Theorem sge0tsms 40597

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

Step	Hyp	Ref	Expression
1		eqid 2622	. . . 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 11974	. . . . . 6 |- < Or RR*
4	3	supex 8369	. . . . 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		elsng 4191	. . . 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 247	. 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 481	. . . . . 6 |- ( ( ph /\ +oo e. ran F ) -> X e. V )
11		sge0tsms.f	. . . . . . 7 |- ( ph -> F : X --> ( 0 [,] +oo ) )
12	11	adantr 481	. . . . . 6 |- ( ( ph /\ +oo e. ran F ) -> F : X --> ( 0 [,] +oo ) )
13		simpr 477	. . . . . 6 |- ( ( ph /\ +oo e. ran F ) -> +oo e. ran F )
14	10, 12, 13	sge0pnfval 40590	. . . . 5 |- ( ( ph /\ +oo e. ran F ) -> (Σ^ ` F ) = +oo )
15		ffn 6045	. . . . . . . . . 10 |- ( F : X --> ( 0 [,] +oo ) -> F Fn X )
16	11, 15	syl 17	. . . . . . . . 9 |- ( ph -> F Fn X )
17	16	adantr 481	. . . . . . . 8 |- ( ( ph /\ +oo e. ran F ) -> F Fn X )
18		fvelrnb 6243	. . . . . . . 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 222	. . . . . 6 |- ( ( ph /\ +oo e. ran F ) -> E. y e. X ( F ` y ) = +oo )
21		iccssxr 12256	. . . . . . . . . . . . . 14 |- ( 0 [,] +oo ) C_ RR*
22		sge0tsms.g	. . . . . . . . . . . . . . 15 |- G = ( RR*s ↾s ( 0 [,] +oo ) )
23		simpr 477	. . . . . . . . . . . . . . 15 |- ( ( ph /\ x e. ( ~P X i^i Fin ) ) -> x e. ( ~P X i^i Fin ) )
24	11	adantr 481	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ x e. ( ~P X i^i Fin ) ) -> F : X --> ( 0 [,] +oo ) )
25		elinel1 3799	. . . . . . . . . . . . . . . . . 18 |- ( x e. ( ~P X i^i Fin ) -> x e. ~P X )
26		elpwi 4168	. . . . . . . . . . . . . . . . . 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 482	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ x e. ( ~P X i^i Fin ) ) -> x C_ X )
29		fssres 6070	. . . . . . . . . . . . . . . 16 |- ( ( F : X --> ( 0 [,] +oo ) /\ x C_ X ) -> ( F |` x ) : x --> ( 0 [,] +oo ) )
30	24, 28, 29	syl2anc 693	. . . . . . . . . . . . . . 15 |- ( ( ph /\ x e. ( ~P X i^i Fin ) ) -> ( F |` x ) : x --> ( 0 [,] +oo ) )
31		elinel2 3800	. . . . . . . . . . . . . . . . 17 |- ( x e. ( ~P X i^i Fin ) -> x e. Fin )
32	31	adantl 482	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ x e. ( ~P X i^i Fin ) ) -> x e. Fin )
33		0red 10041	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ x e. ( ~P X i^i Fin ) ) -> 0 e. RR )
34	30, 32, 33	fdmfifsupp 8285	. . . . . . . . . . . . . . 15 |- ( ( ph /\ x e. ( ~P X i^i Fin ) ) -> ( F |` x ) finSupp 0 )
35	22, 23, 30, 34	gsumge0cl 40588	. . . . . . . . . . . . . 14 |- ( ( ph /\ x e. ( ~P X i^i Fin ) ) -> ( G gsumg ( F |` x ) ) e. ( 0 [,] +oo ) )
36	21, 35	sseldi 3601	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. ( ~P X i^i Fin ) ) -> ( G gsumg ( F |` x ) ) e. RR* )
37	36	ralrimiva 2966	. . . . . . . . . . . 12 |- ( ph -> A. x e. ( ~P X i^i Fin ) ( G gsumg ( F |` x ) ) e. RR* )
38	37	3ad2ant1 1082	. . . . . . . . . . 11 |- ( ( ph /\ y e. X /\ ( F ` y ) = +oo ) -> A. x e. ( ~P X i^i Fin ) ( G gsumg ( F |` x ) ) e. RR* )
39		eqid 2622	. . . . . . . . . . . 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 6392	. . . . . . . . . . 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 4912	. . . . . . . . . . . . . 14 |- ( y e. X -> { y } e. ~P X )
