omeiunltfirp - Mathbox for Glauco Siliprandi

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Theorem omeiunltfirp 47261

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 6895	. . . . 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 485	. . . . . . 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 7079	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐸‘𝑛) ∈ 𝒫 𝑋)
9		fvex 6894	. . . . . . . . 9 ⊢ (𝐸‘𝑛) ∈ V
10	9	elpw 4565	. . . . . . . 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 47245	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑂‘(𝐸‘𝑛)) ∈ (0[,]+∞))
13		eqid 2762	. . . . . 6 ⊢ (𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛))) = (𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))
14	12, 13	fmptd 7109	. . . . 5 ⊢ (𝜑 → (𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛))):𝑍⟶(0[,]+∞))
15	14	adantr 485	. . . 4 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) = +∞) → (𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛))):𝑍⟶(0[,]+∞))
16		simpr 489	. . . 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 485	. . . 4 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) = +∞) → (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) ∈ ℝ)
19	3, 15, 16, 18	sge0pnffigt 47138	. . 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 487	. . . . . . 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 489	. . . . . . . . 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 45797	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (𝒫 𝑍 ∩ Fin) → 𝑧 ⊆ 𝑍)
23	22	resmptd 6041	. . . . . . . . . . 11 ⊢ (𝑧 ∈ (𝒫 𝑍 ∩ Fin) → ((𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛))) ↾ 𝑧) = (𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛))))
24	23	fveq2d 6885	. . . . . . . . . 10 ⊢ (𝑧 ∈ (𝒫 𝑍 ∩ Fin) → (Σ^{^}‘((𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛))) ↾ 𝑧)) = (Σ^{^}‘(𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛)))))
25	24	adantr 485	. . . . . . . . 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 5136	. . . . . . . 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 726	. . . . . . 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 11265	. . . . . . . . 9 ⊢ (𝜑 → (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) ∈ ℝ^*)
29	28	ad2antrr 738	. . . . . . . 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 489	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → 𝑧 ∈ (𝒫 𝑍 ∩ Fin))
31	4	ad2antrr 738	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → 𝑂 ∈ OutMeas)
32	7	ad2antrr 738	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → 𝐸:𝑍⟶𝒫 𝑋)
33	22	adantr 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑛 ∈ 𝑧) → 𝑧 ⊆ 𝑍)
34		simpr 489	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑛 ∈ 𝑧) → 𝑛 ∈ 𝑧)
35	33, 34	sseldd 3937	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑛 ∈ 𝑧) → 𝑛 ∈ 𝑍)
36	35	adantll 726	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → 𝑛 ∈ 𝑍)
37	32, 36	ffvelcdmd 7080	. . . . . . . . . . . . 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 47245	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → (𝑂‘(𝐸‘𝑛)) ∈ (0[,]+∞))
40		eqid 2762	. . . . . . . . . . 11 ⊢ (𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛))) = (𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛)))
41	39, 40	fmptd 7109	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → (𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛))):𝑧⟶(0[,]+∞))
42	30, 41	sge0xrcl 47127	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → (Σ^{^}‘(𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ^*)
43	42	adantr 485	. . . . . . . 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 4154	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ (𝒫 𝑍 ∩ Fin) → 𝑧 ∈ Fin)
45	44	adantl 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → 𝑧 ∈ Fin)
46		rge0ssre 13489	. . . . . . . . . . . . 13 ⊢ (0[,)+∞) ⊆ ℝ
47		0xr 11262	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ ℝ^*
48	47	a1i 11	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → 0 ∈ ℝ^*)
49		pnfxr 11269	. . . . . . . . . . . . . . 15 ⊢ +∞ ∈ ℝ^*
50	49	a1i 11	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → +∞ ∈ ℝ^*)
51	31, 6, 38	omexrcl 47249	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → (𝑂‘(𝐸‘𝑛)) ∈ ℝ^*)
52		iccgelb 13435	. . . . . . . . . . . . . . 15 ⊢ ((0 ∈ ℝ^* ∧ +∞ ∈ ℝ^* ∧ (𝑂‘(𝐸‘𝑛)) ∈ (0[,]+∞)) → 0 ≤ (𝑂‘(𝐸‘𝑛)))
53	48, 50, 39, 52	syl3anc 1397	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → 0 ≤ (𝑂‘(𝐸‘𝑛)))
54	11	ralrimiva 3156	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ∀𝑛 ∈ 𝑍 (𝐸‘𝑛) ⊆ 𝑋)
55		iunss 5008	. . . . . . . . . . . . . . . . . 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 738	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛) ⊆ 𝑋)
58	31, 6, 57	omexrcl 47249	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) ∈ ℝ^*)
59		ssiun2 5011	. . . . . . . . . . . . . . . . 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 47240	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → (𝑂‘(𝐸‘𝑛)) ≤ (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)))
62	17	ltpnfd 13152	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) < +∞)
63	62	ad2antrr 738	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) < +∞)
64	51, 58, 50, 61, 63	xrlelttrd 13191	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → (𝑂‘(𝐸‘𝑛)) < +∞)
65	48, 50, 51, 53, 64	elicod 13428	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → (𝑂‘(𝐸‘𝑛)) ∈ (0[,)+∞))
66	46, 65	sselid 3934	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → (𝑂‘(𝐸‘𝑛)) ∈ ℝ)
67	45, 66	fsumrecl 15792	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → Σ𝑛 ∈ 𝑧 (𝑂‘(𝐸‘𝑛)) ∈ ℝ)
68		omeiunltfirp.y	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑌 ∈ ℝ⁺)
69	68	rpred 13066	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑌 ∈ ℝ)
70	69	adantr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → 𝑌 ∈ ℝ)
71	67, 70	readdcld 11244	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → (Σ𝑛 ∈ 𝑧 (𝑂‘(𝐸‘𝑛)) + 𝑌) ∈ ℝ)
72	71	rexrd 11265	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → (Σ𝑛 ∈ 𝑧 (𝑂‘(𝐸‘𝑛)) + 𝑌) ∈ ℝ^*)
73	72	adantr 485	. . . . . . . 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 489	. . . . . . . 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 7109	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → (𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛))):𝑧⟶(0[,)+∞))
76	45, 75	sge0fsum 47129	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → (Σ^{^}‘(𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛)))) = Σ𝑘 ∈ 𝑧 ((𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛)))‘𝑘))
