12 |- { (/) , { A } } C_ Fin
80		df-ss 3404	. . . . . . . . . . . . 13 |- ( { (/) , { A } } C_ Fin <-> ( { (/) , { A } } i^i Fin ) = { (/) , { A } } )
81	80	biimpi 199	. . . . . . . . . . . 12 |- ( { (/) , { A } } C_ Fin -> ( { (/) , { A } } i^i Fin ) = { (/) , { A } } )
82	79, 81	ax-mp 5	. . . . . . . . . . 11 |- ( { (/) , { A } } i^i Fin ) = { (/) , { A } }
83	75, 82	eqtri 2493	. . . . . . . . . 10 |- ( ~P { A } i^i Fin ) = { (/) , { A } }
84		mpteq1 4476	. . . . . . . . . 10 |- ( ( ~P { A } i^i Fin ) = { (/) , { A } } -> ( x e. ( ~P { A } i^i Fin ) |-> sum_ y e. x ( F ` y ) ) = ( x e. { (/) , { A } } |-> sum_ y e. x ( F ` y ) ) )
85	83, 84	ax-mp 5	. . . . . . . . 9 |- ( x e. ( ~P { A } i^i Fin ) |-> sum_ y e. x ( F ` y ) ) = ( x e. { (/) , { A } } |-> sum_ y e. x ( F ` y ) )
86		0ex 4528	. . . . . . . . . . . . 13 |- (/) e. _V
87	86	a1i 11	. . . . . . . . . . . 12 |- ( T. -> (/) e. _V )
88	1	a1i 11	. . . . . . . . . . . 12 |- ( T. -> { A } e. _V )
89		sumex 13831	. . . . . . . . . . . . 13 |- sum_ y e. (/) ( F ` y ) e. _V
90	89	a1i 11	. . . . . . . . . . . 12 |- ( T. -> sum_ y e. (/) ( F ` y ) e. _V )
91		sumex 13831	. . . . . . . . . . . . 13 |- sum_ y e. { A } ( F ` y ) e. _V
92	91	a1i 11	. . . . . . . . . . . 12 |- ( T. -> sum_ y e. { A } ( F ` y ) e. _V )
93		sumeq1 13832	. . . . . . . . . . . . 13 |- ( x = (/) -> sum_ y e. x ( F ` y ) = sum_ y e. (/) ( F ` y ) )
94	93	adantl 473	. . . . . . . . . . . 12 |- ( ( T. /\ x = (/) ) -> sum_ y e. x ( F ` y ) = sum_ y e. (/) ( F ` y ) )
95		sumeq1 13832	. . . . . . . . . . . . 13 |- ( x = { A } -> sum_ y e. x ( F ` y ) = sum_ y e. { A } ( F ` y ) )
96	95	adantl 473	. . . . . . . . . . . 12 |- ( ( T. /\ x = { A } ) -> sum_ y e. x ( F ` y ) = sum_ y e. { A } ( F ` y ) )
97	87, 88, 90, 92, 94, 96	fmptpr 6105	. . . . . . . . . . 11 |- ( T. -> { <. (/) , sum_ y e. (/) ( F ` y ) >. , <. { A } , sum_ y e. { A } ( F ` y ) >. } = ( x e. { (/) , { A } } |-> sum_ y e. x ( F ` y ) ) )
98	97	trud 1461	. . . . . . . . . 10 |- { <. (/) , sum_ y e. (/) ( F ` y ) >. , <. { A } , sum_ y e. { A } ( F ` y ) >. } = ( x e. { (/) , { A } } |-> sum_ y e. x ( F ` y ) )
99	98	eqcomi 2480	. . . . . . . . 9 |- ( x e. { (/) , { A } } |-> sum_ y e. x ( F ` y ) ) = { <. (/) , sum_ y e. (/) ( F ` y ) >. , <. { A } , sum_ y e. { A } ( F ` y ) >. }
100	85, 99	eqtri 2493	. . . . . . . 8 |- ( x e. ( ~P { A } i^i Fin ) |-> sum_ y e. x ( F ` y ) ) = { <. (/) , sum_ y e. (/) ( F ` y ) >. , <. { A } , sum_ y e. { A } ( F ` y ) >. }
101	100	rneqi 5067	. . . . . . 7 |- ran ( x e. ( ~P { A } i^i Fin ) |-> sum_ y e. x ( F ` y ) ) = ran { <. (/) , sum_ y e. (/) ( F ` y ) >. , <. { A } , sum_ y e. { A } ( F ` y ) >. }
102		rnpropg 5323	. . . . . . . 8 |- ( ( (/) e. _V /\ { A } e. _V ) -> ran { <. (/) , sum_ y e. (/) ( F ` y ) >. , <. { A } , sum_ y e. { A } ( F ` y ) >. } = { sum_ y e. (/) ( F ` y ) , sum_ y e. { A } ( F ` y ) } )
103	86, 1, 102	mp2an 686	. . . . . . 7 |- ran { <. (/) , sum_ y e. (/) ( F ` y ) >. , <. { A } , sum_ y e. { A } ( F ` y ) >. } = { sum_ y e. (/) ( F ` y ) , sum_ y e. { A } ( F ` y ) }
104	101, 103	eqtri 2493	. . . . . 6 |- ran ( x e. ( ~P { A } i^i Fin ) |-> sum_ y e. x ( F ` y ) ) = { sum_ y e. (/) ( F ` y ) , sum_ y e. { A } ( F ` y ) }
105	104	supeq1i 7979	. . . . 5 |- sup ( ran ( x e. ( ~P { A } i^i Fin ) |-> sum_ y e. x ( F ` y ) ) , RR* , < ) = sup ( { sum_ y e. (/) ( F ` y ) , sum_ y e. { A } ( F ` y ) } , RR* , < )
106	105	a1i 11	. . . 4 |- ( ( ph /\ -. ( F ` A ) = +oo ) -> sup ( ran ( x e. ( ~P { A } i^i Fin ) |-> sum_ y e. x ( F ` y ) ) , RR* , < ) = sup ( { sum_ y e. (/) ( F ` y ) , sum_ y e. { A } ( F ` y ) } , RR* , < ) )
107	54, 73, 106	3eqtr4d 2515	. . 3 |- ( ( ph /\ -. ( F ` A ) = +oo ) -> ( F ` A ) = sup ( ran ( x e. ( ~P { A } i^i Fin ) |-> sum_ y e. x ( F ` y ) ) , RR* , < ) )
108	36, 107	eqtr4d 2508	. 2 |- ( ( ph /\ -. ( F ` A ) = +oo ) -> (Σ^ ` F ) = ( F ` A ) )
109	24, 108	pm2.61dan 808	1 |- ( ph -> (Σ^ ` F ) = ( F ` A ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 376   /\ w3a 1007   = wceq 1452   T. wtru 1453   e. wcel 1904   _Vcvv 3031   i^i cin 3389   C_ wss 3390   (/)c0 3722   ifcif 3872   ~Pcpw 3942   {csn 3959   {cpr 3961   <.cop 3965   class class class wbr 4395   |-> cmpt 4454   Or wor 4759   dom cdm 4839   ran crn 4840   Fun wfun 5583   -->wf 5585   ` cfv 5589  (class class class)co 6308   Fincfn 7587   supcsup 7972   CCcc 9555   RRcr 9556   0cc0 9557   +oocpnf 9690   RR*cxr 9692   < clt 9693   <_ cle 9694   [,)cico 11662   [,]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:  sge0snmpt  38339  sge0sup  38347  sge0snmptf  38393  caratheodorylem1  38466