Theorem sge0tsms 38222

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 2451	. . . 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 11440	. . . . . 6 |- < Or RR*
4	3	supex 7977	. . . . 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 3991	. . . 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 236	. 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 467	. . . . . 6 |- ( ( ph /\ +oo e. ran F ) -> X e. V )
11		sge0tsms.f	. . . . . . 7 |- ( ph -> F : X --> ( 0 [,] +oo ) )
12	11	adantr 467	. . . . . 6 |- ( ( ph /\ +oo e. ran F ) -> F : X --> ( 0 [,] +oo ) )
13		simpr 463	. . . . . 6 |- ( ( ph /\ +oo e. ran F ) -> +oo e. ran F )
14	10, 12, 13	sge0pnfval 38215	. . . . 5 |- ( ( ph /\ +oo e. ran F ) -> (Σ^ ` F ) = +oo )
15		ffn 5728	. . . . . . . . . 10 |- ( F : X --> ( 0 [,] +oo ) -> F Fn X )
16	11, 15	syl 17	. . . . . . . . 9 |- ( ph -> F Fn X )
17	16	adantr 467	. . . . . . . 8 |- ( ( ph /\ +oo e. ran F ) -> F Fn X )
18		fvelrnb 5912	. . . . . . . 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 214	. . . . . 6 |- ( ( ph /\ +oo e. ran F ) -> E. y e. X ( F ` y ) = +oo )
21		iccssxr 11717	. . . . . . . . . . . . . 14 |- ( 0 [,] +oo ) C_ RR*
22		sge0tsms.g	. . . . . . . . . . . . . . 15 |- G = ( RR*s ↾s ( 0 [,] +oo ) )
23		simpr 463	. . . . . . . . . . . . . . 15 |- ( ( ph /\ x e. ( ~P X i^i Fin ) ) -> x e. ( ~P X i^i Fin ) )
24	11	adantr 467	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ x e. ( ~P X i^i Fin ) ) -> F : X --> ( 0 [,] +oo ) )
25		elinel1 3619	. . . . . . . . . . . . . . . . . 18 |- ( x e. ( ~P X i^i Fin ) -> x e. ~P X )
26		elpwi 3960	. . . . . . . . . . . . . . . . . 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 468	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ x e. ( ~P X i^i Fin ) ) -> x C_ X )
29		fssres 5749	. . . . . . . . . . . . . . . 16 |- ( ( F : X --> ( 0 [,] +oo ) /\ x C_ X ) -> ( F |` x ) : x --> ( 0 [,] +oo ) )
30	24, 28, 29	syl2anc 667	. . . . . . . . . . . . . . 15 |- ( ( ph /\ x e. ( ~P X i^i Fin ) ) -> ( F |` x ) : x --> ( 0 [,] +oo ) )
31		elinel2 3620	. . . . . . . . . . . . . . . . 17 |- ( x e. ( ~P X i^i Fin ) -> x e. Fin )
32	31	adantl 468	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ x e. ( ~P X i^i Fin ) ) -> x e. Fin )
33		0red 9644	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ x e. ( ~P X i^i Fin ) ) -> 0 e. RR )
34	30, 32, 33	fdmfifsupp 7893	. . . . . . . . . . . . . . 15 |- ( ( ph /\ x e. ( ~P X i^i Fin ) ) -> ( F |` x ) finSupp 0 )
35	22, 23, 30, 34	gsumge0cl 38213	. . . . . . . . . . . . . 14 |- ( ( ph /\ x e. ( ~P X i^i Fin ) ) -> ( G gsumg ( F |` x ) ) e. ( 0 [,] +oo ) )
36	21, 35	sseldi 3430	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. ( ~P X i^i Fin ) ) -> ( G gsumg ( F |` x ) ) e. RR* )
37	36	ralrimiva 2802	. . . . . . . . . . . 12 |- ( ph -> A. x e. ( ~P X i^i Fin ) ( G gsumg ( F |` x ) ) e. RR* )
38	37	3ad2ant1 1029	. . . . . . . . . . 11 |- ( ( ph /\ y e. X /\ ( F ` y ) = +oo ) -> A. x e. ( ~P X i^i Fin ) ( G gsumg ( F |` x ) ) e. RR* )
39		eqid 2451	. . . . . . . . . . . 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 6052	. . . . . . . . . . 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 4645	. . . . . . . . . . . . . 14 |- ( y e. X -> { y } e. ~P X )
43		snfi 7650	. . . . . . . . . . . . . . 15 |- { y } e. Fin
44	43	a1i 11	. . . . . . . . . . . . . 14 |- ( y e. X -> { y } e. Fin )
