Theorem ovn0lem 44908

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 13358	. . 3 ⊢ (0[,]+∞) ⊆ ℝ*
2		ovn0lem.infm	. . 3 ⊢ (𝜑 → inf(𝑀, ℝ*, < ) ∈ (0[,]+∞))
3	1, 2	sselid 3946	. 2 ⊢ (𝜑 → inf(𝑀, ℝ*, < ) ∈ ℝ*)
4		0xr 11212	. . 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 4043	. . . . 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 3982	. . . 4 ⊢ 𝑀 ⊆ ℝ*
9	8	a1i 11	. . 3 ⊢ (𝜑 → 𝑀 ⊆ ℝ*)
10		1re 11165	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℝ
11		0re 11167	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℝ
12	10, 11	pm3.2i 472	. . . . . . . . . . . . 13 ⊢ (1 ∈ ℝ ∧ 0 ∈ ℝ)
13		opelxp 5675	. . . . . . . . . . . . 13 ⊢ (⟨1, 0⟩ ∈ (ℝ × ℝ) ↔ (1 ∈ ℝ ∧ 0 ∈ ℝ))
14	12, 13	mpbir 230	. . . . . . . . . . . 12 ⊢ ⟨1, 0⟩ ∈ (ℝ × ℝ)
15	14	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑙 ∈ 𝑋) → ⟨1, 0⟩ ∈ (ℝ × ℝ))
16		eqid 2732	. . . . . . . . . . 11 ⊢ (𝑙 ∈ 𝑋 ↦ ⟨1, 0⟩) = (𝑙 ∈ 𝑋 ↦ ⟨1, 0⟩)
17	15, 16	fmptd 7068	. . . . . . . . . 10 ⊢ (𝜑 → (𝑙 ∈ 𝑋 ↦ ⟨1, 0⟩):𝑋⟶(ℝ × ℝ))
18		reex 11152	. . . . . . . . . . . . 13 ⊢ ℝ ∈ V
19	18, 18	xpex 7693	. . . . . . . . . . . 12 ⊢ (ℝ × ℝ) ∈ V
20	19	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → (ℝ × ℝ) ∈ V)
21		ovn0lem.x	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ∈ Fin)
22		elmapg 8786	. . . . . . . . . . 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 585	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑙 ∈ 𝑋 ↦ ⟨1, 0⟩) ∈ ((ℝ × ℝ) ↑m 𝑋) ↔ (𝑙 ∈ 𝑋 ↦ ⟨1, 0⟩):𝑋⟶(ℝ × ℝ)))
24	17, 23	mpbird 257	. . . . . . . . 9 ⊢ (𝜑 → (𝑙 ∈ 𝑋 ↦ ⟨1, 0⟩) ∈ ((ℝ × ℝ) ↑m 𝑋))
25	24	adantr 482	. . . . . . . 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 7068	. . . . . . 7 ⊢ (𝜑 → 𝐼:ℕ⟶((ℝ × ℝ) ↑m 𝑋))
28		ovexd 7398	. . . . . . . 8 ⊢ (𝜑 → ((ℝ × ℝ) ↑m 𝑋) ∈ V)
29		nnex 12169	. . . . . . . . 9 ⊢ ℕ ∈ V
30	29	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ℕ ∈ V)
31		elmapg 8786	. . . . . . . 8 ⊢ ((((ℝ × ℝ) ↑m 𝑋) ∈ V ∧ ℕ ∈ V) → (𝐼 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ↔ 𝐼:ℕ⟶((ℝ × ℝ) ↑m 𝑋)))
32	28, 30, 31	syl2anc 585	. . . . . . 7 ⊢ (𝜑 → (𝐼 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ↔ 𝐼:ℕ⟶((ℝ × ℝ) ↑m 𝑋)))
33	27, 32	mpbird 257	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ))
34		ovn0lem.n0	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑋 ≠ ∅)
35		n0 4312	. . . . . . . . . . . 12 ⊢ (𝑋 ≠ ∅ ↔ ∃𝑙 𝑙 ∈ 𝑋)
36	34, 35	sylib 217	. . . . . . . . . . 11 ⊢ (𝜑 → ∃𝑙 𝑙 ∈ 𝑋)
37	36	adantr 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ∃𝑙 𝑙 ∈ 𝑋)
38		nfv 1918	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑙 ∈ 𝑋)
39		nfcv 2903	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘(vol‘(([,) ∘ (𝐼‘𝑗))‘𝑙))
40	21	ad2antrr 725	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑙 ∈ 𝑋) → 𝑋 ∈ Fin)
41	27	ffvelcdmda 7041	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐼‘𝑗) ∈ ((ℝ × ℝ) ↑m 𝑋))
42		elmapi 8795	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐼‘𝑗) ∈ ((ℝ × ℝ) ↑m 𝑋) → (𝐼‘𝑗):𝑋⟶(ℝ × ℝ))
43	41, 42	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐼‘𝑗):𝑋⟶(ℝ × ℝ))
44	43	adantr 482	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐼‘𝑗):𝑋⟶(ℝ × ℝ))
45		simpr 486	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → 𝑘 ∈ 𝑋)
46	44, 45	fvovco 43517	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (([,) ∘ (𝐼‘𝑗))‘𝑘) = ((1st ‘((𝐼‘𝑗)‘𝑘))[,)(2nd ‘((𝐼‘𝑗)‘𝑘))))
47		simpr 486	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ ℕ)
