Theorem sge0tsms 38010

Description: Σ^{^} applied to a nonnegative function (its meaningful domain) is the same as the infinite group sum (that's always convergent, in this case). (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Hypotheses

Ref	Expression
sge0tsms.g	|- G = ( RR*s ↾s ( 0 [,] +oo ) )
sge0tsms.x	|- ( ph -> X e. V )
sge0tsms.f	|- ( ph -> F : X --> ( 0 [,] +oo ) )

Assertion

Ref	Expression
sge0tsms	|- ( ph -> (Σ^ ` F ) e. ( G tsums F ) )

Proof of Theorem sge0tsms

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

Step	Hyp	Ref	Expression
1		eqid 2422	. . . 4 |- sup ( ran ( x e. ( ~P X i^i Fin ) |-> ( G gsumg ( F |` x ) ) ) , RR* , < ) = sup ( ran ( x e. ( ~P X i^i Fin ) |-> ( G gsumg ( F |` x ) ) ) , RR* , < )
2	1	a1i 11	. . 3 |- ( ph -> sup ( ran ( x e. ( ~P X i^i Fin ) |-> ( G gsumg ( F |` x ) ) ) , RR* , < ) = sup ( ran ( x e. ( ~P X i^i Fin ) |-> ( G gsumg ( F |` x ) ) ) , RR* , < ) )
3		xrltso 11441	. . . . . 6 |- < Or RR*
4	3	supex 7980	. . . . 5 |- sup ( ran ( x e. ( ~P X i^i Fin ) |-> ( G gsumg ( F |` x ) ) ) , RR* , < ) e. _V
5	4	a1i 11	. . . 4 |- ( ph -> sup ( ran ( x e. ( ~P X i^i Fin ) |-> ( G gsumg ( F |` x ) ) ) , RR* , < ) e. _V )
6		elsncg 4019	. . . 4 |- ( sup ( ran ( x e. ( ~P X i^i Fin ) |-> ( G gsumg ( F |` x ) ) ) , RR* , < ) e. _V -> ( sup ( ran ( x e. ( ~P X i^i Fin ) |-> ( G gsumg ( F |` x ) ) ) , RR* , < ) e. { sup ( ran ( x e. ( ~P X i^i Fin ) |-> ( G gsumg ( F |` x ) ) ) , RR* , < ) } <-> sup ( ran ( x e. ( ~P X i^i Fin ) |-> ( G gsumg ( F |` x ) ) ) , RR* , < ) = sup ( ran ( x e. ( ~P X i^i Fin ) |-> ( G gsumg ( F |` x ) ) ) , RR* , < ) ) )
7	5, 6	syl 17	. . 3 |- ( ph -> ( sup ( ran ( x e. ( ~P X i^i Fin ) |-> ( G gsumg ( F |` x ) ) ) , RR* , < ) e. { sup ( ran ( x e. ( ~P X i^i Fin ) |-> ( G gsumg ( F |` x ) ) ) , RR* , < ) } <-> sup ( ran ( x e. ( ~P X i^i Fin ) |-> ( G gsumg ( F |` x ) ) ) , RR* , < ) = sup ( ran ( x e. ( ~P X i^i Fin ) |-> ( G gsumg ( F |` x ) ) ) , RR* , < ) ) )
8	2, 7	mpbird 235	. 2 |- ( ph -> sup ( ran ( x e. ( ~P X i^i Fin ) |-> ( G gsumg ( F |` x ) ) ) , RR* , < ) e. { sup ( ran ( x e. ( ~P X i^i Fin ) |-> ( G gsumg ( F |` x ) ) ) , RR* , < ) } )
9		sge0tsms.x	. . . . . . 7 |- ( ph -> X e. V )
10	9	adantr 466	. . . . . 6 |- ( ( ph /\ +oo e. ran F ) -> X e. V )
11		sge0tsms.f	. . . . . . 7 |- ( ph -> F : X --> ( 0 [,] +oo ) )
12	11	adantr 466	. . . . . 6 |- ( ( ph /\ +oo e. ran F ) -> F : X --> ( 0 [,] +oo ) )
13		simpr 462	. . . . . 6 |- ( ( ph /\ +oo e. ran F ) -> +oo e. ran F )
14	10, 12, 13	sge0pnfval 38003	. . . . 5 |- ( ( ph /\ +oo e. ran F ) -> (Σ^ ` F ) = +oo )
15		ffn 5743	. . . . . . . . . 10 |- ( F : X --> ( 0 [,] +oo ) -> F Fn X )
16	11, 15	syl 17	. . . . . . . . 9 |- ( ph -> F Fn X )
17	16	adantr 466	. . . . . . . 8 |- ( ( ph /\ +oo e. ran F ) -> F Fn X )
18		fvelrnb 5925	. . . . . . . 8 |- ( F Fn X -> ( +oo e. ran F <-> E. y e. X ( F ` y ) = +oo ) )
19	17, 18	syl 17	. . . . . . 7 |- ( ( ph /\ +oo e. ran F ) -> ( +oo e. ran F <-> E. y e. X ( F ` y ) = +oo ) )
20	13, 19	mpbid 213	. . . . . 6 |- ( ( ph /\ +oo e. ran F ) -> E. y e. X ( F ` y ) = +oo )
21		iccssxr 11718	. . . . . . . . . . . . . 14 |- ( 0 [,] +oo ) C_ RR*
22		sge0tsms.g	. . . . . . . . . . . . . . 15 |- G = ( RR*s ↾s ( 0 [,] +oo ) )
23		simpr 462	. . . . . . . . . . . . . . 15 |- ( ( ph /\ x e. ( ~P X i^i Fin ) ) -> x e. ( ~P X i^i Fin ) )
24	11	adantr 466	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ x e. ( ~P X i^i Fin ) ) -> F : X --> ( 0 [,] +oo ) )
25		elinel1 3651	. . . . . . . . . . . . . . . . . 18 |- ( x e. ( ~P X i^i Fin ) -> x e. ~P X )
26		elpwi 3988	. . . . . . . . . . . . . . . . . 18 |- ( x e. ~P X -> x C_ X )
27	25, 26	syl 17	. . . . . . . . . . . . . . . . 17 |- ( x e. ( ~P X i^i Fin ) -> x C_ X )
28	27	adantl 467	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ x e. ( ~P X i^i Fin ) ) -> x C_ X )
29		fssres 5763	. . . . . . . . . . . . . . . 16 |- ( ( F : X --> ( 0 [,] +oo ) /\ x C_ X ) -> ( F |` x ) : x --> ( 0 [,] +oo ) )
30	24, 28, 29	syl2anc 665	. . . . . . . . . . . . . . 15 |- ( ( ph /\ x e. ( ~P X i^i Fin ) ) -> ( F |` x ) : x --> ( 0 [,] +oo ) )
31		elinel2 3652	. . . . . . . . . . . . . . . . 17 |- ( x e. ( ~P X i^i Fin ) -> x e. Fin )
32	31	adantl 467	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ x e. ( ~P X i^i Fin ) ) -> x e. Fin )
33		0red 9645	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ x e. ( ~P X i^i Fin ) ) -> 0 e. RR )
34	30, 32, 33	fdmfifsupp 7896	. . . . . . . . . . . . . . 15 |- ( ( ph /\ x e. ( ~P X i^i Fin ) ) -> ( F |` x ) finSupp 0 )
