Theorem sge0seq 46444

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 13417	. . . . . . . 8 ⊢ (0[,)+∞) ⊆ ℝ
4		sge0seq.f	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:𝑍⟶(0[,)+∞))
5	4	ffvelcdmda 7056	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ (0[,)+∞))
6	3, 5	sselid 3944	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ)
7		readdcl 11151	. . . . . . . 8 ⊢ ((𝑘 ∈ ℝ ∧ 𝑖 ∈ ℝ) → (𝑘 + 𝑖) ∈ ℝ)
8	7	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ ℝ ∧ 𝑖 ∈ ℝ)) → (𝑘 + 𝑖) ∈ ℝ)
9	1, 2, 6, 8	seqf 13988	. . . . . 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 6670	. . . . . 6 ⊢ (𝜑 → (𝐺:𝑍⟶ℝ ↔ seq𝑀( + , 𝐹):𝑍⟶ℝ))
13	9, 12	mpbird 257	. . . . 5 ⊢ (𝜑 → 𝐺:𝑍⟶ℝ)
14	13	frnd 6696	. . . 4 ⊢ (𝜑 → ran 𝐺 ⊆ ℝ)
15		ressxr 11218	. . . . 5 ⊢ ℝ ⊆ ℝ*
16	15	a1i 11	. . . 4 ⊢ (𝜑 → ℝ ⊆ ℝ*)
17	14, 16	sstrd 3957	. . 3 ⊢ (𝜑 → ran 𝐺 ⊆ ℝ*)
18	1	fvexi 6872	. . . . 5 ⊢ 𝑍 ∈ V
19	18	a1i 11	. . . 4 ⊢ (𝜑 → 𝑍 ∈ V)
20		icossicc 13397	. . . . . 6 ⊢ (0[,)+∞) ⊆ (0[,]+∞)
21	20	a1i 11	. . . . 5 ⊢ (𝜑 → (0[,)+∞) ⊆ (0[,]+∞))
22	4, 21	fssd 6705	. . . 4 ⊢ (𝜑 → 𝐹:𝑍⟶(0[,]+∞))
23	19, 22	sge0xrcl 46383	. . 3 ⊢ (𝜑 → (Σ^‘𝐹) ∈ ℝ*)
24		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ ran 𝐺) → 𝑧 ∈ ran 𝐺)
25	13	ffnd 6689	. . . . . . . 8 ⊢ (𝜑 → 𝐺 Fn 𝑍)
26		fvelrnb 6921	. . . . . . . 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 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ ran 𝐺) → (𝑧 ∈ ran 𝐺 ↔ ∃𝑗 ∈ 𝑍 (𝐺‘𝑗) = 𝑧))
29	24, 28	mpbid 232	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ ran 𝐺) → ∃𝑗 ∈ 𝑍 (𝐺‘𝑗) = 𝑧)
30	20, 5	sselid 3944	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ (0[,]+∞))
31		elfzuz 13481	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (𝑀...𝑗) → 𝑘 ∈ (ℤ≥‘𝑀))
32	31, 1	eleqtrrdi 2839	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (𝑀...𝑗) → 𝑘 ∈ 𝑍)
33	32	ssriv 3950	. . . . . . . . . . . 12 ⊢ (𝑀...𝑗) ⊆ 𝑍
34	33	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑀...𝑗) ⊆ 𝑍)
35	19, 30, 34	sge0lessmpt 46397	. . . . . . . . . 10 ⊢ (𝜑 → (Σ^‘(𝑘 ∈ (𝑀...𝑗) ↦ (𝐹‘𝑘))) ≤ (Σ^‘(𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘))))
36	35	3ad2ant1 1133	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ (𝐺‘𝑗) = 𝑧) → (Σ^‘(𝑘 ∈ (𝑀...𝑗) ↦ (𝐹‘𝑘))) ≤ (Σ^‘(𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘))))
37		fzfid 13938	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑀...𝑗) ∈ Fin)
38	32, 5	sylan2 593	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑗)) → (𝐹‘𝑘) ∈ (0[,)+∞))
39	37, 38	sge0fsummpt 46388	. . . . . . . . . . . . 13 ⊢ (𝜑 → (Σ^‘(𝑘 ∈ (𝑀...𝑗) ↦ (𝐹‘𝑘))) = Σ𝑘 ∈ (𝑀...𝑗)(𝐹‘𝑘))
40	39	3ad2ant1 1133	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ (𝐺‘𝑗) = 𝑧) → (Σ^‘(𝑘 ∈ (𝑀...𝑗) ↦ (𝐹‘𝑘))) = Σ𝑘 ∈ (𝑀...𝑗)(𝐹‘𝑘))
41		simpll 766	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (𝑀...𝑗)) → 𝜑)
42	32	adantl 481	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (𝑀...𝑗)) → 𝑘 ∈ 𝑍)
43		eqidd 2730	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = (𝐹‘𝑘))
44	41, 42, 43	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (𝑀...𝑗)) → (𝐹‘𝑘) = (𝐹‘𝑘))
45	1	eleq2i 2820	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ 𝑍 ↔ 𝑗 ∈ (ℤ≥‘𝑀))
46	45	biimpi 216	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ 𝑍 → 𝑗 ∈ (ℤ≥‘𝑀))
