Theorem sge0tsms 38336

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 2471	. . . 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 11463	. . . . . 6 |- < Or RR*
4	3	supex 7995	. . . . 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 3983	. . . 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 240	. 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 472	. . . . . 6 |- ( ( ph /\ +oo e. ran F ) -> X e. V )
11		sge0tsms.f	. . . . . . 7 |- ( ph -> F : X --> ( 0 [,] +oo ) )
12	11	adantr 472	. . . . . 6 |- ( ( ph /\ +oo e. ran F ) -> F : X --> ( 0 [,] +oo ) )
13		simpr 468	. . . . . 6 |- ( ( ph /\ +oo e. ran F ) -> +oo e. ran F )
14	10, 12, 13	sge0pnfval 38329	. . . . 5 |- ( ( ph /\ +oo e. ran F ) -> (Σ^ ` F ) = +oo )
15		ffn 5739	. . . . . . . . . 10 |- ( F : X --> ( 0 [,] +oo ) -> F Fn X )
16	11, 15	syl 17	. . . . . . . . 9 |- ( ph -> F Fn X )
17	16	adantr 472	. . . . . . . 8 |- ( ( ph /\ +oo e. ran F ) -> F Fn X )
18		fvelrnb 5926	. . . . . . . 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 215	. . . . . 6 |- ( ( ph /\ +oo e. ran F ) -> E. y e. X ( F ` y ) = +oo )
21		iccssxr 11742	. . . . . . . . . . . . . 14 |- ( 0 [,] +oo ) C_ RR*
22		sge0tsms.g	. . . . . . . . . . . . . . 15 |- G = ( RR*s ↾s ( 0 [,] +oo ) )
23		simpr 468	. . . . . . . . . . . . . . 15 |- ( ( ph /\ x e. ( ~P X i^i Fin ) ) -> x e. ( ~P X i^i Fin ) )
24	11	adantr 472	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ x e. ( ~P X i^i Fin ) ) -> F : X --> ( 0 [,] +oo ) )
25		elinel1 3610	. . . . . . . . . . . . . . . . . 18 |- ( x e. ( ~P X i^i Fin ) -> x e. ~P X )
26		elpwi 3951	. . . . . . . . . . . . . . . . . 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 473	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ x e. ( ~P X i^i Fin ) ) -> x C_ X )
29		fssres 5761	. . . . . . . . . . . . . . . 16 |- ( ( F : X --> ( 0 [,] +oo ) /\ x C_ X ) -> ( F |` x ) : x --> ( 0 [,] +oo ) )
30	24, 28, 29	syl2anc 673	. . . . . . . . . . . . . . 15 |- ( ( ph /\ x e. ( ~P X i^i Fin ) ) -> ( F |` x ) : x --> ( 0 [,] +oo ) )
31		elinel2 3611	. . . . . . . . . . . . . . . . 17 |- ( x e. ( ~P X i^i Fin ) -> x e. Fin )
32	31	adantl 473	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ x e. ( ~P X i^i Fin ) ) -> x e. Fin )
33		0red 9662	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ x e. ( ~P X i^i Fin ) ) -> 0 e. RR )
34	30, 32, 33	fdmfifsupp 7911	. . . . . . . . . . . . . . 15 |- ( ( ph /\ x e. ( ~P X i^i Fin ) ) -> ( F |` x ) finSupp 0 )
35	22, 23, 30, 34	gsumge0cl 38327	. . . . . . . . . . . . . 14 |- ( ( ph /\ x e. ( ~P X i^i Fin ) ) -> ( G gsumg ( F |` x ) ) e. ( 0 [,] +oo ) )
36	21, 35	sseldi 3416	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. ( ~P X i^i Fin ) ) -> ( G gsumg ( F |` x ) ) e. RR* )
37	36	ralrimiva 2809	. . . . . . . . . . . 12 |- ( ph -> A. x e. ( ~P X i^i Fin ) ( G gsumg ( F |` x ) ) e. RR* )
38	37	3ad2ant1 1051	. . . . . . . . . . 11 |- ( ( ph /\ y e. X /\ ( F ` y ) = +oo ) -> A. x e. ( ~P X i^i Fin ) ( G gsumg ( F |` x ) ) e. RR* )
39		eqid 2471	. . . . . . . . . . . 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 6068	. . . . . . . . . . 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 7668	. . . . . . . . . . . . . . 15 |- { y } e. Fin
44	43	a1i 11	. . . . . . . . . . . . . 14 |- ( y e. X -> { y } e. Fin )
