Theorem sge0seq 43602

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	⊢ (𝜑 → 𝑀 ∈ ℤ)
sge0seq.z	⊢ 𝑍 = (ℤ≥‘𝑀)
sge0seq.f	⊢ (𝜑 → 𝐹:𝑍⟶(0[,)+∞))
sge0seq.g	⊢ 𝐺 = seq𝑀( + , 𝐹)

Assertion

Ref	Expression
sge0seq	⊢ (𝜑 → (Σ^‘𝐹) = sup(ran 𝐺, ℝ*, < ))

Proof of Theorem sge0seq

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

Step	Hyp	Ref	Expression
1		sge0seq.z	. . . . . . 7 ⊢ 𝑍 = (ℤ≥‘𝑀)
2		sge0seq.m	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℤ)
3		rge0ssre 13009	. . . . . . . 8 ⊢ (0[,)+∞) ⊆ ℝ
4		sge0seq.f	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:𝑍⟶(0[,)+∞))
5	4	ffvelrnda 6882	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ (0[,)+∞))
6	3, 5	sseldi 3885	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ)
7		readdcl 10777	. . . . . . . 8 ⊢ ((𝑘 ∈ ℝ ∧ 𝑖 ∈ ℝ) → (𝑘 + 𝑖) ∈ ℝ)
8	7	adantl 485	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ ℝ ∧ 𝑖 ∈ ℝ)) → (𝑘 + 𝑖) ∈ ℝ)
9	1, 2, 6, 8	seqf 13562	. . . . . 6 ⊢ (𝜑 → seq𝑀( + , 𝐹):𝑍⟶ℝ)
10		sge0seq.g	. . . . . . . 8 ⊢ 𝐺 = seq𝑀( + , 𝐹)
11	10	a1i 11	. . . . . . 7 ⊢ (𝜑 → 𝐺 = seq𝑀( + , 𝐹))
12	11	feq1d 6508	. . . . . 6 ⊢ (𝜑 → (𝐺:𝑍⟶ℝ ↔ seq𝑀( + , 𝐹):𝑍⟶ℝ))
13	9, 12	mpbird 260	. . . . 5 ⊢ (𝜑 → 𝐺:𝑍⟶ℝ)
14	13	frnd 6531	. . . 4 ⊢ (𝜑 → ran 𝐺 ⊆ ℝ)
15		ressxr 10842	. . . . 5 ⊢ ℝ ⊆ ℝ*
16	15	a1i 11	. . . 4 ⊢ (𝜑 → ℝ ⊆ ℝ*)
17	14, 16	sstrd 3897	. . 3 ⊢ (𝜑 → ran 𝐺 ⊆ ℝ*)
18	1	fvexi 6709	. . . . 5 ⊢ 𝑍 ∈ V
19	18	a1i 11	. . . 4 ⊢ (𝜑 → 𝑍 ∈ V)
20		icossicc 12989	. . . . . 6 ⊢ (0[,)+∞) ⊆ (0[,]+∞)
21	20	a1i 11	. . . . 5 ⊢ (𝜑 → (0[,)+∞) ⊆ (0[,]+∞))
22	4, 21	fssd 6541	. . . 4 ⊢ (𝜑 → 𝐹:𝑍⟶(0[,]+∞))
23	19, 22	sge0xrcl 43541	. . 3 ⊢ (𝜑 → (Σ^‘𝐹) ∈ ℝ*)
24		simpr 488	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ ran 𝐺) → 𝑧 ∈ ran 𝐺)
25	13	ffnd 6524	. . . . . . . 8 ⊢ (𝜑 → 𝐺 Fn 𝑍)
26		fvelrnb 6751	. . . . . . . 8 ⊢ (𝐺 Fn 𝑍 → (𝑧 ∈ ran 𝐺 ↔ ∃𝑗 ∈ 𝑍 (𝐺‘𝑗) = 𝑧))
27	25, 26	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝑧 ∈ ran 𝐺 ↔ ∃𝑗 ∈ 𝑍 (𝐺‘𝑗) = 𝑧))
28	27	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ ran 𝐺) → (𝑧 ∈ ran 𝐺 ↔ ∃𝑗 ∈ 𝑍 (𝐺‘𝑗) = 𝑧))
29	24, 28	mpbid 235	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ ran 𝐺) → ∃𝑗 ∈ 𝑍 (𝐺‘𝑗) = 𝑧)
30	20, 5	sseldi 3885	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ (0[,]+∞))
31		elfzuz 13073	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (𝑀...𝑗) → 𝑘 ∈ (ℤ≥‘𝑀))
32	31, 1	eleqtrrdi 2842	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (𝑀...𝑗) → 𝑘 ∈ 𝑍)
33	32	ssriv 3891	. . . . . . . . . . . 12 ⊢ (𝑀...𝑗) ⊆ 𝑍
34	33	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑀...𝑗) ⊆ 𝑍)
