Theorem sge0seq 47282

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 13513	. . . . . . . 8 ⊢ (0[,)+∞) ⊆ ℝ
4		sge0seq.f	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:𝑍⟶(0[,)+∞))
5	4	ffvelcdmda 7081	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ (0[,)+∞))
6	3, 5	sselid 3932	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ)
7		readdcl 11211	. . . . . . . 8 ⊢ ((𝑘 ∈ ℝ ∧ 𝑖 ∈ ℝ) → (𝑘 + 𝑖) ∈ ℝ)
8	7	adantl 487	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ ℝ ∧ 𝑖 ∈ ℝ)) → (𝑘 + 𝑖) ∈ ℝ)
9	1, 2, 6, 8	seqf 14091	. . . . . 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 6688	. . . . . 6 ⊢ (𝜑 → (𝐺:𝑍⟶ℝ ↔ seq𝑀( + , 𝐹):𝑍⟶ℝ))
13	9, 12	mpbird 260	. . . . 5 ⊢ (𝜑 → 𝐺:𝑍⟶ℝ)
14	13	frnd 6715	. . . 4 ⊢ (𝜑 → ran 𝐺 ⊆ ℝ)
15		ressxr 11281	. . . . 5 ⊢ ℝ ⊆ ℝ*
16	15	a1i 11	. . . 4 ⊢ (𝜑 → ℝ ⊆ ℝ*)
17	14, 16	sstrd 3944	. . 3 ⊢ (𝜑 → ran 𝐺 ⊆ ℝ*)
18	1	fvexi 6896	. . . . 5 ⊢ 𝑍 ∈ V
19	18	a1i 11	. . . 4 ⊢ (𝜑 → 𝑍 ∈ V)
20		icossicc 13493	. . . . . 6 ⊢ (0[,)+∞) ⊆ (0[,]+∞)
21	20	a1i 11	. . . . 5 ⊢ (𝜑 → (0[,)+∞) ⊆ (0[,]+∞))
22	4, 21	fssd 6724	. . . 4 ⊢ (𝜑 → 𝐹:𝑍⟶(0[,]+∞))
23	19, 22	sge0xrcl 47221	. . 3 ⊢ (𝜑 → (Σ^‘𝐹) ∈ ℝ*)
24		simpr 490	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ ran 𝐺) → 𝑧 ∈ ran 𝐺)
25	13	ffnd 6707	. . . . . . . 8 ⊢ (𝜑 → 𝐺 Fn 𝑍)
26		fvelrnb 6942	. . . . . . . 8 ⊢ (𝐺 Fn 𝑍 → (𝑧 ∈ ran 𝐺 ↔ ∃𝑗 ∈ 𝑍 (𝐺‘𝑗) = 𝑧))
27	25, 26	syl 18	. . . . . . 7 ⊢ (𝜑 → (𝑧 ∈ ran 𝐺 ↔ ∃𝑗 ∈ 𝑍 (𝐺‘𝑗) = 𝑧))
28	27	adantr 486	. . . . . 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	sselid 3932	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ (0[,]+∞))
31		elfzuz 13578	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (𝑀...𝑗) → 𝑘 ∈ (ℤ≥‘𝑀))
32	31, 1	eleqtrrdi 2873	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (𝑀...𝑗) → 𝑘 ∈ 𝑍)
33	32	ssriv 3938	. . . . . . . . . . . 12 ⊢ (𝑀...𝑗) ⊆ 𝑍
34	33	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑀...𝑗) ⊆ 𝑍)
35	19, 30, 34	sge0lessmpt 47235	. . . . . . . . . 10 ⊢ (𝜑 → (Σ^‘(𝑘 ∈ (𝑀...𝑗) ↦ (𝐹‘𝑘))) ≤ (Σ^‘(𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘))))
36	35	3ad2ant1 1151	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ (𝐺‘𝑗) = 𝑧) → (Σ^‘(𝑘 ∈ (𝑀...𝑗) ↦ (𝐹‘𝑘))) ≤ (Σ^‘(𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘))))
37		fzfid 14041	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑀...𝑗) ∈ Fin)
38	32, 5	sylan2 605	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑗)) → (𝐹‘𝑘) ∈ (0[,)+∞))
39	37, 38	sge0fsummpt 47226	. . . . . . . . . . . . 13 ⊢ (𝜑 → (Σ^‘(𝑘 ∈ (𝑀...𝑗) ↦ (𝐹‘𝑘))) = Σ𝑘 ∈ (𝑀...𝑗)(𝐹‘𝑘))
40	39	3ad2ant1 1151	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ (𝐺‘𝑗) = 𝑧) → (Σ^‘(𝑘 ∈ (𝑀...𝑗) ↦ (𝐹‘𝑘))) = Σ𝑘 ∈ (𝑀...𝑗)(𝐹‘𝑘))
