Theorem sge0seq 40663

Description: A series of nonnegative reals agrees with the generalized sum of nonnegative reals. (Contributed by Glauco Siliprandi, 3-Mar-2021.)

Hypotheses

Ref	Expression
sge0seq.m	|- ( ph -> M e. ZZ )
sge0seq.z	|- Z = ( ZZ>= ` M )
sge0seq.f	|- ( ph -> F : Z --> ( 0 [,) +oo ) )
sge0seq.g	|- G = seq M ( + , F )

Assertion

Ref	Expression
sge0seq	|- ( ph -> (Σ^ ` F ) = sup ( ran G , RR* , < ) )

Proof of Theorem sge0seq

Dummy variables k i j w z y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sge0seq.z	. . . . . . 7 |- Z = ( ZZ>= ` M )
2		sge0seq.m	. . . . . . 7 |- ( ph -> M e. ZZ )
3		rge0ssre 12280	. . . . . . . 8 |- ( 0 [,) +oo ) C_ RR
4		sge0seq.f	. . . . . . . . 9 |- ( ph -> F : Z --> ( 0 [,) +oo ) )
5	4	ffvelrnda 6359	. . . . . . . 8 |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. ( 0 [,) +oo ) )
6	3, 5	sseldi 3601	. . . . . . 7 |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. RR )
7		readdcl 10019	. . . . . . . 8 |- ( ( k e. RR /\ i e. RR ) -> ( k + i ) e. RR )
8	7	adantl 482	. . . . . . 7 |- ( ( ph /\ ( k e. RR /\ i e. RR ) ) -> ( k + i ) e. RR )
9	1, 2, 6, 8	seqf 12822	. . . . . 6 |- ( ph -> seq M ( + , F ) : Z --> RR )
10		sge0seq.g	. . . . . . . 8 |- G = seq M ( + , F )
11	10	a1i 11	. . . . . . 7 |- ( ph -> G = seq M ( + , F ) )
12	11	feq1d 6030	. . . . . 6 |- ( ph -> ( G : Z --> RR <-> seq M ( + , F ) : Z --> RR ) )
13	9, 12	mpbird 247	. . . . 5 |- ( ph -> G : Z --> RR )
14		frn 6053	. . . . 5 |- ( G : Z --> RR -> ran G C_ RR )
15	13, 14	syl 17	. . . 4 |- ( ph -> ran G C_ RR )
16		ressxr 10083	. . . . 5 |- RR C_ RR*
17	16	a1i 11	. . . 4 |- ( ph -> RR C_ RR* )
18	15, 17	sstrd 3613	. . 3 |- ( ph -> ran G C_ RR* )
19		fvex 6201	. . . . . 6 |- ( ZZ>= ` M ) e. _V
20	1, 19	eqeltri 2697	. . . . 5 |- Z e. _V
21	20	a1i 11	. . . 4 |- ( ph -> Z e. _V )
22		icossicc 12260	. . . . . 6 |- ( 0 [,) +oo ) C_ ( 0 [,] +oo )
23	22	a1i 11	. . . . 5 |- ( ph -> ( 0 [,) +oo ) C_ ( 0 [,] +oo ) )
24	4, 23	fssd 6057	. . . 4 |- ( ph -> F : Z --> ( 0 [,] +oo ) )
25	21, 24	sge0xrcl 40602	. . 3 |- ( ph -> (Σ^ ` F ) e. RR* )
26		simpr 477	. . . . . 6 |- ( ( ph /\ z e. ran G ) -> z e. ran G )
27		ffn 6045	. . . . . . . . 9 |- ( G : Z --> RR -> G Fn Z )
28	13, 27	syl 17	. . . . . . . 8 |- ( ph -> G Fn Z )
29		fvelrnb 6243	. . . . . . . 8 |- ( G Fn Z -> ( z e. ran G <-> E. j e. Z ( G ` j ) = z ) )
30	28, 29	syl 17	. . . . . . 7 |- ( ph -> ( z e. ran G <-> E. j e. Z ( G ` j ) = z ) )
31	30	adantr 481	. . . . . 6 |- ( ( ph /\ z e. ran G ) -> ( z e. ran G <-> E. j e. Z ( G ` j ) = z ) )
