Theorem sge0reuzb 40665

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:   𝑥,𝑘   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 40664	. 2 ⊢ (𝜑 → (Σ^‘(𝑘 ∈ 𝑍 ↦ 𝐵)) = sup(ran (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵), ℝ*, < ))
6		nfv 1843	. . . 4 ⊢ Ⅎ𝑛𝜑
7		eqid 2622	. . . 4 ⊢ (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵) = (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵)
8		nfv 1843	. . . . . 6 ⊢ Ⅎ𝑘 𝑛 ∈ 𝑍
9	1, 8	nfan 1828	. . . . 5 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑛 ∈ 𝑍)
10		fzfid 12772	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑀...𝑛) ∈ Fin)
11		elfzuz 12338	. . . . . . . . 9 ⊢ (𝑘 ∈ (𝑀...𝑛) → 𝑘 ∈ (ℤ≥‘𝑀))
12	11, 3	syl6eleqr 2712	. . . . . . . 8 ⊢ (𝑘 ∈ (𝑀...𝑛) → 𝑘 ∈ 𝑍)
13	12	adantl 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑛)) → 𝑘 ∈ 𝑍)
14		rge0ssre 12280	. . . . . . . 8 ⊢ (0[,)+∞) ⊆ ℝ
15	14, 4	sseldi 3601	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℝ)
16	13, 15	syldan 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑛)) → 𝐵 ∈ ℝ)
17	16	adantlr 751	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ (𝑀...𝑛)) → 𝐵 ∈ ℝ)
18	9, 10, 17	fsumreclf 39808	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → Σ𝑘 ∈ (𝑀...𝑛)𝐵 ∈ ℝ)
19	6, 7, 18	rnmptssd 39385	. . 3 ⊢ (𝜑 → ran (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵) ⊆ ℝ)
20		uzid 11702	. . . . . . . 8 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ≥‘𝑀))
21	2, 20	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘𝑀))
22	21, 3	syl6eleqr 2712	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ 𝑍)
23		eqidd 2623	. . . . . 6 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀...𝑀)𝐵 = Σ𝑘 ∈ (𝑀...𝑀)𝐵)
24		oveq2 6658	. . . . . . . . 9 ⊢ (𝑛 = 𝑀 → (𝑀...𝑛) = (𝑀...𝑀))
25	24	sumeq1d 14431	. . . . . . . 8 ⊢ (𝑛 = 𝑀 → Σ𝑘 ∈ (𝑀...𝑛)𝐵 = Σ𝑘 ∈ (𝑀...𝑀)𝐵)
26	25	eqeq2d 2632	. . . . . . 7 ⊢ (𝑛 = 𝑀 → (Σ𝑘 ∈ (𝑀...𝑀)𝐵 = Σ𝑘 ∈ (𝑀...𝑛)𝐵 ↔ Σ𝑘 ∈ (𝑀...𝑀)𝐵 = Σ𝑘 ∈ (𝑀...𝑀)𝐵))
27	26	rspcev 3309	. . . . . 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 693	. . . . 5 ⊢ (𝜑 → ∃𝑛 ∈ 𝑍 Σ𝑘 ∈ (𝑀...𝑀)𝐵 = Σ𝑘 ∈ (𝑀...𝑛)𝐵)
29		sumex 14418	. . . . . 6 ⊢ Σ𝑘 ∈ (𝑀...𝑀)𝐵 ∈ V
30	29	a1i 11	. . . . 5 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀...𝑀)𝐵 ∈ V)
31	7, 28, 30	elrnmptd 39366	. . . 4 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀...𝑀)𝐵 ∈ ran (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵))
32		ne0i 3921	. . . 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 3203	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ V
37	7	elrnmpt 5372	. . . . . . . . . . . 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 206	. . . . . . . . . 10 ⊢ (𝑦 ∈ ran (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵) → ∃𝑛 ∈ 𝑍 𝑦 = Σ𝑘 ∈ (𝑀...𝑛)𝐵)