41		simpll 779	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (𝑀...𝑗)) → 𝜑)
42	32	adantl 487	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (𝑀...𝑗)) → 𝑘 ∈ 𝑍)
43		eqidd 2763	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = (𝐹‘𝑘))
44	41, 42, 43	syl2anc 596	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (𝑀...𝑗)) → (𝐹‘𝑘) = (𝐹‘𝑘))
45	1	eleq2i 2854	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ 𝑍 ↔ 𝑗 ∈ (ℤ≥‘𝑀))
46	45	bilani 510	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ (ℤ≥‘𝑀))
47	6	recnd 11265	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)
48	41, 42, 47	syl2anc 596	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (𝑀...𝑗)) → (𝐹‘𝑘) ∈ ℂ)
49	44, 46, 48	fsumser 15820	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → Σ𝑘 ∈ (𝑀...𝑗)(𝐹‘𝑘) = (seq𝑀( + , 𝐹)‘𝑗))
50	49	3adant3 1150	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ (𝐺‘𝑗) = 𝑧) → Σ𝑘 ∈ (𝑀...𝑗)(𝐹‘𝑘) = (seq𝑀( + , 𝐹)‘𝑗))
51	40, 50	eqtrd 2797	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ (𝐺‘𝑗) = 𝑧) → (Σ^‘(𝑘 ∈ (𝑀...𝑗) ↦ (𝐹‘𝑘))) = (seq𝑀( + , 𝐹)‘𝑗))
52	10	eqcomi 2771	. . . . . . . . . . . . 13 ⊢ seq𝑀( + , 𝐹) = 𝐺
53	52	fveq1i 6883	. . . . . . . . . . . 12 ⊢ (seq𝑀( + , 𝐹)‘𝑗) = (𝐺‘𝑗)
54	53	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ (𝐺‘𝑗) = 𝑧) → (seq𝑀( + , 𝐹)‘𝑗) = (𝐺‘𝑗))
55		simp3 1156	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ (𝐺‘𝑗) = 𝑧) → (𝐺‘𝑗) = 𝑧)
56	51, 54, 55	3eqtrrd 2802	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ (𝐺‘𝑗) = 𝑧) → 𝑧 = (Σ^‘(𝑘 ∈ (𝑀...𝑗) ↦ (𝐹‘𝑘))))
57	4	feqmptd 6950	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 = (𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘)))
58	57	fveq2d 6886	. . . . . . . . . . 11 ⊢ (𝜑 → (Σ^‘𝐹) = (Σ^‘(𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘))))
59	58	3ad2ant1 1151	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ (𝐺‘𝑗) = 𝑧) → (Σ^‘𝐹) = (Σ^‘(𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘))))
60	56, 59	breq12d 5120	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ (𝐺‘𝑗) = 𝑧) → (𝑧 ≤ (Σ^‘𝐹) ↔ (Σ^‘(𝑘 ∈ (𝑀...𝑗) ↦ (𝐹‘𝑘))) ≤ (Σ^‘(𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘)))))
61	36, 60	mpbird 260	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ (𝐺‘𝑗) = 𝑧) → 𝑧 ≤ (Σ^‘𝐹))
62	61	3exp 1137	. . . . . . 7 ⊢ (𝜑 → (𝑗 ∈ 𝑍 → ((𝐺‘𝑗) = 𝑧 → 𝑧 ≤ (Σ^‘𝐹))))
63	62	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ ran 𝐺) → (𝑗 ∈ 𝑍 → ((𝐺‘𝑗) = 𝑧 → 𝑧 ≤ (Σ^‘𝐹))))
64	63	rexlimdv 3163	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ ran 𝐺) → (∃𝑗 ∈ 𝑍 (𝐺‘𝑗) = 𝑧 → 𝑧 ≤ (Σ^‘𝐹)))
65	29, 64	mpd 16	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ ran 𝐺) → 𝑧 ≤ (Σ^‘𝐹))
66	65	ralrimiva 3156	. . 3 ⊢ (𝜑 → ∀𝑧 ∈ ran 𝐺 𝑧 ≤ (Σ^‘𝐹))
