Theorem sge0reuz 45870

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 6903	. . . 4 ⊢ (ℤ≥‘𝑀) ∈ V
5	3, 4	eqeltrdi 2833	. . 3 ⊢ (𝜑 → 𝑍 ∈ V)
6		sge0reuz.b	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ (0[,)+∞))
7	1, 5, 6	sge0revalmpt 45801	. 2 ⊢ (𝜑 → (Σ^‘(𝑘 ∈ 𝑍 ↦ 𝐵)) = sup(ran (𝑥 ∈ (𝒫 𝑍 ∩ Fin) ↦ Σ𝑘 ∈ 𝑥 𝐵), ℝ*, < ))
8		nfv 1909	. . . . 5 ⊢ Ⅎ𝑥𝜑
9		eqid 2725	. . . . 5 ⊢ (𝑥 ∈ (𝒫 𝑍 ∩ Fin) ↦ Σ𝑘 ∈ 𝑥 𝐵) = (𝑥 ∈ (𝒫 𝑍 ∩ Fin) ↦ Σ𝑘 ∈ 𝑥 𝐵)
10		nfv 1909	. . . . . . . 8 ⊢ Ⅎ𝑘 𝑥 ∈ (𝒫 𝑍 ∩ Fin)
11	1, 10	nfan 1894	. . . . . . 7 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑥 ∈ (𝒫 𝑍 ∩ Fin))
12		elinel2 4188	. . . . . . . 8 ⊢ (𝑥 ∈ (𝒫 𝑍 ∩ Fin) → 𝑥 ∈ Fin)
13	12	adantl 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑍 ∩ Fin)) → 𝑥 ∈ Fin)
14		rge0ssre 13463	. . . . . . . 8 ⊢ (0[,)+∞) ⊆ ℝ
15		simpll 765	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑘 ∈ 𝑥) → 𝜑)
16		elpwinss 44450	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝒫 𝑍 ∩ Fin) → 𝑥 ⊆ 𝑍)
17	16	adantr 479	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑘 ∈ 𝑥) → 𝑥 ⊆ 𝑍)
18		simpr 483	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑘 ∈ 𝑥) → 𝑘 ∈ 𝑥)
19	17, 18	sseldd 3973	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑘 ∈ 𝑥) → 𝑘 ∈ 𝑍)
20	19	adantll 712	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑘 ∈ 𝑥) → 𝑘 ∈ 𝑍)
21	15, 20, 6	syl2anc 582	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑘 ∈ 𝑥) → 𝐵 ∈ (0[,)+∞))
22	14, 21	sselid 3970	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑘 ∈ 𝑥) → 𝐵 ∈ ℝ)
23	11, 13, 22	fsumreclf 44999	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑍 ∩ Fin)) → Σ𝑘 ∈ 𝑥 𝐵 ∈ ℝ)
24	23	rexrd 11292	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑍 ∩ Fin)) → Σ𝑘 ∈ 𝑥 𝐵 ∈ ℝ*)
25	8, 9, 24	rnmptssd 44605	. . . 4 ⊢ (𝜑 → ran (𝑥 ∈ (𝒫 𝑍 ∩ Fin) ↦ Σ𝑘 ∈ 𝑥 𝐵) ⊆ ℝ*)
26		supxrcl 13324	. . . 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 1909	. . . . 5 ⊢ Ⅎ𝑛𝜑
29		eqid 2725	. . . . 5 ⊢ (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵) = (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵)
30		nfv 1909	. . . . . . . 8 ⊢ Ⅎ𝑘 𝑛 ∈ 𝑍
31	1, 30	nfan 1894	. . . . . . 7 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑛 ∈ 𝑍)
32		fzfid 13968	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑀...𝑛) ∈ Fin)
33		elfzuz 13527	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (𝑀...𝑛) → 𝑘 ∈ (ℤ≥‘𝑀))
34	33, 2	eleqtrrdi 2836	. . . . . . . . . 10 ⊢ (𝑘 ∈ (𝑀...𝑛) → 𝑘 ∈ 𝑍)
35	34	adantl 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑛)) → 𝑘 ∈ 𝑍)
36	14, 6	sselid 3970	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℝ)