32	26, 31	mpbid 222	. . . . 5 |- ( ( ph /\ z e. ran G ) -> E. j e. Z ( G ` j ) = z )
33	22, 5	sseldi 3601	. . . . . . . . . . 11 |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. ( 0 [,] +oo ) )
34		elfzuz 12338	. . . . . . . . . . . . . 14 |- ( k e. ( M ... j ) -> k e. ( ZZ>= ` M ) )
35	34, 1	syl6eleqr 2712	. . . . . . . . . . . . 13 |- ( k e. ( M ... j ) -> k e. Z )
36	35	ssriv 3607	. . . . . . . . . . . 12 |- ( M ... j ) C_ Z
37	36	a1i 11	. . . . . . . . . . 11 |- ( ph -> ( M ... j ) C_ Z )
38	21, 33, 37	sge0lessmpt 40616	. . . . . . . . . 10 |- ( ph -> (Σ^ ` ( k e. ( M ... j ) |-> ( F ` k ) ) ) <_ (Σ^ ` ( k e. Z |-> ( F ` k ) ) ) )
39	38	3ad2ant1 1082	. . . . . . . . 9 |- ( ( ph /\ j e. Z /\ ( G ` j ) = z ) -> (Σ^ ` ( k e. ( M ... j ) |-> ( F ` k ) ) ) <_ (Σ^ ` ( k e. Z |-> ( F ` k ) ) ) )
40		fzfid 12772	. . . . . . . . . . . . . 14 |- ( ph -> ( M ... j ) e. Fin )
41	35, 5	sylan2 491	. . . . . . . . . . . . . 14 |- ( ( ph /\ k e. ( M ... j ) ) -> ( F ` k ) e. ( 0 [,) +oo ) )
42	40, 41	sge0fsummpt 40607	. . . . . . . . . . . . 13 |- ( ph -> (Σ^ ` ( k e. ( M ... j ) |-> ( F ` k ) ) ) = sum_ k e. ( M ... j ) ( F ` k ) )
43	42	3ad2ant1 1082	. . . . . . . . . . . 12 |- ( ( ph /\ j e. Z /\ ( G ` j ) = z ) -> (Σ^ ` ( k e. ( M ... j ) |-> ( F ` k ) ) ) = sum_ k e. ( M ... j ) ( F ` k ) )
44		simpll 790	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ j e. Z ) /\ k e. ( M ... j ) ) -> ph )
45	35	adantl 482	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ j e. Z ) /\ k e. ( M ... j ) ) -> k e. Z )
46		eqidd 2623	. . . . . . . . . . . . . . 15 |- ( ( ph /\ k e. Z ) -> ( F ` k ) = ( F ` k ) )
47	44, 45, 46	syl2anc 693	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ j e. Z ) /\ k e. ( M ... j ) ) -> ( F ` k ) = ( F ` k ) )
48	1	eleq2i 2693	. . . . . . . . . . . . . . . 16 |- ( j e. Z <-> j e. ( ZZ>= ` M ) )
49	48	biimpi 206	. . . . . . . . . . . . . . 15 |- ( j e. Z -> j e. ( ZZ>= ` M ) )
50	49	adantl 482	. . . . . . . . . . . . . 14 |- ( ( ph /\ j e. Z ) -> j e. ( ZZ>= ` M ) )
51	6	recnd 10068	. . . . . . . . . . . . . . 15 |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )
52	44, 45, 51	syl2anc 693	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ j e. Z ) /\ k e. ( M ... j ) ) -> ( F ` k ) e. CC )
53	47, 50, 52	fsumser 14461	. . . . . . . . . . . . 13 |- ( ( ph /\ j e. Z ) -> sum_ k e. ( M ... j ) ( F ` k ) = ( seq M ( + , F ) ` j ) )
54	53	3adant3 1081	. . . . . . . . . . . 12 |- ( ( ph /\ j e. Z /\ ( G ` j ) = z ) -> sum_ k e. ( M ... j ) ( F ` k ) = ( seq M ( + , F ) ` j ) )