45	42, 44	elind 3609	. . . . . . . . . . . . 13 |- ( y e. X -> { y } e. ( ~P X i^i Fin ) )
46	45	3ad2ant2 1052	. . . . . . . . . . . 12 |- ( ( ph /\ y e. X /\ ( F ` y ) = +oo ) -> { y } e. ( ~P X i^i Fin ) )
47	11	adantr 472	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ y e. X ) -> F : X --> ( 0 [,] +oo ) )
48		snssi 4107	. . . . . . . . . . . . . . . . . . 19 |- ( y e. X -> { y } C_ X )
49	48	adantl 473	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ y e. X ) -> { y } C_ X )
50	47, 49	fssresd 5762	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ y e. X ) -> ( F |` { y } ) : { y } --> ( 0 [,] +oo ) )
51	50	feqmptd 5932	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ y e. X ) -> ( F |` { y } ) = ( x e. { y } |-> ( ( F |` { y } ) ` x ) ) )
52		fvres 5893	. . . . . . . . . . . . . . . . . 18 |- ( x e. { y } -> ( ( F |` { y } ) ` x ) = ( F ` x ) )
53	52	mpteq2ia 4478	. . . . . . . . . . . . . . . . 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 2505	. . . . . . . . . . . . . . 15 |- ( ( ph /\ y e. X ) -> ( F |` { y } ) = ( x e. { y } |-> ( F ` x ) ) )
56	55	oveq2d 6324	. . . . . . . . . . . . . 14 |- ( ( ph /\ y e. X ) -> ( G gsumg ( F |` { y } ) ) = ( G gsumg ( x e. { y } |-> ( F ` x ) ) ) )
57	56	3adant3 1050	. . . . . . . . . . . . 13 |- ( ( ph /\ y e. X /\ ( F ` y ) = +oo ) -> ( G gsumg ( F |` { y } ) ) = ( G gsumg ( x e. { y } |-> ( F ` x ) ) ) )
58		xrge0cmn 19087	. . . . . . . . . . . . . . . . 17 |- ( RR*s ↾s ( 0 [,] +oo ) ) e. CMnd
59	22, 58	eqeltri 2545	. . . . . . . . . . . . . . . 16 |- G e. CMnd
60		cmnmnd 17523	. . . . . . . . . . . . . . . 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 1031	. . . . . . . . . . . . . 14 |- ( ( ph /\ y e. X /\ ( F ` y ) = +oo ) -> y e. X )
64	11	ffvelrnda 6037	. . . . . . . . . . . . . . 15 |- ( ( ph /\ y e. X ) -> ( F ` y ) e. ( 0 [,] +oo ) )
65	64	3adant3 1050	. . . . . . . . . . . . . 14 |- ( ( ph /\ y e. X /\ ( F ` y ) = +oo ) -> ( F ` y ) e. ( 0 [,] +oo ) )
66		df-ss 3404	. . . . . . . . . . . . . . . . . 18 |- ( ( 0 [,] +oo ) C_ RR* <-> ( ( 0 [,] +oo ) i^i RR* ) = ( 0 [,] +oo ) )
67	21, 66	mpbi 213	. . . . . . . . . . . . . . . . 17 |- ( ( 0 [,] +oo ) i^i RR* ) = ( 0 [,] +oo )
68	67	eqcomi 2480	. . . . . . . . . . . . . . . 16 |- ( 0 [,] +oo ) = ( ( 0 [,] +oo ) i^i RR* )
69		ovex 6336	. . . . . . . . . . . . . . . . 17 |- ( 0 [,] +oo ) e. _V
70		xrsbas 19061	. . . . . . . . . . . . . . . . . 18 |- RR* = ( Base ` RR*s )
71	22, 70	ressbas 15257	. . . . . . . . . . . . . . . . 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 2493	. . . . . . . . . . . . . . 15 |- ( 0 [,] +oo ) = ( Base ` G )
74		fveq2 5879	. . . . . . . . . . . . . . 15 |- ( x = y -> ( F ` x ) = ( F ` y ) )
75	73, 74	gsumsn 17665	. . . . . . . . . . . . . 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 1292	. . . . . . . . . . . . 13 |- ( ( ph /\ y e. X /\ ( F ` y ) = +oo ) -> ( G gsumg ( x e. { y } |-> ( F ` x ) ) ) = ( F ` y ) )
77		simp3 1032	. . . . . . . . . . . . 13 |- ( ( ph /\ y e. X /\ ( F ` y ) = +oo ) -> ( F ` y ) = +oo )
78	57, 76, 77	3eqtrrd 2510	. . . . . . . . . . . 12 |- ( ( ph /\ y e. X /\ ( F ` y ) = +oo ) -> +oo = ( G gsumg ( F |` { y } ) ) )
79		reseq2 5106	. . . . . . . . . . . . . . 15 |- ( x = { y } -> ( F |` x ) = ( F |` { y } ) )