35	22, 23, 30, 34	gsumge0cl 38001	. . . . . . . . . . . . . 14 |- ( ( ph /\ x e. ( ~P X i^i Fin ) ) -> ( G gsumg ( F |` x ) ) e. ( 0 [,] +oo ) )
36	21, 35	sseldi 3462	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. ( ~P X i^i Fin ) ) -> ( G gsumg ( F |` x ) ) e. RR* )
37	36	ralrimiva 2839	. . . . . . . . . . . 12 |- ( ph -> A. x e. ( ~P X i^i Fin ) ( G gsumg ( F |` x ) ) e. RR* )
38	37	3ad2ant1 1026	. . . . . . . . . . 11 |- ( ( ph /\ y e. X /\ ( F ` y ) = +oo ) -> A. x e. ( ~P X i^i Fin ) ( G gsumg ( F |` x ) ) e. RR* )
39		eqid 2422	. . . . . . . . . . . 12 |- ( x e. ( ~P X i^i Fin ) |-> ( G gsumg ( F |` x ) ) ) = ( x e. ( ~P X i^i Fin ) |-> ( G gsumg ( F |` x ) ) )
40	39	rnmptss 6064	. . . . . . . . . . 11 |- ( A. x e. ( ~P X i^i Fin ) ( G gsumg ( F |` x ) ) e. RR* -> ran ( x e. ( ~P X i^i Fin ) |-> ( G gsumg ( F |` x ) ) ) C_ RR* )
41	38, 40	syl 17	. . . . . . . . . 10 |- ( ( ph /\ y e. X /\ ( F ` y ) = +oo ) -> ran ( x e. ( ~P X i^i Fin ) |-> ( G gsumg ( F |` x ) ) ) C_ RR* )
42		snelpwi 4663	. . . . . . . . . . . . . 14 |- ( y e. X -> { y } e. ~P X )
43		snfi 7654	. . . . . . . . . . . . . . 15 |- { y } e. Fin
44	43	a1i 11	. . . . . . . . . . . . . 14 |- ( y e. X -> { y } e. Fin )
45	42, 44	elind 3650	. . . . . . . . . . . . 13 |- ( y e. X -> { y } e. ( ~P X i^i Fin ) )
46	45	3ad2ant2 1027	. . . . . . . . . . . 12 |- ( ( ph /\ y e. X /\ ( F ` y ) = +oo ) -> { y } e. ( ~P X i^i Fin ) )
47	11	adantr 466	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ y e. X ) -> F : X --> ( 0 [,] +oo ) )
48		snssi 4141	. . . . . . . . . . . . . . . . . . 19 |- ( y e. X -> { y } C_ X )
49	48	adantl 467	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ y e. X ) -> { y } C_ X )
50	47, 49	fssresd 5764	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ y e. X ) -> ( F |` { y } ) : { y } --> ( 0 [,] +oo ) )
51	50	feqmptd 5931	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ y e. X ) -> ( F |` { y } ) = ( x e. { y } |-> ( ( F |` { y } ) ` x ) ) )
52		fvres 5892	. . . . . . . . . . . . . . . . . 18 |- ( x e. { y } -> ( ( F |` { y } ) ` x ) = ( F ` x ) )
53	52	mpteq2ia 4503	. . . . . . . . . . . . . . . . 17 |- ( x e. { y } |-> ( ( F |` { y } ) ` x ) ) = ( x e. { y } |-> ( F ` x ) )
54	53	a1i 11	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ y e. X ) -> ( x e. { y } |-> ( ( F |` { y } ) ` x ) ) = ( x e. { y } |-> ( F ` x ) ) )
55	51, 54	eqtrd 2463	. . . . . . . . . . . . . . 15 |- ( ( ph /\ y e. X ) -> ( F |` { y } ) = ( x e. { y } |-> ( F ` x ) ) )
56	55	oveq2d 6318	. . . . . . . . . . . . . 14 |- ( ( ph /\ y e. X ) -> ( G gsumg ( F |` { y } ) ) = ( G gsumg ( x e. { y } |-> ( F ` x ) ) ) )
57	56	3adant3 1025	. . . . . . . . . . . . 13 |- ( ( ph /\ y e. X /\ ( F ` y ) = +oo ) -> ( G gsumg ( F |` { y } ) ) = ( G gsumg ( x e. { y } |-> ( F ` x ) ) ) )
58		xrge0cmn 18998	. . . . . . . . . . . . . . . . 17 |- ( RR*s ↾s ( 0 [,] +oo ) ) e. CMnd
59	22, 58	eqeltri 2506	. . . . . . . . . . . . . . . 16 |- G e. CMnd
60		cmnmnd 17433	. . . . . . . . . . . . . . . 16 |- ( G e. CMnd -> G e. Mnd )
61	59, 60	ax-mp 5	. . . . . . . . . . . . . . 15 |- G e. Mnd
62	61	a1i 11	. . . . . . . . . . . . . 14 |- ( ( ph /\ y e. X /\ ( F ` y ) = +oo ) -> G e. Mnd )
63		simp2 1006	. . . . . . . . . . . . . 14 |- ( ( ph /\ y e. X /\ ( F ` y ) = +oo ) -> y e. X )
64	11	ffvelrnda 6034	. . . . . . . . . . . . . . 15 |- ( ( ph /\ y e. X ) -> ( F ` y ) e. ( 0 [,] +oo ) )
65	64	3adant3 1025	. . . . . . . . . . . . . 14 |- ( ( ph /\ y e. X /\ ( F ` y ) = +oo ) -> ( F ` y ) e. ( 0 [,] +oo ) )
66		df-ss 3450	. . . . . . . . . . . . . . . . . 18 |- ( ( 0 [,] +oo ) C_ RR* <-> ( ( 0 [,] +oo ) i^i RR* ) = ( 0 [,] +oo ) )
67	21, 66	mpbi 211	. . . . . . . . . . . . . . . . 17 |- ( ( 0 [,] +oo ) i^i RR* ) = ( 0 [,] +oo )
68	67	eqcomi 2435	. . . . . . . . . . . . . . . 16 |- ( 0 [,] +oo ) = ( ( 0 [,] +oo ) i^i RR* )
69		ovex 6330	. . . . . . . . . . . . . . . . 17 |- ( 0 [,] +oo ) e. _V
70		xrsbas 18972	. . . . . . . . . . . . . . . . . 18 |- RR* = ( Base ` RR*s )
71	22, 70	ressbas 15167	. . . . . . . . . . . . . . . . 17 |- ( ( 0 [,] +oo ) e. _V -> ( ( 0 [,] +oo ) i^i RR* ) = ( Base ` G ) )
72	69, 71	ax-mp 5	. . . . . . . . . . . . . . . 16 |- ( ( 0 [,] +oo ) i^i RR* ) = ( Base ` G )
73	68, 72	eqtri 2451	. . . . . . . . . . . . . . 15 |- ( 0 [,] +oo ) = ( Base ` G )
74		fveq2 5878	. . . . . . . . . . . . . . 15 |- ( x = y -> ( F ` x ) = ( F ` y ) )
75	73, 74	gsumsn 17575	. . . . . . . . . . . . . 14 |- ( ( G e. Mnd /\ y e. X /\ ( F ` y ) e. ( 0 [,] +oo ) ) -> ( G gsumg ( x e. { y } |-> ( F ` x ) ) ) = ( F ` y ) )
76	62, 63, 65, 75	syl3anc 1264	. . . . . . . . . . . . 13 |- ( ( ph /\ y e. X /\ ( F ` y ) = +oo ) -> ( G gsumg ( x e. { y } |-> ( F ` x ) ) ) = ( F ` y ) )
