Theorem sge0reuz 46633

Description: Value of the generalized sum of nonnegative reals, when the domain is a set of upper integers. (Contributed by Glauco Siliprandi, 8-Apr-2021.)

Hypotheses

Ref	Expression
sge0reuz.k	⊢ Ⅎ𝑘𝜑
sge0reuz.m	⊢ (𝜑 → 𝑀 ∈ ℤ)
sge0reuz.z	⊢ 𝑍 = (ℤ≥‘𝑀)
sge0reuz.b	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ (0[,)+∞))

Assertion

Ref	Expression
sge0reuz	⊢ (𝜑 → (Σ^‘(𝑘 ∈ 𝑍 ↦ 𝐵)) = sup(ran (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵), ℝ*, < ))

Distinct variable groups:   𝐵,𝑛   𝑘,𝑀,𝑛   𝑘,𝑍,𝑛   𝜑,𝑛

Allowed substitution hints:   𝜑(𝑘)   𝐵(𝑘)

Proof of Theorem sge0reuz

Dummy variables 𝑤 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sge0reuz.k	. . 3 ⊢ Ⅎ𝑘𝜑
2		sge0reuz.z	. . . . 5 ⊢ 𝑍 = (ℤ≥‘𝑀)
3	2	a1i 11	. . . 4 ⊢ (𝜑 → 𝑍 = (ℤ≥‘𝑀))
4		fvex 6845	. . . 4 ⊢ (ℤ≥‘𝑀) ∈ V
5	3, 4	eqeltrdi 2842	. . 3 ⊢ (𝜑 → 𝑍 ∈ V)
6		sge0reuz.b	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ (0[,)+∞))
7	1, 5, 6	sge0revalmpt 46564	. 2 ⊢ (𝜑 → (Σ^‘(𝑘 ∈ 𝑍 ↦ 𝐵)) = sup(ran (𝑥 ∈ (𝒫 𝑍 ∩ Fin) ↦ Σ𝑘 ∈ 𝑥 𝐵), ℝ*, < ))
8		nfv 1915	. . . . 5 ⊢ Ⅎ𝑥𝜑
9		eqid 2734	. . . . 5 ⊢ (𝑥 ∈ (𝒫 𝑍 ∩ Fin) ↦ Σ𝑘 ∈ 𝑥 𝐵) = (𝑥 ∈ (𝒫 𝑍 ∩ Fin) ↦ Σ𝑘 ∈ 𝑥 𝐵)
10		nfv 1915	. . . . . . . 8 ⊢ Ⅎ𝑘 𝑥 ∈ (𝒫 𝑍 ∩ Fin)
11	1, 10	nfan 1900	. . . . . . 7 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑥 ∈ (𝒫 𝑍 ∩ Fin))
12		elinel2 4152	. . . . . . . 8 ⊢ (𝑥 ∈ (𝒫 𝑍 ∩ Fin) → 𝑥 ∈ Fin)
13	12	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑍 ∩ Fin)) → 𝑥 ∈ Fin)
14		rge0ssre 13370	. . . . . . . 8 ⊢ (0[,)+∞) ⊆ ℝ
15		simpll 766	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑘 ∈ 𝑥) → 𝜑)
16		elpwinss 45236	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝒫 𝑍 ∩ Fin) → 𝑥 ⊆ 𝑍)
17	16	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑘 ∈ 𝑥) → 𝑥 ⊆ 𝑍)
18		simpr 484	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑘 ∈ 𝑥) → 𝑘 ∈ 𝑥)
19	17, 18	sseldd 3932	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑘 ∈ 𝑥) → 𝑘 ∈ 𝑍)
20	19	adantll 714	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑘 ∈ 𝑥) → 𝑘 ∈ 𝑍)
21	15, 20, 6	syl2anc 584	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑘 ∈ 𝑥) → 𝐵 ∈ (0[,)+∞))
22	14, 21	sselid 3929	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑘 ∈ 𝑥) → 𝐵 ∈ ℝ)
23	11, 13, 22	fsumreclf 45764	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑍 ∩ Fin)) → Σ𝑘 ∈ 𝑥 𝐵 ∈ ℝ)
