omeiunltfirp - Mathbox for Glauco Siliprandi

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Theorem omeiunltfirp 46976

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 6845	. . . . 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 482	. . . . . . 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 7029	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐸‘𝑛) ∈ 𝒫 𝑋)
9		fvex 6844	. . . . . . . . 9 ⊢ (𝐸‘𝑛) ∈ V
10	9	elpw 4536	. . . . . . . 8 ⊢ ((𝐸‘𝑛) ∈ 𝒫 𝑋 ↔ (𝐸‘𝑛) ⊆ 𝑋)
11	8, 10	sylib 220	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐸‘𝑛) ⊆ 𝑋)
12	5, 6, 11	omecl 46960	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑂‘(𝐸‘𝑛)) ∈ (0[,]+∞))
13		eqid 2741	. . . . . 6 ⊢ (𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛))) = (𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))
14	12, 13	fmptd 7059	. . . . 5 ⊢ (𝜑 → (𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛))):𝑍⟶(0[,]+∞))
15	14	adantr 482	. . . 4 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) = +∞) → (𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛))):𝑍⟶(0[,]+∞))
16		simpr 486	. . . 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 482	. . . 4 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) = +∞) → (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) ∈ ℝ)
19	3, 15, 16, 18	sge0pnffigt 46853	. . 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 484	. . . . . . 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 486	. . . . . . . . 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 45512	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (𝒫 𝑍 ∩ Fin) → 𝑧 ⊆ 𝑍)
23	22	resmptd 5999	. . . . . . . . . . 11 ⊢ (𝑧 ∈ (𝒫 𝑍 ∩ Fin) → ((𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛))) ↾ 𝑧) = (𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛))))
24	23	fveq2d 6835	. . . . . . . . . 10 ⊢ (𝑧 ∈ (𝒫 𝑍 ∩ Fin) → (Σ^{^}‘((𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛))) ↾ 𝑧)) = (Σ^{^}‘(𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛)))))
25	24	adantr 482	. . . . . . . . 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 5101	. . . . . . . 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 721	. . . . . . 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 11190	. . . . . . . . 9 ⊢ (𝜑 → (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) ∈ ℝ^*)
29	28	ad2antrr 733	. . . . . . . 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 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → 𝑧 ∈ (𝒫 𝑍 ∩ Fin))
31	4	ad2antrr 733	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → 𝑂 ∈ OutMeas)
32	7	ad2antrr 733	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → 𝐸:𝑍⟶𝒫 𝑋)
33	22	adantr 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑛 ∈ 𝑧) → 𝑧 ⊆ 𝑍)
34		simpr 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑛 ∈ 𝑧) → 𝑛 ∈ 𝑧)
35	33, 34	sseldd 3918	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑛 ∈ 𝑧) → 𝑛 ∈ 𝑍)
36	35	adantll 721	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → 𝑛 ∈ 𝑍)
37	32, 36	ffvelcdmd 7030	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → (𝐸‘𝑛) ∈ 𝒫 𝑋)
38	37, 10	sylib 220	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → (𝐸‘𝑛) ⊆ 𝑋)
39	31, 6, 38	omecl 46960	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → (𝑂‘(𝐸‘𝑛)) ∈ (0[,]+∞))
40		eqid 2741	. . . . . . . . . . 11 ⊢ (𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛))) = (𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛)))
41	39, 40	fmptd 7059	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → (𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛))):𝑧⟶(0[,]+∞))
42	30, 41	sge0xrcl 46842	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → (Σ^{^}‘(𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ^*)
43	42	adantr 482	. . . . . . . 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 4134	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ (𝒫 𝑍 ∩ Fin) → 𝑧 ∈ Fin)