77		simp3 1007	. . . . . . . . . . . . 13 |- ( ( ph /\ y e. X /\ ( F ` y ) = +oo ) -> ( F ` y ) = +oo )
78	57, 76, 77	3eqtrrd 2468	. . . . . . . . . . . 12 |- ( ( ph /\ y e. X /\ ( F ` y ) = +oo ) -> +oo = ( G gsumg ( F |` { y } ) ) )
79		reseq2 5116	. . . . . . . . . . . . . . 15 |- ( x = { y } -> ( F |` x ) = ( F |` { y } ) )
80	79	oveq2d 6318	. . . . . . . . . . . . . 14 |- ( x = { y } -> ( G gsumg ( F |` x ) ) = ( G gsumg ( F |` { y } ) ) )
81	80	eqeq2d 2436	. . . . . . . . . . . . 13 |- ( x = { y } -> ( +oo = ( G gsumg ( F |` x ) ) <-> +oo = ( G gsumg ( F |` { y } ) ) ) )
82	81	rspcev 3182	. . . . . . . . . . . 12 |- ( ( { y } e. ( ~P X i^i Fin ) /\ +oo = ( G gsumg ( F |` { y } ) ) ) -> E. x e. ( ~P X i^i Fin ) +oo = ( G gsumg ( F |` x ) ) )
83	46, 78, 82	syl2anc 665	. . . . . . . . . . 11 |- ( ( ph /\ y e. X /\ ( F ` y ) = +oo ) -> E. x e. ( ~P X i^i Fin ) +oo = ( G gsumg ( F |` x ) ) )
84		pnfxr 11413	. . . . . . . . . . . . 13 |- +oo e. RR*
85	84	a1i 11	. . . . . . . . . . . 12 |- ( ( ph /\ y e. X /\ ( F ` y ) = +oo ) -> +oo e. RR* )
86	39	elrnmpt 5097	. . . . . . . . . . . 12 |- ( +oo e. RR* -> ( +oo e. ran ( x e. ( ~P X i^i Fin ) |-> ( G gsumg ( F |` x ) ) ) <-> E. x e. ( ~P X i^i Fin ) +oo = ( G gsumg ( F |` x ) ) ) )
87	85, 86	syl 17	. . . . . . . . . . 11 |- ( ( ph /\ y e. X /\ ( F ` y ) = +oo ) -> ( +oo e. ran ( x e. ( ~P X i^i Fin ) |-> ( G gsumg ( F |` x ) ) ) <-> E. x e. ( ~P X i^i Fin ) +oo = ( G gsumg ( F |` x ) ) ) )
88	83, 87	mpbird 235	. . . . . . . . . 10 |- ( ( ph /\ y e. X /\ ( F ` y ) = +oo ) -> +oo e. ran ( x e. ( ~P X i^i Fin ) |-> ( G gsumg ( F |` x ) ) ) )
89		supxrpnf 11605	. . . . . . . . . 10 |- ( ( ran ( x e. ( ~P X i^i Fin ) |-> ( G gsumg ( F |` x ) ) ) C_ RR* /\ +oo e. ran ( x e. ( ~P X i^i Fin ) |-> ( G gsumg ( F |` x ) ) ) ) -> sup ( ran ( x e. ( ~P X i^i Fin ) |-> ( G gsumg ( F |` x ) ) ) , RR* , < ) = +oo )
90	41, 88, 89	syl2anc 665	. . . . . . . . 9 |- ( ( ph /\ y e. X /\ ( F ` y ) = +oo ) -> sup ( ran ( x e. ( ~P X i^i Fin ) |-> ( G gsumg ( F |` x ) ) ) , RR* , < ) = +oo )
91	90	3exp 1204	. . . . . . . 8 |- ( ph -> ( y e. X -> ( ( F ` y ) = +oo -> sup ( ran ( x e. ( ~P X i^i Fin ) |-> ( G gsumg ( F |` x ) ) ) , RR* , < ) = +oo ) ) )
92	91	adantr 466	. . . . . . 7 |- ( ( ph /\ +oo e. ran F ) -> ( y e. X -> ( ( F ` y ) = +oo -> sup ( ran ( x e. ( ~P X i^i Fin ) |-> ( G gsumg ( F |` x ) ) ) , RR* , < ) = +oo ) ) )
93	92	rexlimdv 2915	. . . . . 6 |- ( ( ph /\ +oo e. ran F ) -> ( E. y e. X ( F ` y ) = +oo -> sup ( ran ( x e. ( ~P X i^i Fin ) |-> ( G gsumg ( F |` x ) ) ) , RR* , < ) = +oo ) )
94	20, 93	mpd 15	. . . . 5 |- ( ( ph /\ +oo e. ran F ) -> sup ( ran ( x e. ( ~P X i^i Fin ) |-> ( G gsumg ( F |` x ) ) ) , RR* , < ) = +oo )
95	14, 94	eqtr4d 2466	. . . 4 |- ( ( ph /\ +oo e. ran F ) -> (Σ^ ` F ) = sup ( ran ( x e. ( ~P X i^i Fin ) |-> ( G gsumg ( F |` x ) ) ) , RR* , < ) )
96	9	adantr 466	. . . . . 6 |- ( ( ph /\ -. +oo e. ran F ) -> X e. V )
97	11	adantr 466	. . . . . . 7 |- ( ( ph /\ -. +oo e. ran F ) -> F : X --> ( 0 [,] +oo ) )
98		simpr 462	. . . . . . 7 |- ( ( ph /\ -. +oo e. ran F ) -> -. +oo e. ran F )
99	97, 98	fge0iccico 38000	. . . . . 6 |- ( ( ph /\ -. +oo e. ran F ) -> F : X --> ( 0 [,) +oo ) )
100	96, 99	sge0reval 38002	. . . . 5 |- ( ( ph /\ -. +oo e. ran F ) -> (Σ^ ` F ) = sup ( ran ( x e. ( ~P X i^i Fin ) |-> sum_ y e. x ( F ` y ) ) , RR* , < ) )
101	24, 28	feqresmpt 5932	. . . . . . . . . . 11 |- ( ( ph /\ x e. ( ~P X i^i Fin ) ) -> ( F |` x ) = ( y e. x |-> ( F ` y ) ) )
102	101	adantlr 719	. . . . . . . . . 10 |- ( ( ( ph /\ -. +oo e. ran F ) /\ x e. ( ~P X i^i Fin ) ) -> ( F |` x ) = ( y e. x |-> ( F ` y ) ) )
103	102	oveq2d 6318	. . . . . . . . 9 |- ( ( ( ph /\ -. +oo e. ran F ) /\ x e. ( ~P X i^i Fin ) ) -> ( G gsumg ( F |` x ) ) = ( G gsumg ( y e. x |-> ( F ` y ) ) ) )
104	22	fveq2i 5881	. . . . . . . . . . 11 |- ( +g ` G ) = ( +g ` ( RR*s ↾s ( 0 [,] +oo ) ) )
105		eqid 2422	. . . . . . . . . . . . . 14 |- ( RR*s ↾s ( 0 [,] +oo ) ) = ( RR*s ↾s ( 0 [,] +oo ) )
106		xrsadd 18973	. . . . . . . . . . . . . 14 |- +e = ( +g ` RR*s )
107	105, 106	ressplusg 15227	. . . . . . . . . . . . 13 |- ( ( 0 [,] +oo ) e. _V -> +e = ( +g ` ( RR*s ↾s ( 0 [,] +oo ) ) ) )
108	69, 107	ax-mp 5	. . . . . . . . . . . 12 |- +e = ( +g ` ( RR*s ↾s ( 0 [,] +oo ) ) )
109	108	eqcomi 2435	. . . . . . . . . . 11 |- ( +g ` ( RR*s ↾s ( 0 [,] +oo ) ) ) = +e
110	104, 109	eqtr2i 2452	. . . . . . . . . 10 |- +e = ( +g ` G )
111	22	oveq1i 6312	. . . . . . . . . . 11 |- ( G ↾s ( 0 [,) +oo ) ) = ( ( RR*s ↾s ( 0 [,] +oo ) ) ↾s ( 0 [,) +oo ) )