77		eqidd 2763	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑘 ∈ 𝑧) → (𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛))) = (𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛))))
78		2fveq3 6886	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑘 → (𝑂‘(𝐸‘𝑛)) = (𝑂‘(𝐸‘𝑘)))
79	78	adantl 486	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑘 ∈ 𝑧) ∧ 𝑛 = 𝑘) → (𝑂‘(𝐸‘𝑛)) = (𝑂‘(𝐸‘𝑘)))
80		simpr 489	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑘 ∈ 𝑧) → 𝑘 ∈ 𝑧)
81		fvexd 6896	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑘 ∈ 𝑧) → (𝑂‘(𝐸‘𝑘)) ∈ V)
82	77, 79, 80, 81	fvmptd 6997	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑘 ∈ 𝑧) → ((𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛)))‘𝑘) = (𝑂‘(𝐸‘𝑘)))
83	82	sumeq2dv 15760	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → Σ𝑘 ∈ 𝑧 ((𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛)))‘𝑘) = Σ𝑘 ∈ 𝑧 (𝑂‘(𝐸‘𝑘)))
84		2fveq3 6886	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑛 → (𝑂‘(𝐸‘𝑘)) = (𝑂‘(𝐸‘𝑛)))
85	84	cbvsumv 15754	. . . . . . . . . . . 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 2801	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → (Σ^{^}‘(𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛)))) = Σ𝑛 ∈ 𝑧 (𝑂‘(𝐸‘𝑛)))
88	68	adantr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → 𝑌 ∈ ℝ⁺)
89	67, 88	ltaddrpd 13099	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → Σ𝑛 ∈ 𝑧 (𝑂‘(𝐸‘𝑛)) < (Σ𝑛 ∈ 𝑧 (𝑂‘(𝐸‘𝑛)) + 𝑌))
90	87, 89	eqbrtrd 5132	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → (Σ^{^}‘(𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛)))) < (Σ𝑛 ∈ 𝑧 (𝑂‘(𝐸‘𝑛)) + 𝑌))
91	90	adantr 485	. . . . . . . 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 13190	. . . . . . 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 595	. . . . . 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 417	. . . . 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 727	. . . 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 3177	. . 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 487	. . 3 ⊢ ((𝜑 ∧ ¬ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) = +∞) → 𝜑)
99		simpr 489	. . . 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 47128	. . . . 5 ⊢ (𝜑 → ((Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ ↔ ¬ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) = +∞))
102	101	adantr 485	. . . 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 1943	. . . . . 6 ⊢ Ⅎ𝑛𝜑
105		nfcv 2924	. . . . . . . 8 ⊢ Ⅎ𝑛Σ^{^}
106		nfmpt1 5209	. . . . . . . 8 ⊢ Ⅎ𝑛(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))
107	105, 106	nffv 6891	. . . . . . 7 ⊢ Ⅎ𝑛(Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛))))
108		nfcv 2924	. . . . . . 7 ⊢ Ⅎ𝑛ℝ
109	107, 108	nfel 2938	. . . . . 6 ⊢ Ⅎ𝑛(Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ
110	104, 109	nfan 1928	. . . . 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 727	. . . . 5 ⊢ (((𝜑 ∧ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ) ∧ 𝑛 ∈ 𝑍) → (𝑂‘(𝐸‘𝑛)) ∈ (0[,]+∞))
113	68	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ) → 𝑌 ∈ ℝ⁺)
114		simpr 489	. . . . 5 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ) → (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ)
115	110, 111, 112, 113, 114	sge0ltfirpmpt 47150	. . . 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 742	. . . . . . 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 738	. . . . . . 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 763	. . . . . . 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 2924	. . . . . . . . 9 ⊢ Ⅎ𝑛𝐸
120	104, 119, 4, 6, 1, 7	omeiunle 47259	. . . . . . . 8 ⊢ (𝜑 → (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) ≤ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))))
121	120	ad3antrrr 742	. . . . . . 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 489	. . . . . . . 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 778	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ) ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → 𝜑)
124		2fveq3 6886	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑚 → (𝑂‘(𝐸‘𝑛)) = (𝑂‘(𝐸‘𝑚)))
125	124	cbvmptv 5214	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛))) = (𝑚 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑚)))
126	125	fveq2i 6884	. . . . . . . . . . . . . 14 ⊢ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) = (Σ^{^}‘(𝑚 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑚))))
127	126	eleq1i 2853	. . . . . . . . . . . . 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 739	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ) ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → (Σ^{^}‘(𝑚 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑚)))) ∈ ℝ)
130		simpr 489	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ) ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → 𝑧 ∈ (𝒫 𝑍 ∩ Fin))
131	44	adantl 486	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (Σ^{^}‘(𝑚 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑚)))) ∈ ℝ) ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → 𝑧 ∈ Fin)
132	65	adantllr 731	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (Σ^{^}‘(𝑚 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑚)))) ∈ ℝ) ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → (𝑂‘(𝐸‘𝑛)) ∈ (0[,)+∞))
133	131, 132	sge0fsummpt 47132	. . . . . . . . . . 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 850	. . . . . . . . . 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 7427	. . . . . . . . 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 485	. . . . . . . 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 5136	. . . . . . 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 11374	. . . . . 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 417	. . . . 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 3177	. . . 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 595	. 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 824	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 400 = wceq 1569 ∈ wcel 2142 ∀wral 3078 ∃wrex 3088 Vcvv 3454 ∩ cin 3903 ⊆ wss 3904 𝒫 cpw 4561 ∪ cuni 4871 ∪ ciun 4955 class class class wbr 5108 ↦ cmpt 5191 dom cdm 5660 ↾ cres 5662 ⟶wf 6532 ‘cfv 6536 (class class class)co 7412 Fincfn 8941 ℝcr 11105 0cc0 11106 + caddc 11109 +∞cpnf 11246 ℝ^*cxr 11248 < clt 11249 ≤ cle 11250 ℤ_≥cuz 12868 ℝ⁺crp 13022 [,)cico 13380 [,]cicc 13381 Σcsu 15744 Σ^{^}csumge0 47104 OutMeascome 47231

