omeiunltfirp - Mathbox for Glauco Siliprandi

	Mathbox for Glauco Siliprandi		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > omeiunltfirp		Structured version Visualization version GIF version

Theorem omeiunltfirp 47445

Description: If the outer measure of a countable union is not +∞, then it can be arbitrarily approximated by finite sums of outer measures. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

**Hypotheses**
Ref	Expression
omeiunltfirp.o	⊢ (𝜑 → 𝑂 ∈ OutMeas)
omeiunltfirp.x	⊢ 𝑋 = ∪ dom 𝑂
omeiunltfirp.z	⊢ 𝑍 = (ℤ_≥‘𝑁)
omeiunltfirp.e	⊢ (𝜑 → 𝐸:𝑍⟶𝒫 𝑋)
omeiunltfirp.re	⊢ (𝜑 → (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) ∈ ℝ)
omeiunltfirp.y	⊢ (𝜑 → 𝑌 ∈ ℝ⁺)

**Assertion**
Ref	Expression
omeiunltfirp	⊢ (𝜑 → ∃𝑧 ∈ (𝒫 𝑍 ∩ Fin)(𝑂‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) < (Σ𝑛 ∈ 𝑧 (𝑂‘(𝐸‘𝑛)) + 𝑌))

Distinct variable groups: 𝑛,𝐸,𝑧 𝑛,𝑂,𝑧 𝑛,𝑋 𝑧,𝑌 𝑛,𝑍,𝑧 𝜑,𝑛,𝑧 Allowed substitution hints: 𝑁(𝑧, 𝑛) 𝑋(𝑧) 𝑌(𝑛)

Proof of Theorem omeiunltfirp Dummy variables 𝑘 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		omeiunltfirp.z	. . . . . 6 ⊢ 𝑍 = (ℤ_≥‘𝑁)
2	1	fvexi 6888	. . . . 5 ⊢ 𝑍 ∈ V
3	2	a1i 11	. . . 4 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) = +∞) → 𝑍 ∈ V)
4		omeiunltfirp.o	. . . . . . . 8 ⊢ (𝜑 → 𝑂 ∈ OutMeas)
5	4	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑂 ∈ OutMeas)
6		omeiunltfirp.x	. . . . . . 7 ⊢ 𝑋 = ∪ dom 𝑂
7		omeiunltfirp.e	. . . . . . . . 9 ⊢ (𝜑 → 𝐸:𝑍⟶𝒫 𝑋)
8	7	ffvelcdmda 7073	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐸‘𝑛) ∈ 𝒫 𝑋)
9		fvex 6887	. . . . . . . . 9 ⊢ (𝐸‘𝑛) ∈ V
10	9	elpw 4561	. . . . . . . 8 ⊢ ((𝐸‘𝑛) ∈ 𝒫 𝑋 ↔ (𝐸‘𝑛) ⊆ 𝑋)
11	8, 10	sylib 221	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐸‘𝑛) ⊆ 𝑋)
12	5, 6, 11	omecl 47429	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑂‘(𝐸‘𝑛)) ∈ (0[,]+∞))
13		eqid 2760	. . . . . 6 ⊢ (𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛))) = (𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))
14	12, 13	fmptd 7103	. . . . 5 ⊢ (𝜑 → (𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛))):𝑍⟶(0[,]+∞))
15	14	adantr 486	. . . 4 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) = +∞) → (𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛))):𝑍⟶(0[,]+∞))
16		simpr 490	. . . 4 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) = +∞) → (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) = +∞)
17		omeiunltfirp.re	. . . . 5 ⊢ (𝜑 → (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) ∈ ℝ)
18	17	adantr 486	. . . 4 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) = +∞) → (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) ∈ ℝ)
19	3, 15, 16, 18	sge0pnffigt 47322	. . 3 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) = +∞) → ∃𝑧 ∈ (𝒫 𝑍 ∩ Fin)(𝑂‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) < (Σ^{^}‘((𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛))) ↾ 𝑧)))
20		simpl 488	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) < (Σ^{^}‘((𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛))) ↾ 𝑧))) → (𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)))
21		simpr 490	. . . . . . . . 9 ⊢ ((𝑧 ∈ (𝒫 𝑍 ∩ Fin) ∧ (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) < (Σ^{^}‘((𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛))) ↾ 𝑧))) → (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) < (Σ^{^}‘((𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛))) ↾ 𝑧)))
22		elpwinss 45981	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (𝒫 𝑍 ∩ Fin) → 𝑧 ⊆ 𝑍)