55	43, 54	eqtrd 2656	. . . . . . . . . . 11 |- ( ( ph /\ j e. Z /\ ( G ` j ) = z ) -> (Σ^ ` ( k e. ( M ... j ) |-> ( F ` k ) ) ) = ( seq M ( + , F ) ` j ) )
56	10	eqcomi 2631	. . . . . . . . . . . . 13 |- seq M ( + , F ) = G
57	56	fveq1i 6192	. . . . . . . . . . . 12 |- ( seq M ( + , F ) ` j ) = ( G ` j )
58	57	a1i 11	. . . . . . . . . . 11 |- ( ( ph /\ j e. Z /\ ( G ` j ) = z ) -> ( seq M ( + , F ) ` j ) = ( G ` j ) )
59		simp3 1063	. . . . . . . . . . 11 |- ( ( ph /\ j e. Z /\ ( G ` j ) = z ) -> ( G ` j ) = z )
60	55, 58, 59	3eqtrrd 2661	. . . . . . . . . 10 |- ( ( ph /\ j e. Z /\ ( G ` j ) = z ) -> z = (Σ^ ` ( k e. ( M ... j ) |-> ( F ` k ) ) ) )
61	4	feqmptd 6249	. . . . . . . . . . . 12 |- ( ph -> F = ( k e. Z |-> ( F ` k ) ) )
62	61	fveq2d 6195	. . . . . . . . . . 11 |- ( ph -> (Σ^ ` F ) = (Σ^ ` ( k e. Z |-> ( F ` k ) ) ) )
63	62	3ad2ant1 1082	. . . . . . . . . 10 |- ( ( ph /\ j e. Z /\ ( G ` j ) = z ) -> (Σ^ ` F ) = (Σ^ ` ( k e. Z |-> ( F ` k ) ) ) )
64	60, 63	breq12d 4666	. . . . . . . . 9 |- ( ( ph /\ j e. Z /\ ( G ` j ) = z ) -> ( z <_ (Σ^ ` F ) <-> (Σ^ ` ( k e. ( M ... j ) |-> ( F ` k ) ) ) <_ (Σ^ ` ( k e. Z |-> ( F ` k ) ) ) ) )
65	39, 64	mpbird 247	. . . . . . . 8 |- ( ( ph /\ j e. Z /\ ( G ` j ) = z ) -> z <_ (Σ^ ` F ) )
66	65	3exp 1264	. . . . . . 7 |- ( ph -> ( j e. Z -> ( ( G ` j ) = z -> z <_ (Σ^ ` F ) ) ) )
67	66	adantr 481	. . . . . 6 |- ( ( ph /\ z e. ran G ) -> ( j e. Z -> ( ( G ` j ) = z -> z <_ (Σ^ ` F ) ) ) )
68	67	rexlimdv 3030	. . . . 5 |- ( ( ph /\ z e. ran G ) -> ( E. j e. Z ( G ` j ) = z -> z <_ (Σ^ ` F ) ) )
69	32, 68	mpd 15	. . . 4 |- ( ( ph /\ z e. ran G ) -> z <_ (Σ^ ` F ) )
70	69	ralrimiva 2966	. . 3 |- ( ph -> A. z e. ran G z <_ (Σ^ ` F ) )
71		nfv 1843	. . . . . . . 8 |- F/ k ( ( ph /\ z e. RR ) /\ z < (Σ^ ` F ) )
72	20	a1i 11	. . . . . . . 8 |- ( ( ( ph /\ z e. RR ) /\ z < (Σ^ ` F ) ) -> Z e. _V )
73	5	ad4ant14 1293	. . . . . . . 8 |- ( ( ( ( ph /\ z e. RR ) /\ z < (Σ^ ` F ) ) /\ k e. Z ) -> ( F ` k ) e. ( 0 [,) +oo ) )
74		simplr 792	. . . . . . . 8 |- ( ( ( ph /\ z e. RR ) /\ z < (Σ^ ` F ) ) -> z e. RR )
75		simpr 477	. . . . . . . . . 10 |- ( ( ph /\ z < (Σ^ ` F ) ) -> z < (Σ^ ` F ) )
76	62	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ z < (Σ^ ` F ) ) -> (Σ^ ` F ) = (Σ^ ` ( k e. Z |-> ( F ` k ) ) ) )
77	75, 76	breqtrd 4679	. . . . . . . . 9 |- ( ( ph /\ z < (Σ^ ` F ) ) -> z < (Σ^ ` ( k e. Z |-> ( F ` k ) ) ) )
78	77	adantlr 751	. . . . . . . 8 |- ( ( ( ph /\ z e. RR ) /\ z < (Σ^ ` F ) ) -> z < (Σ^ ` ( k e. Z |-> ( F ` k ) ) ) )