37	35, 36	syldan 589	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑛)) → 𝐵 ∈ ℝ)
38	37	adantlr 713	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ (𝑀...𝑛)) → 𝐵 ∈ ℝ)
39	31, 32, 38	fsumreclf 44999	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → Σ𝑘 ∈ (𝑀...𝑛)𝐵 ∈ ℝ)
40	39	rexrd 11292	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → Σ𝑘 ∈ (𝑀...𝑛)𝐵 ∈ ℝ*)
41	28, 29, 40	rnmptssd 44605	. . . 4 ⊢ (𝜑 → ran (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵) ⊆ ℝ*)
42		supxrcl 13324	. . . 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 3467	. . . . . . . 8 ⊢ 𝑦 ∈ V
45	9	elrnmpt 5950	. . . . . . . 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 215	. . . . . 6 ⊢ (𝑦 ∈ ran (𝑥 ∈ (𝒫 𝑍 ∩ Fin) ↦ Σ𝑘 ∈ 𝑥 𝐵) → ∃𝑥 ∈ (𝒫 𝑍 ∩ Fin)𝑦 = Σ𝑘 ∈ 𝑥 𝐵)
48	47	adantl 480	. . . . 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 1130	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑦 = Σ𝑘 ∈ 𝑥 𝐵) → 𝑀 ∈ ℤ)
51	16	3ad2ant2 1131	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑦 = Σ𝑘 ∈ 𝑥 𝐵) → 𝑥 ⊆ 𝑍)
52	13	3adant3 1129	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑦 = Σ𝑘 ∈ 𝑥 𝐵) → 𝑥 ∈ Fin)
53	50, 2, 51, 52	uzfissfz 44743	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑦 = Σ𝑘 ∈ 𝑥 𝐵) → ∃𝑛 ∈ 𝑍 𝑥 ⊆ (𝑀...𝑛))
54		nfv 1909	. . . . . . . . . 10 ⊢ Ⅎ𝑛(𝜑 ∧ 𝑥 ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑦 = Σ𝑘 ∈ 𝑥 𝐵)
55		nfmpt1 5249	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛(𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵)
56	55	nfrn 5946	. . . . . . . . . . 11 ⊢ Ⅎ𝑛ran (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵)
57		nfv 1909	. . . . . . . . . . 11 ⊢ Ⅎ𝑛 𝑦 ≤ 𝑤
58	56, 57	nfrexw 3301	. . . . . . . . . 10 ⊢ Ⅎ𝑛∃𝑤 ∈ ran (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵)𝑦 ≤ 𝑤
59		id 22	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ 𝑍)
60		sumex 15664	. . . . . . . . . . . . . . . 16 ⊢ Σ𝑘 ∈ (𝑀...𝑛)𝐵 ∈ V
61	60	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ 𝑍 → Σ𝑘 ∈ (𝑀...𝑛)𝐵 ∈ V)
62	29	elrnmpt1 5952	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ 𝑍 ∧ Σ𝑘 ∈ (𝑀...𝑛)𝐵 ∈ V) → Σ𝑘 ∈ (𝑀...𝑛)𝐵 ∈ ran (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵))
63	59, 61, 62	syl2anc 582	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ 𝑍 → Σ𝑘 ∈ (𝑀...𝑛)𝐵 ∈ ran (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵))
64	63	3ad2ant2 1131	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 = Σ𝑘 ∈ 𝑥 𝐵) ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ⊆ (𝑀...𝑛)) → Σ𝑘 ∈ (𝑀...𝑛)𝐵 ∈ ran (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵))
65		simplr 767	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 = Σ𝑘 ∈ 𝑥 𝐵) ∧ 𝑥 ⊆ (𝑀...𝑛)) → 𝑦 = Σ𝑘 ∈ 𝑥 𝐵)