35	19, 30, 34	sge0lessmpt 43555	. . . . . . . . . 10 ⊢ (𝜑 → (Σ^‘(𝑘 ∈ (𝑀...𝑗) ↦ (𝐹‘𝑘))) ≤ (Σ^‘(𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘))))
36	35	3ad2ant1 1135	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ (𝐺‘𝑗) = 𝑧) → (Σ^‘(𝑘 ∈ (𝑀...𝑗) ↦ (𝐹‘𝑘))) ≤ (Σ^‘(𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘))))
37		fzfid 13511	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑀...𝑗) ∈ Fin)
38	32, 5	sylan2 596	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑗)) → (𝐹‘𝑘) ∈ (0[,)+∞))
39	37, 38	sge0fsummpt 43546	. . . . . . . . . . . . 13 ⊢ (𝜑 → (Σ^‘(𝑘 ∈ (𝑀...𝑗) ↦ (𝐹‘𝑘))) = Σ𝑘 ∈ (𝑀...𝑗)(𝐹‘𝑘))
40	39	3ad2ant1 1135	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ (𝐺‘𝑗) = 𝑧) → (Σ^‘(𝑘 ∈ (𝑀...𝑗) ↦ (𝐹‘𝑘))) = Σ𝑘 ∈ (𝑀...𝑗)(𝐹‘𝑘))
41		simpll 767	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (𝑀...𝑗)) → 𝜑)
42	32	adantl 485	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (𝑀...𝑗)) → 𝑘 ∈ 𝑍)
43		eqidd 2737	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = (𝐹‘𝑘))
44	41, 42, 43	syl2anc 587	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (𝑀...𝑗)) → (𝐹‘𝑘) = (𝐹‘𝑘))
45	1	eleq2i 2822	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ 𝑍 ↔ 𝑗 ∈ (ℤ≥‘𝑀))
46	45	biimpi 219	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ 𝑍 → 𝑗 ∈ (ℤ≥‘𝑀))
47	46	adantl 485	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ (ℤ≥‘𝑀))
48	6	recnd 10826	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)
49	41, 42, 48	syl2anc 587	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (𝑀...𝑗)) → (𝐹‘𝑘) ∈ ℂ)
50	44, 47, 49	fsumser 15259	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → Σ𝑘 ∈ (𝑀...𝑗)(𝐹‘𝑘) = (seq𝑀( + , 𝐹)‘𝑗))
51	50	3adant3 1134	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ (𝐺‘𝑗) = 𝑧) → Σ𝑘 ∈ (𝑀...𝑗)(𝐹‘𝑘) = (seq𝑀( + , 𝐹)‘𝑗))
52	40, 51	eqtrd 2771	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ (𝐺‘𝑗) = 𝑧) → (Σ^‘(𝑘 ∈ (𝑀...𝑗) ↦ (𝐹‘𝑘))) = (seq𝑀( + , 𝐹)‘𝑗))
53	10	eqcomi 2745	. . . . . . . . . . . . 13 ⊢ seq𝑀( + , 𝐹) = 𝐺
54	53	fveq1i 6696	. . . . . . . . . . . 12 ⊢ (seq𝑀( + , 𝐹)‘𝑗) = (𝐺‘𝑗)
55	54	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ (𝐺‘𝑗) = 𝑧) → (seq𝑀( + , 𝐹)‘𝑗) = (𝐺‘𝑗))
56		simp3 1140	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ (𝐺‘𝑗) = 𝑧) → (𝐺‘𝑗) = 𝑧)
57	52, 55, 56	3eqtrrd 2776	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ (𝐺‘𝑗) = 𝑧) → 𝑧 = (Σ^‘(𝑘 ∈ (𝑀...𝑗) ↦ (𝐹‘𝑘))))
58	4	feqmptd 6758	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 = (𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘)))
59	58	fveq2d 6699	. . . . . . . . . . 11 ⊢ (𝜑 → (Σ^‘𝐹) = (Σ^‘(𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘))))
