Theorem ovn0lem 39455

Description: For any finite dimension, the Lebesgue outer measure of the empty set is zero. This is step (a)(ii) of the proof of Proposition 115D (a) of [Fremlin1] p. 30. (Contributed by Glauco Siliprandi, 11-Oct-2020.)

Hypotheses

Ref	Expression
ovn0lem.x	⊢ (𝜑 → 𝑋 ∈ Fin)
ovn0lem.n0	⊢ (𝜑 → 𝑋 ≠ ∅)
ovn0lem.m	⊢ 𝑀 = {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))))}
ovn0lem.infm	⊢ (𝜑 → inf(𝑀, ℝ*, < ) ∈ (0[,]+∞))
ovn0lem.i	⊢ 𝐼 = (𝑗 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ ⟨1, 0⟩))

Assertion

Ref	Expression
ovn0lem	⊢ (𝜑 → inf(𝑀, ℝ*, < ) = 0)

Distinct variable groups:   𝑖,𝐼,𝑗,𝑘   𝐼,𝑙,𝑗,𝑘   𝑖,𝑋,𝑗,𝑘,𝑧   𝑋,𝑙   𝜑,𝑗,𝑘,𝑙

Allowed substitution hints:   𝜑(𝑧,𝑖)   𝐼(𝑧)   𝑀(𝑧,𝑖,𝑗,𝑘,𝑙)