48	25	elexd 3467	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑙 ∈ 𝑋 ↦ ⟨1, 0⟩) ∈ V)
49	26	fvmpt2 6965	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑗 ∈ ℕ ∧ (𝑙 ∈ 𝑋 ↦ ⟨1, 0⟩) ∈ V) → (𝐼‘𝑗) = (𝑙 ∈ 𝑋 ↦ ⟨1, 0⟩))
50	47, 48, 49	syl2anc 585	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐼‘𝑗) = (𝑙 ∈ 𝑋 ↦ ⟨1, 0⟩))
51	50	adantr 482	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐼‘𝑗) = (𝑙 ∈ 𝑋 ↦ ⟨1, 0⟩))
52		eqidd 2733	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) ∧ 𝑙 = 𝑘) → ⟨1, 0⟩ = ⟨1, 0⟩)
53	14	elexi 3466	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ⟨1, 0⟩ ∈ V
54	53	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ⟨1, 0⟩ ∈ V)
55	51, 52, 45, 54	fvmptd 6961	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐼‘𝑗)‘𝑘) = ⟨1, 0⟩)
56	55	fveq2d 6852	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (1st ‘((𝐼‘𝑗)‘𝑘)) = (1st ‘⟨1, 0⟩))
57	10	elexi 3466	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 1 ∈ V
58	4	elexi 3466	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 0 ∈ V
59	57, 58	op1st 7935	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (1st ‘⟨1, 0⟩) = 1
60	59	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (1st ‘⟨1, 0⟩) = 1)
61	56, 60	eqtrd 2772	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (1st ‘((𝐼‘𝑗)‘𝑘)) = 1)
62	55	fveq2d 6852	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (2nd ‘((𝐼‘𝑗)‘𝑘)) = (2nd ‘⟨1, 0⟩))
63	57, 58	op2nd 7936	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (2nd ‘⟨1, 0⟩) = 0
64	63	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (2nd ‘⟨1, 0⟩) = 0)
65	62, 64	eqtrd 2772	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (2nd ‘((𝐼‘𝑗)‘𝑘)) = 0)
66	61, 65	oveq12d 7381	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((1st ‘((𝐼‘𝑗)‘𝑘))[,)(2nd ‘((𝐼‘𝑗)‘𝑘))) = (1[,)0))
67		0le1 11688	. . . . . . . . . . . . . . . . . . . 20 ⊢ 0 ≤ 1
68		1xr 11224	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 1 ∈ ℝ*
69		ico0 13321	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((1 ∈ ℝ* ∧ 0 ∈ ℝ*) → ((1[,)0) = ∅ ↔ 0 ≤ 1))
70	68, 4, 69	mp2an 691	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((1[,)0) = ∅ ↔ 0 ≤ 1)
71	67, 70	mpbir 230	. . . . . . . . . . . . . . . . . . 19 ⊢ (1[,)0) = ∅
72	71	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (1[,)0) = ∅)
73	46, 66, 72	3eqtrd 2776	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (([,) ∘ (𝐼‘𝑗))‘𝑘) = ∅)
74	73	fveq2d 6852	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)) = (vol‘∅))
75		vol0 44302	. . . . . . . . . . . . . . . . 17 ⊢ (vol‘∅) = 0
76	75	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (vol‘∅) = 0)
77	74, 76	eqtrd 2772	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)) = 0)
78		0cn 11157	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℂ
79	78	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → 0 ∈ ℂ)
80	77, 79	eqeltrd 2833	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)) ∈ ℂ)
81	80	adantlr 714	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑙 ∈ 𝑋) ∧ 𝑘 ∈ 𝑋) → (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)) ∈ ℂ)
82		2fveq3 6853	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑙 → (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑙)))
83		simpr 486	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑙 ∈ 𝑋) → 𝑙 ∈ 𝑋)
84		eleq1w 2816	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑙 → (𝑘 ∈ 𝑋 ↔ 𝑙 ∈ 𝑋))