24	23	rexrd 11180	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑍 ∩ Fin)) → Σ𝑘 ∈ 𝑥 𝐵 ∈ ℝ*)
25	8, 9, 24	rnmptssd 45382	. . . 4 ⊢ (𝜑 → ran (𝑥 ∈ (𝒫 𝑍 ∩ Fin) ↦ Σ𝑘 ∈ 𝑥 𝐵) ⊆ ℝ*)
26		supxrcl 13228	. . . 4 ⊢ (ran (𝑥 ∈ (𝒫 𝑍 ∩ Fin) ↦ Σ𝑘 ∈ 𝑥 𝐵) ⊆ ℝ* → sup(ran (𝑥 ∈ (𝒫 𝑍 ∩ Fin) ↦ Σ𝑘 ∈ 𝑥 𝐵), ℝ*, < ) ∈ ℝ*)
27	25, 26	syl 17	. . 3 ⊢ (𝜑 → sup(ran (𝑥 ∈ (𝒫 𝑍 ∩ Fin) ↦ Σ𝑘 ∈ 𝑥 𝐵), ℝ*, < ) ∈ ℝ*)
28		nfv 1915	. . . . 5 ⊢ Ⅎ𝑛𝜑
29		eqid 2734	. . . . 5 ⊢ (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵) = (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵)
30		nfv 1915	. . . . . . . 8 ⊢ Ⅎ𝑘 𝑛 ∈ 𝑍
31	1, 30	nfan 1900	. . . . . . 7 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑛 ∈ 𝑍)
32		fzfid 13894	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑀...𝑛) ∈ Fin)
33		elfzuz 13434	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (𝑀...𝑛) → 𝑘 ∈ (ℤ≥‘𝑀))
34	33, 2	eleqtrrdi 2845	. . . . . . . . . 10 ⊢ (𝑘 ∈ (𝑀...𝑛) → 𝑘 ∈ 𝑍)
35	34	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑛)) → 𝑘 ∈ 𝑍)
36	14, 6	sselid 3929	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℝ)
37	35, 36	syldan 591	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑛)) → 𝐵 ∈ ℝ)
38	37	adantlr 715	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ (𝑀...𝑛)) → 𝐵 ∈ ℝ)
39	31, 32, 38	fsumreclf 45764	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → Σ𝑘 ∈ (𝑀...𝑛)𝐵 ∈ ℝ)
40	39	rexrd 11180	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → Σ𝑘 ∈ (𝑀...𝑛)𝐵 ∈ ℝ*)
41	28, 29, 40	rnmptssd 45382	. . . 4 ⊢ (𝜑 → ran (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵) ⊆ ℝ*)
42		supxrcl 13228	. . . 4 ⊢ (ran (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵) ⊆ ℝ* → sup(ran (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵), ℝ*, < ) ∈ ℝ*)
43	41, 42	syl 17	. . 3 ⊢ (𝜑 → sup(ran (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵), ℝ*, < ) ∈ ℝ*)
44		vex 3442	. . . . . . . 8 ⊢ 𝑦 ∈ V
45	9	elrnmpt 5905	. . . . . . . 8 ⊢ (𝑦 ∈ V → (𝑦 ∈ ran (𝑥 ∈ (𝒫 𝑍 ∩ Fin) ↦ Σ𝑘 ∈ 𝑥 𝐵) ↔ ∃𝑥 ∈ (𝒫 𝑍 ∩ Fin)𝑦 = Σ𝑘 ∈ 𝑥 𝐵))
46	44, 45	ax-mp 5	. . . . . . 7 ⊢ (𝑦 ∈ ran (𝑥 ∈ (𝒫 𝑍 ∩ Fin) ↦ Σ𝑘 ∈ 𝑥 𝐵) ↔ ∃𝑥 ∈ (𝒫 𝑍 ∩ Fin)𝑦 = Σ𝑘 ∈ 𝑥 𝐵)
47	46	biimpi 216	. . . . . 6 ⊢ (𝑦 ∈ ran (𝑥 ∈ (𝒫 𝑍 ∩ Fin) ↦ Σ𝑘 ∈ 𝑥 𝐵) → ∃𝑥 ∈ (𝒫 𝑍 ∩ Fin)𝑦 = Σ𝑘 ∈ 𝑥 𝐵)
48	47	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ran (𝑥 ∈ (𝒫 𝑍 ∩ Fin) ↦ Σ𝑘 ∈ 𝑥 𝐵)) → ∃𝑥 ∈ (𝒫 𝑍 ∩ Fin)𝑦 = Σ𝑘 ∈ 𝑥 𝐵)