43		snfi 8038	. . . . . . . . . . . . . . 15 |- { y } e. Fin
44	43	a1i 11	. . . . . . . . . . . . . 14 |- ( y e. X -> { y } e. Fin )
45	42, 44	elind 3798	. . . . . . . . . . . . 13 |- ( y e. X -> { y } e. ( ~P X i^i Fin ) )
46	45	3ad2ant2 1083	. . . . . . . . . . . 12 |- ( ( ph /\ y e. X /\ ( F ` y ) = +oo ) -> { y } e. ( ~P X i^i Fin ) )
47	11	adantr 481	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ y e. X ) -> F : X --> ( 0 [,] +oo ) )
48		snssi 4339	. . . . . . . . . . . . . . . . . . 19 |- ( y e. X -> { y } C_ X )
49	48	adantl 482	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ y e. X ) -> { y } C_ X )
50	47, 49	fssresd 6071	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ y e. X ) -> ( F |` { y } ) : { y } --> ( 0 [,] +oo ) )
51	50	feqmptd 6249	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ y e. X ) -> ( F |` { y } ) = ( x e. { y } |-> ( ( F |` { y } ) ` x ) ) )
52		fvres 6207	. . . . . . . . . . . . . . . . . 18 |- ( x e. { y } -> ( ( F |` { y } ) ` x ) = ( F ` x ) )
53	52	mpteq2ia 4740	. . . . . . . . . . . . . . . . 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 2656	. . . . . . . . . . . . . . 15 |- ( ( ph /\ y e. X ) -> ( F |` { y } ) = ( x e. { y } |-> ( F ` x ) ) )
56	55	oveq2d 6666	. . . . . . . . . . . . . 14 |- ( ( ph /\ y e. X ) -> ( G gsumg ( F |` { y } ) ) = ( G gsumg ( x e. { y } |-> ( F ` x ) ) ) )
57	56	3adant3 1081	. . . . . . . . . . . . 13 |- ( ( ph /\ y e. X /\ ( F ` y ) = +oo ) -> ( G gsumg ( F |` { y } ) ) = ( G gsumg ( x e. { y } |-> ( F ` x ) ) ) )
58		xrge0cmn 19788	. . . . . . . . . . . . . . . . 17 |- ( RR*s ↾s ( 0 [,] +oo ) ) e. CMnd
59	22, 58	eqeltri 2697	. . . . . . . . . . . . . . . 16 |- G e. CMnd
60		cmnmnd 18208	. . . . . . . . . . . . . . . 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 1062	. . . . . . . . . . . . . 14 |- ( ( ph /\ y e. X /\ ( F ` y ) = +oo ) -> y e. X )
64	11	ffvelrnda 6359	. . . . . . . . . . . . . . 15 |- ( ( ph /\ y e. X ) -> ( F ` y ) e. ( 0 [,] +oo ) )
65	64	3adant3 1081	. . . . . . . . . . . . . 14 |- ( ( ph /\ y e. X /\ ( F ` y ) = +oo ) -> ( F ` y ) e. ( 0 [,] +oo ) )
66		df-ss 3588	. . . . . . . . . . . . . . . . . 18 |- ( ( 0 [,] +oo ) C_ RR* <-> ( ( 0 [,] +oo ) i^i RR* ) = ( 0 [,] +oo ) )
67	21, 66	mpbi 220	. . . . . . . . . . . . . . . . 17 |- ( ( 0 [,] +oo ) i^i RR* ) = ( 0 [,] +oo )
68	67	eqcomi 2631	. . . . . . . . . . . . . . . 16 |- ( 0 [,] +oo ) = ( ( 0 [,] +oo ) i^i RR* )
69		ovex 6678	. . . . . . . . . . . . . . . . 17 |- ( 0 [,] +oo ) e. _V
70		xrsbas 19762	. . . . . . . . . . . . . . . . . 18 |- RR* = ( Base ` RR*s )
71	22, 70	ressbas 15930	. . . . . . . . . . . . . . . . 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 2644	. . . . . . . . . . . . . . 15 |- ( 0 [,] +oo ) = ( Base ` G )
74		fveq2 6191	. . . . . . . . . . . . . . 15 |- ( x = y -> ( F ` x ) = ( F ` y ) )
75	73, 74	gsumsn 18354	. . . . . . . . . . . . . 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 1326	. . . . . . . . . . . . 13 |- ( ( ph /\ y e. X /\ ( F ` y ) = +oo ) -> ( G gsumg ( x e. { y } |-> ( F ` x ) ) ) = ( F ` y ) )