85	84	anbi2d 630	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑙 → (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) ↔ ((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑙 ∈ 𝑋)))
86	82	eqeq1d 2734	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑙 → ((vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)) = 0 ↔ (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑙)) = 0))
87	85, 86	imbi12d 345	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑙 → ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)) = 0) ↔ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑙 ∈ 𝑋) → (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑙)) = 0)))
88	87, 77	chvarvv 2003	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑙 ∈ 𝑋) → (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑙)) = 0)
89	38, 39, 40, 81, 82, 83, 88	fprod0 43939	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑙 ∈ 𝑋) → ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)) = 0)
90	89	ex 414	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑙 ∈ 𝑋 → ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)) = 0))
91	90	exlimdv 1937	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (∃𝑙 𝑙 ∈ 𝑋 → ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)) = 0))
92	37, 91	mpd 15	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)) = 0)
93	92	mpteq2dva 5211	. . . . . . . 8 ⊢ (𝜑 → (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘))) = (𝑗 ∈ ℕ ↦ 0))
94	93	fveq2d 6852	. . . . . . 7 ⊢ (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)))) = (Σ^‘(𝑗 ∈ ℕ ↦ 0)))
95		nfv 1918	. . . . . . . 8 ⊢ Ⅎ𝑗𝜑
96	95, 30	sge0z 44718	. . . . . . 7 ⊢ (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ 0)) = 0)
97		eqidd 2733	. . . . . . 7 ⊢ (𝜑 → 0 = 0)
98	94, 96, 97	3eqtrrd 2777	. . . . . 6 ⊢ (𝜑 → 0 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)))))
99		fveq1 6847	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝐼 → (𝑖‘𝑗) = (𝐼‘𝑗))
100	99	coeq2d 5824	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝐼 → ([,) ∘ (𝑖‘𝑗)) = ([,) ∘ (𝐼‘𝑗)))
101	100	fveq1d 6850	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝐼 → (([,) ∘ (𝑖‘𝑗))‘𝑘) = (([,) ∘ (𝐼‘𝑗))‘𝑘))
102	101	fveq2d 6852	. . . . . . . . . . 11 ⊢ (𝑖 = 𝐼 → (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)))
103	102	ralrimivw 3144	. . . . . . . . . 10 ⊢ (𝑖 = 𝐼 → ∀𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)))
104	103	prodeq2d 15817	. . . . . . . . 9 ⊢ (𝑖 = 𝐼 → ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)) = ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)))
105	104	mpteq2dv 5213	. . . . . . . 8 ⊢ (𝑖 = 𝐼 → (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))) = (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘))))
106	105	fveq2d 6852	. . . . . . 7 ⊢ (𝑖 = 𝐼 → (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)))))
107	106	rspceeqv 3599	. . . . . 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)))))
108	33, 98, 107	syl2anc 585	. . . . 5 ⊢ (𝜑 → ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)0 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))
109	5, 108	jca 513	. . . 4 ⊢ (𝜑 → (0 ∈ ℝ* ∧ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)0 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))))))
110		eqeq1 2736	. . . . . 6 ⊢ (𝑧 = 0 → (𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))) ↔ 0 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))))))