23	22	resmptd 6031	. . . . . . . . . . 11 ⊢ (𝑧 ∈ (𝒫 𝑍 ∩ Fin) → ((𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛))) ↾ 𝑧) = (𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛))))
24	23	fveq2d 6878	. . . . . . . . . 10 ⊢ (𝑧 ∈ (𝒫 𝑍 ∩ Fin) → (Σ^{^}‘((𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛))) ↾ 𝑧)) = (Σ^{^}‘(𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛)))))
25	24	adantr 486	. . . . . . . . 9 ⊢ ((𝑧 ∈ (𝒫 𝑍 ∩ Fin) ∧ (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) < (Σ^{^}‘((𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛))) ↾ 𝑧))) → (Σ^{^}‘((𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛))) ↾ 𝑧)) = (Σ^{^}‘(𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛)))))
26	21, 25	breqtrd 5131	. . . . . . . 8 ⊢ ((𝑧 ∈ (𝒫 𝑍 ∩ Fin) ∧ (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) < (Σ^{^}‘((𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛))) ↾ 𝑧))) → (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) < (Σ^{^}‘(𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛)))))
27	26	adantll 727	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) < (Σ^{^}‘((𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛))) ↾ 𝑧))) → (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) < (Σ^{^}‘(𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛)))))
28	17	rexrd 11316	. . . . . . . . 9 ⊢ (𝜑 → (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) ∈ ℝ^*)
29	28	ad2antrr 739	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) < (Σ^{^}‘(𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛))))) → (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) ∈ ℝ^*)
30		simpr 490	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → 𝑧 ∈ (𝒫 𝑍 ∩ Fin))
31	4	ad2antrr 739	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → 𝑂 ∈ OutMeas)
32	7	ad2antrr 739	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → 𝐸:𝑍⟶𝒫 𝑋)
33	22	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑛 ∈ 𝑧) → 𝑧 ⊆ 𝑍)
34		simpr 490	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑛 ∈ 𝑧) → 𝑛 ∈ 𝑧)
35	33, 34	sseldd 3932	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑛 ∈ 𝑧) → 𝑛 ∈ 𝑍)
36	35	adantll 727	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → 𝑛 ∈ 𝑍)
37	32, 36	ffvelcdmd 7074	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → (𝐸‘𝑛) ∈ 𝒫 𝑋)
38	37, 10	sylib 221	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → (𝐸‘𝑛) ⊆ 𝑋)
39	31, 6, 38	omecl 47429	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → (𝑂‘(𝐸‘𝑛)) ∈ (0[,]+∞))
40		eqid 2760	. . . . . . . . . . 11 ⊢ (𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛))) = (𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛)))
41	39, 40	fmptd 7103	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → (𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛))):𝑧⟶(0[,]+∞))
42	30, 41	sge0xrcl 47311	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → (Σ^{^}‘(𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ^*)
43	42	adantr 486	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) < (Σ^{^}‘(𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛))))) → (Σ^{^}‘(𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ^*)
44		elinel2 4148	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ (𝒫 𝑍 ∩ Fin) → 𝑧 ∈ Fin)
45	44	adantl 487	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → 𝑧 ∈ Fin)
46		rge0ssre 13542	. . . . . . . . . . . . 13 ⊢ (0[,)+∞) ⊆ ℝ
47		0xr 11313	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ ℝ^*
