Theorem ovn0lem 47339

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 13475	. . 3 ⊢ (0[,]+∞) ⊆ ℝ*
2		ovn0lem.infm	. . 3 ⊢ (𝜑 → inf(𝑀, ℝ*, < ) ∈ (0[,]+∞))
3	1, 2	sselid 3936	. 2 ⊢ (𝜑 → inf(𝑀, ℝ*, < ) ∈ ℝ*)
4		0xr 11273	. . 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 4035	. . . . 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 3984	. . . 4 ⊢ 𝑀 ⊆ ℝ*
9	8	a1i 11	. . 3 ⊢ (𝜑 → 𝑀 ⊆ ℝ*)
10		1re 11225	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℝ
11		0re 11227	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℝ
12	10, 11	pm3.2i 476	. . . . . . . . . . . . 13 ⊢ (1 ∈ ℝ ∧ 0 ∈ ℝ)
13		opelxp 5699	. . . . . . . . . . . . 13 ⊢ (⟨1, 0⟩ ∈ (ℝ × ℝ) ↔ (1 ∈ ℝ ∧ 0 ∈ ℝ))
14	12, 13	mpbir 234	. . . . . . . . . . . 12 ⊢ ⟨1, 0⟩ ∈ (ℝ × ℝ)
15	14	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑙 ∈ 𝑋) → ⟨1, 0⟩ ∈ (ℝ × ℝ))
16		eqid 2765	. . . . . . . . . . 11 ⊢ (𝑙 ∈ 𝑋 ↦ ⟨1, 0⟩) = (𝑙 ∈ 𝑋 ↦ ⟨1, 0⟩)
17	15, 16	fmptd 7113	. . . . . . . . . 10 ⊢ (𝜑 → (𝑙 ∈ 𝑋 ↦ ⟨1, 0⟩):𝑋⟶(ℝ × ℝ))
18		reex 11208	. . . . . . . . . . . . 13 ⊢ ℝ ∈ V
19	18, 18	xpex 7758	. . . . . . . . . . . 12 ⊢ (ℝ × ℝ) ∈ V
20	19	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → (ℝ × ℝ) ∈ V)
21		ovn0lem.x	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ∈ Fin)
22		elmapg 8842	. . . . . . . . . . 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 596	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑙 ∈ 𝑋 ↦ ⟨1, 0⟩) ∈ ((ℝ × ℝ) ↑m 𝑋) ↔ (𝑙 ∈ 𝑋 ↦ ⟨1, 0⟩):𝑋⟶(ℝ × ℝ)))
24	17, 23	mpbird 260	. . . . . . . . 9 ⊢ (𝜑 → (𝑙 ∈ 𝑋 ↦ ⟨1, 0⟩) ∈ ((ℝ × ℝ) ↑m 𝑋))
25	24	adantr 486	. . . . . . . 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 7113	. . . . . . 7 ⊢ (𝜑 → 𝐼:ℕ⟶((ℝ × ℝ) ↑m 𝑋))
28		ovexd 7454	. . . . . . . 8 ⊢ (𝜑 → ((ℝ × ℝ) ↑m 𝑋) ∈ V)
29		nnex 12256	. . . . . . . . 9 ⊢ ℕ ∈ V
30	29	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ℕ ∈ V)
31		elmapg 8842	. . . . . . . 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 596	. . . . . . 7 ⊢ (𝜑 → (𝐼 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ↔ 𝐼:ℕ⟶((ℝ × ℝ) ↑m 𝑋)))
33	27, 32	mpbird 260	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ))
34		ovn0lem.n0	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑋 ≠ ∅)
35		n0 4307	. . . . . . . . . . . 12 ⊢ (𝑋 ≠ ∅ ↔ ∃𝑙 𝑙 ∈ 𝑋)
36	34, 35	sylib 221	. . . . . . . . . . 11 ⊢ (𝜑 → ∃𝑙 𝑙 ∈ 𝑋)
37	36	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ∃𝑙 𝑙 ∈ 𝑋)
38		nfv 1947	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑙 ∈ 𝑋)
39		nfcv 2927	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘(vol‘(([,) ∘ (𝐼‘𝑗))‘𝑙))
40	21	ad2antrr 739	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑙 ∈ 𝑋) → 𝑋 ∈ Fin)
41	27	ffvelcdmda 7083	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐼‘𝑗) ∈ ((ℝ × ℝ) ↑m 𝑋))
42		elmapi 8852	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐼‘𝑗) ∈ ((ℝ × ℝ) ↑m 𝑋) → (𝐼‘𝑗):𝑋⟶(ℝ × ℝ))
43	41, 42	syl 18	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐼‘𝑗):𝑋⟶(ℝ × ℝ))
44	43	adantr 486	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐼‘𝑗):𝑋⟶(ℝ × ℝ))
45		simpr 490	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → 𝑘 ∈ 𝑋)
46	44, 45	fvovco 45971	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (([,) ∘ (𝐼‘𝑗))‘𝑘) = ((1st ‘((𝐼‘𝑗)‘𝑘))[,)(2nd ‘((𝐼‘𝑗)‘𝑘))))
47		simpr 490	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ ℕ)
48	25	elexd 3480	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑙 ∈ 𝑋 ↦ ⟨1, 0⟩) ∈ V)
49	26	fvmpt2 7005	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑗 ∈ ℕ ∧ (𝑙 ∈ 𝑋 ↦ ⟨1, 0⟩) ∈ V) → (𝐼‘𝑗) = (𝑙 ∈ 𝑋 ↦ ⟨1, 0⟩))
50	47, 48, 49	syl2anc 596	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐼‘𝑗) = (𝑙 ∈ 𝑋 ↦ ⟨1, 0⟩))
51	50	adantr 486	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐼‘𝑗) = (𝑙 ∈ 𝑋 ↦ ⟨1, 0⟩))
52		eqidd 2766	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) ∧ 𝑙 = 𝑘) → ⟨1, 0⟩ = ⟨1, 0⟩)