77		simp3 1063	. . . . . . . . . . . . 13 |- ( ( ph /\ y e. X /\ ( F ` y ) = +oo ) -> ( F ` y ) = +oo )
78	57, 76, 77	3eqtrrd 2661	. . . . . . . . . . . 12 |- ( ( ph /\ y e. X /\ ( F ` y ) = +oo ) -> +oo = ( G gsumg ( F |` { y } ) ) )
79		reseq2 5391	. . . . . . . . . . . . . . 15 |- ( x = { y } -> ( F |` x ) = ( F |` { y } ) )
80	79	oveq2d 6666	. . . . . . . . . . . . . 14 |- ( x = { y } -> ( G gsumg ( F |` x ) ) = ( G gsumg ( F |` { y } ) ) )
81	80	eqeq2d 2632	. . . . . . . . . . . . 13 |- ( x = { y } -> ( +oo = ( G gsumg ( F |` x ) ) <-> +oo = ( G gsumg ( F |` { y } ) ) ) )
82	81	rspcev 3309	. . . . . . . . . . . 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 693	. . . . . . . . . . 11 |- ( ( ph /\ y e. X /\ ( F ` y ) = +oo ) -> E. x e. ( ~P X i^i Fin ) +oo = ( G gsumg ( F |` x ) ) )
84		pnfxr 10092	. . . . . . . . . . . . 13 |- +oo e. RR*
85	84	a1i 11	. . . . . . . . . . . 12 |- ( ( ph /\ y e. X /\ ( F ` y ) = +oo ) -> +oo e. RR* )
86	39	elrnmpt 5372	. . . . . . . . . . . 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 247	. . . . . . . . . 10 |- ( ( ph /\ y e. X /\ ( F ` y ) = +oo ) -> +oo e. ran ( x e. ( ~P X i^i Fin ) |-> ( G gsumg ( F |` x ) ) ) )
89		supxrpnf 12148	. . . . . . . . . 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 693	. . . . . . . . 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 1264	. . . . . . . 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 481	. . . . . . 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 3030	. . . . . 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 2659	. . . 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 481	. . . . . 6 |- ( ( ph /\ -. +oo e. ran F ) -> X e. V )
97	11	adantr 481	. . . . . . 7 |- ( ( ph /\ -. +oo e. ran F ) -> F : X --> ( 0 [,] +oo ) )
98		simpr 477	. . . . . . 7 |- ( ( ph /\ -. +oo e. ran F ) -> -. +oo e. ran F )
99	97, 98	fge0iccico 40587	. . . . . 6 |- ( ( ph /\ -. +oo e. ran F ) -> F : X --> ( 0 [,) +oo ) )
100	96, 99	sge0reval 40589	. . . . 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 6250	. . . . . . . . . . 11 |- ( ( ph /\ x e. ( ~P X i^i Fin ) ) -> ( F |` x ) = ( y e. x |-> ( F ` y ) ) )
102	101	adantlr 751	. . . . . . . . . 10 |- ( ( ( ph /\ -. +oo e. ran F ) /\ x e. ( ~P X i^i Fin ) ) -> ( F |` x ) = ( y e. x |-> ( F ` y ) ) )
103	102	oveq2d 6666	. . . . . . . . 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 6194	. . . . . . . . . . 11 |- ( +g ` G ) = ( +g ` ( RR*s ↾s ( 0 [,] +oo ) ) )
105		eqid 2622	. . . . . . . . . . . . . 14 |- ( RR*s ↾s ( 0 [,] +oo ) ) = ( RR*s ↾s ( 0 [,] +oo ) )
106		xrsadd 19763	. . . . . . . . . . . . . 14 |- +e = ( +g ` RR*s )
107	105, 106	ressplusg 15993	. . . . . . . . . . . . 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 2631	. . . . . . . . . . 11 |- ( +g ` ( RR*s ↾s ( 0 [,] +oo ) ) ) = +e
110	104, 109	eqtr2i 2645	. . . . . . . . . 10 |- +e = ( +g ` G )
111	22	oveq1i 6660	. . . . . . . . . . 11 |- ( G ↾s ( 0 [,) +oo ) ) = ( ( RR*s ↾s ( 0 [,] +oo ) ) ↾s ( 0 [,) +oo ) )
112		icossicc 12260	. . . . . . . . . . . . 13 |- ( 0 [,) +oo ) C_ ( 0 [,] +oo )
113	69, 112	pm3.2i 471	. . . . . . . . . . . 12 |- ( ( 0 [,] +oo ) e. _V /\ ( 0 [,) +oo ) C_ ( 0 [,] +oo ) )