48	47	a1i 11	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → 0 ∈ ℝ^*)
49		pnfxr 11320	. . . . . . . . . . . . . . 15 ⊢ +∞ ∈ ℝ^*
50	49	a1i 11	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → +∞ ∈ ℝ^*)
51	31, 6, 38	omexrcl 47433	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → (𝑂‘(𝐸‘𝑛)) ∈ ℝ^*)
52		iccgelb 13488	. . . . . . . . . . . . . . 15 ⊢ ((0 ∈ ℝ^* ∧ +∞ ∈ ℝ^* ∧ (𝑂‘(𝐸‘𝑛)) ∈ (0[,]+∞)) → 0 ≤ (𝑂‘(𝐸‘𝑛)))
53	48, 50, 39, 52	syl3anc 1398	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → 0 ≤ (𝑂‘(𝐸‘𝑛)))
54	11	ralrimiva 3154	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ∀𝑛 ∈ 𝑍 (𝐸‘𝑛) ⊆ 𝑋)
55		iunss 5003	. . . . . . . . . . . . . . . . . 18 ⊢ (∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛) ⊆ 𝑋 ↔ ∀𝑛 ∈ 𝑍 (𝐸‘𝑛) ⊆ 𝑋)
56	54, 55	sylibr 237	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛) ⊆ 𝑋)
57	56	ad2antrr 739	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛) ⊆ 𝑋)
58	31, 6, 57	omexrcl 47433	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) ∈ ℝ^*)
59		ssiun2 5006	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ 𝑍 → (𝐸‘𝑛) ⊆ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))
60	36, 59	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → (𝐸‘𝑛) ⊆ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))
61	31, 6, 57, 60	omessle 47424	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → (𝑂‘(𝐸‘𝑛)) ≤ (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)))
62	17	ltpnfd 13205	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) < +∞)
63	62	ad2antrr 739	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) < +∞)
64	51, 58, 50, 61, 63	xrlelttrd 13244	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → (𝑂‘(𝐸‘𝑛)) < +∞)
65	48, 50, 51, 53, 64	elicod 13481	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → (𝑂‘(𝐸‘𝑛)) ∈ (0[,)+∞))
66	46, 65	sselid 3929	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → (𝑂‘(𝐸‘𝑛)) ∈ ℝ)
67	45, 66	fsumrecl 15853	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → Σ𝑛 ∈ 𝑧 (𝑂‘(𝐸‘𝑛)) ∈ ℝ)
68		omeiunltfirp.y	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑌 ∈ ℝ⁺)
69	68	rpred 13119	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑌 ∈ ℝ)
70	69	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → 𝑌 ∈ ℝ)
71	67, 70	readdcld 11295	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → (Σ𝑛 ∈ 𝑧 (𝑂‘(𝐸‘𝑛)) + 𝑌) ∈ ℝ)
72	71	rexrd 11316	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → (Σ𝑛 ∈ 𝑧 (𝑂‘(𝐸‘𝑛)) + 𝑌) ∈ ℝ^*)
73	72	adantr 486	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) < (Σ^{^}‘(𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛))))) → (Σ𝑛 ∈ 𝑧 (𝑂‘(𝐸‘𝑛)) + 𝑌) ∈ ℝ^*)
74		simpr 490	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) < (Σ^{^}‘(𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛))))) → (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) < (Σ^{^}‘(𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛)))))
75	65, 40	fmptd 7103	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → (𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛))):𝑧⟶(0[,)+∞))
76	45, 75	sge0fsum 47313	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → (Σ^{^}‘(𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛)))) = Σ𝑘 ∈ 𝑧 ((𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛)))‘𝑘))
77		eqidd 2761	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑘 ∈ 𝑧) → (𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛))) = (𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛))))