53	14	elexi 3479	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ⟨1, 0⟩ ∈ V
54	53	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ⟨1, 0⟩ ∈ V)
55	51, 52, 45, 54	fvmptd 7001	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐼‘𝑗)‘𝑘) = ⟨1, 0⟩)
56	55	fveq2d 6889	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (1st ‘((𝐼‘𝑗)‘𝑘)) = (1st ‘⟨1, 0⟩))
57	10	elexi 3479	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 1 ∈ V
58	4	elexi 3479	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 0 ∈ V
59	57, 58	op1st 8000	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (1st ‘⟨1, 0⟩) = 1
60	59	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (1st ‘⟨1, 0⟩) = 1)
61	56, 60	eqtrd 2800	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (1st ‘((𝐼‘𝑗)‘𝑘)) = 1)
62	55	fveq2d 6889	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (2nd ‘((𝐼‘𝑗)‘𝑘)) = (2nd ‘⟨1, 0⟩))
63	57, 58	op2nd 8001	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (2nd ‘⟨1, 0⟩) = 0
64	63	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (2nd ‘⟨1, 0⟩) = 0)
65	62, 64	eqtrd 2800	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (2nd ‘((𝐼‘𝑗)‘𝑘)) = 0)
66	61, 65	oveq12d 7437	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((1st ‘((𝐼‘𝑗)‘𝑘))[,)(2nd ‘((𝐼‘𝑗)‘𝑘))) = (1[,)0))
67		0le1 11754	. . . . . . . . . . . . . . . . . . . 20 ⊢ 0 ≤ 1
68		1xr 11285	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 1 ∈ ℝ*
69		ico0 13436	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((1 ∈ ℝ* ∧ 0 ∈ ℝ*) → ((1[,)0) = ∅ ↔ 0 ≤ 1))
70	68, 4, 69	mp2an 705	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((1[,)0) = ∅ ↔ 0 ≤ 1)
71	67, 70	mpbir 234	. . . . . . . . . . . . . . . . . . 19 ⊢ (1[,)0) = ∅
72	71	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (1[,)0) = ∅)
73	46, 66, 72	3eqtrd 2804	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (([,) ∘ (𝐼‘𝑗))‘𝑘) = ∅)
74	73	fveq2d 6889	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)) = (vol‘∅))
75		vol0 46733	. . . . . . . . . . . . . . . . 17 ⊢ (vol‘∅) = 0
76	75	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (vol‘∅) = 0)
77	74, 76	eqtrd 2800	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)) = 0)
78		0cn 11215	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℂ
79	78	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → 0 ∈ ℂ)
80	77, 79	eqeltrd 2865	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)) ∈ ℂ)
81	80	adantlr 728	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑙 ∈ 𝑋) ∧ 𝑘 ∈ 𝑋) → (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)) ∈ ℂ)
82		2fveq3 6890	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑙 → (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑙)))
83		simpr 490	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑙 ∈ 𝑋) → 𝑙 ∈ 𝑋)
84		eleq1w 2848	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑙 → (𝑘 ∈ 𝑋 ↔ 𝑙 ∈ 𝑋))
85	84	anbi2d 642	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑙 → (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) ↔ ((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑙 ∈ 𝑋)))
86	82	eqeq1d 2767	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑙 → ((vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)) = 0 ↔ (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑙)) = 0))
87	85, 86	imbi12d 347	. . . . . . . . . . . . . 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 2022	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑙 ∈ 𝑋) → (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑙)) = 0)
89	38, 39, 40, 81, 82, 83, 88	fprod0 46372	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑙 ∈ 𝑋) → ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)) = 0)
90	89	ex 418	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑙 ∈ 𝑋 → ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)) = 0))