67		nfv 1947	. . . . . . . 8 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑧 < (Σ^‘𝐹))
68	18	a1i 11	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑧 < (Σ^‘𝐹)) → 𝑍 ∈ V)
69	5	ad4ant14 765	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑧 < (Σ^‘𝐹)) ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ (0[,)+∞))
70		simplr 781	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑧 < (Σ^‘𝐹)) → 𝑧 ∈ ℝ)
71		simpr 490	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 < (Σ^‘𝐹)) → 𝑧 < (Σ^‘𝐹))
72	58	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 < (Σ^‘𝐹)) → (Σ^‘𝐹) = (Σ^‘(𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘))))
73	71, 72	breqtrd 5135	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 < (Σ^‘𝐹)) → 𝑧 < (Σ^‘(𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘))))
74	73	adantlr 728	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑧 < (Σ^‘𝐹)) → 𝑧 < (Σ^‘(𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘))))
75	67, 68, 69, 70, 74	sge0gtfsumgt 47279	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑧 < (Σ^‘𝐹)) → ∃𝑤 ∈ (𝒫 𝑍 ∩ Fin)𝑧 < Σ𝑘 ∈ 𝑤 (𝐹‘𝑘))
76	2	3ad2ant1 1151	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑧 < Σ𝑘 ∈ 𝑤 (𝐹‘𝑘)) → 𝑀 ∈ ℤ)
77		elpwinss 45891	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ (𝒫 𝑍 ∩ Fin) → 𝑤 ⊆ 𝑍)
78	77	3ad2ant2 1152	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑧 < Σ𝑘 ∈ 𝑤 (𝐹‘𝑘)) → 𝑤 ⊆ 𝑍)
79		elinel2 4151	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ (𝒫 𝑍 ∩ Fin) → 𝑤 ∈ Fin)
80	79	3ad2ant2 1152	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑧 < Σ𝑘 ∈ 𝑤 (𝐹‘𝑘)) → 𝑤 ∈ Fin)
81	76, 1, 78, 80	uzfissfz 46164	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑧 < Σ𝑘 ∈ 𝑤 (𝐹‘𝑘)) → ∃𝑗 ∈ 𝑍 𝑤 ⊆ (𝑀...𝑗))
82	81	3adant1r 1196	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑤 ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑧 < Σ𝑘 ∈ 𝑤 (𝐹‘𝑘)) → ∃𝑗 ∈ 𝑍 𝑤 ⊆ (𝑀...𝑗))
83		simpl1r 1244	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑤 ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑧 < Σ𝑘 ∈ 𝑤 (𝐹‘𝑘)) ∧ 𝑤 ⊆ (𝑀...𝑗)) → 𝑧 ∈ ℝ)
84	79	adantl 487	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝒫 𝑍 ∩ Fin)) → 𝑤 ∈ Fin)
85	57, 6	fmpt3d 7113	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ)
86	85	ad2antrr 739	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑘 ∈ 𝑤) → 𝐹:𝑍⟶ℝ)
87	77	sselda 3934	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑤 ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑘 ∈ 𝑤) → 𝑘 ∈ 𝑍)
88	87	adantll 727	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑘 ∈ 𝑤) → 𝑘 ∈ 𝑍)
89	86, 88	ffvelcdmd 7082	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑘 ∈ 𝑤) → (𝐹‘𝑘) ∈ ℝ)
90	84, 89	fsumrecl 15824	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝒫 𝑍 ∩ Fin)) → Σ𝑘 ∈ 𝑤 (𝐹‘𝑘) ∈ ℝ)
