Theorem sge0reuzb 39341

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

Hypotheses

Ref	Expression
sge0reuzb.k	⊢ Ⅎ𝑘𝜑
sge0reuzb.p	⊢ Ⅎ𝑥𝜑
sge0reuzb.m	⊢ (𝜑 → 𝑀 ∈ ℤ)
sge0reuzb.z	⊢ 𝑍 = (ℤ≥‘𝑀)
sge0reuzb.b	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ (0[,)+∞))
sge0reuzb.x	⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑛 ∈ 𝑍 Σ𝑘 ∈ (𝑀...𝑛)𝐵 ≤ 𝑥)

Assertion

Ref	Expression
sge0reuzb	⊢ (𝜑 → (Σ^‘(𝑘 ∈ 𝑍 ↦ 𝐵)) = sup(ran (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵), ℝ, < ))

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

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

Proof of Theorem sge0reuzb

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sge0reuzb.k	. . 3 ⊢ Ⅎ𝑘𝜑
2		sge0reuzb.m	. . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ)
3		sge0reuzb.z	. . 3 ⊢ 𝑍 = (ℤ≥‘𝑀)
4		sge0reuzb.b	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ (0[,)+∞))
5	1, 2, 3, 4	sge0reuz 39340	. 2 ⊢ (𝜑 → (Σ^‘(𝑘 ∈ 𝑍 ↦ 𝐵)) = sup(ran (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵), ℝ*, < ))
6		nfv 1830	. . . 4 ⊢ Ⅎ𝑛𝜑
7		eqid 2610	. . . 4 ⊢ (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵) = (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵)
8		nfv 1830	. . . . . 6 ⊢ Ⅎ𝑘 𝑛 ∈ 𝑍
9	1, 8	nfan 1816	. . . . 5 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑛 ∈ 𝑍)
10		fzfid 12634	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑀...𝑛) ∈ Fin)
11		elfzuz 12209	. . . . . . . . 9 ⊢ (𝑘 ∈ (𝑀...𝑛) → 𝑘 ∈ (ℤ≥‘𝑀))
12	11, 3	syl6eleqr 2699	. . . . . . . 8 ⊢ (𝑘 ∈ (𝑀...𝑛) → 𝑘 ∈ 𝑍)
13	12	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑛)) → 𝑘 ∈ 𝑍)
14		rge0ssre 12151	. . . . . . . 8 ⊢ (0[,)+∞) ⊆ ℝ
15	14, 4	sseldi 3566	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℝ)
16	13, 15	syldan 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑛)) → 𝐵 ∈ ℝ)
17	16	adantlr 747	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ (𝑀...𝑛)) → 𝐵 ∈ ℝ)
18	9, 10, 17	fsumreclf 38643	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → Σ𝑘 ∈ (𝑀...𝑛)𝐵 ∈ ℝ)
19	6, 7, 18	rnmptssd 38380	. . 3 ⊢ (𝜑 → ran (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵) ⊆ ℝ)
20		uzid 11578	. . . . . . . 8 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ≥‘𝑀))
21	2, 20	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘𝑀))
22	21, 3	syl6eleqr 2699	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ 𝑍)
23		eqidd 2611	. . . . . 6 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀...𝑀)𝐵 = Σ𝑘 ∈ (𝑀...𝑀)𝐵)
24		oveq2 6557	. . . . . . . . 9 ⊢ (𝑛 = 𝑀 → (𝑀...𝑛) = (𝑀...𝑀))
25	24	sumeq1d 14279	. . . . . . . 8 ⊢ (𝑛 = 𝑀 → Σ𝑘 ∈ (𝑀...𝑛)𝐵 = Σ𝑘 ∈ (𝑀...𝑀)𝐵)
26	25	eqeq2d 2620	. . . . . . 7 ⊢ (𝑛 = 𝑀 → (Σ𝑘 ∈ (𝑀...𝑀)𝐵 = Σ𝑘 ∈ (𝑀...𝑛)𝐵 ↔ Σ𝑘 ∈ (𝑀...𝑀)𝐵 = Σ𝑘 ∈ (𝑀...𝑀)𝐵))