79	71, 72, 73, 74, 78	sge0gtfsumgt 40660	. . . . . . 7 |- ( ( ( ph /\ z e. RR ) /\ z < (Σ^ ` F ) ) -> E. w e. ( ~P Z i^i Fin ) z < sum_ k e. w ( F ` k ) )
80	2	3ad2ant1 1082	. . . . . . . . . . . . 13 |- ( ( ph /\ w e. ( ~P Z i^i Fin ) /\ z < sum_ k e. w ( F ` k ) ) -> M e. ZZ )
81		elpwinss 39216	. . . . . . . . . . . . . 14 |- ( w e. ( ~P Z i^i Fin ) -> w C_ Z )
82	81	3ad2ant2 1083	. . . . . . . . . . . . 13 |- ( ( ph /\ w e. ( ~P Z i^i Fin ) /\ z < sum_ k e. w ( F ` k ) ) -> w C_ Z )
83		elinel2 3800	. . . . . . . . . . . . . 14 |- ( w e. ( ~P Z i^i Fin ) -> w e. Fin )
84	83	3ad2ant2 1083	. . . . . . . . . . . . 13 |- ( ( ph /\ w e. ( ~P Z i^i Fin ) /\ z < sum_ k e. w ( F ` k ) ) -> w e. Fin )
85	80, 1, 82, 84	uzfissfz 39542	. . . . . . . . . . . 12 |- ( ( ph /\ w e. ( ~P Z i^i Fin ) /\ z < sum_ k e. w ( F ` k ) ) -> E. j e. Z w C_ ( M ... j ) )
86	85	3adant1r 1319	. . . . . . . . . . 11 |- ( ( ( ph /\ z e. RR ) /\ w e. ( ~P Z i^i Fin ) /\ z < sum_ k e. w ( F ` k ) ) -> E. j e. Z w C_ ( M ... j ) )
87		simpl1r 1113	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ z e. RR ) /\ w e. ( ~P Z i^i Fin ) /\ z < sum_ k e. w ( F ` k ) ) /\ w C_ ( M ... j ) ) -> z e. RR )
88	83	adantl 482	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ w e. ( ~P Z i^i Fin ) ) -> w e. Fin )
89	61, 6	fmpt3d 6386	. . . . . . . . . . . . . . . . . . 19 |- ( ph -> F : Z --> RR )
90	89	ad2antrr 762	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ph /\ w e. ( ~P Z i^i Fin ) ) /\ k e. w ) -> F : Z --> RR )
91	81	sselda 3603	. . . . . . . . . . . . . . . . . . 19 |- ( ( w e. ( ~P Z i^i Fin ) /\ k e. w ) -> k e. Z )
92	91	adantll 750	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ph /\ w e. ( ~P Z i^i Fin ) ) /\ k e. w ) -> k e. Z )
93	90, 92	ffvelrnd 6360	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ w e. ( ~P Z i^i Fin ) ) /\ k e. w ) -> ( F ` k ) e. RR )
94	88, 93	fsumrecl 14465	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ w e. ( ~P Z i^i Fin ) ) -> sum_ k e. w ( F ` k ) e. RR )
95	94	ad4ant13 1292	. . . . . . . . . . . . . . 15 |- ( ( ( ( ph /\ z e. RR ) /\ w e. ( ~P Z i^i Fin ) ) /\ w C_ ( M ... j ) ) -> sum_ k e. w ( F ` k ) e. RR )
96	95	3adantl3 1219	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ z e. RR ) /\ w e. ( ~P Z i^i Fin ) /\ z < sum_ k e. w ( F ` k ) ) /\ w C_ ( M ... j ) ) -> sum_ k e. w ( F ` k ) e. RR )
97	35, 6	sylan2 491	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ k e. ( M ... j ) ) -> ( F ` k ) e. RR )
98	40, 97	fsumrecl 14465	. . . . . . . . . . . . . . . 16 |- ( ph -> sum_ k e. ( M ... j ) ( F ` k ) e. RR )