45	42, 44	elind 3618	. . . . . . . . . . . . 13 |- ( y e. X -> { y } e. ( ~P X i^i Fin ) )
46	45	3ad2ant2 1030	. . . . . . . . . . . 12 |- ( ( ph /\ y e. X /\ ( F ` y ) = +oo ) -> { y } e. ( ~P X i^i Fin ) )
47	11	adantr 467	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ y e. X ) -> F : X --> ( 0 [,] +oo ) )
48		snssi 4116	. . . . . . . . . . . . . . . . . . 19 |- ( y e. X -> { y } C_ X )
49	48	adantl 468	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ y e. X ) -> { y } C_ X )
50	47, 49	fssresd 5750	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ y e. X ) -> ( F |` { y } ) : { y } --> ( 0 [,] +oo ) )
51	50	feqmptd 5918	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ y e. X ) -> ( F |` { y } ) = ( x e. { y } |-> ( ( F |` { y } ) ` x ) ) )
52		fvres 5879	. . . . . . . . . . . . . . . . . 18 |- ( x e. { y } -> ( ( F |` { y } ) ` x ) = ( F ` x ) )
53	52	mpteq2ia 4485	. . . . . . . . . . . . . . . . 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 2485	. . . . . . . . . . . . . . 15 |- ( ( ph /\ y e. X ) -> ( F |` { y } ) = ( x e. { y } |-> ( F ` x ) ) )
56	55	oveq2d 6306	. . . . . . . . . . . . . 14 |- ( ( ph /\ y e. X ) -> ( G gsumg ( F |` { y } ) ) = ( G gsumg ( x e. { y } |-> ( F ` x ) ) ) )
57	56	3adant3 1028	. . . . . . . . . . . . 13 |- ( ( ph /\ y e. X /\ ( F ` y ) = +oo ) -> ( G gsumg ( F |` { y } ) ) = ( G gsumg ( x e. { y } |-> ( F ` x ) ) ) )
58		xrge0cmn 19010	. . . . . . . . . . . . . . . . 17 |- ( RR*s ↾s ( 0 [,] +oo ) ) e. CMnd
59	22, 58	eqeltri 2525	. . . . . . . . . . . . . . . 16 |- G e. CMnd
60		cmnmnd 17445	. . . . . . . . . . . . . . . 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 1009	. . . . . . . . . . . . . 14 |- ( ( ph /\ y e. X /\ ( F ` y ) = +oo ) -> y e. X )
64	11	ffvelrnda 6022	. . . . . . . . . . . . . . 15 |- ( ( ph /\ y e. X ) -> ( F ` y ) e. ( 0 [,] +oo ) )
65	64	3adant3 1028	. . . . . . . . . . . . . 14 |- ( ( ph /\ y e. X /\ ( F ` y ) = +oo ) -> ( F ` y ) e. ( 0 [,] +oo ) )
66		df-ss 3418	. . . . . . . . . . . . . . . . . 18 |- ( ( 0 [,] +oo ) C_ RR* <-> ( ( 0 [,] +oo ) i^i RR* ) = ( 0 [,] +oo ) )
67	21, 66	mpbi 212	. . . . . . . . . . . . . . . . 17 |- ( ( 0 [,] +oo ) i^i RR* ) = ( 0 [,] +oo )
68	67	eqcomi 2460	. . . . . . . . . . . . . . . 16 |- ( 0 [,] +oo ) = ( ( 0 [,] +oo ) i^i RR* )
69		ovex 6318	. . . . . . . . . . . . . . . . 17 |- ( 0 [,] +oo ) e. _V
70		xrsbas 18984	. . . . . . . . . . . . . . . . . 18 |- RR* = ( Base ` RR*s )
71	22, 70	ressbas 15179	. . . . . . . . . . . . . . . . 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 2473	. . . . . . . . . . . . . . 15 |- ( 0 [,] +oo ) = ( Base ` G )
74		fveq2 5865	. . . . . . . . . . . . . . 15 |- ( x = y -> ( F ` x ) = ( F ` y ) )
75	73, 74	gsumsn 17587	. . . . . . . . . . . . . 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 1268	. . . . . . . . . . . . 13 |- ( ( ph /\ y e. X /\ ( F ` y ) = +oo ) -> ( G gsumg ( x e. { y } |-> ( F ` x ) ) ) = ( F ` y ) )
77		simp3 1010	. . . . . . . . . . . . 13 |- ( ( ph /\ y e. X /\ ( F ` y ) = +oo ) -> ( F ` y ) = +oo )
78	57, 76, 77	3eqtrrd 2490	. . . . . . . . . . . 12 |- ( ( ph /\ y e. X /\ ( F ` y ) = +oo ) -> +oo = ( G gsumg ( F |` { y } ) ) )
79		reseq2 5100	. . . . . . . . . . . . . . 15 |- ( x = { y } -> ( F |` x ) = ( F |` { y } ) )
80	79	oveq2d 6306	. . . . . . . . . . . . . 14 |- ( x = { y } -> ( G gsumg ( F |` x ) ) = ( G gsumg ( F |` { y } ) ) )