112		icossicc 11722	. . . . . . . . . . . . 13 |- ( 0 [,) +oo ) C_ ( 0 [,] +oo )
113	69, 112	pm3.2i 456	. . . . . . . . . . . 12 |- ( ( 0 [,] +oo ) e. _V /\ ( 0 [,) +oo ) C_ ( 0 [,] +oo ) )
114		ressabs 15176	. . . . . . . . . . . 12 |- ( ( ( 0 [,] +oo ) e. _V /\ ( 0 [,) +oo ) C_ ( 0 [,] +oo ) ) -> ( ( RR*s ↾s ( 0 [,] +oo ) ) ↾s ( 0 [,) +oo ) ) = ( RR*s ↾s ( 0 [,) +oo ) ) )
115	113, 114	ax-mp 5	. . . . . . . . . . 11 |- ( ( RR*s ↾s ( 0 [,] +oo ) ) ↾s ( 0 [,) +oo ) ) = ( RR*s ↾s ( 0 [,) +oo ) )
116	111, 115	eqtr2i 2452	. . . . . . . . . 10 |- ( RR*s ↾s ( 0 [,) +oo ) ) = ( G ↾s ( 0 [,) +oo ) )
117	59	elexi 3091	. . . . . . . . . . 11 |- G e. _V
118	117	a1i 11	. . . . . . . . . 10 |- ( ( ( ph /\ -. +oo e. ran F ) /\ x e. ( ~P X i^i Fin ) ) -> G e. _V )
119		simpr 462	. . . . . . . . . 10 |- ( ( ( ph /\ -. +oo e. ran F ) /\ x e. ( ~P X i^i Fin ) ) -> x e. ( ~P X i^i Fin ) )
120	112	a1i 11	. . . . . . . . . 10 |- ( ( ( ph /\ -. +oo e. ran F ) /\ x e. ( ~P X i^i Fin ) ) -> ( 0 [,) +oo ) C_ ( 0 [,] +oo ) )
121		0xr 9688	. . . . . . . . . . . . 13 |- 0 e. RR*
122	121	a1i 11	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ -. +oo e. ran F ) /\ x e. ( ~P X i^i Fin ) ) /\ y e. x ) -> 0 e. RR* )
123	84	a1i 11	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ -. +oo e. ran F ) /\ x e. ( ~P X i^i Fin ) ) /\ y e. x ) -> +oo e. RR* )
124	97	ad2antrr 730	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ -. +oo e. ran F ) /\ x e. ( ~P X i^i Fin ) ) /\ y e. x ) -> F : X --> ( 0 [,] +oo ) )
125	27	sselda 3464	. . . . . . . . . . . . . . 15 |- ( ( x e. ( ~P X i^i Fin ) /\ y e. x ) -> y e. X )
126	125	adantll 718	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ -. +oo e. ran F ) /\ x e. ( ~P X i^i Fin ) ) /\ y e. x ) -> y e. X )
127	124, 126	ffvelrnd 6035	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ -. +oo e. ran F ) /\ x e. ( ~P X i^i Fin ) ) /\ y e. x ) -> ( F ` y ) e. ( 0 [,] +oo ) )
128	21, 127	sseldi 3462	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ -. +oo e. ran F ) /\ x e. ( ~P X i^i Fin ) ) /\ y e. x ) -> ( F ` y ) e. RR* )
129		iccgelb 11692	. . . . . . . . . . . . 13 |- ( ( 0 e. RR* /\ +oo e. RR* /\ ( F ` y ) e. ( 0 [,] +oo ) ) -> 0 <_ ( F ` y ) )
130	122, 123, 127, 129	syl3anc 1264	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ -. +oo e. ran F ) /\ x e. ( ~P X i^i Fin ) ) /\ y e. x ) -> 0 <_ ( F ` y ) )
131		id 23	. . . . . . . . . . . . . . . . . . . 20 |- ( ( F ` y ) = +oo -> ( F ` y ) = +oo )
132	131	eqcomd 2430	. . . . . . . . . . . . . . . . . . 19 |- ( ( F ` y ) = +oo -> +oo = ( F ` y ) )
133	132	adantl 467	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ph /\ x e. ( ~P X i^i Fin ) ) /\ y e. x ) /\ ( F ` y ) = +oo ) -> +oo = ( F ` y ) )
134		ffun 5745	. . . . . . . . . . . . . . . . . . . . . 22 |- ( F : X --> ( 0 [,] +oo ) -> Fun F )
135	11, 134	syl 17	. . . . . . . . . . . . . . . . . . . . 21 |- ( ph -> Fun F )
136	135	ad2antrr 730	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ph /\ x e. ( ~P X i^i Fin ) ) /\ y e. x ) -> Fun F )
137	23, 125	sylan 473	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ph /\ x e. ( ~P X i^i Fin ) ) /\ y e. x ) -> y e. X )
138		fdm 5747	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( F : X --> ( 0 [,] +oo ) -> dom F = X )
139	11, 138	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ph -> dom F = X )
140	139	eqcomd 2430	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ph -> X = dom F )
141	140	ad2antrr 730	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ph /\ x e. ( ~P X i^i Fin ) ) /\ y e. x ) -> X = dom F )
142	137, 141	eleqtrd 2512	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ph /\ x e. ( ~P X i^i Fin ) ) /\ y e. x ) -> y e. dom F )
143		fvelrn 6027	. . . . . . . . . . . . . . . . . . . 20 |- ( ( Fun F /\ y e. dom F ) -> ( F ` y ) e. ran F )
144	136, 142, 143	syl2anc 665	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ph /\ x e. ( ~P X i^i Fin ) ) /\ y e. x ) -> ( F ` y ) e. ran F )
145	144	adantr 466	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ph /\ x e. ( ~P X i^i Fin ) ) /\ y e. x ) /\ ( F ` y ) = +oo ) -> ( F ` y ) e. ran F )
146	133, 145	eqeltrd 2510	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ph /\ x e. ( ~P X i^i Fin ) ) /\ y e. x ) /\ ( F ` y ) = +oo ) -> +oo e. ran F )
147	146	adantlllr 37222	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ph /\ -. +oo e. ran F ) /\ x e. ( ~P X i^i Fin ) ) /\ y e. x ) /\ ( F ` y ) = +oo ) -> +oo e. ran F )
148	98	ad3antrrr 734	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ph /\ -. +oo e. ran F ) /\ x e. ( ~P X i^i Fin ) ) /\ y e. x ) /\ ( F ` y ) = +oo ) -> -. +oo e. ran F )