27	26	rspcev 3282	. . . . . 6 ⊢ ((𝑀 ∈ 𝑍 ∧ Σ𝑘 ∈ (𝑀...𝑀)𝐵 = Σ𝑘 ∈ (𝑀...𝑀)𝐵) → ∃𝑛 ∈ 𝑍 Σ𝑘 ∈ (𝑀...𝑀)𝐵 = Σ𝑘 ∈ (𝑀...𝑛)𝐵)
28	22, 23, 27	syl2anc 691	. . . . 5 ⊢ (𝜑 → ∃𝑛 ∈ 𝑍 Σ𝑘 ∈ (𝑀...𝑀)𝐵 = Σ𝑘 ∈ (𝑀...𝑛)𝐵)
29		sumex 14266	. . . . . 6 ⊢ Σ𝑘 ∈ (𝑀...𝑀)𝐵 ∈ V
30	29	a1i 11	. . . . 5 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀...𝑀)𝐵 ∈ V)
31	7, 28, 30	elrnmptd 38361	. . . 4 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀...𝑀)𝐵 ∈ ran (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵))
32		ne0i 3880	. . . 4 ⊢ (Σ𝑘 ∈ (𝑀...𝑀)𝐵 ∈ ran (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵) → ran (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵) ≠ ∅)
33	31, 32	syl 17	. . 3 ⊢ (𝜑 → ran (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵) ≠ ∅)
34		sge0reuzb.x	. . . 4 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑛 ∈ 𝑍 Σ𝑘 ∈ (𝑀...𝑛)𝐵 ≤ 𝑥)
35		sge0reuzb.p	. . . . 5 ⊢ Ⅎ𝑥𝜑
36		vex 3176	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ V
37	7	elrnmpt 5293	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ V → (𝑦 ∈ ran (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵) ↔ ∃𝑛 ∈ 𝑍 𝑦 = Σ𝑘 ∈ (𝑀...𝑛)𝐵))
38	36, 37	ax-mp 5	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ran (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵) ↔ ∃𝑛 ∈ 𝑍 𝑦 = Σ𝑘 ∈ (𝑀...𝑛)𝐵)
39	38	biimpi 205	. . . . . . . . . 10 ⊢ (𝑦 ∈ ran (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵) → ∃𝑛 ∈ 𝑍 𝑦 = Σ𝑘 ∈ (𝑀...𝑛)𝐵)
40	39	adantl 481	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ∀𝑛 ∈ 𝑍 Σ𝑘 ∈ (𝑀...𝑛)𝐵 ≤ 𝑥) ∧ 𝑦 ∈ ran (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵)) → ∃𝑛 ∈ 𝑍 𝑦 = Σ𝑘 ∈ (𝑀...𝑛)𝐵)
41		nfv 1830	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛(𝜑 ∧ 𝑥 ∈ ℝ)
42		nfra1 2925	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛∀𝑛 ∈ 𝑍 Σ𝑘 ∈ (𝑀...𝑛)𝐵 ≤ 𝑥
43	41, 42	nfan 1816	. . . . . . . . . . 11 ⊢ Ⅎ𝑛((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ∀𝑛 ∈ 𝑍 Σ𝑘 ∈ (𝑀...𝑛)𝐵 ≤ 𝑥)
44		nfv 1830	. . . . . . . . . . 11 ⊢ Ⅎ𝑛 𝑦 ≤ 𝑥
45		rspa 2914	. . . . . . . . . . . . . 14 ⊢ ((∀𝑛 ∈ 𝑍 Σ𝑘 ∈ (𝑀...𝑛)𝐵 ≤ 𝑥 ∧ 𝑛 ∈ 𝑍) → Σ𝑘 ∈ (𝑀...𝑛)𝐵 ≤ 𝑥)
46		simpr 476	. . . . . . . . . . . . . . . 16 ⊢ ((Σ𝑘 ∈ (𝑀...𝑛)𝐵 ≤ 𝑥 ∧ 𝑦 = Σ𝑘 ∈ (𝑀...𝑛)𝐵) → 𝑦 = Σ𝑘 ∈ (𝑀...𝑛)𝐵)
47		simpl 472	. . . . . . . . . . . . . . . 16 ⊢ ((Σ𝑘 ∈ (𝑀...𝑛)𝐵 ≤ 𝑥 ∧ 𝑦 = Σ𝑘 ∈ (𝑀...𝑛)𝐵) → Σ𝑘 ∈ (𝑀...𝑛)𝐵 ≤ 𝑥)
48	46, 47	eqbrtrd 4605	. . . . . . . . . . . . . . 15 ⊢ ((Σ𝑘 ∈ (𝑀...𝑛)𝐵 ≤ 𝑥 ∧ 𝑦 = Σ𝑘 ∈ (𝑀...𝑛)𝐵) → 𝑦 ≤ 𝑥)
49	48	ex 449	. . . . . . . . . . . . . 14 ⊢ (Σ𝑘 ∈ (𝑀...𝑛)𝐵 ≤ 𝑥 → (𝑦 = Σ𝑘 ∈ (𝑀...𝑛)𝐵 → 𝑦 ≤ 𝑥))
50	45, 49	syl 17	. . . . . . . . . . . . 13 ⊢ ((∀𝑛 ∈ 𝑍 Σ𝑘 ∈ (𝑀...𝑛)𝐵 ≤ 𝑥 ∧ 𝑛 ∈ 𝑍) → (𝑦 = Σ𝑘 ∈ (𝑀...𝑛)𝐵 → 𝑦 ≤ 𝑥))
51	50	ex 449	. . . . . . . . . . . 12 ⊢ (∀𝑛 ∈ 𝑍 Σ𝑘 ∈ (𝑀...𝑛)𝐵 ≤ 𝑥 → (𝑛 ∈ 𝑍 → (𝑦 = Σ𝑘 ∈ (𝑀...𝑛)𝐵 → 𝑦 ≤ 𝑥)))
52	51	adantl 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ∀𝑛 ∈ 𝑍 Σ𝑘 ∈ (𝑀...𝑛)𝐵 ≤ 𝑥) → (𝑛 ∈ 𝑍 → (𝑦 = Σ𝑘 ∈ (𝑀...𝑛)𝐵 → 𝑦 ≤ 𝑥)))