81	80	eqeq2d 2461	. . . . . . . . . . . . 13 |- ( x = { y } -> ( +oo = ( G gsumg ( F |` x ) ) <-> +oo = ( G gsumg ( F |` { y } ) ) ) )
82	81	rspcev 3150	. . . . . . . . . . . 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 667	. . . . . . . . . . 11 |- ( ( ph /\ y e. X /\ ( F ` y ) = +oo ) -> E. x e. ( ~P X i^i Fin ) +oo = ( G gsumg ( F |` x ) ) )
84		pnfxr 11412	. . . . . . . . . . . . 13 |- +oo e. RR*
85	84	a1i 11	. . . . . . . . . . . 12 |- ( ( ph /\ y e. X /\ ( F ` y ) = +oo ) -> +oo e. RR* )
86	39	elrnmpt 5081	. . . . . . . . . . . 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 236	. . . . . . . . . 10 |- ( ( ph /\ y e. X /\ ( F ` y ) = +oo ) -> +oo e. ran ( x e. ( ~P X i^i Fin ) |-> ( G gsumg ( F |` x ) ) ) )
89		supxrpnf 11604	. . . . . . . . . 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 667	. . . . . . . . 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 1207	. . . . . . . 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 467	. . . . . . 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 2877	. . . . . 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 2488	. . . 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 467	. . . . . 6 |- ( ( ph /\ -. +oo e. ran F ) -> X e. V )
97	11	adantr 467	. . . . . . 7 |- ( ( ph /\ -. +oo e. ran F ) -> F : X --> ( 0 [,] +oo ) )
98		simpr 463	. . . . . . 7 |- ( ( ph /\ -. +oo e. ran F ) -> -. +oo e. ran F )
99	97, 98	fge0iccico 38212	. . . . . 6 |- ( ( ph /\ -. +oo e. ran F ) -> F : X --> ( 0 [,) +oo ) )
100	96, 99	sge0reval 38214	. . . . 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 5919	. . . . . . . . . . 11 |- ( ( ph /\ x e. ( ~P X i^i Fin ) ) -> ( F |` x ) = ( y e. x |-> ( F ` y ) ) )
102	101	adantlr 721	. . . . . . . . . 10 |- ( ( ( ph /\ -. +oo e. ran F ) /\ x e. ( ~P X i^i Fin ) ) -> ( F |` x ) = ( y e. x |-> ( F ` y ) ) )
103	102	oveq2d 6306	. . . . . . . . 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 5868	. . . . . . . . . . 11 |- ( +g ` G ) = ( +g ` ( RR*s ↾s ( 0 [,] +oo ) ) )
105		eqid 2451	. . . . . . . . . . . . . 14 |- ( RR*s ↾s ( 0 [,] +oo ) ) = ( RR*s ↾s ( 0 [,] +oo ) )
106		xrsadd 18985	. . . . . . . . . . . . . 14 |- +e = ( +g ` RR*s )
107	105, 106	ressplusg 15239	. . . . . . . . . . . . 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 2460	. . . . . . . . . . 11 |- ( +g ` ( RR*s ↾s ( 0 [,] +oo ) ) ) = +e
110	104, 109	eqtr2i 2474	. . . . . . . . . 10 |- +e = ( +g ` G )
111	22	oveq1i 6300	. . . . . . . . . . 11 |- ( G ↾s ( 0 [,) +oo ) ) = ( ( RR*s ↾s ( 0 [,] +oo ) ) ↾s ( 0 [,) +oo ) )
112		icossicc 11721	. . . . . . . . . . . . 13 |- ( 0 [,) +oo ) C_ ( 0 [,] +oo )
113	69, 112	pm3.2i 457	. . . . . . . . . . . 12 |- ( ( 0 [,] +oo ) e. _V /\ ( 0 [,) +oo ) C_ ( 0 [,] +oo ) )
114		ressabs 15188	. . . . . . . . . . . 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 2474	. . . . . . . . . 10 |- ( RR*s ↾s ( 0 [,) +oo ) ) = ( G ↾s ( 0 [,) +oo ) )
117	59	elexi 3055	. . . . . . . . . . 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 463	. . . . . . . . . 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 9687	. . . . . . . . . . . . 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 732	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ -. +oo e. ran F ) /\ x e. ( ~P X i^i Fin ) ) /\ y e. x ) -> F : X --> ( 0 [,] +oo ) )
125	27	sselda 3432	. . . . . . . . . . . . . . 15 |- ( ( x e. ( ~P X i^i Fin ) /\ y e. x ) -> y e. X )