91	90	ad4ant13 764	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑤 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑤 ⊆ (𝑀...𝑗)) → Σ𝑘 ∈ 𝑤 (𝐹‘𝑘) ∈ ℝ)
92	91	3adantl3 1187	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑤 ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑧 < Σ𝑘 ∈ 𝑤 (𝐹‘𝑘)) ∧ 𝑤 ⊆ (𝑀...𝑗)) → Σ𝑘 ∈ 𝑤 (𝐹‘𝑘) ∈ ℝ)
93	32, 6	sylan2 605	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑗)) → (𝐹‘𝑘) ∈ ℝ)
94	37, 93	fsumrecl 15824	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀...𝑗)(𝐹‘𝑘) ∈ ℝ)
95	94	ad3antrrr 743	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑤 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑤 ⊆ (𝑀...𝑗)) → Σ𝑘 ∈ (𝑀...𝑗)(𝐹‘𝑘) ∈ ℝ)
96	95	3adantl3 1187	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑤 ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑧 < Σ𝑘 ∈ 𝑤 (𝐹‘𝑘)) ∧ 𝑤 ⊆ (𝑀...𝑗)) → Σ𝑘 ∈ (𝑀...𝑗)(𝐹‘𝑘) ∈ ℝ)
97		simpl3 1212	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑤 ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑧 < Σ𝑘 ∈ 𝑤 (𝐹‘𝑘)) ∧ 𝑤 ⊆ (𝑀...𝑗)) → 𝑧 < Σ𝑘 ∈ 𝑤 (𝐹‘𝑘))
98	37	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑤 ⊆ (𝑀...𝑗)) → (𝑀...𝑗) ∈ Fin)
99	93	adantlr 728	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑤 ⊆ (𝑀...𝑗)) ∧ 𝑘 ∈ (𝑀...𝑗)) → (𝐹‘𝑘) ∈ ℝ)
100		0xr 11284	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 0 ∈ ℝ*
101	100	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 0 ∈ ℝ*)
102		pnfxr 11291	. . . . . . . . . . . . . . . . . . . . 21 ⊢ +∞ ∈ ℝ*
103	102	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → +∞ ∈ ℝ*)
104		icogelb 13453	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ (𝐹‘𝑘) ∈ (0[,)+∞)) → 0 ≤ (𝐹‘𝑘))
105	101, 103, 5, 104	syl3anc 1398	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 0 ≤ (𝐹‘𝑘))
106	32, 105	sylan2 605	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑗)) → 0 ≤ (𝐹‘𝑘))
107	106	adantlr 728	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑤 ⊆ (𝑀...𝑗)) ∧ 𝑘 ∈ (𝑀...𝑗)) → 0 ≤ (𝐹‘𝑘))
108		simpr 490	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑤 ⊆ (𝑀...𝑗)) → 𝑤 ⊆ (𝑀...𝑗))
109	98, 99, 107, 108	fsumless 15887	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ⊆ (𝑀...𝑗)) → Σ𝑘 ∈ 𝑤 (𝐹‘𝑘) ≤ Σ𝑘 ∈ (𝑀...𝑗)(𝐹‘𝑘))
110	109	adantlr 728	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑤 ⊆ (𝑀...𝑗)) → Σ𝑘 ∈ 𝑤 (𝐹‘𝑘) ≤ Σ𝑘 ∈ (𝑀...𝑗)(𝐹‘𝑘))
111	110	3ad2antl1 1204	. . . . . . . . . . . . . 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))
112	83, 92, 96, 97, 111	ltletrd 11398	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑤 ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑧 < Σ𝑘 ∈ 𝑤 (𝐹‘𝑘)) ∧ 𝑤 ⊆ (𝑀...𝑗)) → 𝑧 < Σ𝑘 ∈ (𝑀...𝑗)(𝐹‘𝑘))