49		sge0reuz.m	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℤ)
50	49	3ad2ant1 1133	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑦 = Σ𝑘 ∈ 𝑥 𝐵) → 𝑀 ∈ ℤ)
51	16	3ad2ant2 1134	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑦 = Σ𝑘 ∈ 𝑥 𝐵) → 𝑥 ⊆ 𝑍)
52	13	3adant3 1132	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑦 = Σ𝑘 ∈ 𝑥 𝐵) → 𝑥 ∈ Fin)
53	50, 2, 51, 52	uzfissfz 45513	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑦 = Σ𝑘 ∈ 𝑥 𝐵) → ∃𝑛 ∈ 𝑍 𝑥 ⊆ (𝑀...𝑛))
54		nfv 1915	. . . . . . . . . 10 ⊢ Ⅎ𝑛(𝜑 ∧ 𝑥 ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑦 = Σ𝑘 ∈ 𝑥 𝐵)
55		nfmpt1 5195	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛(𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵)
56	55	nfrn 5899	. . . . . . . . . . 11 ⊢ Ⅎ𝑛ran (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵)
57		nfv 1915	. . . . . . . . . . 11 ⊢ Ⅎ𝑛 𝑦 ≤ 𝑤
58	56, 57	nfrexw 3282	. . . . . . . . . 10 ⊢ Ⅎ𝑛∃𝑤 ∈ ran (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵)𝑦 ≤ 𝑤
59		id 22	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ 𝑍)
60		sumex 15609	. . . . . . . . . . . . . . . 16 ⊢ Σ𝑘 ∈ (𝑀...𝑛)𝐵 ∈ V
61	60	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ 𝑍 → Σ𝑘 ∈ (𝑀...𝑛)𝐵 ∈ V)
62	29	elrnmpt1 5907	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ 𝑍 ∧ Σ𝑘 ∈ (𝑀...𝑛)𝐵 ∈ V) → Σ𝑘 ∈ (𝑀...𝑛)𝐵 ∈ ran (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵))
63	59, 61, 62	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ 𝑍 → Σ𝑘 ∈ (𝑀...𝑛)𝐵 ∈ ran (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵))
64	63	3ad2ant2 1134	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 = Σ𝑘 ∈ 𝑥 𝐵) ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ⊆ (𝑀...𝑛)) → Σ𝑘 ∈ (𝑀...𝑛)𝐵 ∈ ran (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵))
65		simplr 768	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 = Σ𝑘 ∈ 𝑥 𝐵) ∧ 𝑥 ⊆ (𝑀...𝑛)) → 𝑦 = Σ𝑘 ∈ 𝑥 𝐵)
66		nfcv 2896	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑘𝑦
67		nfcv 2896	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑘𝑥
68	67	nfsum1 15611	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑘Σ𝑘 ∈ 𝑥 𝐵
69	66, 68	nfeq 2910	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑘 𝑦 = Σ𝑘 ∈ 𝑥 𝐵
70	1, 69	nfan 1900	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑦 = Σ𝑘 ∈ 𝑥 𝐵)