99	98	ad3antrrr 766	. . . . . . . . . . . . . . 15 |- ( ( ( ( ph /\ z e. RR ) /\ w e. ( ~P Z i^i Fin ) ) /\ w C_ ( M ... j ) ) -> sum_ k e. ( M ... j ) ( F ` k ) e. RR )
100	99	3adantl3 1219	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ z e. RR ) /\ w e. ( ~P Z i^i Fin ) /\ z < sum_ k e. w ( F ` k ) ) /\ w C_ ( M ... j ) ) -> sum_ k e. ( M ... j ) ( F ` k ) e. RR )
101		simpl3 1066	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ z e. RR ) /\ w e. ( ~P Z i^i Fin ) /\ z < sum_ k e. w ( F ` k ) ) /\ w C_ ( M ... j ) ) -> z < sum_ k e. w ( F ` k ) )
102	40	adantr 481	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ w C_ ( M ... j ) ) -> ( M ... j ) e. Fin )
103	97	adantlr 751	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ w C_ ( M ... j ) ) /\ k e. ( M ... j ) ) -> ( F ` k ) e. RR )
104		0xr 10086	. . . . . . . . . . . . . . . . . . . . 21 |- 0 e. RR*
105	104	a1i 11	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ k e. Z ) -> 0 e. RR* )
106		pnfxr 10092	. . . . . . . . . . . . . . . . . . . . 21 |- +oo e. RR*
107	106	a1i 11	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ k e. Z ) -> +oo e. RR* )
108		icogelb 12225	. . . . . . . . . . . . . . . . . . . 20 |- ( ( 0 e. RR* /\ +oo e. RR* /\ ( F ` k ) e. ( 0 [,) +oo ) ) -> 0 <_ ( F ` k ) )
109	105, 107, 5, 108	syl3anc 1326	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ k e. Z ) -> 0 <_ ( F ` k ) )
110	35, 109	sylan2 491	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ k e. ( M ... j ) ) -> 0 <_ ( F ` k ) )
111	110	adantlr 751	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ w C_ ( M ... j ) ) /\ k e. ( M ... j ) ) -> 0 <_ ( F ` k ) )
112		simpr 477	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ w C_ ( M ... j ) ) -> w C_ ( M ... j ) )
113	102, 103, 111, 112	fsumless 14528	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ w C_ ( M ... j ) ) -> sum_ k e. w ( F ` k ) <_ sum_ k e. ( M ... j ) ( F ` k ) )
114	113	adantlr 751	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ z e. RR ) /\ w C_ ( M ... j ) ) -> sum_ k e. w ( F ` k ) <_ sum_ k e. ( M ... j ) ( F ` k ) )
115	114	3ad2antl1 1223	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ z e. RR ) /\ w e. ( ~P Z i^i Fin ) /\ z < sum_ k e. w ( F ` k ) ) /\ w C_ ( M ... j ) ) -> sum_ k e. w ( F ` k ) <_ sum_ k e. ( M ... j ) ( F ` k ) )
116	87, 96, 100, 101, 115	ltletrd 10197	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ z e. RR ) /\ w e. ( ~P Z i^i Fin ) /\ z < sum_ k e. w ( F ` k ) ) /\ w C_ ( M ... j ) ) -> z < sum_ k e. ( M ... j ) ( F ` k ) )