126	125	adantll 720	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ -. +oo e. ran F ) /\ x e. ( ~P X i^i Fin ) ) /\ y e. x ) -> y e. X )
127	124, 126	ffvelrnd 6023	. . . . . . . . . . . . 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 3430	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ -. +oo e. ran F ) /\ x e. ( ~P X i^i Fin ) ) /\ y e. x ) -> ( F ` y ) e. RR* )
129		iccgelb 11691	. . . . . . . . . . . . 13 |- ( ( 0 e. RR* /\ +oo e. RR* /\ ( F ` y ) e. ( 0 [,] +oo ) ) -> 0 <_ ( F ` y ) )
130	122, 123, 127, 129	syl3anc 1268	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ -. +oo e. ran F ) /\ x e. ( ~P X i^i Fin ) ) /\ y e. x ) -> 0 <_ ( F ` y ) )
131		id 22	. . . . . . . . . . . . . . . . . . . 20 |- ( ( F ` y ) = +oo -> ( F ` y ) = +oo )
132	131	eqcomd 2457	. . . . . . . . . . . . . . . . . . 19 |- ( ( F ` y ) = +oo -> +oo = ( F ` y ) )
133	132	adantl 468	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ph /\ x e. ( ~P X i^i Fin ) ) /\ y e. x ) /\ ( F ` y ) = +oo ) -> +oo = ( F ` y ) )
134		ffun 5731	. . . . . . . . . . . . . . . . . . . . . 22 |- ( F : X --> ( 0 [,] +oo ) -> Fun F )
135	11, 134	syl 17	. . . . . . . . . . . . . . . . . . . . 21 |- ( ph -> Fun F )
136	135	ad2antrr 732	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ph /\ x e. ( ~P X i^i Fin ) ) /\ y e. x ) -> Fun F )
137	23, 125	sylan 474	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ph /\ x e. ( ~P X i^i Fin ) ) /\ y e. x ) -> y e. X )
138		fdm 5733	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( F : X --> ( 0 [,] +oo ) -> dom F = X )
139	11, 138	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ph -> dom F = X )
140	139	eqcomd 2457	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ph -> X = dom F )
141	140	ad2antrr 732	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ph /\ x e. ( ~P X i^i Fin ) ) /\ y e. x ) -> X = dom F )
142	137, 141	eleqtrd 2531	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ph /\ x e. ( ~P X i^i Fin ) ) /\ y e. x ) -> y e. dom F )
143		fvelrn 6015	. . . . . . . . . . . . . . . . . . . 20 |- ( ( Fun F /\ y e. dom F ) -> ( F ` y ) e. ran F )
144	136, 142, 143	syl2anc 667	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ph /\ x e. ( ~P X i^i Fin ) ) /\ y e. x ) -> ( F ` y ) e. ran F )
145	144	adantr 467	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ph /\ x e. ( ~P X i^i Fin ) ) /\ y e. x ) /\ ( F ` y ) = +oo ) -> ( F ` y ) e. ran F )
146	133, 145	eqeltrd 2529	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ph /\ x e. ( ~P X i^i Fin ) ) /\ y e. x ) /\ ( F ` y ) = +oo ) -> +oo e. ran F )
147	146	adantlllr 37361	. . . . . . . . . . . . . . . 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 736	. . . . . . . . . . . . . . . 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 580	. . . . . . . . . . . . . . 15 |- ( ( ( ( ph /\ -. +oo e. ran F ) /\ x e. ( ~P X i^i Fin ) ) /\ y e. x ) -> -. ( F ` y ) = +oo )
150	149	neqned 2631	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ -. +oo e. ran F ) /\ x e. ( ~P X i^i Fin ) ) /\ y e. x ) -> ( F ` y ) =/= +oo )
151		ge0xrre 37633	. . . . . . . . . . . . . 14 |- ( ( ( F ` y ) e. ( 0 [,] +oo ) /\ ( F ` y ) =/= +oo ) -> ( F ` y ) e. RR )
152	127, 150, 151	syl2anc 667	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ -. +oo e. ran F ) /\ x e. ( ~P X i^i Fin ) ) /\ y e. x ) -> ( F ` y ) e. RR )
153	152	ltpnfd 11423	. . . . . . . . . . . 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 11685	. . . . . . . . . . 11 |- ( ( ( ( ph /\ -. +oo e. ran F ) /\ x e. ( ~P X i^i Fin ) ) /\ y e. x ) -> ( F ` y ) e. ( 0 [,) +oo ) )