71		nfv 1915	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘 𝑥 ⊆ (𝑀...𝑛)
72	70, 71	nfan 1900	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑦 = Σ𝑘 ∈ 𝑥 𝐵) ∧ 𝑥 ⊆ (𝑀...𝑛))
73		fzfid 13894	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑦 = Σ𝑘 ∈ 𝑥 𝐵) ∧ 𝑥 ⊆ (𝑀...𝑛)) → (𝑀...𝑛) ∈ Fin)
74	37	ad4ant14 752	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑦 = Σ𝑘 ∈ 𝑥 𝐵) ∧ 𝑥 ⊆ (𝑀...𝑛)) ∧ 𝑘 ∈ (𝑀...𝑛)) → 𝐵 ∈ ℝ)
75		simplll 774	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑦 = Σ𝑘 ∈ 𝑥 𝐵) ∧ 𝑥 ⊆ (𝑀...𝑛)) ∧ 𝑘 ∈ (𝑀...𝑛)) → 𝜑)
76	34	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑦 = Σ𝑘 ∈ 𝑥 𝐵) ∧ 𝑥 ⊆ (𝑀...𝑛)) ∧ 𝑘 ∈ (𝑀...𝑛)) → 𝑘 ∈ 𝑍)
77		0xr 11177	. . . . . . . . . . . . . . . . . . 19 ⊢ 0 ∈ ℝ*
78	77	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 0 ∈ ℝ*)
79		pnfxr 11184	. . . . . . . . . . . . . . . . . . 19 ⊢ +∞ ∈ ℝ*
80	79	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → +∞ ∈ ℝ*)
81		icogelb 13310	. . . . . . . . . . . . . . . . . 18 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝐵 ∈ (0[,)+∞)) → 0 ≤ 𝐵)
82	78, 80, 6, 81	syl3anc 1373	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 0 ≤ 𝐵)
83	75, 76, 82	syl2anc 584	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑦 = Σ𝑘 ∈ 𝑥 𝐵) ∧ 𝑥 ⊆ (𝑀...𝑛)) ∧ 𝑘 ∈ (𝑀...𝑛)) → 0 ≤ 𝐵)
84		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑦 = Σ𝑘 ∈ 𝑥 𝐵) ∧ 𝑥 ⊆ (𝑀...𝑛)) → 𝑥 ⊆ (𝑀...𝑛))
85	72, 73, 74, 83, 84	fsumlessf 45765	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 = Σ𝑘 ∈ 𝑥 𝐵) ∧ 𝑥 ⊆ (𝑀...𝑛)) → Σ𝑘 ∈ 𝑥 𝐵 ≤ Σ𝑘 ∈ (𝑀...𝑛)𝐵)
86	65, 85	eqbrtrd 5118	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 = Σ𝑘 ∈ 𝑥 𝐵) ∧ 𝑥 ⊆ (𝑀...𝑛)) → 𝑦 ≤ Σ𝑘 ∈ (𝑀...𝑛)𝐵)
87	86	3adant2 1131	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 = Σ𝑘 ∈ 𝑥 𝐵) ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ⊆ (𝑀...𝑛)) → 𝑦 ≤ Σ𝑘 ∈ (𝑀...𝑛)𝐵)
88		breq2 5100	. . . . . . . . . . . . . 14 ⊢ (𝑤 = Σ𝑘 ∈ (𝑀...𝑛)𝐵 → (𝑦 ≤ 𝑤 ↔ 𝑦 ≤ Σ𝑘 ∈ (𝑀...𝑛)𝐵))
89	88	rspcev 3574	. . . . . . . . . . . . 13 ⊢ ((Σ𝑘 ∈ (𝑀...𝑛)𝐵 ∈ ran (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵) ∧ 𝑦 ≤ Σ𝑘 ∈ (𝑀...𝑛)𝐵) → ∃𝑤 ∈ ran (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵)𝑦 ≤ 𝑤)
90	64, 87, 89	syl2anc 584	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 = Σ𝑘 ∈ 𝑥 𝐵) ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ⊆ (𝑀...𝑛)) → ∃𝑤 ∈ ran (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵)𝑦 ≤ 𝑤)