117	116	ex 450	. . . . . . . . . . . 12 |- ( ( ( ph /\ z e. RR ) /\ w e. ( ~P Z i^i Fin ) /\ z < sum_ k e. w ( F ` k ) ) -> ( w C_ ( M ... j ) -> z < sum_ k e. ( M ... j ) ( F ` k ) ) )
118	117	reximdv 3016	. . . . . . . . . . 11 |- ( ( ( ph /\ z e. RR ) /\ w e. ( ~P Z i^i Fin ) /\ z < sum_ k e. w ( F ` k ) ) -> ( E. j e. Z w C_ ( M ... j ) -> E. j e. Z z < sum_ k e. ( M ... j ) ( F ` k ) ) )
119	86, 118	mpd 15	. . . . . . . . . 10 |- ( ( ( ph /\ z e. RR ) /\ w e. ( ~P Z i^i Fin ) /\ z < sum_ k e. w ( F ` k ) ) -> E. j e. Z z < sum_ k e. ( M ... j ) ( F ` k ) )
120	119	3exp 1264	. . . . . . . . 9 |- ( ( ph /\ z e. RR ) -> ( w e. ( ~P Z i^i Fin ) -> ( z < sum_ k e. w ( F ` k ) -> E. j e. Z z < sum_ k e. ( M ... j ) ( F ` k ) ) ) )
121	120	adantr 481	. . . . . . . 8 |- ( ( ( ph /\ z e. RR ) /\ z < (Σ^ ` F ) ) -> ( w e. ( ~P Z i^i Fin ) -> ( z < sum_ k e. w ( F ` k ) -> E. j e. Z z < sum_ k e. ( M ... j ) ( F ` k ) ) ) )
122	121	rexlimdv 3030	. . . . . . 7 |- ( ( ( ph /\ z e. RR ) /\ z < (Σ^ ` F ) ) -> ( E. w e. ( ~P Z i^i Fin ) z < sum_ k e. w ( F ` k ) -> E. j e. Z z < sum_ k e. ( M ... j ) ( F ` k ) ) )
123	79, 122	mpd 15	. . . . . 6 |- ( ( ( ph /\ z e. RR ) /\ z < (Σ^ ` F ) ) -> E. j e. Z z < sum_ k e. ( M ... j ) ( F ` k ) )
124	9	ffnd 6046	. . . . . . . . . . . . . . 15 |- ( ph -> seq M ( + , F ) Fn Z )
125	124	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ph /\ j e. Z ) -> seq M ( + , F ) Fn Z )
126	50, 48	sylibr 224	. . . . . . . . . . . . . 14 |- ( ( ph /\ j e. Z ) -> j e. Z )
127		fnfvelrn 6356	. . . . . . . . . . . . . 14 |- ( ( seq M ( + , F ) Fn Z /\ j e. Z ) -> ( seq M ( + , F ) ` j ) e. ran seq M ( + , F ) )
128	125, 126, 127	syl2anc 693	. . . . . . . . . . . . 13 |- ( ( ph /\ j e. Z ) -> ( seq M ( + , F ) ` j ) e. ran seq M ( + , F ) )
129	10	a1i 11	. . . . . . . . . . . . . . 15 |- ( ( ph /\ j e. Z ) -> G = seq M ( + , F ) )
130	129	rneqd 5353	. . . . . . . . . . . . . 14 |- ( ( ph /\ j e. Z ) -> ran G = ran seq M ( + , F ) )
131	53, 130	eleq12d 2695	. . . . . . . . . . . . 13 |- ( ( ph /\ j e. Z ) -> ( sum_ k e. ( M ... j ) ( F ` k ) e. ran G <-> ( seq M ( + , F ) ` j ) e. ran seq M ( + , F ) ) )
132	128, 131	mpbird 247	. . . . . . . . . . . 12 |- ( ( ph /\ j e. Z ) -> sum_ k e. ( M ... j ) ( F ` k ) e. ran G )
133	132	adantlr 751	. . . . . . . . . . 11 |- ( ( ( ph /\ z e. RR ) /\ j e. Z ) -> sum_ k e. ( M ... j ) ( F ` k ) e. ran G )
134	133	3adant3 1081	. . . . . . . . . 10 |- ( ( ( ph /\ z e. RR ) /\ j e. Z /\ z < sum_ k e. ( M ... j ) ( F ` k ) ) -> sum_ k e. ( M ... j ) ( F ` k ) e. ran G )