60	59	3ad2ant1 1135	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ (𝐺‘𝑗) = 𝑧) → (Σ^‘𝐹) = (Σ^‘(𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘))))
61	57, 60	breq12d 5052	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ (𝐺‘𝑗) = 𝑧) → (𝑧 ≤ (Σ^‘𝐹) ↔ (Σ^‘(𝑘 ∈ (𝑀...𝑗) ↦ (𝐹‘𝑘))) ≤ (Σ^‘(𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘)))))
62	36, 61	mpbird 260	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ (𝐺‘𝑗) = 𝑧) → 𝑧 ≤ (Σ^‘𝐹))
63	62	3exp 1121	. . . . . . 7 ⊢ (𝜑 → (𝑗 ∈ 𝑍 → ((𝐺‘𝑗) = 𝑧 → 𝑧 ≤ (Σ^‘𝐹))))
64	63	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ ran 𝐺) → (𝑗 ∈ 𝑍 → ((𝐺‘𝑗) = 𝑧 → 𝑧 ≤ (Σ^‘𝐹))))
65	64	rexlimdv 3192	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ ran 𝐺) → (∃𝑗 ∈ 𝑍 (𝐺‘𝑗) = 𝑧 → 𝑧 ≤ (Σ^‘𝐹)))
66	29, 65	mpd 15	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ ran 𝐺) → 𝑧 ≤ (Σ^‘𝐹))
67	66	ralrimiva 3095	. . 3 ⊢ (𝜑 → ∀𝑧 ∈ ran 𝐺 𝑧 ≤ (Σ^‘𝐹))
68		nfv 1922	. . . . . . . 8 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑧 < (Σ^‘𝐹))
69	18	a1i 11	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑧 < (Σ^‘𝐹)) → 𝑍 ∈ V)
70	5	ad4ant14 752	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑧 < (Σ^‘𝐹)) ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ (0[,)+∞))
71		simplr 769	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑧 < (Σ^‘𝐹)) → 𝑧 ∈ ℝ)
72		simpr 488	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 < (Σ^‘𝐹)) → 𝑧 < (Σ^‘𝐹))
73	59	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 < (Σ^‘𝐹)) → (Σ^‘𝐹) = (Σ^‘(𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘))))
74	72, 73	breqtrd 5065	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 < (Σ^‘𝐹)) → 𝑧 < (Σ^‘(𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘))))
75	74	adantlr 715	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑧 < (Σ^‘𝐹)) → 𝑧 < (Σ^‘(𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘))))
76	68, 69, 70, 71, 75	sge0gtfsumgt 43599	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑧 < (Σ^‘𝐹)) → ∃𝑤 ∈ (𝒫 𝑍 ∩ Fin)𝑧 < Σ𝑘 ∈ 𝑤 (𝐹‘𝑘))
77	2	3ad2ant1 1135	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑧 < Σ𝑘 ∈ 𝑤 (𝐹‘𝑘)) → 𝑀 ∈ ℤ)
78		elpwinss 42211	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ (𝒫 𝑍 ∩ Fin) → 𝑤 ⊆ 𝑍)
79	78	3ad2ant2 1136	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑧 < Σ𝑘 ∈ 𝑤 (𝐹‘𝑘)) → 𝑤 ⊆ 𝑍)
80		elinel2 4096	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ (𝒫 𝑍 ∩ Fin) → 𝑤 ∈ Fin)
81	80	3ad2ant2 1136	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑧 < Σ𝑘 ∈ 𝑤 (𝐹‘𝑘)) → 𝑤 ∈ Fin)