47	46	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ (ℤ≥‘𝑀))
48	6	recnd 11202	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)
49	41, 42, 48	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (𝑀...𝑗)) → (𝐹‘𝑘) ∈ ℂ)
50	44, 47, 49	fsumser 15696	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → Σ𝑘 ∈ (𝑀...𝑗)(𝐹‘𝑘) = (seq𝑀( + , 𝐹)‘𝑗))
51	50	3adant3 1132	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ (𝐺‘𝑗) = 𝑧) → Σ𝑘 ∈ (𝑀...𝑗)(𝐹‘𝑘) = (seq𝑀( + , 𝐹)‘𝑗))
52	40, 51	eqtrd 2764	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ (𝐺‘𝑗) = 𝑧) → (Σ^‘(𝑘 ∈ (𝑀...𝑗) ↦ (𝐹‘𝑘))) = (seq𝑀( + , 𝐹)‘𝑗))
53	10	eqcomi 2738	. . . . . . . . . . . . 13 ⊢ seq𝑀( + , 𝐹) = 𝐺
54	53	fveq1i 6859	. . . . . . . . . . . 12 ⊢ (seq𝑀( + , 𝐹)‘𝑗) = (𝐺‘𝑗)
55	54	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ (𝐺‘𝑗) = 𝑧) → (seq𝑀( + , 𝐹)‘𝑗) = (𝐺‘𝑗))
56		simp3 1138	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ (𝐺‘𝑗) = 𝑧) → (𝐺‘𝑗) = 𝑧)
57	52, 55, 56	3eqtrrd 2769	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ (𝐺‘𝑗) = 𝑧) → 𝑧 = (Σ^‘(𝑘 ∈ (𝑀...𝑗) ↦ (𝐹‘𝑘))))
58	4	feqmptd 6929	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 = (𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘)))
59	58	fveq2d 6862	. . . . . . . . . . 11 ⊢ (𝜑 → (Σ^‘𝐹) = (Σ^‘(𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘))))
60	59	3ad2ant1 1133	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ (𝐺‘𝑗) = 𝑧) → (Σ^‘𝐹) = (Σ^‘(𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘))))
61	57, 60	breq12d 5120	. . . . . . . . 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 257	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ (𝐺‘𝑗) = 𝑧) → 𝑧 ≤ (Σ^‘𝐹))
63	62	3exp 1119	. . . . . . 7 ⊢ (𝜑 → (𝑗 ∈ 𝑍 → ((𝐺‘𝑗) = 𝑧 → 𝑧 ≤ (Σ^‘𝐹))))
64	63	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ ran 𝐺) → (𝑗 ∈ 𝑍 → ((𝐺‘𝑗) = 𝑧 → 𝑧 ≤ (Σ^‘𝐹))))
65	64	rexlimdv 3132	. . . . 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 3125	. . 3 ⊢ (𝜑 → ∀𝑧 ∈ ran 𝐺 𝑧 ≤ (Σ^‘𝐹))
68		nfv 1914	. . . . . . . 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 768	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑧 < (Σ^‘𝐹)) → 𝑧 ∈ ℝ)
72		simpr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 < (Σ^‘𝐹)) → 𝑧 < (Σ^‘𝐹))
73	59	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 < (Σ^‘𝐹)) → (Σ^‘𝐹) = (Σ^‘(𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘))))
74	72, 73	breqtrd 5133	. . . . . . . . 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 46441	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑧 < (Σ^‘𝐹)) → ∃𝑤 ∈ (𝒫 𝑍 ∩ Fin)𝑧 < Σ𝑘 ∈ 𝑤 (𝐹‘𝑘))
77	2	3ad2ant1 1133	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑧 < Σ𝑘 ∈ 𝑤 (𝐹‘𝑘)) → 𝑀 ∈ ℤ)
78		elpwinss 45043	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ (𝒫 𝑍 ∩ Fin) → 𝑤 ⊆ 𝑍)
79	78	3ad2ant2 1134	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑧 < Σ𝑘 ∈ 𝑤 (𝐹‘𝑘)) → 𝑤 ⊆ 𝑍)
80		elinel2 4165	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ (𝒫 𝑍 ∩ Fin) → 𝑤 ∈ Fin)
81	80	3ad2ant2 1134	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑧 < Σ𝑘 ∈ 𝑤 (𝐹‘𝑘)) → 𝑤 ∈ Fin)
82	77, 1, 79, 81	uzfissfz 45322	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑧 < Σ𝑘 ∈ 𝑤 (𝐹‘𝑘)) → ∃𝑗 ∈ 𝑍 𝑤 ⊆ (𝑀...𝑗))
