ovnsubaddlem2 - Mathbox for Glauco Siliprandi

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Theorem ovnsubaddlem2 46527

Description: (voln*‘𝑋) is subadditive. Proposition 115D (a)(iv) of [Fremlin1] p. 31 . (Contributed by Glauco Siliprandi, 11-Oct-2020.)

**Hypotheses**
Ref	Expression
ovnsubaddlem2.x	⊢ (𝜑 → 𝑋 ∈ Fin)
ovnsubaddlem2.n0	⊢ (𝜑 → 𝑋 ≠ ∅)
ovnsubaddlem2.a	⊢ (𝜑 → 𝐴:ℕ⟶𝒫 (ℝ ↑_m 𝑋))
ovnsubaddlem2.e	⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
ovnsubaddlem2.z	⊢ 𝑍 = (𝑎 ∈ 𝒫 (ℝ ↑_m 𝑋) ↦ {𝑧 ∈ ℝ^* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ)(𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))})
ovnsubaddlem2.c	⊢ 𝐶 = (𝑎 ∈ 𝒫 (ℝ ↑_m 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ) ∣ 𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)})
ovnsubaddlem2.l	⊢ 𝐿 = (ℎ ∈ ((ℝ × ℝ) ↑_m 𝑋) ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑘)))
ovnsubaddlem2.d	⊢ 𝐷 = (𝑎 ∈ 𝒫 (ℝ ↑_m 𝑋) ↦ (𝑒 ∈ ℝ⁺ ↦ {𝑖 ∈ (𝐶‘𝑎) ∣ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +_𝑒 𝑒)}))

**Assertion**
Ref	Expression
ovnsubaddlem2	⊢ (𝜑 → ((voln‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐴‘𝑛)) ≤ ((Σ^{^}‘(𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐴‘𝑛)))) +_𝑒 𝐸))

Distinct variable groups: 𝐴,𝑎,𝑒,𝑖,𝑗,𝑛 𝐴,𝑘,𝑙,𝑎,𝑖,𝑗,𝑛 𝑧,𝐴,𝑎,𝑖,𝑗,𝑘,𝑛 𝐶,𝑎,𝑒,𝑖 𝐷,𝑎,𝑒,𝑖,𝑗,𝑛 𝐷,𝑘 𝐸,𝑎,𝑒,𝑖,𝑗,𝑛 𝑘,𝐸 𝐿,𝑎,𝑒,𝑖,𝑗,𝑛 𝑋,𝑎,𝑒,𝑖,𝑗,𝑛 ℎ,𝑋,𝑘,𝑖,𝑗 𝑋,𝑙 𝑧,𝑋 𝜑,𝑎,𝑒,𝑖,𝑗,𝑛 𝜑,𝑘 Allowed substitution hints: 𝜑(𝑧,ℎ,𝑙) 𝐴(ℎ) 𝐶(𝑧,ℎ,𝑗,𝑘,𝑛,𝑙) 𝐷(𝑧,ℎ,𝑙) 𝐸(𝑧,ℎ,𝑙) 𝐿(𝑧,ℎ,𝑘,𝑙) 𝑍(𝑧,𝑒,ℎ,𝑖,𝑗,𝑘,𝑛,𝑎,𝑙)

Proof of Theorem ovnsubaddlem2 Dummy variables 𝑓 𝑔 𝑚 𝑞 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvex 6920	. . . 4 ⊢ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) ∈ V
2		nnenom 14018	. . . 4 ⊢ ℕ ≈ ω
3	1, 2	axcc3 10476	. . 3 ⊢ ∃𝑔(𝑔 Fn ℕ ∧ ∀𝑛 ∈ ℕ (((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) ≠ ∅ → (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛)))))
4		simprl 771	. . . . . 6 ⊢ ((𝜑 ∧ (𝑔 Fn ℕ ∧ ∀𝑛 ∈ ℕ (((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) ≠ ∅ → (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛)))))) → 𝑔 Fn ℕ)
5		nfv 1912	. . . . . . . . 9 ⊢ Ⅎ𝑛𝜑
6		nfra1 3282	. . . . . . . . 9 ⊢ Ⅎ𝑛∀𝑛 ∈ ℕ (((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) ≠ ∅ → (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))
7	5, 6	nfan 1897	. . . . . . . 8 ⊢ Ⅎ𝑛(𝜑 ∧ ∀𝑛 ∈ ℕ (((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) ≠ ∅ → (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛)))))
8		rspa 3246	. . . . . . . . . . 11 ⊢ ((∀𝑛 ∈ ℕ (((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) ≠ ∅ → (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛)))) ∧ 𝑛 ∈ ℕ) → (((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) ≠ ∅ → (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛)))))
9	8	adantll 714	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ∀𝑛 ∈ ℕ (((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) ≠ ∅ → (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))) ∧ 𝑛 ∈ ℕ) → (((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) ≠ ∅ → (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛)))))
10		ovnsubaddlem2.x	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑋 ∈ Fin)
11	10	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑋 ∈ Fin)
12		ovnsubaddlem2.n0	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑋 ≠ ∅)
13	12	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑋 ≠ ∅)
14		ovnsubaddlem2.a	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐴:ℕ⟶𝒫 (ℝ ↑_m 𝑋))
15	14	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴:ℕ⟶𝒫 (ℝ ↑_m 𝑋))
16		simpr 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
17	15, 16	ffvelcdmd 7105	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴‘𝑛) ∈ 𝒫 (ℝ ↑_m 𝑋))
18		elpwi 4612	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴‘𝑛) ∈ 𝒫 (ℝ ↑_m 𝑋) → (𝐴‘𝑛) ⊆ (ℝ ↑_m 𝑋))
19	17, 18	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴‘𝑛) ⊆ (ℝ ↑_m 𝑋))
20		ovnsubaddlem2.e	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
21	20	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐸 ∈ ℝ⁺)
22		nnnn0 12531	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ₀)
23		2nn 12337	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 2 ∈ ℕ
24	23	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 ∈ ℕ₀ → 2 ∈ ℕ)
25		id 22	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 ∈ ℕ₀ → 𝑛 ∈ ℕ₀)
26		nnexpcl 14112	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((2 ∈ ℕ ∧ 𝑛 ∈ ℕ₀) → (2↑𝑛) ∈ ℕ)
27	24, 25, 26	syl2anc 584	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ ℕ₀ → (2↑𝑛) ∈ ℕ)
28		nnrp 13044	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((2↑𝑛) ∈ ℕ → (2↑𝑛) ∈ ℝ⁺)
29	27, 28	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℕ₀ → (2↑𝑛) ∈ ℝ⁺)
30	22, 29	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℕ → (2↑𝑛) ∈ ℝ⁺)
31	30	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2↑𝑛) ∈ ℝ⁺)
32	21, 31	rpdivcld 13092	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐸 / (2↑𝑛)) ∈ ℝ⁺)
33		ovnsubaddlem2.c	. . . . . . . . . . . . . . . 16 ⊢ 𝐶 = (𝑎 ∈ 𝒫 (ℝ ↑_m 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ) ∣ 𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)})
34		ovnsubaddlem2.l	. . . . . . . . . . . . . . . 16 ⊢ 𝐿 = (ℎ ∈ ((ℝ × ℝ) ↑_m 𝑋) ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑘)))
35		ovnsubaddlem2.d	. . . . . . . . . . . . . . . 16 ⊢ 𝐷 = (𝑎 ∈ 𝒫 (ℝ ↑_m 𝑋) ↦ (𝑒 ∈ ℝ⁺ ↦ {𝑖 ∈ (𝐶‘𝑎) ∣ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +_𝑒 𝑒)}))
36	11, 13, 19, 32, 33, 34, 35	ovncvrrp 46520	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∃𝑖 𝑖 ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))
37		n0 4359	. . . . . . . . . . . . . . 15 ⊢ (((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) ≠ ∅ ↔ ∃𝑖 𝑖 ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))
38	36, 37	sylibr 234	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) ≠ ∅)
39	38	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) ≠ ∅ → (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))) → ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) ≠ ∅)
40		simpr 484	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) ≠ ∅ → (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))) → (((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) ≠ ∅ → (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛)))))