82	77, 1, 79, 81	uzfissfz 42479	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑧 < Σ𝑘 ∈ 𝑤 (𝐹‘𝑘)) → ∃𝑗 ∈ 𝑍 𝑤 ⊆ (𝑀...𝑗))
83	82	3adant1r 1179	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑤 ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑧 < Σ𝑘 ∈ 𝑤 (𝐹‘𝑘)) → ∃𝑗 ∈ 𝑍 𝑤 ⊆ (𝑀...𝑗))
84		simpl1r 1227	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑤 ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑧 < Σ𝑘 ∈ 𝑤 (𝐹‘𝑘)) ∧ 𝑤 ⊆ (𝑀...𝑗)) → 𝑧 ∈ ℝ)
85	80	adantl 485	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝒫 𝑍 ∩ Fin)) → 𝑤 ∈ Fin)
86	58, 6	fmpt3d 6911	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ)
87	86	ad2antrr 726	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑘 ∈ 𝑤) → 𝐹:𝑍⟶ℝ)
88	78	sselda 3887	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑤 ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑘 ∈ 𝑤) → 𝑘 ∈ 𝑍)
89	88	adantll 714	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑘 ∈ 𝑤) → 𝑘 ∈ 𝑍)
90	87, 89	ffvelrnd 6883	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑘 ∈ 𝑤) → (𝐹‘𝑘) ∈ ℝ)
91	85, 90	fsumrecl 15263	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝒫 𝑍 ∩ Fin)) → Σ𝑘 ∈ 𝑤 (𝐹‘𝑘) ∈ ℝ)
92	91	ad4ant13 751	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑤 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑤 ⊆ (𝑀...𝑗)) → Σ𝑘 ∈ 𝑤 (𝐹‘𝑘) ∈ ℝ)
93	92	3adantl3 1170	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑤 ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑧 < Σ𝑘 ∈ 𝑤 (𝐹‘𝑘)) ∧ 𝑤 ⊆ (𝑀...𝑗)) → Σ𝑘 ∈ 𝑤 (𝐹‘𝑘) ∈ ℝ)
94	32, 6	sylan2 596	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑗)) → (𝐹‘𝑘) ∈ ℝ)
95	37, 94	fsumrecl 15263	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀...𝑗)(𝐹‘𝑘) ∈ ℝ)
96	95	ad3antrrr 730	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑤 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑤 ⊆ (𝑀...𝑗)) → Σ𝑘 ∈ (𝑀...𝑗)(𝐹‘𝑘) ∈ ℝ)
97	96	3adantl3 1170	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑤 ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑧 < Σ𝑘 ∈ 𝑤 (𝐹‘𝑘)) ∧ 𝑤 ⊆ (𝑀...𝑗)) → Σ𝑘 ∈ (𝑀...𝑗)(𝐹‘𝑘) ∈ ℝ)
98		simpl3 1195	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑤 ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑧 < Σ𝑘 ∈ 𝑤 (𝐹‘𝑘)) ∧ 𝑤 ⊆ (𝑀...𝑗)) → 𝑧 < Σ𝑘 ∈ 𝑤 (𝐹‘𝑘))
99	37	adantr 484	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑤 ⊆ (𝑀...𝑗)) → (𝑀...𝑗) ∈ Fin)
100	94	adantlr 715	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑤 ⊆ (𝑀...𝑗)) ∧ 𝑘 ∈ (𝑀...𝑗)) → (𝐹‘𝑘) ∈ ℝ)
101		0xr 10845	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 0 ∈ ℝ*
