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

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913  Vcvv 3200   ∩ cin 3573   ⊆ wss 3574  𝒫 cpw 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  ℂcc 9934  ℝcr 9935  0cc0 9936   + caddc 9939  +∞cpnf 10071  ℝ^*cxr 10073   < clt 10074   ≤ cle 10075  ℤcz 11377  ℤ_≥cuz 11687  [,)cico 12177  [,]cicc 12178  ...cfz 12326  seqcseq 12801  Σcsu 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