Theorem ovn0lem 43993

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 13091	. . 3 ⊢ (0[,]+∞) ⊆ ℝ*
2		ovn0lem.infm	. . 3 ⊢ (𝜑 → inf(𝑀, ℝ*, < ) ∈ (0[,]+∞))
3	1, 2	sselid 3915	. 2 ⊢ (𝜑 → inf(𝑀, ℝ*, < ) ∈ ℝ*)
4		0xr 10953	. . 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 4009	. . . . 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 3951	. . . 4 ⊢ 𝑀 ⊆ ℝ*
9	8	a1i 11	. . 3 ⊢ (𝜑 → 𝑀 ⊆ ℝ*)
10		1re 10906	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℝ
11		0re 10908	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℝ
12	10, 11	pm3.2i 470	. . . . . . . . . . . . 13 ⊢ (1 ∈ ℝ ∧ 0 ∈ ℝ)
13		opelxp 5616	. . . . . . . . . . . . 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 2738	. . . . . . . . . . 11 ⊢ (𝑙 ∈ 𝑋 ↦ ⟨1, 0⟩) = (𝑙 ∈ 𝑋 ↦ ⟨1, 0⟩)
17	15, 16	fmptd 6970	. . . . . . . . . 10 ⊢ (𝜑 → (𝑙 ∈ 𝑋 ↦ ⟨1, 0⟩):𝑋⟶(ℝ × ℝ))
18		reex 10893	. . . . . . . . . . . . 13 ⊢ ℝ ∈ V
19	18, 18	xpex 7581	. . . . . . . . . . . 12 ⊢ (ℝ × ℝ) ∈ V
20	19	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → (ℝ × ℝ) ∈ V)
21		ovn0lem.x	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ∈ Fin)
22		elmapg 8586	. . . . . . . . . . 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 583	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑙 ∈ 𝑋 ↦ ⟨1, 0⟩) ∈ ((ℝ × ℝ) ↑m 𝑋) ↔ (𝑙 ∈ 𝑋 ↦ ⟨1, 0⟩):𝑋⟶(ℝ × ℝ)))
24	17, 23	mpbird 256	. . . . . . . . 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 6970	. . . . . . 7 ⊢ (𝜑 → 𝐼:ℕ⟶((ℝ × ℝ) ↑m 𝑋))
28		ovexd 7290	. . . . . . . 8 ⊢ (𝜑 → ((ℝ × ℝ) ↑m 𝑋) ∈ V)
29		nnex 11909	. . . . . . . . 9 ⊢ ℕ ∈ V
30	29	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ℕ ∈ V)
31		elmapg 8586	. . . . . . . 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 583	. . . . . . 7 ⊢ (𝜑 → (𝐼 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ↔ 𝐼:ℕ⟶((ℝ × ℝ) ↑m 𝑋)))
33	27, 32	mpbird 256	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ))
34		ovn0lem.n0	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑋 ≠ ∅)
35		n0 4277	. . . . . . . . . . . 12 ⊢ (𝑋 ≠ ∅ ↔ ∃𝑙 𝑙 ∈ 𝑋)
36	34, 35	sylib 217	. . . . . . . . . . 11 ⊢ (𝜑 → ∃𝑙 𝑙 ∈ 𝑋)
37	36	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ∃𝑙 𝑙 ∈ 𝑋)
38		nfv 1918	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑙 ∈ 𝑋)
39		nfcv 2906	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘(vol‘(([,) ∘ (𝐼‘𝑗))‘𝑙))
40	21	ad2antrr 722	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑙 ∈ 𝑋) → 𝑋 ∈ Fin)
41	27	ffvelrnda 6943	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐼‘𝑗) ∈ ((ℝ × ℝ) ↑m 𝑋))
42		elmapi 8595	. . . . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐼‘𝑗):𝑋⟶(ℝ × ℝ))
45		simpr 484	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → 𝑘 ∈ 𝑋)
46	44, 45	fvovco 42621	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (([,) ∘ (𝐼‘𝑗))‘𝑘) = ((1st ‘((𝐼‘𝑗)‘𝑘))[,)(2nd ‘((𝐼‘𝑗)‘𝑘))))
47		simpr 484	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ ℕ)
48	25	elexd 3442	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑙 ∈ 𝑋 ↦ ⟨1, 0⟩) ∈ V)
49	26	fvmpt2 6868	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑗 ∈ ℕ ∧ (𝑙 ∈ 𝑋 ↦ ⟨1, 0⟩) ∈ V) → (𝐼‘𝑗) = (𝑙 ∈ 𝑋 ↦ ⟨1, 0⟩))
50	47, 48, 49	syl2anc 583	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐼‘𝑗) = (𝑙 ∈ 𝑋 ↦ ⟨1, 0⟩))
51	50	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐼‘𝑗) = (𝑙 ∈ 𝑋 ↦ ⟨1, 0⟩))
52		eqidd 2739	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) ∧ 𝑙 = 𝑘) → ⟨1, 0⟩ = ⟨1, 0⟩)
53	14	elexi 3441	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ⟨1, 0⟩ ∈ V
54	53	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ⟨1, 0⟩ ∈ V)
55	51, 52, 45, 54	fvmptd 6864	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐼‘𝑗)‘𝑘) = ⟨1, 0⟩)