78		2fveq3 6879	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑘 → (𝑂‘(𝐸‘𝑛)) = (𝑂‘(𝐸‘𝑘)))
79	78	adantl 487	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑘 ∈ 𝑧) ∧ 𝑛 = 𝑘) → (𝑂‘(𝐸‘𝑛)) = (𝑂‘(𝐸‘𝑘)))
80		simpr 490	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑘 ∈ 𝑧) → 𝑘 ∈ 𝑧)
81		fvexd 6889	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑘 ∈ 𝑧) → (𝑂‘(𝐸‘𝑘)) ∈ V)
82	77, 79, 80, 81	fvmptd 6990	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑘 ∈ 𝑧) → ((𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛)))‘𝑘) = (𝑂‘(𝐸‘𝑘)))
83	82	sumeq2dv 15822	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → Σ𝑘 ∈ 𝑧 ((𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛)))‘𝑘) = Σ𝑘 ∈ 𝑧 (𝑂‘(𝐸‘𝑘)))
84		2fveq3 6879	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑛 → (𝑂‘(𝐸‘𝑘)) = (𝑂‘(𝐸‘𝑛)))
85	84	cbvsumv 15816	. . . . . . . . . . . 12 ⊢ Σ𝑘 ∈ 𝑧 (𝑂‘(𝐸‘𝑘)) = Σ𝑛 ∈ 𝑧 (𝑂‘(𝐸‘𝑛))
86	85	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → Σ𝑘 ∈ 𝑧 (𝑂‘(𝐸‘𝑘)) = Σ𝑛 ∈ 𝑧 (𝑂‘(𝐸‘𝑛)))
87	76, 83, 86	3eqtrd 2799	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → (Σ^{^}‘(𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛)))) = Σ𝑛 ∈ 𝑧 (𝑂‘(𝐸‘𝑛)))
88	68	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → 𝑌 ∈ ℝ⁺)
89	67, 88	ltaddrpd 13152	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → Σ𝑛 ∈ 𝑧 (𝑂‘(𝐸‘𝑛)) < (Σ𝑛 ∈ 𝑧 (𝑂‘(𝐸‘𝑛)) + 𝑌))
90	87, 89	eqbrtrd 5127	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → (Σ^{^}‘(𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛)))) < (Σ𝑛 ∈ 𝑧 (𝑂‘(𝐸‘𝑛)) + 𝑌))
91	90	adantr 486	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) < (Σ^{^}‘(𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛))))) → (Σ^{^}‘(𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛)))) < (Σ𝑛 ∈ 𝑧 (𝑂‘(𝐸‘𝑛)) + 𝑌))
92	29, 43, 73, 74, 91	xrlttrd 13243	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) < (Σ^{^}‘(𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛))))) → (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) < (Σ𝑛 ∈ 𝑧 (𝑂‘(𝐸‘𝑛)) + 𝑌))
93	20, 27, 92	syl2anc 596	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) < (Σ^{^}‘((𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛))) ↾ 𝑧))) → (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) < (Σ𝑛 ∈ 𝑧 (𝑂‘(𝐸‘𝑛)) + 𝑌))
94	93	ex 418	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → ((𝑂‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) < (Σ^{^}‘((𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛))) ↾ 𝑧)) → (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) < (Σ𝑛 ∈ 𝑧 (𝑂‘(𝐸‘𝑛)) + 𝑌)))
95	94	adantlr 728	. . . 4 ⊢ (((𝜑 ∧ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) = +∞) ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → ((𝑂‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) < (Σ^{^}‘((𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛))) ↾ 𝑧)) → (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) < (Σ𝑛 ∈ 𝑧 (𝑂‘(𝐸‘𝑛)) + 𝑌)))
96	95	reximdva 3175	. . 3 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) = +∞) → (∃𝑧 ∈ (𝒫 𝑍 ∩ Fin)(𝑂‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) < (Σ^{^}‘((𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛))) ↾ 𝑧)) → ∃𝑧 ∈ (𝒫 𝑍 ∩ Fin)(𝑂‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) < (Σ𝑛 ∈ 𝑧 (𝑂‘(𝐸‘𝑛)) + 𝑌)))
97	19, 96	mpd 16	. 2 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) = +∞) → ∃𝑧 ∈ (𝒫 𝑍 ∩ Fin)(𝑂‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) < (Σ𝑛 ∈ 𝑧 (𝑂‘(𝐸‘𝑛)) + 𝑌))
98		simpl 488	. . 3 ⊢ ((𝜑 ∧ ¬ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) = +∞) → 𝜑)
