omeiunltfirp - Mathbox for Glauco Siliprandi

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Theorem omeiunltfirp 45221

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 6902	. . . . 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 481	. . . . . . 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 7083	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐸‘𝑛) ∈ 𝒫 𝑋)
9		fvex 6901	. . . . . . . . 9 ⊢ (𝐸‘𝑛) ∈ V
10	9	elpw 4605	. . . . . . . 8 ⊢ ((𝐸‘𝑛) ∈ 𝒫 𝑋 ↔ (𝐸‘𝑛) ⊆ 𝑋)
11	8, 10	sylib 217	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐸‘𝑛) ⊆ 𝑋)
12	5, 6, 11	omecl 45205	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑂‘(𝐸‘𝑛)) ∈ (0[,]+∞))
13		eqid 2732	. . . . . 6 ⊢ (𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛))) = (𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))
14	12, 13	fmptd 7110	. . . . 5 ⊢ (𝜑 → (𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛))):𝑍⟶(0[,]+∞))
15	14	adantr 481	. . . 4 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) = +∞) → (𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛))):𝑍⟶(0[,]+∞))
16		simpr 485	. . . 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 481	. . . 4 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) = +∞) → (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) ∈ ℝ)
19	3, 15, 16, 18	sge0pnffigt 45098	. . 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 483	. . . . . . 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 485	. . . . . . . . 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 43721	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (𝒫 𝑍 ∩ Fin) → 𝑧 ⊆ 𝑍)
23	22	resmptd 6038	. . . . . . . . . . 11 ⊢ (𝑧 ∈ (𝒫 𝑍 ∩ Fin) → ((𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛))) ↾ 𝑧) = (𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛))))
24	23	fveq2d 6892	. . . . . . . . . 10 ⊢ (𝑧 ∈ (𝒫 𝑍 ∩ Fin) → (Σ^{^}‘((𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛))) ↾ 𝑧)) = (Σ^{^}‘(𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛)))))
25	24	adantr 481	. . . . . . . . 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 5173	. . . . . . . 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 712	. . . . . . 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 11260	. . . . . . . . 9 ⊢ (𝜑 → (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) ∈ ℝ^*)
29	28	ad2antrr 724	. . . . . . . 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 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → 𝑧 ∈ (𝒫 𝑍 ∩ Fin))
31	4	ad2antrr 724	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → 𝑂 ∈ OutMeas)
32	7	ad2antrr 724	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → 𝐸:𝑍⟶𝒫 𝑋)
33	22	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑛 ∈ 𝑧) → 𝑧 ⊆ 𝑍)
34		simpr 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑛 ∈ 𝑧) → 𝑛 ∈ 𝑧)
35	33, 34	sseldd 3982	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑛 ∈ 𝑧) → 𝑛 ∈ 𝑍)
36	35	adantll 712	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → 𝑛 ∈ 𝑍)
37	32, 36	ffvelcdmd 7084	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → (𝐸‘𝑛) ∈ 𝒫 𝑋)
38	37, 10	sylib 217	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → (𝐸‘𝑛) ⊆ 𝑋)
39	31, 6, 38	omecl 45205	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → (𝑂‘(𝐸‘𝑛)) ∈ (0[,]+∞))
40		eqid 2732	. . . . . . . . . . 11 ⊢ (𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛))) = (𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛)))
41	39, 40	fmptd 7110	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → (𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛))):𝑧⟶(0[,]+∞))
42	30, 41	sge0xrcl 45087	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → (Σ^{^}‘(𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ^*)
43	42	adantr 481	. . . . . . . 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 4195	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ (𝒫 𝑍 ∩ Fin) → 𝑧 ∈ Fin)
45	44	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → 𝑧 ∈ Fin)
46		rge0ssre 13429	. . . . . . . . . . . . 13 ⊢ (0[,)+∞) ⊆ ℝ
47		0xr 11257	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ ℝ^*
48	47	a1i 11	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → 0 ∈ ℝ^*)
49		pnfxr 11264	. . . . . . . . . . . . . . 15 ⊢ +∞ ∈ ℝ^*
50	49	a1i 11	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → +∞ ∈ ℝ^*)
51	31, 6, 38	omexrcl 45209	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → (𝑂‘(𝐸‘𝑛)) ∈ ℝ^*)
52		iccgelb 13376	. . . . . . . . . . . . . . 15 ⊢ ((0 ∈ ℝ^* ∧ +∞ ∈ ℝ^* ∧ (𝑂‘(𝐸‘𝑛)) ∈ (0[,]+∞)) → 0 ≤ (𝑂‘(𝐸‘𝑛)))
53	48, 50, 39, 52	syl3anc 1371	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → 0 ≤ (𝑂‘(𝐸‘𝑛)))
54	11	ralrimiva 3146	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ∀𝑛 ∈ 𝑍 (𝐸‘𝑛) ⊆ 𝑋)
55		iunss 5047	. . . . . . . . . . . . . . . . . 18 ⊢ (∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛) ⊆ 𝑋 ↔ ∀𝑛 ∈ 𝑍 (𝐸‘𝑛) ⊆ 𝑋)
56	54, 55	sylibr 233	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛) ⊆ 𝑋)