Proof of Theorem ovn0lem

Step	Hyp	Ref	Expression
1		iccssxr 12127	. . 3 ⊢ (0[,]+∞) ⊆ ℝ*
2		ovn0lem.infm	. . 3 ⊢ (𝜑 → inf(𝑀, ℝ*, < ) ∈ (0[,]+∞))
3	1, 2	sseldi 3566	. 2 ⊢ (𝜑 → inf(𝑀, ℝ*, < ) ∈ ℝ*)
4		0xr 9965	. . 3 ⊢ 0 ∈ ℝ*
5	4	a1i 11	. 2 ⊢ (𝜑 → 0 ∈ ℝ*)
6		ovn0lem.m	. . . . 5 ⊢ 𝑀 = {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))))}
7		ssrab2 3650	. . . . 5 ⊢ {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))))} ⊆ ℝ*
8	6, 7	eqsstri 3598	. . . 4 ⊢ 𝑀 ⊆ ℝ*
9	8	a1i 11	. . 3 ⊢ (𝜑 → 𝑀 ⊆ ℝ*)
10		1re 9918	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℝ
11		0re 9919	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℝ
12	10, 11	pm3.2i 470	. . . . . . . . . . . . 13 ⊢ (1 ∈ ℝ ∧ 0 ∈ ℝ)
13		opelxp 5070	. . . . . . . . . . . . 13 ⊢ (⟨1, 0⟩ ∈ (ℝ × ℝ) ↔ (1 ∈ ℝ ∧ 0 ∈ ℝ))
14	12, 13	mpbir 220	. . . . . . . . . . . 12 ⊢ ⟨1, 0⟩ ∈ (ℝ × ℝ)
15	14	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑙 ∈ 𝑋) → ⟨1, 0⟩ ∈ (ℝ × ℝ))
16		eqid 2610	. . . . . . . . . . 11 ⊢ (𝑙 ∈ 𝑋 ↦ ⟨1, 0⟩) = (𝑙 ∈ 𝑋 ↦ ⟨1, 0⟩)
17	15, 16	fmptd 6292	. . . . . . . . . 10 ⊢ (𝜑 → (𝑙 ∈ 𝑋 ↦ ⟨1, 0⟩):𝑋⟶(ℝ × ℝ))
18		reex 9906	. . . . . . . . . . . . 13 ⊢ ℝ ∈ V
19	18, 18	xpex 6860	. . . . . . . . . . . 12 ⊢ (ℝ × ℝ) ∈ V
20	19	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → (ℝ × ℝ) ∈ V)
21		ovn0lem.x	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ∈ Fin)
22		elmapg 7757	. . . . . . . . . . 11 ⊢ (((ℝ × ℝ) ∈ V ∧ 𝑋 ∈ Fin) → ((𝑙 ∈ 𝑋 ↦ ⟨1, 0⟩) ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↔ (𝑙 ∈ 𝑋 ↦ ⟨1, 0⟩):𝑋⟶(ℝ × ℝ)))
23	20, 21, 22	syl2anc 691	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑙 ∈ 𝑋 ↦ ⟨1, 0⟩) ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↔ (𝑙 ∈ 𝑋 ↦ ⟨1, 0⟩):𝑋⟶(ℝ × ℝ)))
24	17, 23	mpbird 246	. . . . . . . . 9 ⊢ (𝜑 → (𝑙 ∈ 𝑋 ↦ ⟨1, 0⟩) ∈ ((ℝ × ℝ) ↑𝑚 𝑋))
25	24	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑙 ∈ 𝑋 ↦ ⟨1, 0⟩) ∈ ((ℝ × ℝ) ↑𝑚 𝑋))
26		ovn0lem.i	. . . . . . . 8 ⊢ 𝐼 = (𝑗 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ ⟨1, 0⟩))
27	25, 26	fmptd 6292	. . . . . . 7 ⊢ (𝜑 → 𝐼:ℕ⟶((ℝ × ℝ) ↑𝑚 𝑋))
28		ovex 6577	. . . . . . . . 9 ⊢ ((ℝ × ℝ) ↑𝑚 𝑋) ∈ V
29	28	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ((ℝ × ℝ) ↑𝑚 𝑋) ∈ V)
30		nnex 10903	. . . . . . . . 9 ⊢ ℕ ∈ V
31	30	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ℕ ∈ V)
32		elmapg 7757	. . . . . . . 8 ⊢ ((((ℝ × ℝ) ↑𝑚 𝑋) ∈ V ∧ ℕ ∈ V) → (𝐼 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ↔ 𝐼:ℕ⟶((ℝ × ℝ) ↑𝑚 𝑋)))
33	29, 31, 32	syl2anc 691	. . . . . . 7 ⊢ (𝜑 → (𝐼 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ↔ 𝐼:ℕ⟶((ℝ × ℝ) ↑𝑚 𝑋)))
34	27, 33	mpbird 246	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ))
35		ovn0lem.n0	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑋 ≠ ∅)
36		n0 3890	. . . . . . . . . . . 12 ⊢ (𝑋 ≠ ∅ ↔ ∃𝑙 𝑙 ∈ 𝑋)
37	35, 36	sylib 207	. . . . . . . . . . 11 ⊢ (𝜑 → ∃𝑙 𝑙 ∈ 𝑋)
38	37	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ∃𝑙 𝑙 ∈ 𝑋)
39		nfv 1830	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑙 ∈ 𝑋)
40		nfcv 2751	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘(vol‘(([,) ∘ (𝐼‘𝑗))‘𝑙))
41	21	ad2antrr 758	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑙 ∈ 𝑋) → 𝑋 ∈ Fin)
42	27	ffvelrnda 6267	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐼‘𝑗) ∈ ((ℝ × ℝ) ↑𝑚 𝑋))
43		elmapi 7765	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐼‘𝑗) ∈ ((ℝ × ℝ) ↑𝑚 𝑋) → (𝐼‘𝑗):𝑋⟶(ℝ × ℝ))
44	42, 43	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐼‘𝑗):𝑋⟶(ℝ × ℝ))
45	44	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐼‘𝑗):𝑋⟶(ℝ × ℝ))
46		simpr 476	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → 𝑘 ∈ 𝑋)
47	45, 46	fvovco 38376	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (([,) ∘ (𝐼‘𝑗))‘𝑘) = ((1st ‘((𝐼‘𝑗)‘𝑘))[,)(2nd ‘((𝐼‘𝑗)‘𝑘))))
48		simpr 476	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ ℕ)
49	25	elexd 3187	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑙 ∈ 𝑋 ↦ ⟨1, 0⟩) ∈ V)
50	26	fvmpt2 6200	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑗 ∈ ℕ ∧ (𝑙 ∈ 𝑋 ↦ ⟨1, 0⟩) ∈ V) → (𝐼‘𝑗) = (𝑙 ∈ 𝑋 ↦ ⟨1, 0⟩))
51	48, 49, 50	syl2anc 691	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐼‘𝑗) = (𝑙 ∈ 𝑋 ↦ ⟨1, 0⟩))
52	51	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐼‘𝑗) = (𝑙 ∈ 𝑋 ↦ ⟨1, 0⟩))
53		eqidd 2611	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) ∧ 𝑙 = 𝑘) → ⟨1, 0⟩ = ⟨1, 0⟩)