41	39, 40	mpd 15	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) ≠ ∅ → (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))) → (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))
42	41	ex 412	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) ≠ ∅ → (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛)))) → (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛)))))
43	42	adantlr 715	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ∀𝑛 ∈ ℕ (((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) ≠ ∅ → (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))) ∧ 𝑛 ∈ ℕ) → ((((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) ≠ ∅ → (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛)))) → (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛)))))
44	9, 43	mpd 15	. . . . . . . . 9 ⊢ (((𝜑 ∧ ∀𝑛 ∈ ℕ (((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) ≠ ∅ → (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))) ∧ 𝑛 ∈ ℕ) → (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))
45	44	ex 412	. . . . . . . 8 ⊢ ((𝜑 ∧ ∀𝑛 ∈ ℕ (((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) ≠ ∅ → (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))) → (𝑛 ∈ ℕ → (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛)))))
46	7, 45	ralrimi 3255	. . . . . . 7 ⊢ ((𝜑 ∧ ∀𝑛 ∈ ℕ (((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) ≠ ∅ → (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))) → ∀𝑛 ∈ ℕ (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))
47	46	adantrl 716	. . . . . 6 ⊢ ((𝜑 ∧ (𝑔 Fn ℕ ∧ ∀𝑛 ∈ ℕ (((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) ≠ ∅ → (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛)))))) → ∀𝑛 ∈ ℕ (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))
48	4, 47	jca 511	. . . . 5 ⊢ ((𝜑 ∧ (𝑔 Fn ℕ ∧ ∀𝑛 ∈ ℕ (((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) ≠ ∅ → (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛)))))) → (𝑔 Fn ℕ ∧ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛)))))
49	48	ex 412	. . . 4 ⊢ (𝜑 → ((𝑔 Fn ℕ ∧ ∀𝑛 ∈ ℕ (((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) ≠ ∅ → (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))) → (𝑔 Fn ℕ ∧ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))))
50	49	eximdv 1915	. . 3 ⊢ (𝜑 → (∃𝑔(𝑔 Fn ℕ ∧ ∀𝑛 ∈ ℕ (((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) ≠ ∅ → (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))) → ∃𝑔(𝑔 Fn ℕ ∧ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))))
51	3, 50	mpi 20	. 2 ⊢ (𝜑 → ∃𝑔(𝑔 Fn ℕ ∧ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛)))))
52		simpl 482	. . . . 5 ⊢ ((𝜑 ∧ (𝑔 Fn ℕ ∧ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))) → 𝜑)
53		simprl 771	. . . . 5 ⊢ ((𝜑 ∧ (𝑔 Fn ℕ ∧ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))) → 𝑔 Fn ℕ)
54		simprr 773	. . . . 5 ⊢ ((𝜑 ∧ (𝑔 Fn ℕ ∧ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))) → ∀𝑛 ∈ ℕ (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))
55		nnf1oxpnn 45138	. . . . . 6 ⊢ ∃𝑓 𝑓:ℕ–1-1-onto→(ℕ × ℕ)
56		simpl1 1190	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑔 Fn ℕ ∧ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛)))) ∧ 𝑓:ℕ–1-1-onto→(ℕ × ℕ)) → 𝜑)