57	56	ad2antrr 724	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛) ⊆ 𝑋)
58	31, 6, 57	omexrcl 45209	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) ∈ ℝ^*)
59		ssiun2 5049	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ 𝑍 → (𝐸‘𝑛) ⊆ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))
60	36, 59	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → (𝐸‘𝑛) ⊆ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))
61	31, 6, 57, 60	omessle 45200	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → (𝑂‘(𝐸‘𝑛)) ≤ (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)))
62	17	ltpnfd 13097	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) < +∞)
63	62	ad2antrr 724	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) < +∞)
64	51, 58, 50, 61, 63	xrlelttrd 13135	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → (𝑂‘(𝐸‘𝑛)) < +∞)
65	48, 50, 51, 53, 64	elicod 13370	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → (𝑂‘(𝐸‘𝑛)) ∈ (0[,)+∞))
66	46, 65	sselid 3979	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → (𝑂‘(𝐸‘𝑛)) ∈ ℝ)
67	45, 66	fsumrecl 15676	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → Σ𝑛 ∈ 𝑧 (𝑂‘(𝐸‘𝑛)) ∈ ℝ)
68		omeiunltfirp.y	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑌 ∈ ℝ⁺)
69	68	rpred 13012	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑌 ∈ ℝ)
70	69	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → 𝑌 ∈ ℝ)
71	67, 70	readdcld 11239	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → (Σ𝑛 ∈ 𝑧 (𝑂‘(𝐸‘𝑛)) + 𝑌) ∈ ℝ)
72	71	rexrd 11260	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → (Σ𝑛 ∈ 𝑧 (𝑂‘(𝐸‘𝑛)) + 𝑌) ∈ ℝ^*)
73	72	adantr 481	. . . . . . . 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 485	. . . . . . . 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 7110	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → (𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛))):𝑧⟶(0[,)+∞))
76	45, 75	sge0fsum 45089	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → (Σ^{^}‘(𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛)))) = Σ𝑘 ∈ 𝑧 ((𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛)))‘𝑘))
77		eqidd 2733	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑘 ∈ 𝑧) → (𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛))) = (𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛))))
78		2fveq3 6893	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑘 → (𝑂‘(𝐸‘𝑛)) = (𝑂‘(𝐸‘𝑘)))
79	78	adantl 482	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑘 ∈ 𝑧) ∧ 𝑛 = 𝑘) → (𝑂‘(𝐸‘𝑛)) = (𝑂‘(𝐸‘𝑘)))
80		simpr 485	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑘 ∈ 𝑧) → 𝑘 ∈ 𝑧)
81		fvexd 6903	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑘 ∈ 𝑧) → (𝑂‘(𝐸‘𝑘)) ∈ V)
82	77, 79, 80, 81	fvmptd 7002	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑘 ∈ 𝑧) → ((𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛)))‘𝑘) = (𝑂‘(𝐸‘𝑘)))
83	82	sumeq2dv 15645	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → Σ𝑘 ∈ 𝑧 ((𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛)))‘𝑘) = Σ𝑘 ∈ 𝑧 (𝑂‘(𝐸‘𝑘)))
84		2fveq3 6893	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑛 → (𝑂‘(𝐸‘𝑘)) = (𝑂‘(𝐸‘𝑛)))
85	84	cbvsumv 15638	. . . . . . . . . . . 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 2776	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → (Σ^{^}‘(𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛)))) = Σ𝑛 ∈ 𝑧 (𝑂‘(𝐸‘𝑛)))
88	68	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → 𝑌 ∈ ℝ⁺)
89	67, 88	ltaddrpd 13045	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → Σ𝑛 ∈ 𝑧 (𝑂‘(𝐸‘𝑛)) < (Σ𝑛 ∈ 𝑧 (𝑂‘(𝐸‘𝑛)) + 𝑌))
90	87, 89	eqbrtrd 5169	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → (Σ^{^}‘(𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛)))) < (Σ𝑛 ∈ 𝑧 (𝑂‘(𝐸‘𝑛)) + 𝑌))
91	90	adantr 481	. . . . . . . 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 13134	. . . . . . 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 584	. . . . . 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 413	. . . . 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 713	. . . 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 3168	. . 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 15	. 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 483	. . 3 ⊢ ((𝜑 ∧ ¬ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) = +∞) → 𝜑)
99		simpr 485	. . . 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 45088	. . . . 5 ⊢ (𝜑 → ((Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ ↔ ¬ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) = +∞))
102	101	adantr 481	. . . 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 256	. . 3 ⊢ ((𝜑 ∧ ¬ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) = +∞) → (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ)
104		nfv 1917	. . . . . 6 ⊢ Ⅎ𝑛𝜑
105		nfcv 2903	. . . . . . . 8 ⊢ Ⅎ𝑛Σ^{^}
106		nfmpt1 5255	. . . . . . . 8 ⊢ Ⅎ𝑛(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))
107	105, 106	nffv 6898	. . . . . . 7 ⊢ Ⅎ𝑛(Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛))))