155		eqid 2451	. . . . . . . . . . 11 |- ( y e. x |-> ( F ` y ) ) = ( y e. x |-> ( F ` y ) )
156	154, 155	fmptd 6046	. . . . . . . . . 10 |- ( ( ( ph /\ -. +oo e. ran F ) /\ x e. ( ~P X i^i Fin ) ) -> ( y e. x |-> ( F ` y ) ) : x --> ( 0 [,) +oo ) )
157		0e0icopnf 11742	. . . . . . . . . . 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 3428	. . . . . . . . . . . 12 |- ( y e. ( 0 [,] +oo ) -> y e. RR* )
160		xaddid2 11533	. . . . . . . . . . . . 13 |- ( y e. RR* -> ( 0 +e y ) = y )
161		xaddid1 11532	. . . . . . . . . . . . 13 |- ( y e. RR* -> ( y +e 0 ) = y )
162	160, 161	jca 535	. . . . . . . . . . . 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 468	. . . . . . . . . 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 16519	. . . . . . . . 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 19024	. . . . . . . . . . . . 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 2451	. . . . . . . . . . . 12 |- (ℂfld ↾s ( 0 [,) +oo ) ) = (ℂfld ↾s ( 0 [,) +oo ) )
169	119, 167, 156, 168	gsumsubm 16620	. . . . . . . . . . 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 2452	. . . . . . . . . . 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 3048	. . . . . . . . . . . . . 14 |- x e. _V
172	171	mptex 6136	. . . . . . . . . . . . 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 6318	. . . . . . . . . . . . 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 6318	. . . . . . . . . . . . 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 11740	. . . . . . . . . . . . . . . . 17 |- ( 0 [,) +oo ) C_ RR
179		ax-resscn 9596	. . . . . . . . . . . . . . . . 17 |- RR C_ CC
180	178, 179	sstri 3441	. . . . . . . . . . . . . . . 16 |- ( 0 [,) +oo ) C_ CC
181		cnfldbas 18974	. . . . . . . . . . . . . . . . 17 |- CC = ( Base ` ℂfld )
182	168, 181	ressbas2 15180	. . . . . . . . . . . . . . . 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 2460	. . . . . . . . . . . . . 14 |- ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) = ( 0 [,) +oo )
185	112, 21	sstri 3441	. . . . . . . . . . . . . . 15 |- ( 0 [,) +oo ) C_ RR*
186		eqid 2451	. . . . . . . . . . . . . . . 16 |- ( RR*s ↾s ( 0 [,) +oo ) ) = ( RR*s ↾s ( 0 [,) +oo ) )
187	186, 70	ressbas2 15180	. . . . . . . . . . . . . . 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 2473	. . . . . . . . . . . . 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 19038	. . . . . . . . . . . . . . 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 459	. . . . . . . . . . . . . 14 |- ( ( s e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) /\ t e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) ) -> s e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) )
194		simpr 463	. . . . . . . . . . . . . 14 |- ( ( s e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) /\ t e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) ) -> t e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) )
195		eqid 2451	. . . . . . . . . . . . . . 15 |- ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) = ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) )
196		eqid 2451	. . . . . . . . . . . . . . 15 |- ( +g ` (ℂfld ↾s ( 0 [,) +oo ) ) ) = ( +g ` (ℂfld ↾s ( 0 [,) +oo ) ) )