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1824 ax-4 1838 ax-5 1939 ax-6 1996 ax-7 2037 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-rep 5237 ax-sep 5256 ax-nul 5268 ax-pow 5335 ax-pr 5403 ax-un 7734 ax-inf2 9608 ax-ac2 10453 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184

This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1103 df-3an 1104 df-tru 1572 df-fal 1582 df-ex 1809 df-nf 1813 df-sb 2096 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3368 df-reu 3369 df-rab 3416 df-v 3456 df-sbc 3744 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4487 df-pw 4563 df-sn 4589 df-pr 4591 df-op 4595 df-uni 4872 df-int 4912 df-iun 4957 df-br 5109 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5555 df-eprel 5560 df-po 5568 df-so 5569 df-fr 5613 df-se 5614 df-we 5615 df-xp 5666 df-rel 5667 df-cnv 5668 df-co 5669 df-dm 5670 df-rn 5671 df-res 5672 df-ima 5673 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7861 df-1st 7984 df-2nd 7985 df-frecs 8276 df-wrecs 8307 df-recs 8356 df-rdg 8395 df-1o 8451 df-oadd 8455 df-omul 8456 df-er 8692 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-sup 9400 df-oi 9470 df-card 9932 df-acn 9935 df-ac 10107 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11449 df-neg 11450 df-div 11878 df-nn 12240 df-2 12309 df-3 12310 df-n0 12511 df-z 12598 df-uz 12869 df-rp 13023 df-ico 13384 df-icc 13385 df-fz 13542 df-fzo 13690 df-seq 14045 df-exp 14105 df-hash 14374 df-cj 15157 df-re 15158 df-im 15159 df-sqrt 15293 df-abs 15294 df-clim 15546 df-sum 15745 df-sumge0 47105 df-ome 47232

This theorem is used by: carageniuncllem2 47264

W3C validator