102	101	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 0 ∈ ℝ*)
103		pnfxr 10852	. . . . . . . . . . . . . . . . . . . . 21 ⊢ +∞ ∈ ℝ*
104	103	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → +∞ ∈ ℝ*)
105		icogelb 12951	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ (𝐹‘𝑘) ∈ (0[,)+∞)) → 0 ≤ (𝐹‘𝑘))
106	102, 104, 5, 105	syl3anc 1373	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 0 ≤ (𝐹‘𝑘))
107	32, 106	sylan2 596	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑗)) → 0 ≤ (𝐹‘𝑘))
108	107	adantlr 715	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑤 ⊆ (𝑀...𝑗)) ∧ 𝑘 ∈ (𝑀...𝑗)) → 0 ≤ (𝐹‘𝑘))
109		simpr 488	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑤 ⊆ (𝑀...𝑗)) → 𝑤 ⊆ (𝑀...𝑗))
110	99, 100, 108, 109	fsumless 15323	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ⊆ (𝑀...𝑗)) → Σ𝑘 ∈ 𝑤 (𝐹‘𝑘) ≤ Σ𝑘 ∈ (𝑀...𝑗)(𝐹‘𝑘))
111	110	adantlr 715	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑤 ⊆ (𝑀...𝑗)) → Σ𝑘 ∈ 𝑤 (𝐹‘𝑘) ≤ Σ𝑘 ∈ (𝑀...𝑗)(𝐹‘𝑘))
112	111	3ad2antl1 1187	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑤 ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑧 < Σ𝑘 ∈ 𝑤 (𝐹‘𝑘)) ∧ 𝑤 ⊆ (𝑀...𝑗)) → Σ𝑘 ∈ 𝑤 (𝐹‘𝑘) ≤ Σ𝑘 ∈ (𝑀...𝑗)(𝐹‘𝑘))
113	84, 93, 97, 98, 112	ltletrd 10957	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑤 ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑧 < Σ𝑘 ∈ 𝑤 (𝐹‘𝑘)) ∧ 𝑤 ⊆ (𝑀...𝑗)) → 𝑧 < Σ𝑘 ∈ (𝑀...𝑗)(𝐹‘𝑘))
114	113	ex 416	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑤 ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑧 < Σ𝑘 ∈ 𝑤 (𝐹‘𝑘)) → (𝑤 ⊆ (𝑀...𝑗) → 𝑧 < Σ𝑘 ∈ (𝑀...𝑗)(𝐹‘𝑘)))
115	114	reximdv 3182	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑤 ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑧 < Σ𝑘 ∈ 𝑤 (𝐹‘𝑘)) → (∃𝑗 ∈ 𝑍 𝑤 ⊆ (𝑀...𝑗) → ∃𝑗 ∈ 𝑍 𝑧 < Σ𝑘 ∈ (𝑀...𝑗)(𝐹‘𝑘)))
116	83, 115	mpd 15	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑤 ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑧 < Σ𝑘 ∈ 𝑤 (𝐹‘𝑘)) → ∃𝑗 ∈ 𝑍 𝑧 < Σ𝑘 ∈ (𝑀...𝑗)(𝐹‘𝑘))
117	116	3exp 1121	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → (𝑤 ∈ (𝒫 𝑍 ∩ Fin) → (𝑧 < Σ𝑘 ∈ 𝑤 (𝐹‘𝑘) → ∃𝑗 ∈ 𝑍 𝑧 < Σ𝑘 ∈ (𝑀...𝑗)(𝐹‘𝑘))))
118	117	adantr 484	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑧 < (Σ^‘𝐹)) → (𝑤 ∈ (𝒫 𝑍 ∩ Fin) → (𝑧 < Σ𝑘 ∈ 𝑤 (𝐹‘𝑘) → ∃𝑗 ∈ 𝑍 𝑧 < Σ𝑘 ∈ (𝑀...𝑗)(𝐹‘𝑘))))
119	118	rexlimdv 3192	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑧 < (Σ^‘𝐹)) → (∃𝑤 ∈ (𝒫 𝑍 ∩ Fin)𝑧 < Σ𝑘 ∈ 𝑤 (𝐹‘𝑘) → ∃𝑗 ∈ 𝑍 𝑧 < Σ𝑘 ∈ (𝑀...𝑗)(𝐹‘𝑘)))
120	76, 119	mpd 15	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑧 < (Σ^‘𝐹)) → ∃𝑗 ∈ 𝑍 𝑧 < Σ𝑘 ∈ (𝑀...𝑗)(𝐹‘𝑘))