54	14	elexi 3186	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ⟨1, 0⟩ ∈ V
55	54	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ⟨1, 0⟩ ∈ V)
56	52, 53, 46, 55	fvmptd 6197	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐼‘𝑗)‘𝑘) = ⟨1, 0⟩)
57	56	fveq2d 6107	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (1st ‘((𝐼‘𝑗)‘𝑘)) = (1st ‘⟨1, 0⟩))
58	10	elexi 3186	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 1 ∈ V
59	4	elexi 3186	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 0 ∈ V
60	58, 59	op1st 7067	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (1st ‘⟨1, 0⟩) = 1
61	60	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (1st ‘⟨1, 0⟩) = 1)
62	57, 61	eqtrd 2644	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (1st ‘((𝐼‘𝑗)‘𝑘)) = 1)
63	56	fveq2d 6107	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (2nd ‘((𝐼‘𝑗)‘𝑘)) = (2nd ‘⟨1, 0⟩))
64	58, 59	op2nd 7068	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (2nd ‘⟨1, 0⟩) = 0
65	64	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (2nd ‘⟨1, 0⟩) = 0)
66	63, 65	eqtrd 2644	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (2nd ‘((𝐼‘𝑗)‘𝑘)) = 0)
67	62, 66	oveq12d 6567	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((1st ‘((𝐼‘𝑗)‘𝑘))[,)(2nd ‘((𝐼‘𝑗)‘𝑘))) = (1[,)0))
68		0le1 10430	. . . . . . . . . . . . . . . . . . . 20 ⊢ 0 ≤ 1
69	10	rexri 9976	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 1 ∈ ℝ*
70		ico0 12092	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((1 ∈ ℝ* ∧ 0 ∈ ℝ*) → ((1[,)0) = ∅ ↔ 0 ≤ 1))
71	69, 4, 70	mp2an 704	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((1[,)0) = ∅ ↔ 0 ≤ 1)
72	68, 71	mpbir 220	. . . . . . . . . . . . . . . . . . 19 ⊢ (1[,)0) = ∅
73	72	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (1[,)0) = ∅)
74	47, 67, 73	3eqtrd 2648	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (([,) ∘ (𝐼‘𝑗))‘𝑘) = ∅)
75	74	fveq2d 6107	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)) = (vol‘∅))
76		vol0 38851	. . . . . . . . . . . . . . . . 17 ⊢ (vol‘∅) = 0
77	76	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (vol‘∅) = 0)
78	75, 77	eqtrd 2644	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)) = 0)
79		0cn 9911	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℂ
80	79	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → 0 ∈ ℂ)
81	78, 80	eqeltrd 2688	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)) ∈ ℂ)
82	81	adantlr 747	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑙 ∈ 𝑋) ∧ 𝑘 ∈ 𝑋) → (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)) ∈ ℂ)
83		fveq2 6103	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑙 → (([,) ∘ (𝐼‘𝑗))‘𝑘) = (([,) ∘ (𝐼‘𝑗))‘𝑙))
84	83	fveq2d 6107	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑙 → (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑙)))
85		simpr 476	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑙 ∈ 𝑋) → 𝑙 ∈ 𝑋)
86		eleq1 2676	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑙 → (𝑘 ∈ 𝑋 ↔ 𝑙 ∈ 𝑋))
87	86	anbi2d 736	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑙 → (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) ↔ ((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑙 ∈ 𝑋)))
88	84	eqeq1d 2612	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑙 → ((vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)) = 0 ↔ (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑙)) = 0))
89	87, 88	imbi12d 333	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑙 → ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)) = 0) ↔ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑙 ∈ 𝑋) → (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑙)) = 0)))
90	89, 78	chvarv 2251	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑙 ∈ 𝑋) → (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑙)) = 0)
91	39, 40, 41, 82, 84, 85, 90	fprod0 38663	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑙 ∈ 𝑋) → ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)) = 0)
92	91	ex 449	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑙 ∈ 𝑋 → ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)) = 0))
93	92	exlimdv 1848	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (∃𝑙 𝑙 ∈ 𝑋 → ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)) = 0))
94	38, 93	mpd 15	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)) = 0)
