omeiunltfirp - Mathbox for Glauco Siliprandi

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Theorem omeiunltfirp 40733

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		fvex 6201	. . . . . 6 ⊢ (ℤ_≥‘𝑁) ∈ V
3	1, 2	eqeltri 2697	. . . . 5 ⊢ 𝑍 ∈ V
4	3	a1i 11	. . . 4 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) = +∞) → 𝑍 ∈ V)
5		omeiunltfirp.o	. . . . . . . 8 ⊢ (𝜑 → 𝑂 ∈ OutMeas)
6	5	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑂 ∈ OutMeas)
7		omeiunltfirp.x	. . . . . . 7 ⊢ 𝑋 = ∪ dom 𝑂
8		omeiunltfirp.e	. . . . . . . . 9 ⊢ (𝜑 → 𝐸:𝑍⟶𝒫 𝑋)
9	8	ffvelrnda 6359	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐸‘𝑛) ∈ 𝒫 𝑋)
10		fvex 6201	. . . . . . . . 9 ⊢ (𝐸‘𝑛) ∈ V
11	10	elpw 4164	. . . . . . . 8 ⊢ ((𝐸‘𝑛) ∈ 𝒫 𝑋 ↔ (𝐸‘𝑛) ⊆ 𝑋)
12	9, 11	sylib 208	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐸‘𝑛) ⊆ 𝑋)
13	6, 7, 12	omecl 40717	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑂‘(𝐸‘𝑛)) ∈ (0[,]+∞))
14		eqid 2622	. . . . . 6 ⊢ (𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛))) = (𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))
15	13, 14	fmptd 6385	. . . . 5 ⊢ (𝜑 → (𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛))):𝑍⟶(0[,]+∞))
16	15	adantr 481	. . . 4 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) = +∞) → (𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛))):𝑍⟶(0[,]+∞))
17		simpr 477	. . . 4 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) = +∞) → (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) = +∞)
18		omeiunltfirp.re	. . . . 5 ⊢ (𝜑 → (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) ∈ ℝ)
19	18	adantr 481	. . . 4 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) = +∞) → (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) ∈ ℝ)
20	4, 16, 17, 19	sge0pnffigt 40613	. . 3 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) = +∞) → ∃𝑧 ∈ (𝒫 𝑍 ∩ Fin)(𝑂‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) < (Σ^{^}‘((𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛))) ↾ 𝑧)))
21		simpl 473	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) < (Σ^{^}‘((𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛))) ↾ 𝑧))) → (𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)))
22		simpr 477	. . . . . . . . 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)))
23		elpwinss 39216	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (𝒫 𝑍 ∩ Fin) → 𝑧 ⊆ 𝑍)
24	23	resmptd 5452	. . . . . . . . . . 11 ⊢ (𝑧 ∈ (𝒫 𝑍 ∩ Fin) → ((𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛))) ↾ 𝑧) = (𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛))))
25	24	fveq2d 6195	. . . . . . . . . 10 ⊢ (𝑧 ∈ (𝒫 𝑍 ∩ Fin) → (Σ^{^}‘((𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛))) ↾ 𝑧)) = (Σ^{^}‘(𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛)))))
26	25	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)))))
27	22, 26	breqtrd 4679	. . . . . . . 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)))))
28	27	adantll 750	. . . . . . 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)))))
29	18	rexrd 10089	. . . . . . . . 9 ⊢ (𝜑 → (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) ∈ ℝ^*)
30	29	ad2antrr 762	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) < (Σ^{^}‘(𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛))))) → (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) ∈ ℝ^*)
31		simpr 477	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → 𝑧 ∈ (𝒫 𝑍 ∩ Fin))
32	5	ad2antrr 762	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → 𝑂 ∈ OutMeas)
33	8	ad2antrr 762	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → 𝐸:𝑍⟶𝒫 𝑋)
34	23	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑛 ∈ 𝑧) → 𝑧 ⊆ 𝑍)
35		simpr 477	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑛 ∈ 𝑧) → 𝑛 ∈ 𝑧)
36	34, 35	sseldd 3604	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ (𝒫 𝑍 ∩ Fin) ∧ 𝑛 ∈ 𝑧) → 𝑛 ∈ 𝑍)
37	36	adantll 750	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → 𝑛 ∈ 𝑍)
38	33, 37	ffvelrnd 6360	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → (𝐸‘𝑛) ∈ 𝒫 𝑋)
39	38, 11	sylib 208	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → (𝐸‘𝑛) ⊆ 𝑋)
40	32, 7, 39	omecl 40717	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → (𝑂‘(𝐸‘𝑛)) ∈ (0[,]+∞))