121	9	ffnd 6524	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → seq𝑀( + , 𝐹) Fn 𝑍)
122	121	adantr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → seq𝑀( + , 𝐹) Fn 𝑍)
123	47, 45	sylibr 237	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ 𝑍)
124		fnfvelrn 6879	. . . . . . . . . . . . . 14 ⊢ ((seq𝑀( + , 𝐹) Fn 𝑍 ∧ 𝑗 ∈ 𝑍) → (seq𝑀( + , 𝐹)‘𝑗) ∈ ran seq𝑀( + , 𝐹))
125	122, 123, 124	syl2anc 587	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (seq𝑀( + , 𝐹)‘𝑗) ∈ ran seq𝑀( + , 𝐹))
126	10	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝐺 = seq𝑀( + , 𝐹))
127	126	rneqd 5792	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ran 𝐺 = ran seq𝑀( + , 𝐹))
128	50, 127	eleq12d 2825	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (Σ𝑘 ∈ (𝑀...𝑗)(𝐹‘𝑘) ∈ ran 𝐺 ↔ (seq𝑀( + , 𝐹)‘𝑗) ∈ ran seq𝑀( + , 𝐹)))
129	125, 128	mpbird 260	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → Σ𝑘 ∈ (𝑀...𝑗)(𝐹‘𝑘) ∈ ran 𝐺)
130	129	adantlr 715	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) → Σ𝑘 ∈ (𝑀...𝑗)(𝐹‘𝑘) ∈ ran 𝐺)
131	130	3adant3 1134	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑗 ∈ 𝑍 ∧ 𝑧 < Σ𝑘 ∈ (𝑀...𝑗)(𝐹‘𝑘)) → Σ𝑘 ∈ (𝑀...𝑗)(𝐹‘𝑘) ∈ ran 𝐺)
132		simp3 1140	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑗 ∈ 𝑍 ∧ 𝑧 < Σ𝑘 ∈ (𝑀...𝑗)(𝐹‘𝑘)) → 𝑧 < Σ𝑘 ∈ (𝑀...𝑗)(𝐹‘𝑘))
133		breq2 5043	. . . . . . . . . . 11 ⊢ (𝑦 = Σ𝑘 ∈ (𝑀...𝑗)(𝐹‘𝑘) → (𝑧 < 𝑦 ↔ 𝑧 < Σ𝑘 ∈ (𝑀...𝑗)(𝐹‘𝑘)))
134	133	rspcev 3527	. . . . . . . . . 10 ⊢ ((Σ𝑘 ∈ (𝑀...𝑗)(𝐹‘𝑘) ∈ ran 𝐺 ∧ 𝑧 < Σ𝑘 ∈ (𝑀...𝑗)(𝐹‘𝑘)) → ∃𝑦 ∈ ran 𝐺 𝑧 < 𝑦)
135	131, 132, 134	syl2anc 587	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑗 ∈ 𝑍 ∧ 𝑧 < Σ𝑘 ∈ (𝑀...𝑗)(𝐹‘𝑘)) → ∃𝑦 ∈ ran 𝐺 𝑧 < 𝑦)
136	135	3exp 1121	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → (𝑗 ∈ 𝑍 → (𝑧 < Σ𝑘 ∈ (𝑀...𝑗)(𝐹‘𝑘) → ∃𝑦 ∈ ran 𝐺 𝑧 < 𝑦)))
137	136	rexlimdv 3192	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → (∃𝑗 ∈ 𝑍 𝑧 < Σ𝑘 ∈ (𝑀...𝑗)(𝐹‘𝑘) → ∃𝑦 ∈ ran 𝐺 𝑧 < 𝑦))
138	137	adantr 484	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑧 < (Σ^‘𝐹)) → (∃𝑗 ∈ 𝑍 𝑧 < Σ𝑘 ∈ (𝑀...𝑗)(𝐹‘𝑘) → ∃𝑦 ∈ ran 𝐺 𝑧 < 𝑦))
139	120, 138	mpd 15	. . . . 5 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑧 < (Σ^‘𝐹)) → ∃𝑦 ∈ ran 𝐺 𝑧 < 𝑦)
140	139	ex 416	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → (𝑧 < (Σ^‘𝐹) → ∃𝑦 ∈ ran 𝐺 𝑧 < 𝑦))
141	140	ralrimiva 3095	. . 3 ⊢ (𝜑 → ∀𝑧 ∈ ℝ (𝑧 < (Σ^‘𝐹) → ∃𝑦 ∈ ran 𝐺 𝑧 < 𝑦))
142		supxr2 12869	. . 3 ⊢ (((ran 𝐺 ⊆ ℝ* ∧ (Σ^‘𝐹) ∈ ℝ*) ∧ (∀𝑧 ∈ ran 𝐺 𝑧 ≤ (Σ^‘𝐹) ∧ ∀𝑧 ∈ ℝ (𝑧 < (Σ^‘𝐹) → ∃𝑦 ∈ ran 𝐺 𝑧 < 𝑦))) → sup(ran 𝐺, ℝ*, < ) = (Σ^‘𝐹))
143	17, 23, 67, 141, 142	syl22anc 839	. 2 ⊢ (𝜑 → sup(ran 𝐺, ℝ*, < ) = (Σ^‘𝐹))