80	79	oveq2d 6324	. . . . . . . . . . . . . 14 |- ( x = { y } -> ( G gsumg ( F |` x ) ) = ( G gsumg ( F |` { y } ) ) )
81	80	eqeq2d 2481	. . . . . . . . . . . . 13 |- ( x = { y } -> ( +oo = ( G gsumg ( F |` x ) ) <-> +oo = ( G gsumg ( F |` { y } ) ) ) )
82	81	rspcev 3136	. . . . . . . . . . . 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 673	. . . . . . . . . . 11 |- ( ( ph /\ y e. X /\ ( F ` y ) = +oo ) -> E. x e. ( ~P X i^i Fin ) +oo = ( G gsumg ( F |` x ) ) )
84		pnfxr 11435	. . . . . . . . . . . . 13 |- +oo e. RR*
85	84	a1i 11	. . . . . . . . . . . 12 |- ( ( ph /\ y e. X /\ ( F ` y ) = +oo ) -> +oo e. RR* )
86	39	elrnmpt 5087	. . . . . . . . . . . 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 240	. . . . . . . . . 10 |- ( ( ph /\ y e. X /\ ( F ` y ) = +oo ) -> +oo e. ran ( x e. ( ~P X i^i Fin ) |-> ( G gsumg ( F |` x ) ) ) )
89		supxrpnf 11629	. . . . . . . . . 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 673	. . . . . . . . 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 1230	. . . . . . . 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 472	. . . . . . 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 2870	. . . . . 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 2508	. . . 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 472	. . . . . 6 |- ( ( ph /\ -. +oo e. ran F ) -> X e. V )
97	11	adantr 472	. . . . . . 7 |- ( ( ph /\ -. +oo e. ran F ) -> F : X --> ( 0 [,] +oo ) )
98		simpr 468	. . . . . . 7 |- ( ( ph /\ -. +oo e. ran F ) -> -. +oo e. ran F )
99	97, 98	fge0iccico 38326	. . . . . 6 |- ( ( ph /\ -. +oo e. ran F ) -> F : X --> ( 0 [,) +oo ) )
100	96, 99	sge0reval 38328	. . . . 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 5933	. . . . . . . . . . 11 |- ( ( ph /\ x e. ( ~P X i^i Fin ) ) -> ( F |` x ) = ( y e. x |-> ( F ` y ) ) )
102	101	adantlr 729	. . . . . . . . . 10 |- ( ( ( ph /\ -. +oo e. ran F ) /\ x e. ( ~P X i^i Fin ) ) -> ( F |` x ) = ( y e. x |-> ( F ` y ) ) )
103	102	oveq2d 6324	. . . . . . . . 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 5882	. . . . . . . . . . 11 |- ( +g ` G ) = ( +g ` ( RR*s ↾s ( 0 [,] +oo ) ) )
105		eqid 2471	. . . . . . . . . . . . . 14 |- ( RR*s ↾s ( 0 [,] +oo ) ) = ( RR*s ↾s ( 0 [,] +oo ) )
106		xrsadd 19062	. . . . . . . . . . . . . 14 |- +e = ( +g ` RR*s )
107	105, 106	ressplusg 15317	. . . . . . . . . . . . 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 2480	. . . . . . . . . . 11 |- ( +g ` ( RR*s ↾s ( 0 [,] +oo ) ) ) = +e
110	104, 109	eqtr2i 2494	. . . . . . . . . 10 |- +e = ( +g ` G )
111	22	oveq1i 6318	. . . . . . . . . . 11 |- ( G ↾s ( 0 [,) +oo ) ) = ( ( RR*s ↾s ( 0 [,] +oo ) ) ↾s ( 0 [,) +oo ) )
112		icossicc 11746	. . . . . . . . . . . . 13 |- ( 0 [,) +oo ) C_ ( 0 [,] +oo )
113	69, 112	pm3.2i 462	. . . . . . . . . . . 12 |- ( ( 0 [,] +oo ) e. _V /\ ( 0 [,) +oo ) C_ ( 0 [,] +oo ) )
114		ressabs 15266	. . . . . . . . . . . 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 2494	. . . . . . . . . 10 |- ( RR*s ↾s ( 0 [,) +oo ) ) = ( G ↾s ( 0 [,) +oo ) )
117	59	elexi 3041	. . . . . . . . . . 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 468	. . . . . . . . . 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 9705	. . . . . . . . . . . . 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 740	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ -. +oo e. ran F ) /\ x e. ( ~P X i^i Fin ) ) /\ y e. x ) -> F : X --> ( 0 [,] +oo ) )