41		eqid 2622	. . . . . . . . . . 11 ⊢ (𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛))) = (𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛)))
42	40, 41	fmptd 6385	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → (𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛))):𝑧⟶(0[,]+∞))
43	31, 42	sge0xrcl 40602	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → (Σ^{^}‘(𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ^*)
44	43	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^*)
45		elinel2 3800	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ (𝒫 𝑍 ∩ Fin) → 𝑧 ∈ Fin)
46	45	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → 𝑧 ∈ Fin)
47		rge0ssre 12280	. . . . . . . . . . . . 13 ⊢ (0[,)+∞) ⊆ ℝ
48		0xr 10086	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ ℝ^*
49	48	a1i 11	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → 0 ∈ ℝ^*)
50		pnfxr 10092	. . . . . . . . . . . . . . 15 ⊢ +∞ ∈ ℝ^*
51	50	a1i 11	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → +∞ ∈ ℝ^*)
52	32, 7, 39	omexrcl 40721	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → (𝑂‘(𝐸‘𝑛)) ∈ ℝ^*)
53		iccgelb 12230	. . . . . . . . . . . . . . 15 ⊢ ((0 ∈ ℝ^* ∧ +∞ ∈ ℝ^* ∧ (𝑂‘(𝐸‘𝑛)) ∈ (0[,]+∞)) → 0 ≤ (𝑂‘(𝐸‘𝑛)))
54	49, 51, 40, 53	syl3anc 1326	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → 0 ≤ (𝑂‘(𝐸‘𝑛)))
55	12	ralrimiva 2966	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ∀𝑛 ∈ 𝑍 (𝐸‘𝑛) ⊆ 𝑋)
56		iunss 4561	. . . . . . . . . . . . . . . . . 18 ⊢ (∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛) ⊆ 𝑋 ↔ ∀𝑛 ∈ 𝑍 (𝐸‘𝑛) ⊆ 𝑋)
57	55, 56	sylibr 224	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛) ⊆ 𝑋)
58	57	ad2antrr 762	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛) ⊆ 𝑋)
59	32, 7, 58	omexrcl 40721	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) ∈ ℝ^*)
60		ssiun2 4563	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ 𝑍 → (𝐸‘𝑛) ⊆ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))
61	37, 60	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → (𝐸‘𝑛) ⊆ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))
62	32, 7, 58, 61	omessle 40712	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → (𝑂‘(𝐸‘𝑛)) ≤ (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)))
63	18	ltpnfd 11955	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) < +∞)
64	63	ad2antrr 762	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) < +∞)
65	52, 59, 51, 62, 64	xrlelttrd 11991	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → (𝑂‘(𝐸‘𝑛)) < +∞)
66	49, 51, 52, 54, 65	elicod 12224	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → (𝑂‘(𝐸‘𝑛)) ∈ (0[,)+∞))
67	47, 66	sseldi 3601	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → (𝑂‘(𝐸‘𝑛)) ∈ ℝ)
68	46, 67	fsumrecl 14465	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → Σ𝑛 ∈ 𝑧 (𝑂‘(𝐸‘𝑛)) ∈ ℝ)
69		omeiunltfirp.y	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑌 ∈ ℝ⁺)
70	69	rpred 11872	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑌 ∈ ℝ)
71	70	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → 𝑌 ∈ ℝ)
72	68, 71	readdcld 10069	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → (Σ𝑛 ∈ 𝑧 (𝑂‘(𝐸‘𝑛)) + 𝑌) ∈ ℝ)
73	72	rexrd 10089	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → (Σ𝑛 ∈ 𝑧 (𝑂‘(𝐸‘𝑛)) + 𝑌) ∈ ℝ^*)
74	73	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^*)
75		simpr 477	. . . . . . . 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)))))
76	66, 41	fmptd 6385	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → (𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛))):𝑧⟶(0[,)+∞))
77	46, 76	sge0fsum 40604	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → (Σ^{^}‘(𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛)))) = Σ𝑘 ∈ 𝑧 ((𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛)))‘𝑘))
78		eqidd 2623	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑘 ∈ 𝑧) → (𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛))) = (𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛))))
79		fveq2 6191	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑘 → (𝐸‘𝑛) = (𝐸‘𝑘))
80	79	fveq2d 6195	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑘 → (𝑂‘(𝐸‘𝑛)) = (𝑂‘(𝐸‘𝑘)))
81	80	adantl 482	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑘 ∈ 𝑧) ∧ 𝑛 = 𝑘) → (𝑂‘(𝐸‘𝑛)) = (𝑂‘(𝐸‘𝑘)))