66		nfcv 2892	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑘𝑦
67		nfcv 2892	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑘𝑥
68	67	nfsum1 15666	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑘Σ𝑘 ∈ 𝑥 𝐵
69	66, 68	nfeq 2906	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑘 𝑦 = Σ𝑘 ∈ 𝑥 𝐵
70	1, 69	nfan 1894	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑦 = Σ𝑘 ∈ 𝑥 𝐵)
71		nfv 1909	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘 𝑥 ⊆ (𝑀...𝑛)
72	70, 71	nfan 1894	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑦 = Σ𝑘 ∈ 𝑥 𝐵) ∧ 𝑥 ⊆ (𝑀...𝑛))
73		fzfid 13968	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑦 = Σ𝑘 ∈ 𝑥 𝐵) ∧ 𝑥 ⊆ (𝑀...𝑛)) → (𝑀...𝑛) ∈ Fin)
74	37	ad4ant14 750	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑦 = Σ𝑘 ∈ 𝑥 𝐵) ∧ 𝑥 ⊆ (𝑀...𝑛)) ∧ 𝑘 ∈ (𝑀...𝑛)) → 𝐵 ∈ ℝ)
75		simplll 773	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑦 = Σ𝑘 ∈ 𝑥 𝐵) ∧ 𝑥 ⊆ (𝑀...𝑛)) ∧ 𝑘 ∈ (𝑀...𝑛)) → 𝜑)
76	34	adantl 480	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑦 = Σ𝑘 ∈ 𝑥 𝐵) ∧ 𝑥 ⊆ (𝑀...𝑛)) ∧ 𝑘 ∈ (𝑀...𝑛)) → 𝑘 ∈ 𝑍)
77		0xr 11289	. . . . . . . . . . . . . . . . . . 19 ⊢ 0 ∈ ℝ*
78	77	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 0 ∈ ℝ*)
79		pnfxr 11296	. . . . . . . . . . . . . . . . . . 19 ⊢ +∞ ∈ ℝ*
80	79	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → +∞ ∈ ℝ*)
81		icogelb 13405	. . . . . . . . . . . . . . . . . 18 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝐵 ∈ (0[,)+∞)) → 0 ≤ 𝐵)
82	78, 80, 6, 81	syl3anc 1368	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 0 ≤ 𝐵)
83	75, 76, 82	syl2anc 582	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑦 = Σ𝑘 ∈ 𝑥 𝐵) ∧ 𝑥 ⊆ (𝑀...𝑛)) ∧ 𝑘 ∈ (𝑀...𝑛)) → 0 ≤ 𝐵)
84		simpr 483	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑦 = Σ𝑘 ∈ 𝑥 𝐵) ∧ 𝑥 ⊆ (𝑀...𝑛)) → 𝑥 ⊆ (𝑀...𝑛))
85	72, 73, 74, 83, 84	fsumlessf 45000	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 = Σ𝑘 ∈ 𝑥 𝐵) ∧ 𝑥 ⊆ (𝑀...𝑛)) → Σ𝑘 ∈ 𝑥 𝐵 ≤ Σ𝑘 ∈ (𝑀...𝑛)𝐵)
86	65, 85	eqbrtrd 5163	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 = Σ𝑘 ∈ 𝑥 𝐵) ∧ 𝑥 ⊆ (𝑀...𝑛)) → 𝑦 ≤ Σ𝑘 ∈ (𝑀...𝑛)𝐵)
87	86	3adant2 1128	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 = Σ𝑘 ∈ 𝑥 𝐵) ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ⊆ (𝑀...𝑛)) → 𝑦 ≤ Σ𝑘 ∈ (𝑀...𝑛)𝐵)
88		breq2 5145	. . . . . . . . . . . . . 14 ⊢ (𝑤 = Σ𝑘 ∈ (𝑀...𝑛)𝐵 → (𝑦 ≤ 𝑤 ↔ 𝑦 ≤ Σ𝑘 ∈ (𝑀...𝑛)𝐵))
89	88	rspcev 3601	. . . . . . . . . . . . 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 582	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 = Σ𝑘 ∈ 𝑥 𝐵) ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ⊆ (𝑀...𝑛)) → ∃𝑤 ∈ ran (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵)𝑦 ≤ 𝑤)