125	27	sselda 3418	. . . . . . . . . . . . . . 15 |- ( ( x e. ( ~P X i^i Fin ) /\ y e. x ) -> y e. X )
126	125	adantll 728	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ -. +oo e. ran F ) /\ x e. ( ~P X i^i Fin ) ) /\ y e. x ) -> y e. X )
127	124, 126	ffvelrnd 6038	. . . . . . . . . . . . 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 3416	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ -. +oo e. ran F ) /\ x e. ( ~P X i^i Fin ) ) /\ y e. x ) -> ( F ` y ) e. RR* )
129		iccgelb 11716	. . . . . . . . . . . . 13 |- ( ( 0 e. RR* /\ +oo e. RR* /\ ( F ` y ) e. ( 0 [,] +oo ) ) -> 0 <_ ( F ` y ) )
130	122, 123, 127, 129	syl3anc 1292	. . . . . . . . . . . 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 2477	. . . . . . . . . . . . . . . . . . 19 |- ( ( F ` y ) = +oo -> +oo = ( F ` y ) )
133	132	adantl 473	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ph /\ x e. ( ~P X i^i Fin ) ) /\ y e. x ) /\ ( F ` y ) = +oo ) -> +oo = ( F ` y ) )
134		ffun 5742	. . . . . . . . . . . . . . . . . . . . . 22 |- ( F : X --> ( 0 [,] +oo ) -> Fun F )
135	11, 134	syl 17	. . . . . . . . . . . . . . . . . . . . 21 |- ( ph -> Fun F )
136	135	ad2antrr 740	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ph /\ x e. ( ~P X i^i Fin ) ) /\ y e. x ) -> Fun F )
137	23, 125	sylan 479	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ph /\ x e. ( ~P X i^i Fin ) ) /\ y e. x ) -> y e. X )
138		fdm 5745	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( F : X --> ( 0 [,] +oo ) -> dom F = X )
139	11, 138	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ph -> dom F = X )
140	139	eqcomd 2477	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ph -> X = dom F )
141	140	ad2antrr 740	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ph /\ x e. ( ~P X i^i Fin ) ) /\ y e. x ) -> X = dom F )
142	137, 141	eleqtrd 2551	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ph /\ x e. ( ~P X i^i Fin ) ) /\ y e. x ) -> y e. dom F )
143		fvelrn 6030	. . . . . . . . . . . . . . . . . . . 20 |- ( ( Fun F /\ y e. dom F ) -> ( F ` y ) e. ran F )
144	136, 142, 143	syl2anc 673	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ph /\ x e. ( ~P X i^i Fin ) ) /\ y e. x ) -> ( F ` y ) e. ran F )
145	144	adantr 472	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ph /\ x e. ( ~P X i^i Fin ) ) /\ y e. x ) /\ ( F ` y ) = +oo ) -> ( F ` y ) e. ran F )
146	133, 145	eqeltrd 2549	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ph /\ x e. ( ~P X i^i Fin ) ) /\ y e. x ) /\ ( F ` y ) = +oo ) -> +oo e. ran F )
147	146	adantlllr 37424	. . . . . . . . . . . . . . . 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 744	. . . . . . . . . . . . . . . 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 586	. . . . . . . . . . . . . . 15 |- ( ( ( ( ph /\ -. +oo e. ran F ) /\ x e. ( ~P X i^i Fin ) ) /\ y e. x ) -> -. ( F ` y ) = +oo )
150	149	neqned 2650	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ -. +oo e. ran F ) /\ x e. ( ~P X i^i Fin ) ) /\ y e. x ) -> ( F ` y ) =/= +oo )
151		ge0xrre 37729	. . . . . . . . . . . . . 14 |- ( ( ( F ` y ) e. ( 0 [,] +oo ) /\ ( F ` y ) =/= +oo ) -> ( F ` y ) e. RR )