56	55	fveq2d 6760	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (1st ‘((𝐼‘𝑗)‘𝑘)) = (1st ‘⟨1, 0⟩))
57	10	elexi 3441	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 1 ∈ V
58	4	elexi 3441	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 0 ∈ V
59	57, 58	op1st 7812	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (1st ‘⟨1, 0⟩) = 1
60	59	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (1st ‘⟨1, 0⟩) = 1)
61	56, 60	eqtrd 2778	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (1st ‘((𝐼‘𝑗)‘𝑘)) = 1)
62	55	fveq2d 6760	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (2nd ‘((𝐼‘𝑗)‘𝑘)) = (2nd ‘⟨1, 0⟩))
63	57, 58	op2nd 7813	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (2nd ‘⟨1, 0⟩) = 0
64	63	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (2nd ‘⟨1, 0⟩) = 0)
65	62, 64	eqtrd 2778	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (2nd ‘((𝐼‘𝑗)‘𝑘)) = 0)
66	61, 65	oveq12d 7273	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((1st ‘((𝐼‘𝑗)‘𝑘))[,)(2nd ‘((𝐼‘𝑗)‘𝑘))) = (1[,)0))
67		0le1 11428	. . . . . . . . . . . . . . . . . . . 20 ⊢ 0 ≤ 1
68		1xr 10965	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 1 ∈ ℝ*
69		ico0 13054	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((1 ∈ ℝ* ∧ 0 ∈ ℝ*) → ((1[,)0) = ∅ ↔ 0 ≤ 1))
70	68, 4, 69	mp2an 688	. . . . . . . . . . . . . . . . . . . 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 2782	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (([,) ∘ (𝐼‘𝑗))‘𝑘) = ∅)
74	73	fveq2d 6760	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)) = (vol‘∅))
75		vol0 43390	. . . . . . . . . . . . . . . . 17 ⊢ (vol‘∅) = 0
76	75	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (vol‘∅) = 0)
77	74, 76	eqtrd 2778	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)) = 0)
78		0cn 10898	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℂ
79	78	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → 0 ∈ ℂ)
80	77, 79	eqeltrd 2839	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)) ∈ ℂ)
81	80	adantlr 711	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑙 ∈ 𝑋) ∧ 𝑘 ∈ 𝑋) → (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)) ∈ ℂ)
82		2fveq3 6761	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑙 → (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑙)))
83		simpr 484	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑙 ∈ 𝑋) → 𝑙 ∈ 𝑋)
84		eleq1w 2821	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑙 → (𝑘 ∈ 𝑋 ↔ 𝑙 ∈ 𝑋))
85	84	anbi2d 628	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑙 → (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) ↔ ((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑙 ∈ 𝑋)))
86	82	eqeq1d 2740	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑙 → ((vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)) = 0 ↔ (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑙)) = 0))
87	85, 86	imbi12d 344	. . . . . . . . . . . . . 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 43027	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑙 ∈ 𝑋) → ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)) = 0)
90	89	ex 412	. . . . . . . . . . 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 5170	. . . . . . . 8 ⊢ (𝜑 → (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘))) = (𝑗 ∈ ℕ ↦ 0))
94	93	fveq2d 6760	. . . . . . 7 ⊢ (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)))) = (Σ^‘(𝑗 ∈ ℕ ↦ 0)))
95		nfv 1918	. . . . . . . 8 ⊢ Ⅎ𝑗𝜑
96	95, 30	sge0z 43803	. . . . . . 7 ⊢ (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ 0)) = 0)
97		eqidd 2739	. . . . . . 7 ⊢ (𝜑 → 0 = 0)
98	94, 96, 97	3eqtrrd 2783	. . . . . 6 ⊢ (𝜑 → 0 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)))))