45	44	adantl 483	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → 𝑧 ∈ Fin)
46		rge0ssre 13404	. . . . . . . . . . . . 13 ⊢ (0[,)+∞) ⊆ ℝ
47		0xr 11187	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ ℝ^*
48	47	a1i 11	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → 0 ∈ ℝ^*)
49		pnfxr 11194	. . . . . . . . . . . . . . 15 ⊢ +∞ ∈ ℝ^*
50	49	a1i 11	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → +∞ ∈ ℝ^*)
51	31, 6, 38	omexrcl 46964	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → (𝑂‘(𝐸‘𝑛)) ∈ ℝ^*)
52		iccgelb 13350	. . . . . . . . . . . . . . 15 ⊢ ((0 ∈ ℝ^* ∧ +∞ ∈ ℝ^* ∧ (𝑂‘(𝐸‘𝑛)) ∈ (0[,]+∞)) → 0 ≤ (𝑂‘(𝐸‘𝑛)))
53	48, 50, 39, 52	syl3anc 1380	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → 0 ≤ (𝑂‘(𝐸‘𝑛)))
54	11	ralrimiva 3133	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ∀𝑛 ∈ 𝑍 (𝐸‘𝑛) ⊆ 𝑋)
55		iunss 4977	. . . . . . . . . . . . . . . . . 18 ⊢ (∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛) ⊆ 𝑋 ↔ ∀𝑛 ∈ 𝑍 (𝐸‘𝑛) ⊆ 𝑋)
56	54, 55	sylibr 236	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛) ⊆ 𝑋)
57	56	ad2antrr 733	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛) ⊆ 𝑋)
58	31, 6, 57	omexrcl 46964	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) ∈ ℝ^*)
59		ssiun2 4980	. . . . . . . . . . . . . . . . 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 46955	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → (𝑂‘(𝐸‘𝑛)) ≤ (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)))
62	17	ltpnfd 13067	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) < +∞)
63	62	ad2antrr 733	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) < +∞)
64	51, 58, 50, 61, 63	xrlelttrd 13106	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → (𝑂‘(𝐸‘𝑛)) < +∞)
65	48, 50, 51, 53, 64	elicod 13343	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → (𝑂‘(𝐸‘𝑛)) ∈ (0[,)+∞))
66	46, 65	sselid 3915	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → (𝑂‘(𝐸‘𝑛)) ∈ ℝ)
67	45, 66	fsumrecl 15691	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → Σ𝑛 ∈ 𝑧 (𝑂‘(𝐸‘𝑛)) ∈ ℝ)
68		omeiunltfirp.y	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑌 ∈ ℝ⁺)
69	68	rpred 12981	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑌 ∈ ℝ)
70	69	adantr 482	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → 𝑌 ∈ ℝ)
71	67, 70	readdcld 11169	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → (Σ𝑛 ∈ 𝑧 (𝑂‘(𝐸‘𝑛)) + 𝑌) ∈ ℝ)
72	71	rexrd 11190	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → (Σ𝑛 ∈ 𝑧 (𝑂‘(𝐸‘𝑛)) + 𝑌) ∈ ℝ^*)
73	72	adantr 482	. . . . . . . 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 486	. . . . . . . 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 7059	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → (𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛))):𝑧⟶(0[,)+∞))
76	45, 75	sge0fsum 46844	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → (Σ^{^}‘(𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛)))) = Σ𝑘 ∈ 𝑧 ((𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛)))‘𝑘))
77		eqidd 2742	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑘 ∈ 𝑧) → (𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛))) = (𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛))))
78		2fveq3 6836	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑘 → (𝑂‘(𝐸‘𝑛)) = (𝑂‘(𝐸‘𝑘)))
79	78	adantl 483	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑘 ∈ 𝑧) ∧ 𝑛 = 𝑘) → (𝑂‘(𝐸‘𝑛)) = (𝑂‘(𝐸‘𝑘)))
80		simpr 486	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑘 ∈ 𝑧) → 𝑘 ∈ 𝑧)
81		fvexd 6846	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑘 ∈ 𝑧) → (𝑂‘(𝐸‘𝑘)) ∈ V)
82	77, 79, 80, 81	fvmptd 6947	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑘 ∈ 𝑧) → ((𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛)))‘𝑘) = (𝑂‘(𝐸‘𝑘)))
83	82	sumeq2dv 15659	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → Σ𝑘 ∈ 𝑧 ((𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛)))‘𝑘) = Σ𝑘 ∈ 𝑧 (𝑂‘(𝐸‘𝑘)))
84		2fveq3 6836	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑛 → (𝑂‘(𝐸‘𝑘)) = (𝑂‘(𝐸‘𝑛)))