152	127, 150, 151	syl2anc 673	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ -. +oo e. ran F ) /\ x e. ( ~P X i^i Fin ) ) /\ y e. x ) -> ( F ` y ) e. RR )
153	152	ltpnfd 11446	. . . . . . . . . . . 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 11710	. . . . . . . . . . 11 |- ( ( ( ( ph /\ -. +oo e. ran F ) /\ x e. ( ~P X i^i Fin ) ) /\ y e. x ) -> ( F ` y ) e. ( 0 [,) +oo ) )
155		eqid 2471	. . . . . . . . . . 11 |- ( y e. x |-> ( F ` y ) ) = ( y e. x |-> ( F ` y ) )
156	154, 155	fmptd 6061	. . . . . . . . . 10 |- ( ( ( ph /\ -. +oo e. ran F ) /\ x e. ( ~P X i^i Fin ) ) -> ( y e. x |-> ( F ` y ) ) : x --> ( 0 [,) +oo ) )
157		0e0icopnf 11768	. . . . . . . . . . 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 3414	. . . . . . . . . . . 12 |- ( y e. ( 0 [,] +oo ) -> y e. RR* )
160		xaddid2 11557	. . . . . . . . . . . . 13 |- ( y e. RR* -> ( 0 +e y ) = y )
161		xaddid1 11556	. . . . . . . . . . . . 13 |- ( y e. RR* -> ( y +e 0 ) = y )
162	160, 161	jca 541	. . . . . . . . . . . 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 473	. . . . . . . . . 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 16597	. . . . . . . . 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 19101	. . . . . . . . . . . . 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 2471	. . . . . . . . . . . 12 |- (ℂfld ↾s ( 0 [,) +oo ) ) = (ℂfld ↾s ( 0 [,) +oo ) )
169	119, 167, 156, 168	gsumsubm 16698	. . . . . . . . . . 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 2472	. . . . . . . . . . 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 3034	. . . . . . . . . . . . . 14 |- x e. _V
172	171	mptex 6152	. . . . . . . . . . . . 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 6336	. . . . . . . . . . . . 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 6336	. . . . . . . . . . . . 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 11766	. . . . . . . . . . . . . . . . 17 |- ( 0 [,) +oo ) C_ RR
179		ax-resscn 9614	. . . . . . . . . . . . . . . . 17 |- RR C_ CC
180	178, 179	sstri 3427	. . . . . . . . . . . . . . . 16 |- ( 0 [,) +oo ) C_ CC
181		cnfldbas 19051	. . . . . . . . . . . . . . . . 17 |- CC = ( Base ` ℂfld )
182	168, 181	ressbas2 15258	. . . . . . . . . . . . . . . 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 2480	. . . . . . . . . . . . . 14 |- ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) = ( 0 [,) +oo )
185	112, 21	sstri 3427	. . . . . . . . . . . . . . 15 |- ( 0 [,) +oo ) C_ RR*
186		eqid 2471	. . . . . . . . . . . . . . . 16 |- ( RR*s ↾s ( 0 [,) +oo ) ) = ( RR*s ↾s ( 0 [,) +oo ) )
187	186, 70	ressbas2 15258	. . . . . . . . . . . . . . 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 2493	. . . . . . . . . . . . 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 19115	. . . . . . . . . . . . . . 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 464	. . . . . . . . . . . . . 14 |- ( ( s e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) /\ t e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) ) -> s e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) )
194		simpr 468	. . . . . . . . . . . . . 14 |- ( ( s e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) /\ t e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) ) -> t e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) )