113	112	ex 418	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑤 ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑧 < Σ𝑘 ∈ 𝑤 (𝐹‘𝑘)) → (𝑤 ⊆ (𝑀...𝑗) → 𝑧 < Σ𝑘 ∈ (𝑀...𝑗)(𝐹‘𝑘)))
114	113	reximdv 3179	. . . . . . . . . . 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)))
115	82, 114	mpd 16	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑤 ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑧 < Σ𝑘 ∈ 𝑤 (𝐹‘𝑘)) → ∃𝑗 ∈ 𝑍 𝑧 < Σ𝑘 ∈ (𝑀...𝑗)(𝐹‘𝑘))
116	115	3exp 1137	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → (𝑤 ∈ (𝒫 𝑍 ∩ Fin) → (𝑧 < Σ𝑘 ∈ 𝑤 (𝐹‘𝑘) → ∃𝑗 ∈ 𝑍 𝑧 < Σ𝑘 ∈ (𝑀...𝑗)(𝐹‘𝑘))))
117	116	adantr 486	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑧 < (Σ^‘𝐹)) → (𝑤 ∈ (𝒫 𝑍 ∩ Fin) → (𝑧 < Σ𝑘 ∈ 𝑤 (𝐹‘𝑘) → ∃𝑗 ∈ 𝑍 𝑧 < Σ𝑘 ∈ (𝑀...𝑗)(𝐹‘𝑘))))
118	117	rexlimdv 3163	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑧 < (Σ^‘𝐹)) → (∃𝑤 ∈ (𝒫 𝑍 ∩ Fin)𝑧 < Σ𝑘 ∈ 𝑤 (𝐹‘𝑘) → ∃𝑗 ∈ 𝑍 𝑧 < Σ𝑘 ∈ (𝑀...𝑗)(𝐹‘𝑘)))
119	75, 118	mpd 16	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑧 < (Σ^‘𝐹)) → ∃𝑗 ∈ 𝑍 𝑧 < Σ𝑘 ∈ (𝑀...𝑗)(𝐹‘𝑘))
120	9	ffnd 6707	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → seq𝑀( + , 𝐹) Fn 𝑍)
121	120	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → seq𝑀( + , 𝐹) Fn 𝑍)
122	46, 45	sylibr 237	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ 𝑍)
123		fnfvelrn 7077	. . . . . . . . . . . . . 14 ⊢ ((seq𝑀( + , 𝐹) Fn 𝑍 ∧ 𝑗 ∈ 𝑍) → (seq𝑀( + , 𝐹)‘𝑗) ∈ ran seq𝑀( + , 𝐹))
124	121, 122, 123	syl2anc 596	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (seq𝑀( + , 𝐹)‘𝑗) ∈ ran seq𝑀( + , 𝐹))
125	10	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝐺 = seq𝑀( + , 𝐹))
126	125	rneqd 5926	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ran 𝐺 = ran seq𝑀( + , 𝐹))
127	49, 126	eleq12d 2856	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (Σ𝑘 ∈ (𝑀...𝑗)(𝐹‘𝑘) ∈ ran 𝐺 ↔ (seq𝑀( + , 𝐹)‘𝑗) ∈ ran seq𝑀( + , 𝐹)))
128	124, 127	mpbird 260	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → Σ𝑘 ∈ (𝑀...𝑗)(𝐹‘𝑘) ∈ ran 𝐺)
129	128	adantlr 728	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) → Σ𝑘 ∈ (𝑀...𝑗)(𝐹‘𝑘) ∈ ran 𝐺)
130	129	3adant3 1150	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑗 ∈ 𝑍 ∧ 𝑧 < Σ𝑘 ∈ (𝑀...𝑗)(𝐹‘𝑘)) → Σ𝑘 ∈ (𝑀...𝑗)(𝐹‘𝑘) ∈ ran 𝐺)
131		simp3 1156	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑗 ∈ 𝑍 ∧ 𝑧 < Σ𝑘 ∈ (𝑀...𝑗)(𝐹‘𝑘)) → 𝑧 < Σ𝑘 ∈ (𝑀...𝑗)(𝐹‘𝑘))
132		breq2 5111	. . . . . . . . . . 11 ⊢ (𝑦 = Σ𝑘 ∈ (𝑀...𝑗)(𝐹‘𝑘) → (𝑧 < 𝑦 ↔ 𝑧 < Σ𝑘 ∈ (𝑀...𝑗)(𝐹‘𝑘)))
133	132	rspcev 3579	. . . . . . . . . 10 ⊢ ((Σ𝑘 ∈ (𝑀...𝑗)(𝐹‘𝑘) ∈ ran 𝐺 ∧ 𝑧 < Σ𝑘 ∈ (𝑀...𝑗)(𝐹‘𝑘)) → ∃𝑦 ∈ ran 𝐺 𝑧 < 𝑦)
134	130, 131, 133	syl2anc 596	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑗 ∈ 𝑍 ∧ 𝑧 < Σ𝑘 ∈ (𝑀...𝑗)(𝐹‘𝑘)) → ∃𝑦 ∈ ran 𝐺 𝑧 < 𝑦)