91	90	3exp 1119	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 = Σ𝑘 ∈ 𝑥 𝐵) → (𝑛 ∈ 𝑍 → (𝑥 ⊆ (𝑀...𝑛) → ∃𝑤 ∈ ran (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵)𝑦 ≤ 𝑤)))
92	91	3adant2 1131	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑦 = Σ𝑘 ∈ 𝑥 𝐵) → (𝑛 ∈ 𝑍 → (𝑥 ⊆ (𝑀...𝑛) → ∃𝑤 ∈ ran (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵)𝑦 ≤ 𝑤)))
93	54, 58, 92	rexlimd 3241	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑦 = Σ𝑘 ∈ 𝑥 𝐵) → (∃𝑛 ∈ 𝑍 𝑥 ⊆ (𝑀...𝑛) → ∃𝑤 ∈ ran (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵)𝑦 ≤ 𝑤))
94	53, 93	mpd 15	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑦 = Σ𝑘 ∈ 𝑥 𝐵) → ∃𝑤 ∈ ran (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵)𝑦 ≤ 𝑤)
95	94	3exp 1119	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (𝒫 𝑍 ∩ Fin) → (𝑦 = Σ𝑘 ∈ 𝑥 𝐵 → ∃𝑤 ∈ ran (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵)𝑦 ≤ 𝑤)))
96	95	rexlimdv 3133	. . . . . 6 ⊢ (𝜑 → (∃𝑥 ∈ (𝒫 𝑍 ∩ Fin)𝑦 = Σ𝑘 ∈ 𝑥 𝐵 → ∃𝑤 ∈ ran (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵)𝑦 ≤ 𝑤))
97	96	imp 406	. . . . 5 ⊢ ((𝜑 ∧ ∃𝑥 ∈ (𝒫 𝑍 ∩ Fin)𝑦 = Σ𝑘 ∈ 𝑥 𝐵) → ∃𝑤 ∈ ran (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵)𝑦 ≤ 𝑤)
98	48, 97	syldan 591	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ran (𝑥 ∈ (𝒫 𝑍 ∩ Fin) ↦ Σ𝑘 ∈ 𝑥 𝐵)) → ∃𝑤 ∈ ran (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵)𝑦 ≤ 𝑤)
99	25, 41, 98	suplesup2 45562	. . 3 ⊢ (𝜑 → sup(ran (𝑥 ∈ (𝒫 𝑍 ∩ Fin) ↦ Σ𝑘 ∈ 𝑥 𝐵), ℝ*, < ) ≤ sup(ran (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵), ℝ*, < ))
100	29	elrnmpt 5905	. . . . . . . . . 10 ⊢ (𝑦 ∈ V → (𝑦 ∈ ran (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵) ↔ ∃𝑛 ∈ 𝑍 𝑦 = Σ𝑘 ∈ (𝑀...𝑛)𝐵))
101	44, 100	ax-mp 5	. . . . . . . . 9 ⊢ (𝑦 ∈ ran (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵) ↔ ∃𝑛 ∈ 𝑍 𝑦 = Σ𝑘 ∈ (𝑀...𝑛)𝐵)
102	101	biimpi 216	. . . . . . . 8 ⊢ (𝑦 ∈ ran (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵) → ∃𝑛 ∈ 𝑍 𝑦 = Σ𝑘 ∈ (𝑀...𝑛)𝐵)
103	102	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ran (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵)) → ∃𝑛 ∈ 𝑍 𝑦 = Σ𝑘 ∈ (𝑀...𝑛)𝐵)
104	34	ssriv 3935	. . . . . . . . . . . . . . 15 ⊢ (𝑀...𝑛) ⊆ 𝑍
105		ovex 7389	. . . . . . . . . . . . . . . 16 ⊢ (𝑀...𝑛) ∈ V