83	82	3adant1r 1178	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑤 ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑧 < Σ𝑘 ∈ 𝑤 (𝐹‘𝑘)) → ∃𝑗 ∈ 𝑍 𝑤 ⊆ (𝑀...𝑗))
84		simpl1r 1226	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑤 ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑧 < Σ𝑘 ∈ 𝑤 (𝐹‘𝑘)) ∧ 𝑤 ⊆ (𝑀...𝑗)) → 𝑧 ∈ ℝ)
85	80	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝒫 𝑍 ∩ Fin)) → 𝑤 ∈ Fin)
86	58, 6	fmpt3d 7088	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ)
87	86	ad2antrr 726	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑘 ∈ 𝑤) → 𝐹:𝑍⟶ℝ)
88	78	sselda 3946	. . . . . . . . . . . . . . . . . . 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	ffvelcdmd 7057	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑘 ∈ 𝑤) → (𝐹‘𝑘) ∈ ℝ)
91	85, 90	fsumrecl 15700	. . . . . . . . . . . . . . . 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 1169	. . . . . . . . . . . . . 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 593	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑗)) → (𝐹‘𝑘) ∈ ℝ)
95	37, 94	fsumrecl 15700	. . . . . . . . . . . . . . . 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 1169	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑤 ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑧 < Σ𝑘 ∈ 𝑤 (𝐹‘𝑘)) ∧ 𝑤 ⊆ (𝑀...𝑗)) → Σ𝑘 ∈ (𝑀...𝑗)(𝐹‘𝑘) ∈ ℝ)
98		simpl3 1194	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑤 ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑧 < Σ𝑘 ∈ 𝑤 (𝐹‘𝑘)) ∧ 𝑤 ⊆ (𝑀...𝑗)) → 𝑧 < Σ𝑘 ∈ 𝑤 (𝐹‘𝑘))
99	37	adantr 480	. . . . . . . . . . . . . . . . 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 11221	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 0 ∈ ℝ*
102	101	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 0 ∈ ℝ*)
103		pnfxr 11228	. . . . . . . . . . . . . . . . . . . . 21 ⊢ +∞ ∈ ℝ*
104	103	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → +∞ ∈ ℝ*)
105		icogelb 13357	. . . . . . . . . . . . . . . . . . . 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 593	. . . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑤 ⊆ (𝑀...𝑗)) → 𝑤 ⊆ (𝑀...𝑗))
110	99, 100, 108, 109	fsumless 15762	. . . . . . . . . . . . . . . 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 1186	. . . . . . . . . . . . . 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 11334	. . . . . . . . . . . . 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 412	. . . . . . . . . . . 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 3148	. . . . . . . . . . 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 1119	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → (𝑤 ∈ (𝒫 𝑍 ∩ Fin) → (𝑧 < Σ𝑘 ∈ 𝑤 (𝐹‘𝑘) → ∃𝑗 ∈ 𝑍 𝑧 < Σ𝑘 ∈ (𝑀...𝑗)(𝐹‘𝑘))))
118	117	adantr 480	. . . . . . . 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 3132	. . . . . . 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 6689	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → seq𝑀( + , 𝐹) Fn 𝑍)
122	121	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → seq𝑀( + , 𝐹) Fn 𝑍)
123	47, 45	sylibr 234	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ 𝑍)
124		fnfvelrn 7052	. . . . . . . . . . . . . 14 ⊢ ((seq𝑀( + , 𝐹) Fn 𝑍 ∧ 𝑗 ∈ 𝑍) → (seq𝑀( + , 𝐹)‘𝑗) ∈ ran seq𝑀( + , 𝐹))