135	134	3exp 1137	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → (𝑗 ∈ 𝑍 → (𝑧 < Σ𝑘 ∈ (𝑀...𝑗)(𝐹‘𝑘) → ∃𝑦 ∈ ran 𝐺 𝑧 < 𝑦)))
136	135	rexlimdv 3163	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → (∃𝑗 ∈ 𝑍 𝑧 < Σ𝑘 ∈ (𝑀...𝑗)(𝐹‘𝑘) → ∃𝑦 ∈ ran 𝐺 𝑧 < 𝑦))
137	136	adantr 486	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑧 < (Σ^‘𝐹)) → (∃𝑗 ∈ 𝑍 𝑧 < Σ𝑘 ∈ (𝑀...𝑗)(𝐹‘𝑘) → ∃𝑦 ∈ ran 𝐺 𝑧 < 𝑦))
138	119, 137	mpd 16	. . . . 5 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑧 < (Σ^‘𝐹)) → ∃𝑦 ∈ ran 𝐺 𝑧 < 𝑦)
139	138	ex 418	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → (𝑧 < (Σ^‘𝐹) → ∃𝑦 ∈ ran 𝐺 𝑧 < 𝑦))
140	139	ralrimiva 3156	. . 3 ⊢ (𝜑 → ∀𝑧 ∈ ℝ (𝑧 < (Σ^‘𝐹) → ∃𝑦 ∈ ran 𝐺 𝑧 < 𝑦))
141		supxr2 13370	. . 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))
142	17, 23, 66, 140, 141	syl22anc 852	. 2 ⊢ (𝜑 → sup(ran 𝐺, ℝ*, < ) = (Σ^‘𝐹))
143	142	eqcomd 2768	1 ⊢ (𝜑 → (Σ^‘𝐹) = sup(ran 𝐺, ℝ*, < ))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3078  ∃wrex 3088  Vcvv 3453   ∩ cin 3901   ⊆ wss 3902  𝒫 cpw 4560   class class class wbr 5107   ↦ cmpt 5190  ran crn 5660   Fn wfn 6532  ⟶wf 6533  ‘cfv 6537  (class class class)co 7417  Fincfn 8956  supcsup 9414  ℂcc 11126  ℝcr 11127  0cc0 11128   + caddc 11131  +∞cpnf 11268  ℝ^*cxr 11270   < clt 11271   ≤ cle 11272  ℤcz 12619  ℤ_≥cuz 12891  [,)cico 13404  [,]cicc 13405  ...cfz 13565  seqcseq 14069  Σcsu 15777  Σ^{^}csumge0 47198

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740  ax-inf2 9624  ax-cnex 11184  ax-resscn 11185  ax-1cn 11186  ax-icn 11187  ax-addcl 11188  ax-addrcl 11189  ax-mulcl 11190  ax-mulrcl 11191  ax-mulcom 11192  ax-addass 11193  ax-mulass 11194  ax-distr 11195  ax-i2m1 11196  ax-1ne0 11197  ax-1rid 11198  ax-rnegex 11199  ax-rrecex 11200  ax-cnre 11201  ax-pre-lttri 11202  ax-pre-lttrn 11203  ax-pre-ltadd 11204  ax-pre-mulgt0 11205  ax-pre-sup 11206

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-se 5613  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7374  df-ov 7420  df-oprab 7421  df-mpo 7422  df-om 7867  df-1st 7990  df-2nd 7991  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-1o 8459  df-er 8700  df-en 8957  df-dom 8958  df-sdom 8959  df-fin 8960  df-sup 9416  df-oi 9486  df-card 9948  df-pnf 11273  df-mnf 11274  df-xr 11275  df-ltxr 11276  df-le 11277  df-sub 11471  df-neg 11472  df-div 11900  df-nn 12262  df-2 12331  df-3 12332  df-n0 12533  df-z 12620  df-uz 12892  df-rp 13047  df-ico 13408  df-icc 13409  df-fz 13566  df-fzo 13714  df-seq 14070  df-exp 14130  df-hash 14399  df-cj 15190  df-re 15191  df-im 15192  df-sqrt 15326  df-abs 15327  df-clim 15579  df-sum 15778  df-sumge0 47199

This theorem is used by:  voliunsge0lem  47308  ovolval2  47480