149	147, 148	pm2.65da 578	. . . . . . . . . . . . . . 15 |- ( ( ( ( ph /\ -. +oo e. ran F ) /\ x e. ( ~P X i^i Fin ) ) /\ y e. x ) -> -. ( F ` y ) = +oo )
150	149	neqned 2627	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ -. +oo e. ran F ) /\ x e. ( ~P X i^i Fin ) ) /\ y e. x ) -> ( F ` y ) =/= +oo )
151		ge0xrre 37464	. . . . . . . . . . . . . 14 |- ( ( ( F ` y ) e. ( 0 [,] +oo ) /\ ( F ` y ) =/= +oo ) -> ( F ` y ) e. RR )
152	127, 150, 151	syl2anc 665	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ -. +oo e. ran F ) /\ x e. ( ~P X i^i Fin ) ) /\ y e. x ) -> ( F ` y ) e. RR )
153	152	ltpnfd 11424	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ -. +oo e. ran F ) /\ x e. ( ~P X i^i Fin ) ) /\ y e. x ) -> ( F ` y ) < +oo )
154	122, 123, 128, 130, 153	elicod 11686	. . . . . . . . . . 11 |- ( ( ( ( ph /\ -. +oo e. ran F ) /\ x e. ( ~P X i^i Fin ) ) /\ y e. x ) -> ( F ` y ) e. ( 0 [,) +oo ) )
155		eqid 2422	. . . . . . . . . . 11 |- ( y e. x |-> ( F ` y ) ) = ( y e. x |-> ( F ` y ) )
156	154, 155	fmptd 6058	. . . . . . . . . 10 |- ( ( ( ph /\ -. +oo e. ran F ) /\ x e. ( ~P X i^i Fin ) ) -> ( y e. x |-> ( F ` y ) ) : x --> ( 0 [,) +oo ) )
157		0e0icopnf 11743	. . . . . . . . . . 11 |- 0 e. ( 0 [,) +oo )
158	157	a1i 11	. . . . . . . . . 10 |- ( ( ( ph /\ -. +oo e. ran F ) /\ x e. ( ~P X i^i Fin ) ) -> 0 e. ( 0 [,) +oo ) )
159	21	sseli 3460	. . . . . . . . . . . 12 |- ( y e. ( 0 [,] +oo ) -> y e. RR* )
160		xaddid2 11534	. . . . . . . . . . . . 13 |- ( y e. RR* -> ( 0 +e y ) = y )
161		xaddid1 11533	. . . . . . . . . . . . 13 |- ( y e. RR* -> ( y +e 0 ) = y )
162	160, 161	jca 534	. . . . . . . . . . . 12 |- ( y e. RR* -> ( ( 0 +e y ) = y /\ ( y +e 0 ) = y ) )
163	159, 162	syl 17	. . . . . . . . . . 11 |- ( y e. ( 0 [,] +oo ) -> ( ( 0 +e y ) = y /\ ( y +e 0 ) = y ) )
164	163	adantl 467	. . . . . . . . . 10 |- ( ( ( ( ph /\ -. +oo e. ran F ) /\ x e. ( ~P X i^i Fin ) ) /\ y e. ( 0 [,] +oo ) ) -> ( ( 0 +e y ) = y /\ ( y +e 0 ) = y ) )
165	73, 110, 116, 118, 119, 120, 156, 158, 164	gsumress 16507	. . . . . . . . 9 |- ( ( ( ph /\ -. +oo e. ran F ) /\ x e. ( ~P X i^i Fin ) ) -> ( G gsumg ( y e. x |-> ( F ` y ) ) ) = ( ( RR*s ↾s ( 0 [,) +oo ) ) gsumg ( y e. x |-> ( F ` y ) ) ) )
166		rege0subm 19012	. . . . . . . . . . . . 13 |- ( 0 [,) +oo ) e. (SubMnd ` ℂfld )
167	166	a1i 11	. . . . . . . . . . . 12 |- ( ( ( ph /\ -. +oo e. ran F ) /\ x e. ( ~P X i^i Fin ) ) -> ( 0 [,) +oo ) e. (SubMnd ` ℂfld ) )
168		eqid 2422	. . . . . . . . . . . 12 |- (ℂfld ↾s ( 0 [,) +oo ) ) = (ℂfld ↾s ( 0 [,) +oo ) )
169	119, 167, 156, 168	gsumsubm 16608	. . . . . . . . . . 11 |- ( ( ( ph /\ -. +oo e. ran F ) /\ x e. ( ~P X i^i Fin ) ) -> (ℂfld gsumg ( y e. x |-> ( F ` y ) ) ) = ( (ℂfld ↾s ( 0 [,) +oo ) ) gsumg ( y e. x |-> ( F ` y ) ) ) )
170		eqidd 2423	. . . . . . . . . . 11 |- ( ( ( ph /\ -. +oo e. ran F ) /\ x e. ( ~P X i^i Fin ) ) -> ( (ℂfld ↾s ( 0 [,) +oo ) ) gsumg ( y e. x |-> ( F ` y ) ) ) = ( (ℂfld ↾s ( 0 [,) +oo ) ) gsumg ( y e. x |-> ( F ` y ) ) ) )
171		vex 3084	. . . . . . . . . . . . . 14 |- x e. _V
172	171	mptex 6148	. . . . . . . . . . . . 13 |- ( y e. x |-> ( F ` y ) ) e. _V
173	172	a1i 11	. . . . . . . . . . . 12 |- ( ( ( ph /\ -. +oo e. ran F ) /\ x e. ( ~P X i^i Fin ) ) -> ( y e. x |-> ( F ` y ) ) e. _V )
174		ovex 6330	. . . . . . . . . . . . 13 |- (ℂfld ↾s ( 0 [,) +oo ) ) e. _V
175	174	a1i 11	. . . . . . . . . . . 12 |- ( ( ( ph /\ -. +oo e. ran F ) /\ x e. ( ~P X i^i Fin ) ) -> (ℂfld ↾s ( 0 [,) +oo ) ) e. _V )
176		ovex 6330	. . . . . . . . . . . . 13 |- ( RR*s ↾s ( 0 [,) +oo ) ) e. _V
177	176	a1i 11	. . . . . . . . . . . 12 |- ( ( ( ph /\ -. +oo e. ran F ) /\ x e. ( ~P X i^i Fin ) ) -> ( RR*s ↾s ( 0 [,) +oo ) ) e. _V )
178		rge0ssre 11741	. . . . . . . . . . . . . . . . 17 |- ( 0 [,) +oo ) C_ RR
179		ax-resscn 9597	. . . . . . . . . . . . . . . . 17 |- RR C_ CC
180	178, 179	sstri 3473	. . . . . . . . . . . . . . . 16 |- ( 0 [,) +oo ) C_ CC
181		cnfldbas 18962	. . . . . . . . . . . . . . . . 17 |- CC = ( Base ` ℂfld )
182	168, 181	ressbas2 15168	. . . . . . . . . . . . . . . 16 |- ( ( 0 [,) +oo ) C_ CC -> ( 0 [,) +oo ) = ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) )
183	180, 182	ax-mp 5	. . . . . . . . . . . . . . 15 |- ( 0 [,) +oo ) = ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) )
184	183	eqcomi 2435	. . . . . . . . . . . . . 14 |- ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) = ( 0 [,) +oo )
185	112, 21	sstri 3473	. . . . . . . . . . . . . . 15 |- ( 0 [,) +oo ) C_ RR*
186		eqid 2422	. . . . . . . . . . . . . . . 16 |- ( RR*s ↾s ( 0 [,) +oo ) ) = ( RR*s ↾s ( 0 [,) +oo ) )