57		simpl2 1191	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑔 Fn ℕ ∧ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛)))) ∧ 𝑓:ℕ–1-1-onto→(ℕ × ℕ)) → 𝑔 Fn ℕ)
58		fveq2 6907	. . . . . . . . . . . . . 14 ⊢ (𝑞 = 𝑛 → (𝑔‘𝑞) = (𝑔‘𝑛))
59		2fveq3 6912	. . . . . . . . . . . . . . 15 ⊢ (𝑞 = 𝑛 → (𝐷‘(𝐴‘𝑞)) = (𝐷‘(𝐴‘𝑛)))
60		oveq2 7439	. . . . . . . . . . . . . . . 16 ⊢ (𝑞 = 𝑛 → (2↑𝑞) = (2↑𝑛))
61	60	oveq2d 7447	. . . . . . . . . . . . . . 15 ⊢ (𝑞 = 𝑛 → (𝐸 / (2↑𝑞)) = (𝐸 / (2↑𝑛)))
62	59, 61	fveq12d 6914	. . . . . . . . . . . . . 14 ⊢ (𝑞 = 𝑛 → ((𝐷‘(𝐴‘𝑞))‘(𝐸 / (2↑𝑞))) = ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))
63	58, 62	eleq12d 2833	. . . . . . . . . . . . 13 ⊢ (𝑞 = 𝑛 → ((𝑔‘𝑞) ∈ ((𝐷‘(𝐴‘𝑞))‘(𝐸 / (2↑𝑞))) ↔ (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛)))))
64	63	cbvralvw 3235	. . . . . . . . . . . 12 ⊢ (∀𝑞 ∈ ℕ (𝑔‘𝑞) ∈ ((𝐷‘(𝐴‘𝑞))‘(𝐸 / (2↑𝑞))) ↔ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))
65	64	biimpri 228	. . . . . . . . . . 11 ⊢ (∀𝑛 ∈ ℕ (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) → ∀𝑞 ∈ ℕ (𝑔‘𝑞) ∈ ((𝐷‘(𝐴‘𝑞))‘(𝐸 / (2↑𝑞))))
66	65	3ad2ant3 1134	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 Fn ℕ ∧ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛)))) → ∀𝑞 ∈ ℕ (𝑔‘𝑞) ∈ ((𝐷‘(𝐴‘𝑞))‘(𝐸 / (2↑𝑞))))
67	66	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑔 Fn ℕ ∧ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛)))) ∧ 𝑓:ℕ–1-1-onto→(ℕ × ℕ)) → ∀𝑞 ∈ ℕ (𝑔‘𝑞) ∈ ((𝐷‘(𝐴‘𝑞))‘(𝐸 / (2↑𝑞))))
68		simpr 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑔 Fn ℕ ∧ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛)))) ∧ 𝑓:ℕ–1-1-onto→(ℕ × ℕ)) → 𝑓:ℕ–1-1-onto→(ℕ × ℕ))
69	10	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓:ℕ–1-1-onto→(ℕ × ℕ)) → 𝑋 ∈ Fin)
70	69	3ad2antl1 1184	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔 Fn ℕ ∧ ∀𝑞 ∈ ℕ (𝑔‘𝑞) ∈ ((𝐷‘(𝐴‘𝑞))‘(𝐸 / (2↑𝑞)))) ∧ 𝑓:ℕ–1-1-onto→(ℕ × ℕ)) → 𝑋 ∈ Fin)
71	12	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓:ℕ–1-1-onto→(ℕ × ℕ)) → 𝑋 ≠ ∅)
72	71	3ad2antl1 1184	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔 Fn ℕ ∧ ∀𝑞 ∈ ℕ (𝑔‘𝑞) ∈ ((𝐷‘(𝐴‘𝑞))‘(𝐸 / (2↑𝑞)))) ∧ 𝑓:ℕ–1-1-onto→(ℕ × ℕ)) → 𝑋 ≠ ∅)
73	14	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓:ℕ–1-1-onto→(ℕ × ℕ)) → 𝐴:ℕ⟶𝒫 (ℝ ↑_m 𝑋))
74	73	3ad2antl1 1184	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔 Fn ℕ ∧ ∀𝑞 ∈ ℕ (𝑔‘𝑞) ∈ ((𝐷‘(𝐴‘𝑞))‘(𝐸 / (2↑𝑞)))) ∧ 𝑓:ℕ–1-1-onto→(ℕ × ℕ)) → 𝐴:ℕ⟶𝒫 (ℝ ↑_m 𝑋))
75	20	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓:ℕ–1-1-onto→(ℕ × ℕ)) → 𝐸 ∈ ℝ⁺)
76	75	3ad2antl1 1184	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔 Fn ℕ ∧ ∀𝑞 ∈ ℕ (𝑔‘𝑞) ∈ ((𝐷‘(𝐴‘𝑞))‘(𝐸 / (2↑𝑞)))) ∧ 𝑓:ℕ–1-1-onto→(ℕ × ℕ)) → 𝐸 ∈ ℝ⁺)
77		ovnsubaddlem2.z	. . . . . . . . . 10 ⊢ 𝑍 = (𝑎 ∈ 𝒫 (ℝ ↑_m 𝑋) ↦ {𝑧 ∈ ℝ^* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ)(𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))})
78		coeq2 5872	. . . . . . . . . . . . . . 15 ⊢ (ℎ = 𝑖 → ([,) ∘ ℎ) = ([,) ∘ 𝑖))
79	78	fveq1d 6909	. . . . . . . . . . . . . 14 ⊢ (ℎ = 𝑖 → (([,) ∘ ℎ)‘𝑘) = (([,) ∘ 𝑖)‘𝑘))
80	79	fveq2d 6911	. . . . . . . . . . . . 13 ⊢ (ℎ = 𝑖 → (vol‘(([,) ∘ ℎ)‘𝑘)) = (vol‘(([,) ∘ 𝑖)‘𝑘)))
81	80	prodeq2ad 45548	. . . . . . . . . . . 12 ⊢ (ℎ = 𝑖 → ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑘)) = ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ 𝑖)‘𝑘)))