82		simpr 477	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑘 ∈ 𝑧) → 𝑘 ∈ 𝑧)
83		fvexd 6203	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑘 ∈ 𝑧) → (𝑂‘(𝐸‘𝑘)) ∈ V)
84	78, 81, 82, 83	fvmptd 6288	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑘 ∈ 𝑧) → ((𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛)))‘𝑘) = (𝑂‘(𝐸‘𝑘)))
85	84	sumeq2dv 14433	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → Σ𝑘 ∈ 𝑧 ((𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛)))‘𝑘) = Σ𝑘 ∈ 𝑧 (𝑂‘(𝐸‘𝑘)))
86		fveq2 6191	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑛 → (𝐸‘𝑘) = (𝐸‘𝑛))
87	86	fveq2d 6195	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑛 → (𝑂‘(𝐸‘𝑘)) = (𝑂‘(𝐸‘𝑛)))
88	87	cbvsumv 14426	. . . . . . . . . . . 12 ⊢ Σ𝑘 ∈ 𝑧 (𝑂‘(𝐸‘𝑘)) = Σ𝑛 ∈ 𝑧 (𝑂‘(𝐸‘𝑛))
89	88	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → Σ𝑘 ∈ 𝑧 (𝑂‘(𝐸‘𝑘)) = Σ𝑛 ∈ 𝑧 (𝑂‘(𝐸‘𝑛)))
90	77, 85, 89	3eqtrd 2660	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → (Σ^{^}‘(𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛)))) = Σ𝑛 ∈ 𝑧 (𝑂‘(𝐸‘𝑛)))
91	69	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → 𝑌 ∈ ℝ⁺)
92	68, 91	ltaddrpd 11905	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → Σ𝑛 ∈ 𝑧 (𝑂‘(𝐸‘𝑛)) < (Σ𝑛 ∈ 𝑧 (𝑂‘(𝐸‘𝑛)) + 𝑌))
93	90, 92	eqbrtrd 4675	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → (Σ^{^}‘(𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛)))) < (Σ𝑛 ∈ 𝑧 (𝑂‘(𝐸‘𝑛)) + 𝑌))
94	93	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))
95	30, 44, 74, 75, 94	xrlttrd 11990	. . . . . . 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))
96	21, 28, 95	syl2anc 693	. . . . . 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))
97	96	ex 450	. . . . 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)))
98	97	adantlr 751	. . . 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)))
99	98	reximdva 3017	. . 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)))
100	20, 99	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))
101		simpl 473	. . 3 ⊢ ((𝜑 ∧ ¬ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) = +∞) → 𝜑)
102		simpr 477	. . . 4 ⊢ ((𝜑 ∧ ¬ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) = +∞) → ¬ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) = +∞)
103	3	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ V)
104	103, 15	sge0repnf 40603	. . . . 5 ⊢ (𝜑 → ((Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ ↔ ¬ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) = +∞))
105	104	adantr 481	. . . 4 ⊢ ((𝜑 ∧ ¬ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) = +∞) → ((Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ ↔ ¬ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) = +∞))
106	102, 105	mpbird 247	. . 3 ⊢ ((𝜑 ∧ ¬ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) = +∞) → (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ)
107		nfv 1843	. . . . . 6 ⊢ Ⅎ𝑛𝜑
108		nfcv 2764	. . . . . . . 8 ⊢ Ⅎ𝑛Σ^{^}
109		nfmpt1 4747	. . . . . . . 8 ⊢ Ⅎ𝑛(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))
110	108, 109	nffv 6198	. . . . . . 7 ⊢ Ⅎ𝑛(Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛))))
111		nfcv 2764	. . . . . . 7 ⊢ Ⅎ𝑛ℝ
112	110, 111	nfel 2777	. . . . . 6 ⊢ Ⅎ𝑛(Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ
113	107, 112	nfan 1828	. . . . 5 ⊢ Ⅎ𝑛(𝜑 ∧ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ)
114	3	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ) → 𝑍 ∈ V)
115	13	adantlr 751	. . . . 5 ⊢ (((𝜑 ∧ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ) ∧ 𝑛 ∈ 𝑍) → (𝑂‘(𝐸‘𝑛)) ∈ (0[,]+∞))
116	69	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ) → 𝑌 ∈ ℝ⁺)
117		simpr 477	. . . . 5 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ) → (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ)
118	113, 114, 115, 116, 117	sge0ltfirpmpt 40625	. . . 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))
119	18	ad3antrrr 766	. . . . . . 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)
120	117	ad2antrr 762	. . . . . . 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)
121	72	ad4ant13 1292	. . . . . . 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)
122		nfcv 2764	. . . . . . . . 9 ⊢ Ⅎ𝑛𝐸