187	186, 70	ressbas2 15168	. . . . . . . . . . . . . . 15 |- ( ( 0 [,) +oo ) C_ RR* -> ( 0 [,) +oo ) = ( Base ` ( RR*s ↾s ( 0 [,) +oo ) ) ) )
188	185, 187	ax-mp 5	. . . . . . . . . . . . . 14 |- ( 0 [,) +oo ) = ( Base ` ( RR*s ↾s ( 0 [,) +oo ) ) )
189	184, 188	eqtri 2451	. . . . . . . . . . . . 13 |- ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) = ( Base ` ( RR*s ↾s ( 0 [,) +oo ) ) )
190	189	a1i 11	. . . . . . . . . . . 12 |- ( ( ( ph /\ -. +oo e. ran F ) /\ x e. ( ~P X i^i Fin ) ) -> ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) = ( Base ` ( RR*s ↾s ( 0 [,) +oo ) ) ) )
191		rge0srg 19025	. . . . . . . . . . . . . . 15 |- (ℂfld ↾s ( 0 [,) +oo ) ) e. SRing
192	191	a1i 11	. . . . . . . . . . . . . 14 |- ( ( s e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) /\ t e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) ) -> (ℂfld ↾s ( 0 [,) +oo ) ) e. SRing )
193		simpl 458	. . . . . . . . . . . . . 14 |- ( ( s e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) /\ t e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) ) -> s e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) )
194		simpr 462	. . . . . . . . . . . . . 14 |- ( ( s e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) /\ t e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) ) -> t e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) )
195		eqid 2422	. . . . . . . . . . . . . . 15 |- ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) = ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) )
196		eqid 2422	. . . . . . . . . . . . . . 15 |- ( +g ` (ℂfld ↾s ( 0 [,) +oo ) ) ) = ( +g ` (ℂfld ↾s ( 0 [,) +oo ) ) )
197	195, 196	srgacl 17745	. . . . . . . . . . . . . 14 |- ( ( (ℂfld ↾s ( 0 [,) +oo ) ) e. SRing /\ s e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) /\ t e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) ) -> ( s ( +g ` (ℂfld ↾s ( 0 [,) +oo ) ) ) t ) e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) )
198	192, 193, 194, 197	syl3anc 1264	. . . . . . . . . . . . 13 |- ( ( s e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) /\ t e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) ) -> ( s ( +g ` (ℂfld ↾s ( 0 [,) +oo ) ) ) t ) e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) )
199	198	adantl 467	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ -. +oo e. ran F ) /\ x e. ( ~P X i^i Fin ) ) /\ ( s e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) /\ t e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) ) ) -> ( s ( +g ` (ℂfld ↾s ( 0 [,) +oo ) ) ) t ) e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) )
200	178	a1i 11	. . . . . . . . . . . . . . . 16 |- ( s e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) -> ( 0 [,) +oo ) C_ RR )
201		id 23	. . . . . . . . . . . . . . . . 17 |- ( s e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) -> s e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) )
202	201, 184	syl6eleq 2520	. . . . . . . . . . . . . . . 16 |- ( s e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) -> s e. ( 0 [,) +oo ) )
203	200, 202	sseldd 3465	. . . . . . . . . . . . . . 15 |- ( s e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) -> s e. RR )
204	203	adantr 466	. . . . . . . . . . . . . 14 |- ( ( s e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) /\ t e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) ) -> s e. RR )
205	178	a1i 11	. . . . . . . . . . . . . . . 16 |- ( t e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) -> ( 0 [,) +oo ) C_ RR )
206		id 23	. . . . . . . . . . . . . . . . 17 |- ( t e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) -> t e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) )
207	206, 184	syl6eleq 2520	. . . . . . . . . . . . . . . 16 |- ( t e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) -> t e. ( 0 [,) +oo ) )
208	205, 207	sseldd 3465	. . . . . . . . . . . . . . 15 |- ( t e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) -> t e. RR )
209	208	adantl 467	. . . . . . . . . . . . . 14 |- ( ( s e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) /\ t e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) ) -> t e. RR )
210		rexadd 11526	. . . . . . . . . . . . . . . 16 |- ( ( s e. RR /\ t e. RR ) -> ( s +e t ) = ( s + t ) )
211	210	eqcomd 2430	. . . . . . . . . . . . . . 15 |- ( ( s e. RR /\ t e. RR ) -> ( s + t ) = ( s +e t ) )
212	166	elexi 3091	. . . . . . . . . . . . . . . . . . . 20 |- ( 0 [,) +oo ) e. _V
213		cnfldadd 18963	. . . . . . . . . . . . . . . . . . . . 21 |- + = ( +g ` ℂfld )