195		eqid 2471	. . . . . . . . . . . . . . 15 |- ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) = ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) )
196		eqid 2471	. . . . . . . . . . . . . . 15 |- ( +g ` (ℂfld ↾s ( 0 [,) +oo ) ) ) = ( +g ` (ℂfld ↾s ( 0 [,) +oo ) ) )
197	195, 196	srgacl 17835	. . . . . . . . . . . . . 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 1292	. . . . . . . . . . . . 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 473	. . . . . . . . . . . 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 2559	. . . . . . . . . . . . . . . 16 |- ( s e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) -> s e. ( 0 [,) +oo ) )
203	200, 202	sseldd 3419	. . . . . . . . . . . . . . 15 |- ( s e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) -> s e. RR )
204	203	adantr 472	. . . . . . . . . . . . . 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 2559	. . . . . . . . . . . . . . . 16 |- ( t e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) -> t e. ( 0 [,) +oo ) )
208	205, 207	sseldd 3419	. . . . . . . . . . . . . . 15 |- ( t e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) -> t e. RR )
209	208	adantl 473	. . . . . . . . . . . . . 14 |- ( ( s e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) /\ t e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) ) -> t e. RR )
210		rexadd 11548	. . . . . . . . . . . . . . . 16 |- ( ( s e. RR /\ t e. RR ) -> ( s +e t ) = ( s + t ) )
211	210	eqcomd 2477	. . . . . . . . . . . . . . 15 |- ( ( s e. RR /\ t e. RR ) -> ( s + t ) = ( s +e t ) )
212	166	elexi 3041	. . . . . . . . . . . . . . . . . . . 20 |- ( 0 [,) +oo ) e. _V
213		cnfldadd 19052	. . . . . . . . . . . . . . . . . . . . 21 |- + = ( +g ` ℂfld )
214	168, 213	ressplusg 15317	. . . . . . . . . . . . . . . . . . . 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 2495	. . . . . . . . . . . . . . . . . 18 |- ( +g ` (ℂfld ↾s ( 0 [,) +oo ) ) ) = ( +g ` ℂfld )
217	216, 213	eqtr4i 2496	. . . . . . . . . . . . . . . . 17 |- ( +g ` (ℂfld ↾s ( 0 [,) +oo ) ) ) = +
218	217	oveqi 6321	. . . . . . . . . . . . . . . 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 15317	. . . . . . . . . . . . . . . . . . 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 2480	. . . . . . . . . . . . . . . . 17 |- ( +g ` ( RR*s ↾s ( 0 [,) +oo ) ) ) = +e
223	222	oveqi 6321	. . . . . . . . . . . . . . . 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 2515	. . . . . . . . . . . . . 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 673	. . . . . . . . . . . . 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 473	. . . . . . . . . . . 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 5625	. . . . . . . . . . . . 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 2559	. . . . . . . . . . . . . 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 2809	. . . . . . . . . . . . 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 6068	. . . . . . . . . . . . 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 16595	. . . . . . . . . . 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 2509	. . . . . . . . . 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 473	. . . . . . . . . . 11 |- ( ( ( ph /\ -. +oo e. ran F ) /\ x e. ( ~P X i^i Fin ) ) -> x e. Fin )