99		fveq1 6755	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝐼 → (𝑖‘𝑗) = (𝐼‘𝑗))
100	99	coeq2d 5760	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝐼 → ([,) ∘ (𝑖‘𝑗)) = ([,) ∘ (𝐼‘𝑗)))
101	100	fveq1d 6758	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝐼 → (([,) ∘ (𝑖‘𝑗))‘𝑘) = (([,) ∘ (𝐼‘𝑗))‘𝑘))
102	101	fveq2d 6760	. . . . . . . . . . 11 ⊢ (𝑖 = 𝐼 → (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)))
103	102	ralrimivw 3108	. . . . . . . . . 10 ⊢ (𝑖 = 𝐼 → ∀𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)))
104	103	prodeq2d 15560	. . . . . . . . 9 ⊢ (𝑖 = 𝐼 → ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)) = ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)))
105	104	mpteq2dv 5172	. . . . . . . 8 ⊢ (𝑖 = 𝐼 → (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))) = (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘))))
106	105	fveq2d 6760	. . . . . . 7 ⊢ (𝑖 = 𝐼 → (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)))))
107	106	rspceeqv 3567	. . . . . 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 583	. . . . 5 ⊢ (𝜑 → ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)0 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))
109	5, 108	jca 511	. . . 4 ⊢ (𝜑 → (0 ∈ ℝ* ∧ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)0 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))))))
110		eqeq1 2742	. . . . . 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 3225	. . . . 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 3620	. . . 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 12997	. . 3 ⊢ ((𝑀 ⊆ ℝ* ∧ 0 ∈ 𝑀) → inf(𝑀, ℝ*, < ) ≤ 0)
115	9, 113, 114	syl2anc 583	. 2 ⊢ (𝜑 → inf(𝑀, ℝ*, < ) ≤ 0)
116		pnfxr 10960	. . . 4 ⊢ +∞ ∈ ℝ*
117	116	a1i 11	. . 3 ⊢ (𝜑 → +∞ ∈ ℝ*)
118		iccgelb 13064	. . 3 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ inf(𝑀, ℝ*, < ) ∈ (0[,]+∞)) → 0 ≤ inf(𝑀, ℝ*, < ))
119	5, 117, 2, 118	syl3anc 1369	. 2 ⊢ (𝜑 → 0 ≤ inf(𝑀, ℝ*, < ))
120	3, 5, 115, 119	xrletrid 12818	1 ⊢ (𝜑 → inf(𝑀, ℝ*, < ) = 0)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539  ∃wex 1783   ∈ wcel 2108   ≠ wne 2942  ∃wrex 3064  {crab 3067  Vcvv 3422   ⊆ wss 3883  ∅c0 4253  ⟨cop 4564   class class class wbr 5070   ↦ cmpt 5153   × cxp 5578   ∘ ccom 5584  ⟶wf 6414  ‘cfv 6418  (class class class)co 7255  1^st c1st 7802  2^nd c2nd 7803   ↑_m cmap 8573  Fincfn 8691  infcinf 9130  ℂcc 10800  ℝcr 10801  0cc0 10802  1c1 10803  +∞cpnf 10937  ℝ^*cxr 10939   < clt 10940   ≤ cle 10941  ℕcn 11903  [,)cico 13010  [,]cicc 13011  ∏cprod 15543  volcvol 24532  Σ^{^}csumge0 43790

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-inf2 9329  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879  ax-pre-sup 10880

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-se 5536  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-isom 6427  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-of 7511  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-2o 8268  df-er 8456  df-map 8575  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-sup 9131  df-inf 9132  df-oi 9199  df-dju 9590  df-card 9628  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-div 11563  df-nn 11904  df-2 11966  df-3 11967  df-n0 12164  df-z 12250  df-uz 12512  df-q 12618  df-rp 12660  df-xadd 12778  df-ioo 13012  df-ico 13014  df-icc 13015  df-fz 13169  df-fzo 13312  df-fl 13440  df-seq 13650  df-exp 13711  df-hash 13973  df-cj 14738  df-re 14739  df-im 14740  df-sqrt 14874  df-abs 14875  df-clim 15125  df-sum 15326  df-prod 15544  df-xmet 20503  df-met 20504  df-ovol 24533  df-vol 24534  df-sumge0 43791

This theorem is referenced by:  ovn0  43994