106	105	elpw 4556	. . . . . . . . . . . . . . 15 ⊢ ((𝑀...𝑛) ∈ 𝒫 𝑍 ↔ (𝑀...𝑛) ⊆ 𝑍)
107	104, 106	mpbir 231	. . . . . . . . . . . . . 14 ⊢ (𝑀...𝑛) ∈ 𝒫 𝑍
108		fzfi 13893	. . . . . . . . . . . . . 14 ⊢ (𝑀...𝑛) ∈ Fin
109	107, 108	elini 4149	. . . . . . . . . . . . 13 ⊢ (𝑀...𝑛) ∈ (𝒫 𝑍 ∩ Fin)
110	109	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑦 = Σ𝑘 ∈ (𝑀...𝑛)𝐵 → (𝑀...𝑛) ∈ (𝒫 𝑍 ∩ Fin))
111		id 22	. . . . . . . . . . . 12 ⊢ (𝑦 = Σ𝑘 ∈ (𝑀...𝑛)𝐵 → 𝑦 = Σ𝑘 ∈ (𝑀...𝑛)𝐵)
112		sumeq1 15610	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝑀...𝑛) → Σ𝑘 ∈ 𝑥 𝐵 = Σ𝑘 ∈ (𝑀...𝑛)𝐵)
113	112	rspceeqv 3597	. . . . . . . . . . . 12 ⊢ (((𝑀...𝑛) ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑦 = Σ𝑘 ∈ (𝑀...𝑛)𝐵) → ∃𝑥 ∈ (𝒫 𝑍 ∩ Fin)𝑦 = Σ𝑘 ∈ 𝑥 𝐵)
114	110, 111, 113	syl2anc 584	. . . . . . . . . . 11 ⊢ (𝑦 = Σ𝑘 ∈ (𝑀...𝑛)𝐵 → ∃𝑥 ∈ (𝒫 𝑍 ∩ Fin)𝑦 = Σ𝑘 ∈ 𝑥 𝐵)
115	44	a1i 11	. . . . . . . . . . 11 ⊢ (𝑦 = Σ𝑘 ∈ (𝑀...𝑛)𝐵 → 𝑦 ∈ V)
116	9, 114, 115	elrnmptd 5910	. . . . . . . . . 10 ⊢ (𝑦 = Σ𝑘 ∈ (𝑀...𝑛)𝐵 → 𝑦 ∈ ran (𝑥 ∈ (𝒫 𝑍 ∩ Fin) ↦ Σ𝑘 ∈ 𝑥 𝐵))
117	116	2a1i 12	. . . . . . . . 9 ⊢ (𝜑 → (𝑛 ∈ 𝑍 → (𝑦 = Σ𝑘 ∈ (𝑀...𝑛)𝐵 → 𝑦 ∈ ran (𝑥 ∈ (𝒫 𝑍 ∩ Fin) ↦ Σ𝑘 ∈ 𝑥 𝐵))))
118	117	rexlimdv 3133	. . . . . . . 8 ⊢ (𝜑 → (∃𝑛 ∈ 𝑍 𝑦 = Σ𝑘 ∈ (𝑀...𝑛)𝐵 → 𝑦 ∈ ran (𝑥 ∈ (𝒫 𝑍 ∩ Fin) ↦ Σ𝑘 ∈ 𝑥 𝐵)))
119	118	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ran (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵)) → (∃𝑛 ∈ 𝑍 𝑦 = Σ𝑘 ∈ (𝑀...𝑛)𝐵 → 𝑦 ∈ ran (𝑥 ∈ (𝒫 𝑍 ∩ Fin) ↦ Σ𝑘 ∈ 𝑥 𝐵)))
120	103, 119	mpd 15	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ran (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵)) → 𝑦 ∈ ran (𝑥 ∈ (𝒫 𝑍 ∩ Fin) ↦ Σ𝑘 ∈ 𝑥 𝐵))
121	120	ralrimiva 3126	. . . . 5 ⊢ (𝜑 → ∀𝑦 ∈ ran (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵)𝑦 ∈ ran (𝑥 ∈ (𝒫 𝑍 ∩ Fin) ↦ Σ𝑘 ∈ 𝑥 𝐵))
122		dfss3 3920	. . . . 5 ⊢ (ran (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵) ⊆ ran (𝑥 ∈ (𝒫 𝑍 ∩ Fin) ↦ Σ𝑘 ∈ 𝑥 𝐵) ↔ ∀𝑦 ∈ ran (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵)𝑦 ∈ ran (𝑥 ∈ (𝒫 𝑍 ∩ Fin) ↦ Σ𝑘 ∈ 𝑥 𝐵))
123	121, 122	sylibr 234	. . . 4 ⊢ (𝜑 → ran (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵) ⊆ ran (𝑥 ∈ (𝒫 𝑍 ∩ Fin) ↦ Σ𝑘 ∈ 𝑥 𝐵))
124		supxrss 13245	. . . 4 ⊢ ((ran (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵) ⊆ ran (𝑥 ∈ (𝒫 𝑍 ∩ Fin) ↦ Σ𝑘 ∈ 𝑥 𝐵) ∧ ran (𝑥 ∈ (𝒫 𝑍 ∩ Fin) ↦ Σ𝑘 ∈ 𝑥 𝐵) ⊆ ℝ*) → sup(ran (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵), ℝ*, < ) ≤ sup(ran (𝑥 ∈ (𝒫 𝑍 ∩ Fin) ↦ Σ𝑘 ∈ 𝑥 𝐵), ℝ*, < ))