91	90	3exp 1116	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 = Σ𝑘 ∈ 𝑥 𝐵) → (𝑛 ∈ 𝑍 → (𝑥 ⊆ (𝑀...𝑛) → ∃𝑤 ∈ ran (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵)𝑦 ≤ 𝑤)))
92	91	3adant2 1128	. . . . . . . . . 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 3254	. . . . . . . . 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 1116	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (𝒫 𝑍 ∩ Fin) → (𝑦 = Σ𝑘 ∈ 𝑥 𝐵 → ∃𝑤 ∈ ran (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵)𝑦 ≤ 𝑤)))
96	95	rexlimdv 3143	. . . . . 6 ⊢ (𝜑 → (∃𝑥 ∈ (𝒫 𝑍 ∩ Fin)𝑦 = Σ𝑘 ∈ 𝑥 𝐵 → ∃𝑤 ∈ ran (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵)𝑦 ≤ 𝑤))
97	96	imp 405	. . . . 5 ⊢ ((𝜑 ∧ ∃𝑥 ∈ (𝒫 𝑍 ∩ Fin)𝑦 = Σ𝑘 ∈ 𝑥 𝐵) → ∃𝑤 ∈ ran (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵)𝑦 ≤ 𝑤)
98	48, 97	syldan 589	. . . 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 44793	. . 3 ⊢ (𝜑 → sup(ran (𝑥 ∈ (𝒫 𝑍 ∩ Fin) ↦ Σ𝑘 ∈ 𝑥 𝐵), ℝ*, < ) ≤ sup(ran (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵), ℝ*, < ))
100	29	elrnmpt 5950	. . . . . . . . . 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 215	. . . . . . . 8 ⊢ (𝑦 ∈ ran (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵) → ∃𝑛 ∈ 𝑍 𝑦 = Σ𝑘 ∈ (𝑀...𝑛)𝐵)
103	102	adantl 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ran (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵)) → ∃𝑛 ∈ 𝑍 𝑦 = Σ𝑘 ∈ (𝑀...𝑛)𝐵)
104	34	ssriv 3976	. . . . . . . . . . . . . . 15 ⊢ (𝑀...𝑛) ⊆ 𝑍
105		ovex 7447	. . . . . . . . . . . . . . . 16 ⊢ (𝑀...𝑛) ∈ V
106	105	elpw 4600	. . . . . . . . . . . . . . 15 ⊢ ((𝑀...𝑛) ∈ 𝒫 𝑍 ↔ (𝑀...𝑛) ⊆ 𝑍)
107	104, 106	mpbir 230	. . . . . . . . . . . . . 14 ⊢ (𝑀...𝑛) ∈ 𝒫 𝑍
108		fzfi 13967	. . . . . . . . . . . . . 14 ⊢ (𝑀...𝑛) ∈ Fin
109	107, 108	elini 4185	. . . . . . . . . . . . 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 15665	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝑀...𝑛) → Σ𝑘 ∈ 𝑥 𝐵 = Σ𝑘 ∈ (𝑀...𝑛)𝐵)
113	112	rspceeqv 3623	. . . . . . . . . . . 12 ⊢ (((𝑀...𝑛) ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑦 = Σ𝑘 ∈ (𝑀...𝑛)𝐵) → ∃𝑥 ∈ (𝒫 𝑍 ∩ Fin)𝑦 = Σ𝑘 ∈ 𝑥 𝐵)
114	110, 111, 113	syl2anc 582	. . . . . . . . . . 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 5955	. . . . . . . . . 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 3143	. . . . . . . 8 ⊢ (𝜑 → (∃𝑛 ∈ 𝑍 𝑦 = Σ𝑘 ∈ (𝑀...𝑛)𝐵 → 𝑦 ∈ ran (𝑥 ∈ (𝒫 𝑍 ∩ Fin) ↦ Σ𝑘 ∈ 𝑥 𝐵)))
119	118	adantr 479	. . . . . . 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 3136	. . . . 5 ⊢ (𝜑 → ∀𝑦 ∈ ran (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵)𝑦 ∈ ran (𝑥 ∈ (𝒫 𝑍 ∩ Fin) ↦ Σ𝑘 ∈ 𝑥 𝐵))