85	84	cbvsumv 15653	. . . . . . . . . . . 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 2780	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → (Σ^{^}‘(𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛)))) = Σ𝑛 ∈ 𝑧 (𝑂‘(𝐸‘𝑛)))
88	68	adantr 482	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → 𝑌 ∈ ℝ⁺)
89	67, 88	ltaddrpd 13014	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → Σ𝑛 ∈ 𝑧 (𝑂‘(𝐸‘𝑛)) < (Σ𝑛 ∈ 𝑧 (𝑂‘(𝐸‘𝑛)) + 𝑌))
90	87, 89	eqbrtrd 5097	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → (Σ^{^}‘(𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛)))) < (Σ𝑛 ∈ 𝑧 (𝑂‘(𝐸‘𝑛)) + 𝑌))
91	90	adantr 482	. . . . . . . 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 13105	. . . . . . 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 591	. . . . . 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 414	. . . . 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 722	. . . 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 3154	. . 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 484	. . 3 ⊢ ((𝜑 ∧ ¬ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) = +∞) → 𝜑)
99		simpr 486	. . . 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 46843	. . . . 5 ⊢ (𝜑 → ((Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ ↔ ¬ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) = +∞))
102	101	adantr 482	. . . 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 259	. . 3 ⊢ ((𝜑 ∧ ¬ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) = +∞) → (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ)
104		nfv 1922	. . . . . 6 ⊢ Ⅎ𝑛𝜑
105		nfcv 2903	. . . . . . . 8 ⊢ Ⅎ𝑛Σ^{^}
106		nfmpt1 5174	. . . . . . . 8 ⊢ Ⅎ𝑛(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))
107	105, 106	nffv 6841	. . . . . . 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 1907	. . . . 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 722	. . . . 5 ⊢ (((𝜑 ∧ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ) ∧ 𝑛 ∈ 𝑍) → (𝑂‘(𝐸‘𝑛)) ∈ (0[,]+∞))
113	68	adantr 482	. . . . 5 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ) → 𝑌 ∈ ℝ⁺)
114		simpr 486	. . . . 5 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ) → (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ)
115	110, 111, 112, 113, 114	sge0ltfirpmpt 46865	. . . 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 737	. . . . . . 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 733	. . . . . . 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 758	. . . . . . 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 46974	. . . . . . . 8 ⊢ (𝜑 → (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) ≤ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))))
121	120	ad3antrrr 737	. . . . . . 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 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)))) < ((Σ^{^}‘(𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛)))) + 𝑌))
123		simpll 773	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ) ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → 𝜑)
124		2fveq3 6836	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑚 → (𝑂‘(𝐸‘𝑛)) = (𝑂‘(𝐸‘𝑚)))
125	124	cbvmptv 5179	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛))) = (𝑚 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑚)))
126	125	fveq2i 6834	. . . . . . . . . . . . . 14 ⊢ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) = (Σ^{^}‘(𝑚 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑚))))
127	126	eleq1i 2832	. . . . . . . . . . . . 13 ⊢ ((Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ ↔ (Σ^{^}‘(𝑚 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑚)))) ∈ ℝ)
128	127	biimpi 218	. . . . . . . . . . . 12 ⊢ ((Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ → (Σ^{^}‘(𝑚 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑚)))) ∈ ℝ)