99		simpr 490	. . . 4 ⊢ ((𝜑 ∧ ¬ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) = +∞) → ¬ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) = +∞)
100	2	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ V)
101	100, 14	sge0repnf 47312	. . . . 5 ⊢ (𝜑 → ((Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ ↔ ¬ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) = +∞))
102	101	adantr 486	. . . 4 ⊢ ((𝜑 ∧ ¬ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) = +∞) → ((Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ ↔ ¬ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) = +∞))
103	99, 102	mpbird 260	. . 3 ⊢ ((𝜑 ∧ ¬ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) = +∞) → (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ)
104		nfv 1947	. . . . . 6 ⊢ Ⅎ𝑛𝜑
105		nfcv 2922	. . . . . . . 8 ⊢ Ⅎ𝑛Σ^{^}
106		nfmpt1 5204	. . . . . . . 8 ⊢ Ⅎ𝑛(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))
107	105, 106	nffv 6884	. . . . . . 7 ⊢ Ⅎ𝑛(Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛))))
108		nfcv 2922	. . . . . . 7 ⊢ Ⅎ𝑛ℝ
109	107, 108	nfel 2936	. . . . . 6 ⊢ Ⅎ𝑛(Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ
110	104, 109	nfan 1932	. . . . 5 ⊢ Ⅎ𝑛(𝜑 ∧ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ)
111	2	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ) → 𝑍 ∈ V)
112	12	adantlr 728	. . . . 5 ⊢ (((𝜑 ∧ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ) ∧ 𝑛 ∈ 𝑍) → (𝑂‘(𝐸‘𝑛)) ∈ (0[,]+∞))
113	68	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ) → 𝑌 ∈ ℝ⁺)
114		simpr 490	. . . . 5 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ) → (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ)
115	110, 111, 112, 113, 114	sge0ltfirpmpt 47334	. . . 4 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ) → ∃𝑧 ∈ (𝒫 𝑍 ∩ Fin)(Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) < ((Σ^{^}‘(𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛)))) + 𝑌))
116	17	ad3antrrr 743	. . . . . . 7 ⊢ ((((𝜑 ∧ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ) ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) < ((Σ^{^}‘(𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛)))) + 𝑌)) → (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) ∈ ℝ)
117	114	ad2antrr 739	. . . . . . 7 ⊢ ((((𝜑 ∧ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ) ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) < ((Σ^{^}‘(𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛)))) + 𝑌)) → (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ)
118	71	ad4ant13 764	. . . . . . 7 ⊢ ((((𝜑 ∧ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ) ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) < ((Σ^{^}‘(𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛)))) + 𝑌)) → (Σ𝑛 ∈ 𝑧 (𝑂‘(𝐸‘𝑛)) + 𝑌) ∈ ℝ)
119		nfcv 2922	. . . . . . . . 9 ⊢ Ⅎ𝑛𝐸
120	104, 119, 4, 6, 1, 7	omeiunle 47443	. . . . . . . 8 ⊢ (𝜑 → (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) ≤ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))))
121	120	ad3antrrr 743	. . . . . . 7 ⊢ ((((𝜑 ∧ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ) ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) < ((Σ^{^}‘(𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛)))) + 𝑌)) → (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) ≤ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))))