122		dfss3 3960	. . . . 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 233	. . . 4 ⊢ (𝜑 → ran (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵) ⊆ ran (𝑥 ∈ (𝒫 𝑍 ∩ Fin) ↦ Σ𝑘 ∈ 𝑥 𝐵))
124		supxrss 13341	. . . 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 582	. . 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 13164	. 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 2765	1 ⊢ (𝜑 → (Σ^‘(𝑘 ∈ 𝑍 ↦ 𝐵)) = sup(ran (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵), ℝ*, < ))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084   = wceq 1533  Ⅎwnf 1777   ∈ wcel 2098  ∀wral 3051  ∃wrex 3060  Vcvv 3463   ∩ cin 3938   ⊆ wss 3939  𝒫 cpw 4596   class class class wbr 5141   ↦ cmpt 5224  ran crn 5671  ‘cfv 6541  (class class class)co 7414  Fincfn 8960  supcsup 9461  ℝcr 11135  0cc0 11136  +∞cpnf 11273  ℝ^*cxr 11275   < clt 11276   ≤ cle 11277  ℤcz 12586  ℤ_≥cuz 12850  [,)cico 13356  ...cfz 13514  Σcsu 15662  Σ^{^}csumge0 45785

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5357  ax-pr 5421  ax-un 7736  ax-inf2 9662  ax-cnex 11192  ax-resscn 11193  ax-1cn 11194  ax-icn 11195  ax-addcl 11196  ax-addrcl 11197  ax-mulcl 11198  ax-mulrcl 11199  ax-mulcom 11200  ax-addass 11201  ax-mulass 11202  ax-distr 11203  ax-i2m1 11204  ax-1ne0 11205  ax-1rid 11206  ax-rnegex 11207  ax-rrecex 11208  ax-cnre 11209  ax-pre-lttri 11210  ax-pre-lttrn 11211  ax-pre-ltadd 11212  ax-pre-mulgt0 11213  ax-pre-sup 11214

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-pss 3958  df-nul 4317  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-int 4943  df-iun 4991  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5568  df-eprel 5574  df-po 5582  df-so 5583  df-fr 5625  df-se 5626  df-we 5627  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-rn 5681  df-res 5682  df-ima 5683  df-pred 6298  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-isom 6550  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7867  df-1st 7989  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8388  df-rdg 8427  df-1o 8483  df-er 8721  df-en 8961  df-dom 8962  df-sdom 8963  df-fin 8964  df-sup 9463  df-oi 9531  df-card 9960  df-pnf 11278  df-mnf 11279  df-xr 11280  df-ltxr 11281  df-le 11282  df-sub 11474  df-neg 11475  df-div 11900  df-nn 12241  df-2 12303  df-3 12304  df-n0 12501  df-z 12587  df-uz 12851  df-rp 13005  df-ico 13360  df-icc 13361  df-fz 13515  df-fzo 13658  df-seq 13997  df-exp 14057  df-hash 14320  df-cj 15076  df-re 15077  df-im 15078  df-sqrt 15212  df-abs 15213  df-clim 15462  df-sum 15663  df-sumge0 45786

This theorem is referenced by:  sge0reuzb  45871