197	195, 196	srgacl 17757	. . . . . . . . . . . . . 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 1268	. . . . . . . . . . . . 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 468	. . . . . . . . . . . 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 22	. . . . . . . . . . . . . . . . 17 |- ( s e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) -> s e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) )
202	201, 184	syl6eleq 2539	. . . . . . . . . . . . . . . 16 |- ( s e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) -> s e. ( 0 [,) +oo ) )
203	200, 202	sseldd 3433	. . . . . . . . . . . . . . 15 |- ( s e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) -> s e. RR )
204	203	adantr 467	. . . . . . . . . . . . . 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 22	. . . . . . . . . . . . . . . . 17 |- ( t e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) -> t e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) )
207	206, 184	syl6eleq 2539	. . . . . . . . . . . . . . . 16 |- ( t e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) -> t e. ( 0 [,) +oo ) )
208	205, 207	sseldd 3433	. . . . . . . . . . . . . . 15 |- ( t e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) -> t e. RR )
209	208	adantl 468	. . . . . . . . . . . . . 14 |- ( ( s e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) /\ t e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) ) -> t e. RR )
210		rexadd 11525	. . . . . . . . . . . . . . . 16 |- ( ( s e. RR /\ t e. RR ) -> ( s +e t ) = ( s + t ) )
211	210	eqcomd 2457	. . . . . . . . . . . . . . 15 |- ( ( s e. RR /\ t e. RR ) -> ( s + t ) = ( s +e t ) )
212	166	elexi 3055	. . . . . . . . . . . . . . . . . . . 20 |- ( 0 [,) +oo ) e. _V
213		cnfldadd 18975	. . . . . . . . . . . . . . . . . . . . 21 |- + = ( +g ` ℂfld )
214	168, 213	ressplusg 15239	. . . . . . . . . . . . . . . . . . . 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 2475	. . . . . . . . . . . . . . . . . 18 |- ( +g ` (ℂfld ↾s ( 0 [,) +oo ) ) ) = ( +g ` ℂfld )
217	216, 213	eqtr4i 2476	. . . . . . . . . . . . . . . . 17 |- ( +g ` (ℂfld ↾s ( 0 [,) +oo ) ) ) = +
218	217	oveqi 6303	. . . . . . . . . . . . . . . 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 15239	. . . . . . . . . . . . . . . . . . 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 2460	. . . . . . . . . . . . . . . . 17 |- ( +g ` ( RR*s ↾s ( 0 [,) +oo ) ) ) = +e
223	222	oveqi 6303	. . . . . . . . . . . . . . . 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 2495	. . . . . . . . . . . . . 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 667	. . . . . . . . . . . . 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 468	. . . . . . . . . . . 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 5618	. . . . . . . . . . . . 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 2539	. . . . . . . . . . . . . 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 2802	. . . . . . . . . . . . 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 6052	. . . . . . . . . . . . 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 16517	. . . . . . . . . . 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 2489	. . . . . . . . . 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 468	. . . . . . . . . . 11 |- ( ( ( ph /\ -. +oo e. ran F ) /\ x e. ( ~P X i^i Fin ) ) -> x e. Fin )
237	152	recnd 9669	. . . . . . . . . . 11 |- ( ( ( ( ph /\ -. +oo e. ran F ) /\ x e. ( ~P X i^i Fin ) ) /\ y e. x ) -> ( F ` y ) e. CC )