114		ressabs 15939	. . . . . . . . . . . 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 2645	. . . . . . . . . 10 |- ( RR*s ↾s ( 0 [,) +oo ) ) = ( G ↾s ( 0 [,) +oo ) )
117	59	elexi 3213	. . . . . . . . . . 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 477	. . . . . . . . . 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 10086	. . . . . . . . . . . . 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 762	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ -. +oo e. ran F ) /\ x e. ( ~P X i^i Fin ) ) /\ y e. x ) -> F : X --> ( 0 [,] +oo ) )
125	27	sselda 3603	. . . . . . . . . . . . . . 15 |- ( ( x e. ( ~P X i^i Fin ) /\ y e. x ) -> y e. X )
126	125	adantll 750	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ -. +oo e. ran F ) /\ x e. ( ~P X i^i Fin ) ) /\ y e. x ) -> y e. X )
127	124, 126	ffvelrnd 6360	. . . . . . . . . . . . 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 3601	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ -. +oo e. ran F ) /\ x e. ( ~P X i^i Fin ) ) /\ y e. x ) -> ( F ` y ) e. RR* )
129		iccgelb 12230	. . . . . . . . . . . . 13 |- ( ( 0 e. RR* /\ +oo e. RR* /\ ( F ` y ) e. ( 0 [,] +oo ) ) -> 0 <_ ( F ` y ) )
130	122, 123, 127, 129	syl3anc 1326	. . . . . . . . . . . 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 2628	. . . . . . . . . . . . . . . . . . 19 |- ( ( F ` y ) = +oo -> +oo = ( F ` y ) )
133	132	adantl 482	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ph /\ x e. ( ~P X i^i Fin ) ) /\ y e. x ) /\ ( F ` y ) = +oo ) -> +oo = ( F ` y ) )
134		ffun 6048	. . . . . . . . . . . . . . . . . . . . . 22 |- ( F : X --> ( 0 [,] +oo ) -> Fun F )
135	11, 134	syl 17	. . . . . . . . . . . . . . . . . . . . 21 |- ( ph -> Fun F )
136	135	ad2antrr 762	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ph /\ x e. ( ~P X i^i Fin ) ) /\ y e. x ) -> Fun F )
137	23, 125	sylan 488	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ph /\ x e. ( ~P X i^i Fin ) ) /\ y e. x ) -> y e. X )
138		fdm 6051	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( F : X --> ( 0 [,] +oo ) -> dom F = X )
139	11, 138	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ph -> dom F = X )
140	139	eqcomd 2628	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ph -> X = dom F )
141	140	ad2antrr 762	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ph /\ x e. ( ~P X i^i Fin ) ) /\ y e. x ) -> X = dom F )
142	137, 141	eleqtrd 2703	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ph /\ x e. ( ~P X i^i Fin ) ) /\ y e. x ) -> y e. dom F )
143		fvelrn 6352	. . . . . . . . . . . . . . . . . . . 20 |- ( ( Fun F /\ y e. dom F ) -> ( F ` y ) e. ran F )
144	136, 142, 143	syl2anc 693	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ph /\ x e. ( ~P X i^i Fin ) ) /\ y e. x ) -> ( F ` y ) e. ran F )
145	144	adantr 481	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ph /\ x e. ( ~P X i^i Fin ) ) /\ y e. x ) /\ ( F ` y ) = +oo ) -> ( F ` y ) e. ran F )
146	133, 145	eqeltrd 2701	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ph /\ x e. ( ~P X i^i Fin ) ) /\ y e. x ) /\ ( F ` y ) = +oo ) -> +oo e. ran F )