111	110	rexbidv 3172	. . . . 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))))))
112	111, 6	elrab2 3652	. . . 4 ⊢ (0 ∈ 𝑀 ↔ (0 ∈ ℝ* ∧ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)0 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))))))
113	109, 112	sylibr 233	. . 3 ⊢ (𝜑 → 0 ∈ 𝑀)
114		infxrlb 13264	. . 3 ⊢ ((𝑀 ⊆ ℝ* ∧ 0 ∈ 𝑀) → inf(𝑀, ℝ*, < ) ≤ 0)
115	9, 113, 114	syl2anc 585	. 2 ⊢ (𝜑 → inf(𝑀, ℝ*, < ) ≤ 0)
116		pnfxr 11219	. . . 4 ⊢ +∞ ∈ ℝ*
117	116	a1i 11	. . 3 ⊢ (𝜑 → +∞ ∈ ℝ*)
118		iccgelb 13331	. . 3 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ inf(𝑀, ℝ*, < ) ∈ (0[,]+∞)) → 0 ≤ inf(𝑀, ℝ*, < ))
119	5, 117, 2, 118	syl3anc 1372	. 2 ⊢ (𝜑 → 0 ≤ inf(𝑀, ℝ*, < ))
120	3, 5, 115, 119	xrletrid 13085	1 ⊢ (𝜑 → inf(𝑀, ℝ*, < ) = 0)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542  ∃wex 1782   ∈ wcel 2107   ≠ wne 2940  ∃wrex 3070  {crab 3406  Vcvv 3447   ⊆ wss 3914  ∅c0 4288  ⟨cop 4598   class class class wbr 5111   ↦ cmpt 5194   × cxp 5637   ∘ ccom 5643  ⟶wf 6498  ‘cfv 6502  (class class class)co 7363  1^st c1st 7925  2^nd c2nd 7926   ↑_m cmap 8773  Fincfn 8891  infcinf 9387  ℂcc 11059  ℝcr 11060  0cc0 11061  1c1 11062  +∞cpnf 11196  ℝ^*cxr 11198   < clt 11199   ≤ cle 11200  ℕcn 12163  [,)cico 13277  [,]cicc 13278  ∏cprod 15800  volcvol 24865  Σ^{^}csumge0 44705

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2703  ax-rep 5248  ax-sep 5262  ax-nul 5269  ax-pow 5326  ax-pr 5390  ax-un 7678  ax-inf2 9587  ax-cnex 11117  ax-resscn 11118  ax-1cn 11119  ax-icn 11120  ax-addcl 11121  ax-addrcl 11122  ax-mulcl 11123  ax-mulrcl 11124  ax-mulcom 11125  ax-addass 11126  ax-mulass 11127  ax-distr 11128  ax-i2m1 11129  ax-1ne0 11130  ax-1rid 11131  ax-rnegex 11132  ax-rrecex 11133  ax-cnre 11134  ax-pre-lttri 11135  ax-pre-lttrn 11136  ax-pre-ltadd 11137  ax-pre-mulgt0 11138  ax-pre-sup 11139

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  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 3352  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4289  df-if 4493  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4872  df-int 4914  df-iun 4962  df-br 5112  df-opab 5174  df-mpt 5195  df-tr 5229  df-id 5537  df-eprel 5543  df-po 5551  df-so 5552  df-fr 5594  df-se 5595  df-we 5596  df-xp 5645  df-rel 5646  df-cnv 5647  df-co 5648  df-dm 5649  df-rn 5650  df-res 5651  df-ima 5652  df-pred 6259  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6504  df-fn 6505  df-f 6506  df-f1 6507  df-fo 6508  df-f1o 6509  df-fv 6510  df-isom 6511  df-riota 7319  df-ov 7366  df-oprab 7367  df-mpo 7368  df-of 7623  df-om 7809  df-1st 7927  df-2nd 7928  df-frecs 8218  df-wrecs 8249  df-recs 8323  df-rdg 8362  df-1o 8418  df-2o 8419  df-er 8656  df-map 8775  df-en 8892  df-dom 8893  df-sdom 8894  df-fin 8895  df-sup 9388  df-inf 9389  df-oi 9456  df-dju 9847  df-card 9885  df-pnf 11201  df-mnf 11202  df-xr 11203  df-ltxr 11204  df-le 11205  df-sub 11397  df-neg 11398  df-div 11823  df-nn 12164  df-2 12226  df-3 12227  df-n0 12424  df-z 12510  df-uz 12774  df-q 12884  df-rp 12926  df-xadd 13044  df-ioo 13279  df-ico 13281  df-icc 13282  df-fz 13436  df-fzo 13579  df-fl 13708  df-seq 13918  df-exp 13979  df-hash 14242  df-cj 14997  df-re 14998  df-im 14999  df-sqrt 15133  df-abs 15134  df-clim 15383  df-sum 15584  df-prod 15801  df-xmet 20827  df-met 20828  df-ovol 24866  df-vol 24867  df-sumge0 44706

This theorem is referenced by:  ovn0  44909