125	122, 123, 124	syl2anc 584	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (seq𝑀( + , 𝐹)‘𝑗) ∈ ran seq𝑀( + , 𝐹))
126	10	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝐺 = seq𝑀( + , 𝐹))
127	126	rneqd 5902	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ran 𝐺 = ran seq𝑀( + , 𝐹))
128	50, 127	eleq12d 2822	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (Σ𝑘 ∈ (𝑀...𝑗)(𝐹‘𝑘) ∈ ran 𝐺 ↔ (seq𝑀( + , 𝐹)‘𝑗) ∈ ran seq𝑀( + , 𝐹)))
129	125, 128	mpbird 257	. . . . . . . . . . . 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 1132	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑗 ∈ 𝑍 ∧ 𝑧 < Σ𝑘 ∈ (𝑀...𝑗)(𝐹‘𝑘)) → Σ𝑘 ∈ (𝑀...𝑗)(𝐹‘𝑘) ∈ ran 𝐺)
132		simp3 1138	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑗 ∈ 𝑍 ∧ 𝑧 < Σ𝑘 ∈ (𝑀...𝑗)(𝐹‘𝑘)) → 𝑧 < Σ𝑘 ∈ (𝑀...𝑗)(𝐹‘𝑘))
133		breq2 5111	. . . . . . . . . . 11 ⊢ (𝑦 = Σ𝑘 ∈ (𝑀...𝑗)(𝐹‘𝑘) → (𝑧 < 𝑦 ↔ 𝑧 < Σ𝑘 ∈ (𝑀...𝑗)(𝐹‘𝑘)))
134	133	rspcev 3588	. . . . . . . . . 10 ⊢ ((Σ𝑘 ∈ (𝑀...𝑗)(𝐹‘𝑘) ∈ ran 𝐺 ∧ 𝑧 < Σ𝑘 ∈ (𝑀...𝑗)(𝐹‘𝑘)) → ∃𝑦 ∈ ran 𝐺 𝑧 < 𝑦)
135	131, 132, 134	syl2anc 584	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑗 ∈ 𝑍 ∧ 𝑧 < Σ𝑘 ∈ (𝑀...𝑗)(𝐹‘𝑘)) → ∃𝑦 ∈ ran 𝐺 𝑧 < 𝑦)
136	135	3exp 1119	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → (𝑗 ∈ 𝑍 → (𝑧 < Σ𝑘 ∈ (𝑀...𝑗)(𝐹‘𝑘) → ∃𝑦 ∈ ran 𝐺 𝑧 < 𝑦)))
137	136	rexlimdv 3132	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → (∃𝑗 ∈ 𝑍 𝑧 < Σ𝑘 ∈ (𝑀...𝑗)(𝐹‘𝑘) → ∃𝑦 ∈ ran 𝐺 𝑧 < 𝑦))
138	137	adantr 480	. . . . . 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 412	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → (𝑧 < (Σ^‘𝐹) → ∃𝑦 ∈ ran 𝐺 𝑧 < 𝑦))
141	140	ralrimiva 3125	. . 3 ⊢ (𝜑 → ∀𝑧 ∈ ℝ (𝑧 < (Σ^‘𝐹) → ∃𝑦 ∈ ran 𝐺 𝑧 < 𝑦))
142		supxr2 13274	. . 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 838	. 2 ⊢ (𝜑 → sup(ran 𝐺, ℝ*, < ) = (Σ^‘𝐹))
144	143	eqcomd 2735	1 ⊢ (𝜑 → (Σ^‘𝐹) = sup(ran 𝐺, ℝ*, < ))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053  Vcvv 3447   ∩ cin 3913   ⊆ wss 3914  𝒫 cpw 4563   class class class wbr 5107   ↦ cmpt 5188  ran crn 5639   Fn wfn 6506  ⟶wf 6507  ‘cfv 6511  (class class class)co 7387  Fincfn 8918  supcsup 9391  ℂcc 11066  ℝcr 11067  0cc0 11068   + caddc 11071  +∞cpnf 11205  ℝ^*cxr 11207   < clt 11208   ≤ cle 11209  ℤcz 12529  ℤ_≥cuz 12793  [,)cico 13308  [,]cicc 13309  ...cfz 13468  seqcseq 13966  Σcsu 15652  Σ^{^}csumge0 46360

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-inf2 9594  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145  ax-pre-sup 11146

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-se 5592  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-isom 6520  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-1o 8434  df-er 8671  df-en 8919  df-dom 8920  df-sdom 8921  df-fin 8922  df-sup 9393  df-oi 9463  df-card 9892  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-div 11836  df-nn 12187  df-2 12249  df-3 12250  df-n0 12443  df-z 12530  df-uz 12794  df-rp 12952  df-ico 13312  df-icc 13313  df-fz 13469  df-fzo 13616  df-seq 13967  df-exp 14027  df-hash 14296  df-cj 15065  df-re 15066  df-im 15067  df-sqrt 15201  df-abs 15202  df-clim 15454  df-sum 15653  df-sumge0 46361

This theorem is referenced by:  voliunsge0lem  46470  ovolval2  46642