147	146	adantlllr 39199	. . . . . . . . . . . . . . . 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 766	. . . . . . . . . . . . . . . 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 600	. . . . . . . . . . . . . . 15 |- ( ( ( ( ph /\ -. +oo e. ran F ) /\ x e. ( ~P X i^i Fin ) ) /\ y e. x ) -> -. ( F ` y ) = +oo )
150	149	neqned 2801	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ -. +oo e. ran F ) /\ x e. ( ~P X i^i Fin ) ) /\ y e. x ) -> ( F ` y ) =/= +oo )
151		ge0xrre 39758	. . . . . . . . . . . . . 14 |- ( ( ( F ` y ) e. ( 0 [,] +oo ) /\ ( F ` y ) =/= +oo ) -> ( F ` y ) e. RR )
152	127, 150, 151	syl2anc 693	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ -. +oo e. ran F ) /\ x e. ( ~P X i^i Fin ) ) /\ y e. x ) -> ( F ` y ) e. RR )
153	152	ltpnfd 11955	. . . . . . . . . . . 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 12224	. . . . . . . . . . 11 |- ( ( ( ( ph /\ -. +oo e. ran F ) /\ x e. ( ~P X i^i Fin ) ) /\ y e. x ) -> ( F ` y ) e. ( 0 [,) +oo ) )
155		eqid 2622	. . . . . . . . . . 11 |- ( y e. x |-> ( F ` y ) ) = ( y e. x |-> ( F ` y ) )
156	154, 155	fmptd 6385	. . . . . . . . . 10 |- ( ( ( ph /\ -. +oo e. ran F ) /\ x e. ( ~P X i^i Fin ) ) -> ( y e. x |-> ( F ` y ) ) : x --> ( 0 [,) +oo ) )
157		0e0icopnf 12282	. . . . . . . . . . 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 3599	. . . . . . . . . . . 12 |- ( y e. ( 0 [,] +oo ) -> y e. RR* )
160		xaddid2 12073	. . . . . . . . . . . . 13 |- ( y e. RR* -> ( 0 +e y ) = y )
161		xaddid1 12072	. . . . . . . . . . . . 13 |- ( y e. RR* -> ( y +e 0 ) = y )
162	160, 161	jca 554	. . . . . . . . . . . 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 482	. . . . . . . . . 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 17276	. . . . . . . . 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 19802	. . . . . . . . . . . . 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 2622	. . . . . . . . . . . 12 |- (ℂfld ↾s ( 0 [,) +oo ) ) = (ℂfld ↾s ( 0 [,) +oo ) )
169	119, 167, 156, 168	gsumsubm 17373	. . . . . . . . . . 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 2623	. . . . . . . . . . 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 3203	. . . . . . . . . . . . . 14 |- x e. _V
172	171	mptex 6486	. . . . . . . . . . . . 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		ovexd 6680	. . . . . . . . . . . 12 |- ( ( ( ph /\ -. +oo e. ran F ) /\ x e. ( ~P X i^i Fin ) ) -> (ℂfld ↾s ( 0 [,) +oo ) ) e. _V )
175		ovexd 6680	. . . . . . . . . . . 12 |- ( ( ( ph /\ -. +oo e. ran F ) /\ x e. ( ~P X i^i Fin ) ) -> ( RR*s ↾s ( 0 [,) +oo ) ) e. _V )
176		rge0ssre 12280	. . . . . . . . . . . . . . . . 17 |- ( 0 [,) +oo ) C_ RR
177		ax-resscn 9993	. . . . . . . . . . . . . . . . 17 |- RR C_ CC
178	176, 177	sstri 3612	. . . . . . . . . . . . . . . 16 |- ( 0 [,) +oo ) C_ CC
179		cnfldbas 19750	. . . . . . . . . . . . . . . . 17 |- CC = ( Base ` ℂfld )
180	168, 179	ressbas2 15931	. . . . . . . . . . . . . . . 16 |- ( ( 0 [,) +oo ) C_ CC -> ( 0 [,) +oo ) = ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) )
181	178, 180	ax-mp 5	. . . . . . . . . . . . . . 15 |- ( 0 [,) +oo ) = ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) )
182	181	eqcomi 2631	. . . . . . . . . . . . . 14 |- ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) = ( 0 [,) +oo )