95	94	mpteq2dva 4672	. . . . . . . 8 ⊢ (𝜑 → (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘))) = (𝑗 ∈ ℕ ↦ 0))
96	95	fveq2d 6107	. . . . . . 7 ⊢ (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)))) = (Σ^‘(𝑗 ∈ ℕ ↦ 0)))
97		nfv 1830	. . . . . . . 8 ⊢ Ⅎ𝑗𝜑
98	97, 31	sge0z 39268	. . . . . . 7 ⊢ (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ 0)) = 0)
99		eqidd 2611	. . . . . . 7 ⊢ (𝜑 → 0 = 0)
100	96, 98, 99	3eqtrrd 2649	. . . . . 6 ⊢ (𝜑 → 0 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)))))
101		fveq1 6102	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 𝐼 → (𝑖‘𝑗) = (𝐼‘𝑗))
102	101	coeq2d 5206	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝐼 → ([,) ∘ (𝑖‘𝑗)) = ([,) ∘ (𝐼‘𝑗)))
103	102	fveq1d 6105	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝐼 → (([,) ∘ (𝑖‘𝑗))‘𝑘) = (([,) ∘ (𝐼‘𝑗))‘𝑘))
104	103	fveq2d 6107	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝐼 → (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)))
105	104	ralrimivw 2950	. . . . . . . . . . 11 ⊢ (𝑖 = 𝐼 → ∀𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)))
106	105	prodeq2d 14491	. . . . . . . . . 10 ⊢ (𝑖 = 𝐼 → ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)) = ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)))
107	106	mpteq2dv 4673	. . . . . . . . 9 ⊢ (𝑖 = 𝐼 → (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))) = (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘))))
108	107	fveq2d 6107	. . . . . . . 8 ⊢ (𝑖 = 𝐼 → (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)))))
109	108	eqeq2d 2620	. . . . . . 7 ⊢ (𝑖 = 𝐼 → (0 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))) ↔ 0 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘))))))
110	109	rspcev 3282	. . . . . 6 ⊢ ((𝐼 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 0 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘))))) → ∃𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)0 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))
111	34, 100, 110	syl2anc 691	. . . . 5 ⊢ (𝜑 → ∃𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)0 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))
112	5, 111	jca 553	. . . 4 ⊢ (𝜑 → (0 ∈ ℝ* ∧ ∃𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)0 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))))))
113		eqeq1 2614	. . . . . 6 ⊢ (𝑧 = 0 → (𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))) ↔ 0 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))))))
114	113	rexbidv 3034	. . . . 5 ⊢ (𝑧 = 0 → (∃𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))) ↔ ∃𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)0 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))))))
115	114, 6	elrab2 3333	. . . 4 ⊢ (0 ∈ 𝑀 ↔ (0 ∈ ℝ* ∧ ∃𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)0 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))))))
116	112, 115	sylibr 223	. . 3 ⊢ (𝜑 → 0 ∈ 𝑀)
117		infxrlb 12036	. . 3 ⊢ ((𝑀 ⊆ ℝ* ∧ 0 ∈ 𝑀) → inf(𝑀, ℝ*, < ) ≤ 0)
118	9, 116, 117	syl2anc 691	. 2 ⊢ (𝜑 → inf(𝑀, ℝ*, < ) ≤ 0)
119		pnfxr 9971	. . . 4 ⊢ +∞ ∈ ℝ*
120	119	a1i 11	. . 3 ⊢ (𝜑 → +∞ ∈ ℝ*)
121		iccgelb 12101	. . 3 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ inf(𝑀, ℝ*, < ) ∈ (0[,]+∞)) → 0 ≤ inf(𝑀, ℝ*, < ))
122	5, 120, 2, 121	syl3anc 1318	. 2 ⊢ (𝜑 → 0 ≤ inf(𝑀, ℝ*, < ))
123	3, 5, 118, 122	xrletrid 11862	1 ⊢ (𝜑 → inf(𝑀, ℝ*, < ) = 0)

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977   ≠ wne 2780  ∃wrex 2897  {crab 2900  Vcvv 3173   ⊆ wss 3540  ∅c0 3874  ⟨cop 4131   class class class wbr 4583   ↦ cmpt 4643   × cxp 5036   ∘ ccom 5042  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  1^st c1st 7057  2^nd c2nd 7058   ↑_𝑚 cmap 7744  Fincfn 7841  infcinf 8230  ℂcc 9813  ℝcr 9814  0cc0 9815  1c1 9816  +∞cpnf 9950  ℝ^*cxr 9952   < clt 9953   ≤ cle 9954  ℕcn 10897  [,)cico 12048  [,]cicc 12049  ∏cprod 14474  volcvol 23039  Σ^{^}csumge0 39255

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-q 11665  df-rp 11709  df-xadd 11823  df-ioo 12050  df-ico 12052  df-icc 12053  df-fz 12198  df-fzo 12335  df-fl 12455  df-seq 12664  df-exp 12723  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-sum 14265  df-prod 14475  df-xmet 19560  df-met 19561  df-ovol 23040  df-vol 23041  df-sumge0 39256

This theorem is referenced by:  ovn0  39456