122		simpr 490	. . . . . . . 8 ⊢ ((((𝜑 ∧ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ) ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) < ((Σ^{^}‘(𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛)))) + 𝑌)) → (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) < ((Σ^{^}‘(𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛)))) + 𝑌))
123		simpll 779	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ) ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → 𝜑)
124		2fveq3 6879	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑚 → (𝑂‘(𝐸‘𝑛)) = (𝑂‘(𝐸‘𝑚)))
125	124	cbvmptv 5209	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛))) = (𝑚 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑚)))
126	125	fveq2i 6877	. . . . . . . . . . . . . 14 ⊢ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) = (Σ^{^}‘(𝑚 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑚))))
127	126	eleq1i 2851	. . . . . . . . . . . . 13 ⊢ ((Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ ↔ (Σ^{^}‘(𝑚 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑚)))) ∈ ℝ)
128	127	biimpi 219	. . . . . . . . . . . 12 ⊢ ((Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ → (Σ^{^}‘(𝑚 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑚)))) ∈ ℝ)
129	128	ad2antlr 740	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ) ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → (Σ^{^}‘(𝑚 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑚)))) ∈ ℝ)
130		simpr 490	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ) ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → 𝑧 ∈ (𝒫 𝑍 ∩ Fin))
131	44	adantl 487	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (Σ^{^}‘(𝑚 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑚)))) ∈ ℝ) ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → 𝑧 ∈ Fin)
132	65	adantllr 732	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (Σ^{^}‘(𝑚 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑚)))) ∈ ℝ) ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → (𝑂‘(𝐸‘𝑛)) ∈ (0[,)+∞))
133	131, 132	sge0fsummpt 47316	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (Σ^{^}‘(𝑚 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑚)))) ∈ ℝ) ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → (Σ^{^}‘(𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛)))) = Σ𝑛 ∈ 𝑧 (𝑂‘(𝐸‘𝑛)))
134	123, 129, 130, 133	syl21anc 851	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ) ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → (Σ^{^}‘(𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛)))) = Σ𝑛 ∈ 𝑧 (𝑂‘(𝐸‘𝑛)))
135	134	oveq1d 7424	. . . . . . . . 9 ⊢ (((𝜑 ∧ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ) ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → ((Σ^{^}‘(𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛)))) + 𝑌) = (Σ𝑛 ∈ 𝑧 (𝑂‘(𝐸‘𝑛)) + 𝑌))
136	135	adantr 486	. . . . . . . 8 ⊢ ((((𝜑 ∧ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ) ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) < ((Σ^{^}‘(𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛)))) + 𝑌)) → ((Σ^{^}‘(𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛)))) + 𝑌) = (Σ𝑛 ∈ 𝑧 (𝑂‘(𝐸‘𝑛)) + 𝑌))
137	122, 136	breqtrd 5131	. . . . . . 7 ⊢ ((((𝜑 ∧ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ) ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) < ((Σ^{^}‘(𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛)))) + 𝑌)) → (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) < (Σ𝑛 ∈ 𝑧 (𝑂‘(𝐸‘𝑛)) + 𝑌))