129	128	ad2antlr 734	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ) ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → (Σ^{^}‘(𝑚 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑚)))) ∈ ℝ)
130		simpr 486	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ) ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → 𝑧 ∈ (𝒫 𝑍 ∩ Fin))
131	44	adantl 483	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (Σ^{^}‘(𝑚 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑚)))) ∈ ℝ) ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → 𝑧 ∈ Fin)
132	65	adantllr 726	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (Σ^{^}‘(𝑚 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑚)))) ∈ ℝ) ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → (𝑂‘(𝐸‘𝑛)) ∈ (0[,)+∞))
133	131, 132	sge0fsummpt 46847	. . . . . . . . . . 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 844	. . . . . . . . . 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 7375	. . . . . . . . 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 482	. . . . . . . 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 5101	. . . . . . 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 11299	. . . . . 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 414	. . . . 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 3154	. . . 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 591	. 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 819	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 208 ∧ wa 397 = wceq 1548 ∈ wcel 2121 ∀wral 3055 ∃wrex 3065 Vcvv 3433 ∩ cin 3884 ⊆ wss 3885 𝒫 cpw 4532 ∪ cuni 4841 ∪ ciun 4924 class class class wbr 5075 ↦ cmpt 5156 dom cdm 5621 ↾ cres 5623 ⟶wf 6485 ‘cfv 6489 (class class class)co 7360 Fincfn 8887 ℝcr 11032 0cc0 11033 + caddc 11036 +∞cpnf 11171 ℝ^*cxr 11173 < clt 11174 ≤ cle 11175 ℤ_≥cuz 12783 ℝ⁺crp 12937 [,)cico 13295 [,]cicc 13296 Σcsu 15643 Σ^{^}csumge0 46819 OutMeascome 46946

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1975 ax-7 2016 ax-8 2123 ax-9 2131 ax-10 2154 ax-11 2170 ax-12 2191 ax-ext 2713 ax-rep 5202 ax-sep 5221 ax-nul 5231 ax-pow 5297 ax-pr 5365 ax-un 7682 ax-inf2 9557 ax-ac2 10380 ax-cnex 11089 ax-resscn 11090 ax-1cn 11091 ax-icn 11092 ax-addcl 11093 ax-addrcl 11094 ax-mulcl 11095 ax-mulrcl 11096 ax-mulcom 11097 ax-addass 11098 ax-mulass 11099 ax-distr 11100 ax-i2m1 11101 ax-1ne0 11102 ax-1rid 11103 ax-rnegex 11104 ax-rrecex 11105 ax-cnre 11106 ax-pre-lttri 11107 ax-pre-lttrn 11108 ax-pre-ltadd 11109 ax-pre-mulgt0 11110 ax-pre-sup 11111

This theorem depends on definitions: df-bi 209 df-an 398 df-or 855 df-3or 1094 df-3an 1095 df-tru 1551 df-fal 1561 df-ex 1788 df-nf 1792 df-sb 2075 df-mo 2545 df-eu 2575 df-clab 2720 df-cleq 2733 df-clel 2816 df-nfc 2890 df-ne 2937 df-nel 3041 df-ral 3056 df-rex 3066 df-rmo 3346 df-reu 3347 df-rab 3394 df-v 3435 df-sbc 3726 df-csb 3834 df-dif 3888 df-un 3890 df-in 3892 df-ss 3902 df-pss 3905 df-nul 4265 df-if 4458 df-pw 4534 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4842 df-int 4881 df-iun 4926 df-br 5076 df-opab 5138 df-mpt 5157 df-tr 5183 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-se 5575 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6256 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-isom 6498 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-om 7811 df-1st 7935 df-2nd 7936 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-oadd 8403 df-omul 8404 df-er 8637 df-map 8769 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-sup 9349 df-oi 9419 df-card 9858 df-acn 9861 df-ac 10033 df-pnf 11176 df-mnf 11177 df-xr 11178 df-ltxr 11179 df-le 11180 df-sub 11374 df-neg 11375 df-div 11803 df-nn 12170 df-2 12239 df-3 12240 df-n0 12433 df-z 12520 df-uz 12784 df-rp 12938 df-ico 13299 df-icc 13300 df-fz 13457 df-fzo 13604 df-seq 13959 df-exp 14019 df-hash 14288 df-cj 15056 df-re 15057 df-im 15058 df-sqrt 15192 df-abs 15193 df-clim 15445 df-sum 15644 df-sumge0 46820 df-ome 46947

This theorem is referenced by: carageniuncllem2 46979

W3C validator