214	168, 213	ressplusg 15227	. . . . . . . . . . . . . . . . . . . 20 |- ( ( 0 [,) +oo ) e. _V -> + = ( +g ` (ℂfld ↾s ( 0 [,) +oo ) ) ) )
215	212, 214	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 |- + = ( +g ` (ℂfld ↾s ( 0 [,) +oo ) ) )
216	215, 213	eqtr3i 2453	. . . . . . . . . . . . . . . . . 18 |- ( +g ` (ℂfld ↾s ( 0 [,) +oo ) ) ) = ( +g ` ℂfld )
217	216, 213	eqtr4i 2454	. . . . . . . . . . . . . . . . 17 |- ( +g ` (ℂfld ↾s ( 0 [,) +oo ) ) ) = +
218	217	oveqi 6315	. . . . . . . . . . . . . . . 16 |- ( s ( +g ` (ℂfld ↾s ( 0 [,) +oo ) ) ) t ) = ( s + t )
219	218	a1i 11	. . . . . . . . . . . . . . 15 |- ( ( s e. RR /\ t e. RR ) -> ( s ( +g ` (ℂfld ↾s ( 0 [,) +oo ) ) ) t ) = ( s + t ) )
220	186, 106	ressplusg 15227	. . . . . . . . . . . . . . . . . . 19 |- ( ( 0 [,) +oo ) e. _V -> +e = ( +g ` ( RR*s ↾s ( 0 [,) +oo ) ) ) )
221	212, 220	ax-mp 5	. . . . . . . . . . . . . . . . . 18 |- +e = ( +g ` ( RR*s ↾s ( 0 [,) +oo ) ) )
222	221	eqcomi 2435	. . . . . . . . . . . . . . . . 17 |- ( +g ` ( RR*s ↾s ( 0 [,) +oo ) ) ) = +e
223	222	oveqi 6315	. . . . . . . . . . . . . . . 16 |- ( s ( +g ` ( RR*s ↾s ( 0 [,) +oo ) ) ) t ) = ( s +e t )
224	223	a1i 11	. . . . . . . . . . . . . . 15 |- ( ( s e. RR /\ t e. RR ) -> ( s ( +g ` ( RR*s ↾s ( 0 [,) +oo ) ) ) t ) = ( s +e t ) )
225	211, 219, 224	3eqtr4d 2473	. . . . . . . . . . . . . 14 |- ( ( s e. RR /\ t e. RR ) -> ( s ( +g ` (ℂfld ↾s ( 0 [,) +oo ) ) ) t ) = ( s ( +g ` ( RR*s ↾s ( 0 [,) +oo ) ) ) t ) )
226	204, 209, 225	syl2anc 665	. . . . . . . . . . . . 13 |- ( ( s e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) /\ t e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) ) -> ( s ( +g ` (ℂfld ↾s ( 0 [,) +oo ) ) ) t ) = ( s ( +g ` ( RR*s ↾s ( 0 [,) +oo ) ) ) t ) )
227	226	adantl 467	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ -. +oo e. ran F ) /\ x e. ( ~P X i^i Fin ) ) /\ ( s e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) /\ t e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) ) ) -> ( s ( +g ` (ℂfld ↾s ( 0 [,) +oo ) ) ) t ) = ( s ( +g ` ( RR*s ↾s ( 0 [,) +oo ) ) ) t ) )
228		funmpt 5634	. . . . . . . . . . . . 13 |- Fun ( y e. x |-> ( F ` y ) )
229	228	a1i 11	. . . . . . . . . . . 12 |- ( ( ( ph /\ -. +oo e. ran F ) /\ x e. ( ~P X i^i Fin ) ) -> Fun ( y e. x |-> ( F ` y ) ) )
230	154, 183	syl6eleq 2520	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ -. +oo e. ran F ) /\ x e. ( ~P X i^i Fin ) ) /\ y e. x ) -> ( F ` y ) e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) )
231	230	ralrimiva 2839	. . . . . . . . . . . . 13 |- ( ( ( ph /\ -. +oo e. ran F ) /\ x e. ( ~P X i^i Fin ) ) -> A. y e. x ( F ` y ) e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) )
232	155	rnmptss 6064	. . . . . . . . . . . . 13 |- ( A. y e. x ( F ` y ) e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) -> ran ( y e. x |-> ( F ` y ) ) C_ ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) )
233	231, 232	syl 17	. . . . . . . . . . . 12 |- ( ( ( ph /\ -. +oo e. ran F ) /\ x e. ( ~P X i^i Fin ) ) -> ran ( y e. x |-> ( F ` y ) ) C_ ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) )
234	173, 175, 177, 190, 199, 227, 229, 233	gsumpropd2 16505	. . . . . . . . . . 11 |- ( ( ( ph /\ -. +oo e. ran F ) /\ x e. ( ~P X i^i Fin ) ) -> ( (ℂfld ↾s ( 0 [,) +oo ) ) gsumg ( y e. x |-> ( F ` y ) ) ) = ( ( RR*s ↾s ( 0 [,) +oo ) ) gsumg ( y e. x |-> ( F ` y ) ) ) )
235	169, 170, 234	3eqtrd 2467	. . . . . . . . . 10 |- ( ( ( ph /\ -. +oo e. ran F ) /\ x e. ( ~P X i^i Fin ) ) -> (ℂfld gsumg ( y e. x |-> ( F ` y ) ) ) = ( ( RR*s ↾s ( 0 [,) +oo ) ) gsumg ( y e. x |-> ( F ` y ) ) ) )
236	31	adantl 467	. . . . . . . . . . 11 |- ( ( ( ph /\ -. +oo e. ran F ) /\ x e. ( ~P X i^i Fin ) ) -> x e. Fin )
237	152	recnd 9670	. . . . . . . . . . 11 |- ( ( ( ( ph /\ -. +oo e. ran F ) /\ x e. ( ~P X i^i Fin ) ) /\ y e. x ) -> ( F ` y ) e. CC )
238	236, 237	gsumfsum 19022	. . . . . . . . . 10 |- ( ( ( ph /\ -. +oo e. ran F ) /\ x e. ( ~P X i^i Fin ) ) -> (ℂfld gsumg ( y e. x |-> ( F ` y ) ) ) = sum_ y e. x ( F ` y ) )
239	235, 238	eqtr3d 2465	. . . . . . . . 9 |- ( ( ( ph /\ -. +oo e. ran F ) /\ x e. ( ~P X i^i Fin ) ) -> ( ( RR*s ↾s ( 0 [,) +oo ) ) gsumg ( y e. x |-> ( F ` y ) ) ) = sum_ y e. x ( F ` y ) )
240	103, 165, 239	3eqtrrd 2468	. . . . . . . 8 |- ( ( ( ph /\ -. +oo e. ran F ) /\ x e. ( ~P X i^i Fin ) ) -> sum_ y e. x ( F ` y ) = ( G gsumg ( F |` x ) ) )
241	240	mpteq2dva 4507	. . . . . . 7 |- ( ( ph /\ -. +oo e. ran F ) -> ( x e. ( ~P X i^i Fin ) |-> sum_ y e. x ( F ` y ) ) = ( x e. ( ~P X i^i Fin ) |-> ( G gsumg ( F |` x ) ) ) )
242	241	rneqd 5078	. . . . . 6 |- ( ( ph /\ -. +oo e. ran F ) -> ran ( x e. ( ~P X i^i Fin ) |-> sum_ y e. x ( F ` y ) ) = ran ( x e. ( ~P X i^i Fin ) |-> ( G gsumg ( F |` x ) ) ) )