135		simp3 1063	. . . . . . . . . 10 |- ( ( ( ph /\ z e. RR ) /\ j e. Z /\ z < sum_ k e. ( M ... j ) ( F ` k ) ) -> z < sum_ k e. ( M ... j ) ( F ` k ) )
136		breq2 4657	. . . . . . . . . . 11 |- ( y = sum_ k e. ( M ... j ) ( F ` k ) -> ( z < y <-> z < sum_ k e. ( M ... j ) ( F ` k ) ) )
137	136	rspcev 3309	. . . . . . . . . 10 |- ( ( sum_ k e. ( M ... j ) ( F ` k ) e. ran G /\ z < sum_ k e. ( M ... j ) ( F ` k ) ) -> E. y e. ran G z < y )
138	134, 135, 137	syl2anc 693	. . . . . . . . 9 |- ( ( ( ph /\ z e. RR ) /\ j e. Z /\ z < sum_ k e. ( M ... j ) ( F ` k ) ) -> E. y e. ran G z < y )
139	138	3exp 1264	. . . . . . . 8 |- ( ( ph /\ z e. RR ) -> ( j e. Z -> ( z < sum_ k e. ( M ... j ) ( F ` k ) -> E. y e. ran G z < y ) ) )
140	139	rexlimdv 3030	. . . . . . 7 |- ( ( ph /\ z e. RR ) -> ( E. j e. Z z < sum_ k e. ( M ... j ) ( F ` k ) -> E. y e. ran G z < y ) )
141	140	adantr 481	. . . . . 6 |- ( ( ( ph /\ z e. RR ) /\ z < (Σ^ ` F ) ) -> ( E. j e. Z z < sum_ k e. ( M ... j ) ( F ` k ) -> E. y e. ran G z < y ) )
142	123, 141	mpd 15	. . . . 5 |- ( ( ( ph /\ z e. RR ) /\ z < (Σ^ ` F ) ) -> E. y e. ran G z < y )
143	142	ex 450	. . . 4 |- ( ( ph /\ z e. RR ) -> ( z < (Σ^ ` F ) -> E. y e. ran G z < y ) )
144	143	ralrimiva 2966	. . 3 |- ( ph -> A. z e. RR ( z < (Σ^ ` F ) -> E. y e. ran G z < y ) )
145		supxr2 12144	. . 3 |- ( ( ( ran G C_ RR* /\ (Σ^ ` F ) e. RR* ) /\ ( A. z e. ran G z <_ (Σ^ ` F ) /\ A. z e. RR ( z < (Σ^ ` F ) -> E. y e. ran G z < y ) ) ) -> sup ( ran G , RR* , < ) = (Σ^ ` F ) )
146	18, 25, 70, 144, 145	syl22anc 1327	. 2 |- ( ph -> sup ( ran G , RR* , < ) = (Σ^ ` F ) )
147	146	eqcomd 2628	1 |- ( ph -> (Σ^ ` F ) = sup ( ran G , RR* , < ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   _Vcvv 3200   i^i cin 3573   C_ wss 3574   ~Pcpw 4158   class class class wbr 4653   |-> cmpt 4729   ran crn 5115   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   ZZcz 11377   ZZ>=cuz 11687   [,)cico 12177   [,]cicc 12178   ...cfz 12326   seqcseq 12801   sum_csu 14416  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

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-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-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  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-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  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-sumge0 40580

This theorem is referenced by:  voliunsge0lem  40689  ovolval2  40858