125	123, 25, 124	syl2anc 584	. . 3 ⊢ (𝜑 → sup(ran (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵), ℝ*, < ) ≤ sup(ran (𝑥 ∈ (𝒫 𝑍 ∩ Fin) ↦ Σ𝑘 ∈ 𝑥 𝐵), ℝ*, < ))
126	27, 43, 99, 125	xrletrid 13067	. 2 ⊢ (𝜑 → sup(ran (𝑥 ∈ (𝒫 𝑍 ∩ Fin) ↦ Σ𝑘 ∈ 𝑥 𝐵), ℝ*, < ) = sup(ran (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵), ℝ*, < ))
127	7, 126	eqtrd 2769	1 ⊢ (𝜑 → (Σ^‘(𝑘 ∈ 𝑍 ↦ 𝐵)) = sup(ran (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵), ℝ*, < ))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541  Ⅎwnf 1784   ∈ wcel 2113  ∀wral 3049  ∃wrex 3058  Vcvv 3438   ∩ cin 3898   ⊆ wss 3899  𝒫 cpw 4552   class class class wbr 5096   ↦ cmpt 5177  ran crn 5623  ‘cfv 6490  (class class class)co 7356  Fincfn 8881  supcsup 9341  ℝcr 11023  0cc0 11024  +∞cpnf 11161  ℝ^*cxr 11163   < clt 11164   ≤ cle 11165  ℤcz 12486  ℤ_≥cuz 12749  [,)cico 13261  ...cfz 13421  Σcsu 15607  Σ^{^}csumge0 46548

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678  ax-inf2 9548  ax-cnex 11080  ax-resscn 11081  ax-1cn 11082  ax-icn 11083  ax-addcl 11084  ax-addrcl 11085  ax-mulcl 11086  ax-mulrcl 11087  ax-mulcom 11088  ax-addass 11089  ax-mulass 11090  ax-distr 11091  ax-i2m1 11092  ax-1ne0 11093  ax-1rid 11094  ax-rnegex 11095  ax-rrecex 11096  ax-cnre 11097  ax-pre-lttri 11098  ax-pre-lttrn 11099  ax-pre-ltadd 11100  ax-pre-mulgt0 11101  ax-pre-sup 11102

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-nel 3035  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-int 4901  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-se 5576  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-isom 6499  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-er 8633  df-en 8882  df-dom 8883  df-sdom 8884  df-fin 8885  df-sup 9343  df-oi 9413  df-card 9849  df-pnf 11166  df-mnf 11167  df-xr 11168  df-ltxr 11169  df-le 11170  df-sub 11364  df-neg 11365  df-div 11793  df-nn 12144  df-2 12206  df-3 12207  df-n0 12400  df-z 12487  df-uz 12750  df-rp 12904  df-ico 13265  df-icc 13266  df-fz 13422  df-fzo 13569  df-seq 13923  df-exp 13983  df-hash 14252  df-cj 15020  df-re 15021  df-im 15022  df-sqrt 15156  df-abs 15157  df-clim 15409  df-sum 15608  df-sumge0 46549

This theorem is referenced by:  sge0reuzb  46634