237	152	recnd 9687	. . . . . . . . . . 11 |- ( ( ( ( ph /\ -. +oo e. ran F ) /\ x e. ( ~P X i^i Fin ) ) /\ y e. x ) -> ( F ` y ) e. CC )
238	236, 237	gsumfsum 19111	. . . . . . . . . 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 2507	. . . . . . . . 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 2510	. . . . . . . 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 4482	. . . . . . 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 5068	. . . . . 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 7978	. . . . 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 2505	. . . 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 808	. . 3 |- ( ph -> (Σ^ ` F ) = sup ( ran ( x e. ( ~P X i^i Fin ) |-> ( G gsumg ( F |` x ) ) ) , RR* , < ) )
246	22, 9, 11, 1	xrge0tsms 21930	. . 3 |- ( ph -> ( G tsums F ) = { sup ( ran ( x e. ( ~P X i^i Fin ) |-> ( G gsumg ( F |` x ) ) ) , RR* , < ) } )
247	245, 246	eleq12d 2543	. 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 240	1 |- ( ph -> (Σ^ ` F ) e. ( G tsums F ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 189   /\ wa 376   /\ w3a 1007   = wceq 1452   e. wcel 1904   =/= wne 2641   A.wral 2756   E.wrex 2757   _Vcvv 3031   i^i cin 3389   C_ wss 3390   ~Pcpw 3942   {csn 3959   class class class wbr 4395   |-> cmpt 4454   dom cdm 4839   ran crn 4840   |` cres 4841   Fun wfun 5583   Fn wfn 5584   -->wf 5585   ` cfv 5589  (class class class)co 6308   Fincfn 7587   supcsup 7972   CCcc 9555   RRcr 9556   0cc0 9557   + caddc 9560   +oocpnf 9690   RR*cxr 9692   < clt 9693   <_ cle 9694   +ecxad 11430   [,)cico 11662   [,]cicc 11663   sum_csu 13829   Basecbs 15199   ↾_s cress 15200   +g cplusg 15268   gsum_g cgsu 15417   RR*scxrs 15476   Mndcmnd 16613  SubMndcsubmnd 16659  CMndccmn 17508  SRingcsrg 17817  ℂ_fldccnfld 19047   tsums ctsu 21218  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  ax-addf 9636  ax-mulf 9637

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-iin 4272  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-of 6550  df-om 6712  df-1st 6812  df-2nd 6813  df-supp 6934  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-map 7492  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fsupp 7902  df-fi 7943  df-sup 7974  df-inf 7975  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-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-q 11288  df-rp 11326  df-xadd 11433  df-ioo 11664  df-ioc 11665  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-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-starv 15283  df-tset 15287  df-ple 15288  df-ds 15290  df-unif 15291  df-rest 15399  df-topn 15400  df-0g 15418  df-gsum 15419  df-topgen 15420  df-ordt 15477  df-xrs 15478  df-mre 15570  df-mrc 15571  df-acs 15573  df-ps 16524  df-tsr 16525  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-submnd 16661  df-grp 16751  df-minusg 16752  df-mulg 16754  df-cntz 17049  df-cmn 17510  df-abl 17511  df-mgp 17802  df-ur 17814  df-srg 17818  df-ring 17860  df-cring 17861  df-fbas 19044  df-fg 19045  df-cnfld 19048  df-top 19998  df-bases 19999  df-topon 20000  df-topsp 20001  df-ntr 20112  df-nei 20191  df-cn 20320  df-haus 20408  df-fil 20939  df-fm 21031  df-flim 21032  df-flf 21033  df-tsms 21219  df-sumge0 38319

This theorem is referenced by: (None)