183	112, 21	sstri 3612	. . . . . . . . . . . . . . 15 |- ( 0 [,) +oo ) C_ RR*
184		eqid 2622	. . . . . . . . . . . . . . . 16 |- ( RR*s ↾s ( 0 [,) +oo ) ) = ( RR*s ↾s ( 0 [,) +oo ) )
185	184, 70	ressbas2 15931	. . . . . . . . . . . . . . 15 |- ( ( 0 [,) +oo ) C_ RR* -> ( 0 [,) +oo ) = ( Base ` ( RR*s ↾s ( 0 [,) +oo ) ) ) )
186	183, 185	ax-mp 5	. . . . . . . . . . . . . 14 |- ( 0 [,) +oo ) = ( Base ` ( RR*s ↾s ( 0 [,) +oo ) ) )
187	182, 186	eqtri 2644	. . . . . . . . . . . . 13 |- ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) = ( Base ` ( RR*s ↾s ( 0 [,) +oo ) ) )
188	187	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 ) ) ) )
189		rge0srg 19817	. . . . . . . . . . . . . . 15 |- (ℂfld ↾s ( 0 [,) +oo ) ) e. SRing
190	189	a1i 11	. . . . . . . . . . . . . 14 |- ( ( s e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) /\ t e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) ) -> (ℂfld ↾s ( 0 [,) +oo ) ) e. SRing )
191		simpl 473	. . . . . . . . . . . . . 14 |- ( ( s e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) /\ t e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) ) -> s e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) )
192		simpr 477	. . . . . . . . . . . . . 14 |- ( ( s e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) /\ t e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) ) -> t e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) )
193		eqid 2622	. . . . . . . . . . . . . . 15 |- ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) = ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) )
194		eqid 2622	. . . . . . . . . . . . . . 15 |- ( +g ` (ℂfld ↾s ( 0 [,) +oo ) ) ) = ( +g ` (ℂfld ↾s ( 0 [,) +oo ) ) )
195	193, 194	srgacl 18524	. . . . . . . . . . . . . 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 ) ) ) )
196	190, 191, 192, 195	syl3anc 1326	. . . . . . . . . . . . 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 ) ) ) )
197	196	adantl 482	. . . . . . . . . . . 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 ) ) ) )
198	176	a1i 11	. . . . . . . . . . . . . . . 16 |- ( s e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) -> ( 0 [,) +oo ) C_ RR )
199		id 22	. . . . . . . . . . . . . . . . 17 |- ( s e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) -> s e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) )
200	199, 182	syl6eleq 2711	. . . . . . . . . . . . . . . 16 |- ( s e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) -> s e. ( 0 [,) +oo ) )
201	198, 200	sseldd 3604	. . . . . . . . . . . . . . 15 |- ( s e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) -> s e. RR )
202	201	adantr 481	. . . . . . . . . . . . . 14 |- ( ( s e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) /\ t e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) ) -> s e. RR )
203	176	a1i 11	. . . . . . . . . . . . . . . 16 |- ( t e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) -> ( 0 [,) +oo ) C_ RR )
204		id 22	. . . . . . . . . . . . . . . . 17 |- ( t e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) -> t e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) )