243	242	supeq1d 7963	. . . . 5 |- ( ( ph /\ -. +oo e. ran F ) -> sup ( ran ( x e. ( ~P X i^i Fin ) |-> sum_ y e. x ( F ` y ) ) , RR* , < ) = sup ( ran ( x e. ( ~P X i^i Fin ) |-> ( G gsumg ( F |` x ) ) ) , RR* , < ) )
244	100, 243	eqtrd 2463	. . . 4 |- ( ( ph /\ -. +oo e. ran F ) -> (Σ^ ` F ) = sup ( ran ( x e. ( ~P X i^i Fin ) |-> ( G gsumg ( F |` x ) ) ) , RR* , < ) )
245	95, 244	pm2.61dan 798	. . 3 |- ( ph -> (Σ^ ` F ) = sup ( ran ( x e. ( ~P X i^i Fin ) |-> ( G gsumg ( F |` x ) ) ) , RR* , < ) )
246	22, 9, 11, 1	xrge0tsms 21839	. . 3 |- ( ph -> ( G tsums F ) = { sup ( ran ( x e. ( ~P X i^i Fin ) |-> ( G gsumg ( F |` x ) ) ) , RR* , < ) } )
247	245, 246	eleq12d 2504	. 2 |- ( ph -> ( (Σ^ ` F ) e. ( G tsums F ) <-> sup ( ran ( x e. ( ~P X i^i Fin ) |-> ( G gsumg ( F |` x ) ) ) , RR* , < ) e. { sup ( ran ( x e. ( ~P X i^i Fin ) |-> ( G gsumg ( F |` x ) ) ) , RR* , < ) } ) )
248	8, 247	mpbird 235	1 |- ( ph -> (Σ^ ` F ) e. ( G tsums F ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 187   /\ wa 370   /\ w3a 982   = wceq 1437   e. wcel 1868   =/= wne 2618   A.wral 2775   E.wrex 2776   _Vcvv 3081   i^i cin 3435   C_ wss 3436   ~Pcpw 3979   {csn 3996   class class class wbr 4420   |-> cmpt 4479   dom cdm 4850   ran crn 4851   |` cres 4852   Fun wfun 5592   Fn wfn 5593   -->wf 5594   ` cfv 5598  (class class class)co 6302   Fincfn 7574   supcsup 7957   CCcc 9538   RRcr 9539   0cc0 9540   + caddc 9543   +oocpnf 9673   RR*cxr 9675   < clt 9676   <_ cle 9677   +ecxad 11408   [,)cico 11638   [,]cicc 11639   sum_csu 13740   Basecbs 15109   ↾_s cress 15110   +g cplusg 15178   gsum_g cgsu 15327   RR*scxrs 15386   Mndcmnd 16523  SubMndcsubmnd 16569  CMndccmn 17418  SRingcsrg 17727  ℂ_fldccnfld 18958   tsums ctsu 21127  Σ^{^}csumge0 37992

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4552  ax-pow 4599  ax-pr 4657  ax-un 6594  ax-inf2 8149  ax-cnex 9596  ax-resscn 9597  ax-1cn 9598  ax-icn 9599  ax-addcl 9600  ax-addrcl 9601  ax-mulcl 9602  ax-mulrcl 9603  ax-mulcom 9604  ax-addass 9605  ax-mulass 9606  ax-distr 9607  ax-i2m1 9608  ax-1ne0 9609  ax-1rid 9610  ax-rnegex 9611  ax-rrecex 9612  ax-cnre 9613  ax-pre-lttri 9614  ax-pre-lttrn 9615  ax-pre-ltadd 9616  ax-pre-mulgt0 9617  ax-pre-sup 9618  ax-addf 9619  ax-mulf 9620

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-int 4253  df-iun 4298  df-iin 4299  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4761  df-id 4765  df-po 4771  df-so 4772  df-fr 4809  df-se 4810  df-we 4811  df-xp 4856  df-rel 4857  df-cnv 4858  df-co 4859  df-dm 4860  df-rn 4861  df-res 4862  df-ima 4863  df-pred 5396  df-ord 5442  df-on 5443  df-lim 5444  df-suc 5445  df-iota 5562  df-fun 5600  df-fn 5601  df-f 5602  df-f1 5603  df-fo 5604  df-f1o 5605  df-fv 5606  df-isom 5607  df-riota 6264  df-ov 6305  df-oprab 6306  df-mpt2 6307  df-of 6542  df-om 6704  df-1st 6804  df-2nd 6805  df-supp 6923  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-oadd 7191  df-er 7368  df-map 7479  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-fsupp 7887  df-fi 7928  df-sup 7959  df-inf 7960  df-oi 8028  df-card 8375  df-pnf 9678  df-mnf 9679  df-xr 9680  df-ltxr 9681  df-le 9682  df-sub 9863  df-neg 9864  df-div 10271  df-nn 10611  df-2 10669  df-3 10670  df-4 10671  df-5 10672  df-6 10673  df-7 10674  df-8 10675  df-9 10676  df-10 10677  df-n0 10871  df-z 10939  df-dec 11053  df-uz 11161  df-q 11266  df-rp 11304  df-xadd 11411  df-ioo 11640  df-ioc 11641  df-ico 11642  df-icc 11643  df-fz 11786  df-fzo 11917  df-seq 12214  df-exp 12273  df-hash 12516  df-cj 13151  df-re 13152  df-im 13153  df-sqrt 13287  df-abs 13288  df-clim 13540  df-sum 13741  df-struct 15111  df-ndx 15112  df-slot 15113  df-base 15114  df-sets 15115  df-ress 15116  df-plusg 15191  df-mulr 15192  df-starv 15193  df-tset 15197  df-ple 15198  df-ds 15200  df-unif 15201  df-rest 15309  df-topn 15310  df-0g 15328  df-gsum 15329  df-topgen 15330  df-ordt 15387  df-xrs 15388  df-mre 15480  df-mrc 15481  df-acs 15483  df-ps 16434  df-tsr 16435  df-mgm 16476  df-sgrp 16515  df-mnd 16525  df-submnd 16571  df-grp 16661  df-minusg 16662  df-mulg 16664  df-cntz 16959  df-cmn 17420  df-abl 17421  df-mgp 17712  df-ur 17724  df-srg 17728  df-ring 17770  df-cring 17771  df-fbas 18955  df-fg 18956  df-cnfld 18959  df-top 19908  df-bases 19909  df-topon 19910  df-topsp 19911  df-ntr 20022  df-nei 20101  df-cn 20230  df-haus 20318  df-fil 20848  df-fm 20940  df-flim 20941  df-flf 20942  df-tsms 21128  df-sumge0 37993

This theorem is referenced by: (None)