40	39	adantl 482	. . . . . . . . 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 1843	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛(𝜑 ∧ 𝑥 ∈ ℝ)
42		nfra1 2941	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛∀𝑛 ∈ 𝑍 Σ𝑘 ∈ (𝑀...𝑛)𝐵 ≤ 𝑥
43	41, 42	nfan 1828	. . . . . . . . . . 11 ⊢ Ⅎ𝑛((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ∀𝑛 ∈ 𝑍 Σ𝑘 ∈ (𝑀...𝑛)𝐵 ≤ 𝑥)
44		nfv 1843	. . . . . . . . . . 11 ⊢ Ⅎ𝑛 𝑦 ≤ 𝑥
45		rspa 2930	. . . . . . . . . . . . . 14 ⊢ ((∀𝑛 ∈ 𝑍 Σ𝑘 ∈ (𝑀...𝑛)𝐵 ≤ 𝑥 ∧ 𝑛 ∈ 𝑍) → Σ𝑘 ∈ (𝑀...𝑛)𝐵 ≤ 𝑥)
46		simpr 477	. . . . . . . . . . . . . . . 16 ⊢ ((Σ𝑘 ∈ (𝑀...𝑛)𝐵 ≤ 𝑥 ∧ 𝑦 = Σ𝑘 ∈ (𝑀...𝑛)𝐵) → 𝑦 = Σ𝑘 ∈ (𝑀...𝑛)𝐵)
47		simpl 473	. . . . . . . . . . . . . . . 16 ⊢ ((Σ𝑘 ∈ (𝑀...𝑛)𝐵 ≤ 𝑥 ∧ 𝑦 = Σ𝑘 ∈ (𝑀...𝑛)𝐵) → Σ𝑘 ∈ (𝑀...𝑛)𝐵 ≤ 𝑥)
48	46, 47	eqbrtrd 4675	. . . . . . . . . . . . . . 15 ⊢ ((Σ𝑘 ∈ (𝑀...𝑛)𝐵 ≤ 𝑥 ∧ 𝑦 = Σ𝑘 ∈ (𝑀...𝑛)𝐵) → 𝑦 ≤ 𝑥)
49	48	ex 450	. . . . . . . . . . . . . 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 450	. . . . . . . . . . . 12 ⊢ (∀𝑛 ∈ 𝑍 Σ𝑘 ∈ (𝑀...𝑛)𝐵 ≤ 𝑥 → (𝑛 ∈ 𝑍 → (𝑦 = Σ𝑘 ∈ (𝑀...𝑛)𝐵 → 𝑦 ≤ 𝑥)))
52	51	adantl 482	. . . . . . . . . . 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 3026	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ∀𝑛 ∈ 𝑍 Σ𝑘 ∈ (𝑀...𝑛)𝐵 ≤ 𝑥) → (∃𝑛 ∈ 𝑍 𝑦 = Σ𝑘 ∈ (𝑀...𝑛)𝐵 → 𝑦 ≤ 𝑥))
54	53	adantr 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 → 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 2966	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ∀𝑛 ∈ 𝑍 Σ𝑘 ∈ (𝑀...𝑛)𝐵 ≤ 𝑥) → ∀𝑦 ∈ ran (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵)𝑦 ≤ 𝑥)
57	56	ex 450	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∀𝑛 ∈ 𝑍 Σ𝑘 ∈ (𝑀...𝑛)𝐵 ≤ 𝑥 → ∀𝑦 ∈ ran (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵)𝑦 ≤ 𝑥))
58	57	ex 450	. . . . 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 3012	. . . 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 12157	. . 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 1326	. 2 ⊢ (𝜑 → sup(ran (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵), ℝ*, < ) = sup(ran (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵), ℝ, < ))
63	5, 62	eqtrd 2656	1 ⊢ (𝜑 → (Σ^‘(𝑘 ∈ 𝑍 ↦ 𝐵)) = sup(ran (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵), ℝ, < ))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483  Ⅎwnf 1708   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912  ∃wrex 2913  Vcvv 3200   ⊆ wss 3574  ∅c0 3915   class class class wbr 4653   ↦ cmpt 4729  ran crn 5115  ‘cfv 5888  (class class class)co 6650  supcsup 8346  ℝcr 9935  0cc0 9936  +∞cpnf 10071  ℝ^*cxr 10073   < clt 10074   ≤ cle 10075  ℤcz 11377  ℤ_≥cuz 11687  [,)cico 12177  ...cfz 12326  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:  meaiuninclem  40697