123	107, 122, 5, 7, 1, 8	omeiunle 40731	. . . . . . . 8 ⊢ (𝜑 → (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) ≤ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))))
124	123	ad3antrrr 766	. . . . . . 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)))))
125		simpr 477	. . . . . . . 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))
126		simpll 790	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ) ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → 𝜑)
127		fveq2 6191	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑚 → (𝐸‘𝑛) = (𝐸‘𝑚))
128	127	fveq2d 6195	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑚 → (𝑂‘(𝐸‘𝑛)) = (𝑂‘(𝐸‘𝑚)))
129	128	cbvmptv 4750	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛))) = (𝑚 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑚)))
130	129	fveq2i 6194	. . . . . . . . . . . . . 14 ⊢ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) = (Σ^{^}‘(𝑚 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑚))))
131	130	eleq1i 2692	. . . . . . . . . . . . 13 ⊢ ((Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ ↔ (Σ^{^}‘(𝑚 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑚)))) ∈ ℝ)
132	131	biimpi 206	. . . . . . . . . . . 12 ⊢ ((Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ → (Σ^{^}‘(𝑚 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑚)))) ∈ ℝ)
133	132	ad2antlr 763	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ) ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → (Σ^{^}‘(𝑚 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑚)))) ∈ ℝ)
134		simpr 477	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ) ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → 𝑧 ∈ (𝒫 𝑍 ∩ Fin))
135	45	adantl 482	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (Σ^{^}‘(𝑚 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑚)))) ∈ ℝ) ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → 𝑧 ∈ Fin)
136	66	adantllr 755	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (Σ^{^}‘(𝑚 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑚)))) ∈ ℝ) ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) ∧ 𝑛 ∈ 𝑧) → (𝑂‘(𝐸‘𝑛)) ∈ (0[,)+∞))
137	135, 136	sge0fsummpt 40607	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (Σ^{^}‘(𝑚 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑚)))) ∈ ℝ) ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → (Σ^{^}‘(𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛)))) = Σ𝑛 ∈ 𝑧 (𝑂‘(𝐸‘𝑛)))
138	126, 133, 134, 137	syl21anc 1325	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ) ∧ 𝑧 ∈ (𝒫 𝑍 ∩ Fin)) → (Σ^{^}‘(𝑛 ∈ 𝑧 ↦ (𝑂‘(𝐸‘𝑛)))) = Σ𝑛 ∈ 𝑧 (𝑂‘(𝐸‘𝑛)))
139	138	oveq1d 6665	. . . . . . . . 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))
140	139	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))
141	125, 140	breqtrd 4679	. . . . . . 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))
142	119, 120, 121, 124, 141	lelttrd 10195	. . . . . 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))
143	142	ex 450	. . . . 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)))
144	143	reximdva 3017	. . . 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)))
145	118, 144	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))
146	101, 106, 145	syl2anc 693	. 2 ⊢ ((𝜑 ∧ ¬ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) = +∞) → ∃𝑧 ∈ (𝒫 𝑍 ∩ Fin)(𝑂‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) < (Σ𝑛 ∈ 𝑧 (𝑂‘(𝐸‘𝑛)) + 𝑌))
147	100, 146	pm2.61dan 832	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 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ∃wrex 2913 Vcvv 3200 ∩ cin 3573 ⊆ wss 3574 𝒫 cpw 4158 ∪ cuni 4436 ∪ ciun 4520 class class class wbr 4653 ↦ cmpt 4729 dom cdm 5114 ↾ cres 5116 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 Fincfn 7955 ℝcr 9935 0cc0 9936 + caddc 9939 +∞cpnf 10071 ℝ^*cxr 10073 < clt 10074 ≤ cle 10075 ℤ_≥cuz 11687 ℝ⁺crp 11832 [,)cico 12177 [,]cicc 12178 Σcsu 14416 Σ^{^}csumge0 40579 OutMeascome 40703

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-ac2 9285 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-omul 7565 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-oi 8415 df-card 8765 df-acn 8768 df-ac 8939 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 df-ome 40704

This theorem is referenced by: carageniuncllem2 40736

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