108		nfcv 2903	. . . . . . 7 ⊢ Ⅎ𝑛ℝ
109	107, 108	nfel 2917	. . . . . 6 ⊢ Ⅎ𝑛(Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ
110	104, 109	nfan 1902	. . . . 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 713	. . . . 5 ⊢ (((𝜑 ∧ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ) ∧ 𝑛 ∈ 𝑍) → (𝑂‘(𝐸‘𝑛)) ∈ (0[,]+∞))
113	68	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ) → 𝑌 ∈ ℝ⁺)
114		simpr 485	. . . . 5 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ) → (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ)
115	110, 111, 112, 113, 114	sge0ltfirpmpt 45110	. . . 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 728	. . . . . . 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 724	. . . . . . 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 749	. . . . . . 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 2903	. . . . . . . . 9 ⊢ Ⅎ𝑛𝐸
120	104, 119, 4, 6, 1, 7	omeiunle 45219	. . . . . . . 8 ⊢ (𝜑 → (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) ≤ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))))
121	120	ad3antrrr 728	. . . . . . 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 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)))) < ((Σ^{^}‘(𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛)))) + 𝑌))
123		simpll 765	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ) ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → 𝜑)
124		2fveq3 6893	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑚 → (𝑂‘(𝐸‘𝑛)) = (𝑂‘(𝐸‘𝑚)))
125	124	cbvmptv 5260	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛))) = (𝑚 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑚)))
126	125	fveq2i 6891	. . . . . . . . . . . . . 14 ⊢ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) = (Σ^{^}‘(𝑚 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑚))))
127	126	eleq1i 2824	. . . . . . . . . . . . 13 ⊢ ((Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ ↔ (Σ^{^}‘(𝑚 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑚)))) ∈ ℝ)
128	127	biimpi 215	. . . . . . . . . . . 12 ⊢ ((Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ → (Σ^{^}‘(𝑚 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑚)))) ∈ ℝ)
129	128	ad2antlr 725	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ) ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → (Σ^{^}‘(𝑚 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑚)))) ∈ ℝ)
130		simpr 485	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ) ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → 𝑧 ∈ (𝒫 𝑍 ∩ Fin))
131	44	adantl 482	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (Σ^{^}‘(𝑚 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑚)))) ∈ ℝ) ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → 𝑧 ∈ Fin)
132	65	adantllr 717	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (Σ^{^}‘(𝑚 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑚)))) ∈ ℝ) ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → (𝑂‘(𝐸‘𝑛)) ∈ (0[,)+∞))
133	131, 132	sge0fsummpt 45092	. . . . . . . . . . 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 836	. . . . . . . . . 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 7420	. . . . . . . . 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 481	. . . . . . . 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 5173	. . . . . . 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 11368	. . . . . 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 413	. . . . 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 3168	. . . 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 15	. . 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 584	. 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 811	1 ⊢ (𝜑 → ∃𝑧 ∈ (𝒫 𝑍 ∩ Fin)(𝑂‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) < (Σ𝑛 ∈ 𝑧 (𝑂‘(𝐸‘𝑛)) + 𝑌))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∀wral 3061 ∃wrex 3070 Vcvv 3474 ∩ cin 3946 ⊆ wss 3947 𝒫 cpw 4601 ∪ cuni 4907 ∪ ciun 4996 class class class wbr 5147 ↦ cmpt 5230 dom cdm 5675 ↾ cres 5677 ⟶wf 6536 ‘cfv 6540 (class class class)co 7405 Fincfn 8935 ℝcr 11105 0cc0 11106 + caddc 11109 +∞cpnf 11241 ℝ^*cxr 11243 < clt 11244 ≤ cle 11245 ℤ_≥cuz 12818 ℝ⁺crp 12970 [,)cico 13322 [,]cicc 13323 Σcsu 15628 Σ^{^}csumge0 45064 OutMeascome 45191

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-inf2 9632 ax-ac2 10454 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 theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-oadd 8466 df-omul 8467 df-er 8699 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-sup 9433 df-oi 9501 df-card 9930 df-acn 9933 df-ac 10107 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-n0 12469 df-z 12555 df-uz 12819 df-rp 12971 df-ico 13326 df-icc 13327 df-fz 13481 df-fzo 13624 df-seq 13963 df-exp 14024 df-hash 14287 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-clim 15428 df-sum 15629 df-sumge0 45065 df-ome 45192

This theorem is referenced by: carageniuncllem2 45224

W3C validator