138	116, 117, 118, 121, 137	lelttrd 11425	. . . . . 6 ⊢ ((((𝜑 ∧ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ) ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) < ((Σ^{^}‘(𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛)))) + 𝑌)) → (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) < (Σ𝑛 ∈ 𝑧 (𝑂‘(𝐸‘𝑛)) + 𝑌))
139	138	ex 418	. . . . 5 ⊢ (((𝜑 ∧ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ) ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → ((Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) < ((Σ^{^}‘(𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛)))) + 𝑌) → (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) < (Σ𝑛 ∈ 𝑧 (𝑂‘(𝐸‘𝑛)) + 𝑌)))
140	139	reximdva 3175	. . . 4 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ) → (∃𝑧 ∈ (𝒫 𝑍 ∩ Fin)(Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) < ((Σ^{^}‘(𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛)))) + 𝑌) → ∃𝑧 ∈ (𝒫 𝑍 ∩ Fin)(𝑂‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) < (Σ𝑛 ∈ 𝑧 (𝑂‘(𝐸‘𝑛)) + 𝑌)))
141	115, 140	mpd 16	. . 3 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ) → ∃𝑧 ∈ (𝒫 𝑍 ∩ Fin)(𝑂‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) < (Σ𝑛 ∈ 𝑧 (𝑂‘(𝐸‘𝑛)) + 𝑌))
142	98, 103, 141	syl2anc 596	. 2 ⊢ ((𝜑 ∧ ¬ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) = +∞) → ∃𝑧 ∈ (𝒫 𝑍 ∩ Fin)(𝑂‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) < (Σ𝑛 ∈ 𝑧 (𝑂‘(𝐸‘𝑛)) + 𝑌))
143	97, 142	pm2.61dan 825	1 ⊢ (𝜑 → ∃𝑧 ∈ (𝒫 𝑍 ∩ Fin)(𝑂‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) < (Σ𝑛 ∈ 𝑧 (𝑂‘(𝐸‘𝑛)) + 𝑌))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ∀wral 3076 ∃wrex 3086 Vcvv 3450 ∩ cin 3898 ⊆ wss 3899 𝒫 cpw 4557 ∪ cuni 4867 ∪ ciun 4951 class class class wbr 5103 ↦ cmpt 5186 dom cdm 5648 ↾ cres 5650 ⟶wf 6524 ‘cfv 6528 (class class class)co 7409 Fincfn 8952 ℝcr 11156 0cc0 11157 + caddc 11160 +∞cpnf 11297 ℝ^*cxr 11299 < clt 11300 ≤ cle 11301 ℤ_≥cuz 12920 ℝ⁺crp 13075 [,)cico 13433 [,]cicc 13434 Σcsu 15806 Σ^{^}csumge0 47288 OutMeascome 47415

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7735 ax-inf2 9620 ax-cnex 11213 ax-resscn 11214 ax-1cn 11215 ax-icn 11216 ax-addcl 11217 ax-addrcl 11218 ax-mulcl 11219 ax-mulrcl 11220 ax-mulcom 11221 ax-addass 11222 ax-mulass 11223 ax-distr 11224 ax-i2m1 11225 ax-1ne0 11226 ax-1rid 11227 ax-rnegex 11228 ax-rrecex 11229 ax-cnre 11230 ax-pre-lttri 11231 ax-pre-lttrn 11232 ax-pre-ltadd 11233 ax-pre-mulgt0 11234 ax-pre-sup 11235

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5543 df-eprel 5548 df-po 5556 df-so 5557 df-fr 5601 df-se 5602 df-we 5603 df-xp 5654 df-rel 5655 df-cnv 5656 df-co 5657 df-dm 5658 df-rn 5659 df-res 5660 df-ima 5661 df-pred 6294 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6484 df-fun 6530 df-fn 6531 df-f 6532 df-f1 6533 df-fo 6534 df-f1o 6535 df-fv 6536 df-isom 6537 df-riota 7366 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7862 df-1st 7985 df-2nd 7986 df-frecs 8278 df-wrecs 8309 df-recs 8358 df-rdg 8397 df-1o 8455 df-oadd 8459 df-omul 8460 df-er 8696 df-map 8828 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-sup 9412 df-oi 9482 df-card 9977 df-acn 9980 df-pnf 11302 df-mnf 11303 df-xr 11304 df-ltxr 11305 df-le 11306 df-sub 11500 df-neg 11501 df-div 11929 df-nn 12291 df-2 12360 df-3 12361 df-n0 12562 df-z 12649 df-uz 12921 df-rp 13076 df-ico 13437 df-icc 13438 df-fz 13595 df-fzo 13743 df-seq 14099 df-exp 14159 df-hash 14428 df-cj 15219 df-re 15220 df-im 15221 df-sqrt 15355 df-abs 15356 df-clim 15608 df-sum 15807 df-sumge0 47289 df-ome 47416

This theorem is used by: carageniuncllem2 47448

W3C validator