205	204, 182	syl6eleq 2711	. . . . . . . . . . . . . . . 16 |- ( t e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) -> t e. ( 0 [,) +oo ) )
206	203, 205	sseldd 3604	. . . . . . . . . . . . . . 15 |- ( t e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) -> t e. RR )
207	206	adantl 482	. . . . . . . . . . . . . 14 |- ( ( s e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) /\ t e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) ) -> t e. RR )
208		rexadd 12063	. . . . . . . . . . . . . . . 16 |- ( ( s e. RR /\ t e. RR ) -> ( s +e t ) = ( s + t ) )
209	208	eqcomd 2628	. . . . . . . . . . . . . . 15 |- ( ( s e. RR /\ t e. RR ) -> ( s + t ) = ( s +e t ) )
210	166	elexi 3213	. . . . . . . . . . . . . . . . . . . 20 |- ( 0 [,) +oo ) e. _V
211		cnfldadd 19751	. . . . . . . . . . . . . . . . . . . . 21 |- + = ( +g ` ℂfld )
212	168, 211	ressplusg 15993	. . . . . . . . . . . . . . . . . . . 20 |- ( ( 0 [,) +oo ) e. _V -> + = ( +g ` (ℂfld ↾s ( 0 [,) +oo ) ) ) )
213	210, 212	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 |- + = ( +g ` (ℂfld ↾s ( 0 [,) +oo ) ) )
214	213, 211	eqtr3i 2646	. . . . . . . . . . . . . . . . . 18 |- ( +g ` (ℂfld ↾s ( 0 [,) +oo ) ) ) = ( +g ` ℂfld )
215	214, 211	eqtr4i 2647	. . . . . . . . . . . . . . . . 17 |- ( +g ` (ℂfld ↾s ( 0 [,) +oo ) ) ) = +
216	215	oveqi 6663	. . . . . . . . . . . . . . . 16 |- ( s ( +g ` (ℂfld ↾s ( 0 [,) +oo ) ) ) t ) = ( s + t )
217	216	a1i 11	. . . . . . . . . . . . . . 15 |- ( ( s e. RR /\ t e. RR ) -> ( s ( +g ` (ℂfld ↾s ( 0 [,) +oo ) ) ) t ) = ( s + t ) )
218	184, 106	ressplusg 15993	. . . . . . . . . . . . . . . . . . 19 |- ( ( 0 [,) +oo ) e. _V -> +e = ( +g ` ( RR*s ↾s ( 0 [,) +oo ) ) ) )
219	210, 218	ax-mp 5	. . . . . . . . . . . . . . . . . 18 |- +e = ( +g ` ( RR*s ↾s ( 0 [,) +oo ) ) )
220	219	eqcomi 2631	. . . . . . . . . . . . . . . . 17 |- ( +g ` ( RR*s ↾s ( 0 [,) +oo ) ) ) = +e
221	220	oveqi 6663	. . . . . . . . . . . . . . . 16 |- ( s ( +g ` ( RR*s ↾s ( 0 [,) +oo ) ) ) t ) = ( s +e t )
222	221	a1i 11	. . . . . . . . . . . . . . 15 |- ( ( s e. RR /\ t e. RR ) -> ( s ( +g ` ( RR*s ↾s ( 0 [,) +oo ) ) ) t ) = ( s +e t ) )
223	209, 217, 222	3eqtr4d 2666	. . . . . . . . . . . . . 14 |- ( ( s e. RR /\ t e. RR ) -> ( s ( +g ` (ℂfld ↾s ( 0 [,) +oo ) ) ) t ) = ( s ( +g ` ( RR*s ↾s ( 0 [,) +oo ) ) ) t ) )
224	202, 207, 223	syl2anc 693	. . . . . . . . . . . . 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 ) )
225	224	adantl 482	. . . . . . . . . . . 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 ) )
226		funmpt 5926	. . . . . . . . . . . . 13 |- Fun ( y e. x |-> ( F ` y ) )
227	226	a1i 11	. . . . . . . . . . . 12 |- ( ( ( ph /\ -. +oo e. ran F ) /\ x e. ( ~P X i^i Fin ) ) -> Fun ( y e. x |-> ( F ` y ) ) )
228	154, 181	syl6eleq 2711	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ -. +oo e. ran F ) /\ x e. ( ~P X i^i Fin ) ) /\ y e. x ) -> ( F ` y ) e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) )
229	228	ralrimiva 2966	. . . . . . . . . . . . 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 ) ) ) )
230	155	rnmptss 6392	. . . . . . . . . . . . 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 ) ) ) )
231	229, 230	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 ) ) ) )