53	43, 44, 52	rexlimd 3008	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ∀𝑛 ∈ 𝑍 Σ𝑘 ∈ (𝑀...𝑛)𝐵 ≤ 𝑥) → (∃𝑛 ∈ 𝑍 𝑦 = Σ𝑘 ∈ (𝑀...𝑛)𝐵 → 𝑦 ≤ 𝑥))
54	53	adantr 480	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ∀𝑛 ∈ 𝑍 Σ𝑘 ∈ (𝑀...𝑛)𝐵 ≤ 𝑥) ∧ 𝑦 ∈ ran (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵)) → (∃𝑛 ∈ 𝑍 𝑦 = Σ𝑘 ∈ (𝑀...𝑛)𝐵 → 𝑦 ≤ 𝑥))
55	40, 54	mpd 15	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ∀𝑛 ∈ 𝑍 Σ𝑘 ∈ (𝑀...𝑛)𝐵 ≤ 𝑥) ∧ 𝑦 ∈ ran (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵)) → 𝑦 ≤ 𝑥)
56	55	ralrimiva 2949	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ∀𝑛 ∈ 𝑍 Σ𝑘 ∈ (𝑀...𝑛)𝐵 ≤ 𝑥) → ∀𝑦 ∈ ran (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵)𝑦 ≤ 𝑥)
57	56	ex 449	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∀𝑛 ∈ 𝑍 Σ𝑘 ∈ (𝑀...𝑛)𝐵 ≤ 𝑥 → ∀𝑦 ∈ ran (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵)𝑦 ≤ 𝑥))
58	57	ex 449	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ ℝ → (∀𝑛 ∈ 𝑍 Σ𝑘 ∈ (𝑀...𝑛)𝐵 ≤ 𝑥 → ∀𝑦 ∈ ran (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵)𝑦 ≤ 𝑥)))
59	35, 58	reximdai 2995	. . . 4 ⊢ (𝜑 → (∃𝑥 ∈ ℝ ∀𝑛 ∈ 𝑍 Σ𝑘 ∈ (𝑀...𝑛)𝐵 ≤ 𝑥 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵)𝑦 ≤ 𝑥))
60	34, 59	mpd 15	. . 3 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵)𝑦 ≤ 𝑥)
61		supxrre 12029	. . 3 ⊢ ((ran (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵) ⊆ ℝ ∧ ran (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵) ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵)𝑦 ≤ 𝑥) → sup(ran (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵), ℝ*, < ) = sup(ran (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵), ℝ, < ))
62	19, 33, 60, 61	syl3anc 1318	. 2 ⊢ (𝜑 → sup(ran (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵), ℝ*, < ) = sup(ran (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵), ℝ, < ))
63	5, 62	eqtrd 2644	1 ⊢ (𝜑 → (Σ^‘(𝑘 ∈ 𝑍 ↦ 𝐵)) = sup(ran (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵), ℝ, < ))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475  Ⅎwnf 1699   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  Vcvv 3173   ⊆ wss 3540  ∅c0 3874   class class class wbr 4583   ↦ cmpt 4643  ran crn 5039  ‘cfv 5804  (class class class)co 6549  supcsup 8229  ℝcr 9814  0cc0 9815  +∞cpnf 9950  ℝ^*cxr 9952   < clt 9953   ≤ cle 9954  ℤcz 11254  ℤ_≥cuz 11563  [,)cico 12048  ...cfz 12197  Σcsu 14264  Σ^{^}csumge0 39255

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-oi 8298  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-rp 11709  df-ico 12052  df-icc 12053  df-fz 12198  df-fzo 12335  df-seq 12664  df-exp 12723  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-sum 14265  df-sumge0 39256

This theorem is referenced by:  meaiuninclem  39373