82	81	cbvmptv 5261	. . . . . . . . . . 11 ⊢ (ℎ ∈ ((ℝ × ℝ) ↑_m 𝑋) ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑘))) = (𝑖 ∈ ((ℝ × ℝ) ↑_m 𝑋) ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ 𝑖)‘𝑘)))
83	34, 82	eqtri 2763	. . . . . . . . . 10 ⊢ 𝐿 = (𝑖 ∈ ((ℝ × ℝ) ↑_m 𝑋) ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ 𝑖)‘𝑘)))
84	64	biimpi 216	. . . . . . . . . . . . 13 ⊢ (∀𝑞 ∈ ℕ (𝑔‘𝑞) ∈ ((𝐷‘(𝐴‘𝑞))‘(𝐸 / (2↑𝑞))) → ∀𝑛 ∈ ℕ (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))
85	84	3ad2ant3 1134	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑔 Fn ℕ ∧ ∀𝑞 ∈ ℕ (𝑔‘𝑞) ∈ ((𝐷‘(𝐴‘𝑞))‘(𝐸 / (2↑𝑞)))) → ∀𝑛 ∈ ℕ (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))
86	85	ad2antrr 726	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑔 Fn ℕ ∧ ∀𝑞 ∈ ℕ (𝑔‘𝑞) ∈ ((𝐷‘(𝐴‘𝑞))‘(𝐸 / (2↑𝑞)))) ∧ 𝑓:ℕ–1-1-onto→(ℕ × ℕ)) ∧ 𝑛 ∈ ℕ) → ∀𝑛 ∈ ℕ (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))
87		simpr 484	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑔 Fn ℕ ∧ ∀𝑞 ∈ ℕ (𝑔‘𝑞) ∈ ((𝐷‘(𝐴‘𝑞))‘(𝐸 / (2↑𝑞)))) ∧ 𝑓:ℕ–1-1-onto→(ℕ × ℕ)) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
88		rspa 3246	. . . . . . . . . . 11 ⊢ ((∀𝑛 ∈ ℕ (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) ∧ 𝑛 ∈ ℕ) → (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))
89	86, 87, 88	syl2anc 584	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑔 Fn ℕ ∧ ∀𝑞 ∈ ℕ (𝑔‘𝑞) ∈ ((𝐷‘(𝐴‘𝑞))‘(𝐸 / (2↑𝑞)))) ∧ 𝑓:ℕ–1-1-onto→(ℕ × ℕ)) ∧ 𝑛 ∈ ℕ) → (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))
90		simpr 484	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔 Fn ℕ ∧ ∀𝑞 ∈ ℕ (𝑔‘𝑞) ∈ ((𝐷‘(𝐴‘𝑞))‘(𝐸 / (2↑𝑞)))) ∧ 𝑓:ℕ–1-1-onto→(ℕ × ℕ)) → 𝑓:ℕ–1-1-onto→(ℕ × ℕ))
91		2fveq3 6912	. . . . . . . . . . . . 13 ⊢ (𝑞 = 𝑚 → (1^st ‘(𝑓‘𝑞)) = (1^st ‘(𝑓‘𝑚)))
92	91	fveq2d 6911	. . . . . . . . . . . 12 ⊢ (𝑞 = 𝑚 → (𝑔‘(1^st ‘(𝑓‘𝑞))) = (𝑔‘(1^st ‘(𝑓‘𝑚))))
93		2fveq3 6912	. . . . . . . . . . . 12 ⊢ (𝑞 = 𝑚 → (2^nd ‘(𝑓‘𝑞)) = (2^nd ‘(𝑓‘𝑚)))
94	92, 93	fveq12d 6914	. . . . . . . . . . 11 ⊢ (𝑞 = 𝑚 → ((𝑔‘(1^st ‘(𝑓‘𝑞)))‘(2^nd ‘(𝑓‘𝑞))) = ((𝑔‘(1^st ‘(𝑓‘𝑚)))‘(2^nd ‘(𝑓‘𝑚))))
95	94	cbvmptv 5261	. . . . . . . . . 10 ⊢ (𝑞 ∈ ℕ ↦ ((𝑔‘(1^st ‘(𝑓‘𝑞)))‘(2^nd ‘(𝑓‘𝑞)))) = (𝑚 ∈ ℕ ↦ ((𝑔‘(1^st ‘(𝑓‘𝑚)))‘(2^nd ‘(𝑓‘𝑚))))
96	70, 72, 74, 76, 77, 33, 83, 35, 89, 90, 95	ovnsubaddlem1 46526	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑔 Fn ℕ ∧ ∀𝑞 ∈ ℕ (𝑔‘𝑞) ∈ ((𝐷‘(𝐴‘𝑞))‘(𝐸 / (2↑𝑞)))) ∧ 𝑓:ℕ–1-1-onto→(ℕ × ℕ)) → ((voln‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐴‘𝑛)) ≤ ((Σ^{^}‘(𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐴‘𝑛)))) +_𝑒 𝐸))
97	56, 57, 67, 68, 96	syl31anc 1372	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑔 Fn ℕ ∧ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛)))) ∧ 𝑓:ℕ–1-1-onto→(ℕ × ℕ)) → ((voln‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐴‘𝑛)) ≤ ((Σ^{^}‘(𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐴‘𝑛)))) +_𝑒 𝐸))
98	97	ex 412	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 Fn ℕ ∧ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛)))) → (𝑓:ℕ–1-1-onto→(ℕ × ℕ) → ((voln‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐴‘𝑛)) ≤ ((Σ^{^}‘(𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐴‘𝑛)))) +_𝑒 𝐸)))