232	173, 174, 175, 188, 197, 225, 227, 231	gsumpropd2 17274	. . . . . . . . . . 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 ) ) ) )
233	169, 170, 232	3eqtrd 2660	. . . . . . . . . 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 ) ) ) )
234	31	adantl 482	. . . . . . . . . . 11 |- ( ( ( ph /\ -. +oo e. ran F ) /\ x e. ( ~P X i^i Fin ) ) -> x e. Fin )
235	152	recnd 10068	. . . . . . . . . . 11 |- ( ( ( ( ph /\ -. +oo e. ran F ) /\ x e. ( ~P X i^i Fin ) ) /\ y e. x ) -> ( F ` y ) e. CC )
236	234, 235	gsumfsum 19813	. . . . . . . . . 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 ) )
237	233, 236	eqtr3d 2658	. . . . . . . . 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 ) )
238	103, 165, 237	3eqtrrd 2661	. . . . . . . 8 |- ( ( ( ph /\ -. +oo e. ran F ) /\ x e. ( ~P X i^i Fin ) ) -> sum_ y e. x ( F ` y ) = ( G gsumg ( F |` x ) ) )
239	238	mpteq2dva 4744	. . . . . . 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 ) ) ) )
240	239	rneqd 5353	. . . . . 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 ) ) ) )
241	240	supeq1d 8352	. . . . 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* , < ) )
242	100, 241	eqtrd 2656	. . . 4 |- ( ( ph /\ -. +oo e. ran F ) -> (Σ^ ` F ) = sup ( ran ( x e. ( ~P X i^i Fin ) |-> ( G gsumg ( F |` x ) ) ) , RR* , < ) )
243	95, 242	pm2.61dan 832	. . 3 |- ( ph -> (Σ^ ` F ) = sup ( ran ( x e. ( ~P X i^i Fin ) |-> ( G gsumg ( F |` x ) ) ) , RR* , < ) )
244	22, 9, 11, 1	xrge0tsms 22637	. . 3 |- ( ph -> ( G tsums F ) = { sup ( ran ( x e. ( ~P X i^i Fin ) |-> ( G gsumg ( F |` x ) ) ) , RR* , < ) } )
245	243, 244	eleq12d 2695	. 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* , < ) } ) )
246	8, 245	mpbird 247	1 |- ( ph -> (Σ^ ` F ) e. ( G tsums F ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   _Vcvv 3200   i^i cin 3573   C_ wss 3574   ~Pcpw 4158   {csn 4177   class class class wbr 4653   |-> cmpt 4729   dom cdm 5114   ran crn 5115   |` cres 5116   Fun wfun 5882   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   Fincfn 7955   supcsup 8346   CCcc 9934   RRcr 9935   0cc0 9936   + caddc 9939   +oocpnf 10071   RR*cxr 10073   < clt 10074   <_ cle 10075   +ecxad 11944   [,)cico 12177   [,]cicc 12178   sum_csu 14416   Basecbs 15857   ↾_s cress 15858   +g cplusg 15941   gsum_g cgsu 16101   RR*scxrs 16160   Mndcmnd 17294  SubMndcsubmnd 17334  CMndccmn 18193  SRingcsrg 18505  ℂ_fldccnfld 19746   tsums ctsu 21929  Σ^{^}csumge0 40579

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xadd 11947  df-ioo 12179  df-ioc 12180  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-sum 14417  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-ordt 16161  df-xrs 16162  df-mre 16246  df-mrc 16247  df-acs 16249  df-ps 17200  df-tsr 17201  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-grp 17425  df-minusg 17426  df-mulg 17541  df-cntz 17750  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-srg 18506  df-ring 18549  df-cring 18550  df-fbas 19743  df-fg 19744  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-ntr 20824  df-nei 20902  df-cn 21031  df-haus 21119  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-tsms 21930  df-sumge0 40580

This theorem is referenced by: (None)