144	143	eqcomd 2742	1 ⊢ (𝜑 → (Σ^‘𝐹) = sup(ran 𝐺, ℝ*, < ))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1089   = wceq 1543   ∈ wcel 2112  ∀wral 3051  ∃wrex 3052  Vcvv 3398   ∩ cin 3852   ⊆ wss 3853  𝒫 cpw 4499   class class class wbr 5039   ↦ cmpt 5120  ran crn 5537   Fn wfn 6353  ⟶wf 6354  ‘cfv 6358  (class class class)co 7191  Fincfn 8604  supcsup 9034  ℂcc 10692  ℝcr 10693  0cc0 10694   + caddc 10697  +∞cpnf 10829  ℝ^*cxr 10831   < clt 10832   ≤ cle 10833  ℤcz 12141  ℤ_≥cuz 12403  [,)cico 12902  [,]cicc 12903  ...cfz 13060  seqcseq 13539  Σcsu 15214  Σ^{^}csumge0 43518

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-rep 5164  ax-sep 5177  ax-nul 5184  ax-pow 5243  ax-pr 5307  ax-un 7501  ax-inf2 9234  ax-cnex 10750  ax-resscn 10751  ax-1cn 10752  ax-icn 10753  ax-addcl 10754  ax-addrcl 10755  ax-mulcl 10756  ax-mulrcl 10757  ax-mulcom 10758  ax-addass 10759  ax-mulass 10760  ax-distr 10761  ax-i2m1 10762  ax-1ne0 10763  ax-1rid 10764  ax-rnegex 10765  ax-rrecex 10766  ax-cnre 10767  ax-pre-lttri 10768  ax-pre-lttrn 10769  ax-pre-ltadd 10770  ax-pre-mulgt0 10771  ax-pre-sup 10772

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ne 2933  df-nel 3037  df-ral 3056  df-rex 3057  df-reu 3058  df-rmo 3059  df-rab 3060  df-v 3400  df-sbc 3684  df-csb 3799  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-pss 3872  df-nul 4224  df-if 4426  df-pw 4501  df-sn 4528  df-pr 4530  df-tp 4532  df-op 4534  df-uni 4806  df-int 4846  df-iun 4892  df-br 5040  df-opab 5102  df-mpt 5121  df-tr 5147  df-id 5440  df-eprel 5445  df-po 5453  df-so 5454  df-fr 5494  df-se 5495  df-we 5496  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-pred 6140  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6316  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-isom 6367  df-riota 7148  df-ov 7194  df-oprab 7195  df-mpo 7196  df-om 7623  df-1st 7739  df-2nd 7740  df-wrecs 8025  df-recs 8086  df-rdg 8124  df-1o 8180  df-er 8369  df-en 8605  df-dom 8606  df-sdom 8607  df-fin 8608  df-sup 9036  df-oi 9104  df-card 9520  df-pnf 10834  df-mnf 10835  df-xr 10836  df-ltxr 10837  df-le 10838  df-sub 11029  df-neg 11030  df-div 11455  df-nn 11796  df-2 11858  df-3 11859  df-n0 12056  df-z 12142  df-uz 12404  df-rp 12552  df-ico 12906  df-icc 12907  df-fz 13061  df-fzo 13204  df-seq 13540  df-exp 13601  df-hash 13862  df-cj 14627  df-re 14628  df-im 14629  df-sqrt 14763  df-abs 14764  df-clim 15014  df-sum 15215  df-sumge0 43519

This theorem is referenced by:  voliunsge0lem  43628  ovolval2  43800