99	98	exlimdv 1931	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 Fn ℕ ∧ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛)))) → (∃𝑓 𝑓:ℕ–1-1-onto→(ℕ × ℕ) → ((voln‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐴‘𝑛)) ≤ ((Σ^{^}‘(𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐴‘𝑛)))) +_𝑒 𝐸)))
100	55, 99	mpi 20	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 Fn ℕ ∧ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛)))) → ((voln‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐴‘𝑛)) ≤ ((Σ^{^}‘(𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐴‘𝑛)))) +_𝑒 𝐸))
101	52, 53, 54, 100	syl3anc 1370	. . . 4 ⊢ ((𝜑 ∧ (𝑔 Fn ℕ ∧ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))) → ((voln‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐴‘𝑛)) ≤ ((Σ^{^}‘(𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐴‘𝑛)))) +_𝑒 𝐸))
102	101	ex 412	. . 3 ⊢ (𝜑 → ((𝑔 Fn ℕ ∧ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛)))) → ((voln‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐴‘𝑛)) ≤ ((Σ^{^}‘(𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐴‘𝑛)))) +_𝑒 𝐸)))
103	102	exlimdv 1931	. 2 ⊢ (𝜑 → (∃𝑔(𝑔 Fn ℕ ∧ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛)))) → ((voln‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐴‘𝑛)) ≤ ((Σ^{^}‘(𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐴‘𝑛)))) +_𝑒 𝐸)))
104	51, 103	mpd 15	1 ⊢ (𝜑 → ((voln‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐴‘𝑛)) ≤ ((Σ^{^}‘(𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐴‘𝑛)))) +_𝑒 𝐸))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∃wex 1776 ∈ wcel 2106 ≠ wne 2938 ∀wral 3059 ∃wrex 3068 {crab 3433 ⊆ wss 3963 ∅c0 4339 𝒫 cpw 4605 ∪ ciun 4996 class class class wbr 5148 ↦ cmpt 5231 × cxp 5687 ∘ ccom 5693 Fn wfn 6558 ⟶wf 6559 –1-1-onto→wf1o 6562 ‘cfv 6563 (class class class)co 7431 1^st c1st 8011 2^nd c2nd 8012 ↑_m cmap 8865 Xcixp 8936 Fincfn 8984 ℝcr 11152 ℝ^*cxr 11292 ≤ cle 11294 / cdiv 11918 ℕcn 12264 2c2 12319 ℕ₀cn0 12524 ℝ⁺crp 13032 +_𝑒 cxad 13150 [,)cico 13386 ↑cexp 14099 ∏cprod 15936 volcvol 25512 Σ^{^}csumge0 46318 voln*covoln 46492

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cc 10473 ax-ac2 10501 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-disj 5116 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-pm 8868 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fi 9449 df-sup 9480 df-inf 9481 df-oi 9548 df-dju 9939 df-card 9977 df-acn 9980 df-ac 10154 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-z 12612 df-uz 12877 df-q 12989 df-rp 13033 df-xneg 13152 df-xadd 13153 df-xmul 13154 df-ioo 13388 df-ico 13390 df-icc 13391 df-fz 13545 df-fzo 13692 df-fl 13829 df-seq 14040 df-exp 14100 df-hash 14367 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-clim 15521 df-rlim 15522 df-sum 15720 df-prod 15937 df-rest 17469 df-topgen 17490 df-psmet 21374 df-xmet 21375 df-met 21376 df-bl 21377 df-mopn 21378 df-top 22916 df-topon 22933 df-bases 22969 df-cmp 23411 df-ovol 25513 df-vol 25514 df-sumge0 46319 df-ovoln 46493

This theorem is referenced by: ovnsubadd 46528

W3C validator