238	236, 237	gsumfsum 19034	. . . . . . . . . 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 2487	. . . . . . . . 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 2490	. . . . . . . 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 4489	. . . . . . 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 5062	. . . . . 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 7960	. . . . 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 2485	. . . 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 800	. . 3 |- ( ph -> (Σ^ ` F ) = sup ( ran ( x e. ( ~P X i^i Fin ) |-> ( G gsumg ( F |` x ) ) ) , RR* , < ) )
246	22, 9, 11, 1	xrge0tsms 21852	. . 3 |- ( ph -> ( G tsums F ) = { sup ( ran ( x e. ( ~P X i^i Fin ) |-> ( G gsumg ( F |` x ) ) ) , RR* , < ) } )
247	245, 246	eleq12d 2523	. 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 236	1 |- ( ph -> (Σ^ ` F ) e. ( G tsums F ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 188   /\ wa 371   /\ w3a 985   = wceq 1444   e. wcel 1887   =/= wne 2622   A.wral 2737   E.wrex 2738   _Vcvv 3045   i^i cin 3403   C_ wss 3404   ~Pcpw 3951   {csn 3968   class class class wbr 4402   |-> cmpt 4461   dom cdm 4834   ran crn 4835   |` cres 4836   Fun wfun 5576   Fn wfn 5577   -->wf 5578   ` cfv 5582  (class class class)co 6290   Fincfn 7569   supcsup 7954   CCcc 9537   RRcr 9538   0cc0 9539   + caddc 9542   +oocpnf 9672   RR*cxr 9674   < clt 9675   <_ cle 9676   +ecxad 11407   [,)cico 11637   [,]cicc 11638   sum_csu 13752   Basecbs 15121   ↾_s cress 15122   +g cplusg 15190   gsum_g cgsu 15339   RR*scxrs 15398   Mndcmnd 16535  SubMndcsubmnd 16581  CMndccmn 17430  SRingcsrg 17739  ℂ_fldccnfld 18970   tsums ctsu 21140  Σ^{^}csumge0 38204

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-inf2 8146  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617  ax-addf 9618  ax-mulf 9619

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-iin 4281  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-of 6531  df-om 6693  df-1st 6793  df-2nd 6794  df-supp 6915  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-oadd 7186  df-er 7363  df-map 7474  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-fsupp 7884  df-fi 7925  df-sup 7956  df-inf 7957  df-oi 8025  df-card 8373  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-q 11265  df-rp 11303  df-xadd 11410  df-ioo 11639  df-ioc 11640  df-ico 11641  df-icc 11642  df-fz 11785  df-fzo 11916  df-seq 12214  df-exp 12273  df-hash 12516  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-clim 13552  df-sum 13753  df-struct 15123  df-ndx 15124  df-slot 15125  df-base 15126  df-sets 15127  df-ress 15128  df-plusg 15203  df-mulr 15204  df-starv 15205  df-tset 15209  df-ple 15210  df-ds 15212  df-unif 15213  df-rest 15321  df-topn 15322  df-0g 15340  df-gsum 15341  df-topgen 15342  df-ordt 15399  df-xrs 15400  df-mre 15492  df-mrc 15493  df-acs 15495  df-ps 16446  df-tsr 16447  df-mgm 16488  df-sgrp 16527  df-mnd 16537  df-submnd 16583  df-grp 16673  df-minusg 16674  df-mulg 16676  df-cntz 16971  df-cmn 17432  df-abl 17433  df-mgp 17724  df-ur 17736  df-srg 17740  df-ring 17782  df-cring 17783  df-fbas 18967  df-fg 18968  df-cnfld 18971  df-top 19921  df-bases 19922  df-topon 19923  df-topsp 19924  df-ntr 20035  df-nei 20114  df-cn 20243  df-haus 20331  df-fil 20861  df-fm 20953  df-flim 20954  df-flf 20955  df-tsms 21141  df-sumge0 38205

This theorem is referenced by: (None)