91	90	exlimdv 1966	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (∃𝑙 𝑙 ∈ 𝑋 → ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)) = 0))
92	37, 91	mpd 16	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)) = 0)
93	92	mpteq2dva 5206	. . . . . . . 8 ⊢ (𝜑 → (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘))) = (𝑗 ∈ ℕ ↦ 0))
94	93	fveq2d 6889	. . . . . . 7 ⊢ (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)))) = (Σ^‘(𝑗 ∈ ℕ ↦ 0)))
95		nfv 1947	. . . . . . . 8 ⊢ Ⅎ𝑗𝜑
96	95, 30	sge0z 47149	. . . . . . 7 ⊢ (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ 0)) = 0)
97		eqidd 2766	. . . . . . 7 ⊢ (𝜑 → 0 = 0)
98	94, 96, 97	3eqtrrd 2805	. . . . . 6 ⊢ (𝜑 → 0 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)))))
99		fveq1 6884	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝐼 → (𝑖‘𝑗) = (𝐼‘𝑗))
100	99	coeq2d 5850	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝐼 → ([,) ∘ (𝑖‘𝑗)) = ([,) ∘ (𝐼‘𝑗)))
101	100	fveq1d 6887	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝐼 → (([,) ∘ (𝑖‘𝑗))‘𝑘) = (([,) ∘ (𝐼‘𝑗))‘𝑘))
102	101	fveq2d 6889	. . . . . . . . . . 11 ⊢ (𝑖 = 𝐼 → (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)))
103	102	ralrimivw 3163	. . . . . . . . . 10 ⊢ (𝑖 = 𝐼 → ∀𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)))
104	103	prodeq2d 16000	. . . . . . . . 9 ⊢ (𝑖 = 𝐼 → ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)) = ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)))
105	104	mpteq2dv 5207	. . . . . . . 8 ⊢ (𝑖 = 𝐼 → (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))) = (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘))))
106	105	fveq2d 6889	. . . . . . 7 ⊢ (𝑖 = 𝐼 → (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)))))
107	106	rspceeqv 3606	. . . . . 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 596	. . . . 5 ⊢ (𝜑 → ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)0 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))
109	5, 108	jca 521	. . . 4 ⊢ (𝜑 → (0 ∈ ℝ* ∧ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)0 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))))))
110		eqeq1 2769	. . . . . 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 3191	. . . . 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 3656	. . . 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 237	. . 3 ⊢ (𝜑 → 0 ∈ 𝑀)
114		infxrlb 13379	. . 3 ⊢ ((𝑀 ⊆ ℝ* ∧ 0 ∈ 𝑀) → inf(𝑀, ℝ*, < ) ≤ 0)
115	9, 113, 114	syl2anc 596	. 2 ⊢ (𝜑 → inf(𝑀, ℝ*, < ) ≤ 0)
116		pnfxr 11280	. . . 4 ⊢ +∞ ∈ ℝ*
117	116	a1i 11	. . 3 ⊢ (𝜑 → +∞ ∈ ℝ*)
118		iccgelb 13447	. . 3 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ inf(𝑀, ℝ*, < ) ∈ (0[,]+∞)) → 0 ≤ inf(𝑀, ℝ*, < ))
119	5, 117, 2, 118	syl3anc 1398	. 2 ⊢ (𝜑 → 0 ≤ inf(𝑀, ℝ*, < ))
120	3, 5, 115, 119	xrletrid 13198	1 ⊢ (𝜑 → inf(𝑀, ℝ*, < ) = 0)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2146   ≠ wne 2960  ∃wrex 3091  {crab 3418  Vcvv 3457   ⊆ wss 3906  ∅c0 4286  ⟨cop 4597   class class class wbr 5111   ↦ cmpt 5194   × cxp 5661   ∘ ccom 5667  ⟶wf 6536  ‘cfv 6540  (class class class)co 7419  1^st c1st 7990  2^nd c2nd 7991   ↑_m cmap 8830  Fincfn 8949  infcinf 9408  ℂcc 11115  ℝcr 11116  0cc0 11117  1c1 11118  +∞cpnf 11257  ℝ^*cxr 11259   < clt 11260   ≤ cle 11261  ℕcn 12250  [,)cico 13392  [,]cicc 13393  ∏cprod 15982  volcvol 25675  Σ^{^}csumge0 47136

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742  ax-inf2 9617  ax-cnex 11173  ax-resscn 11174  ax-1cn 11175  ax-icn 11176  ax-addcl 11177  ax-addrcl 11178  ax-mulcl 11179  ax-mulrcl 11180  ax-mulcom 11181  ax-addass 11182  ax-mulass 11183  ax-distr 11184  ax-i2m1 11185  ax-1ne0 11186  ax-1rid 11187  ax-rnegex 11188  ax-rrecex 11189  ax-cnre 11190  ax-pre-lttri 11191  ax-pre-lttrn 11192  ax-pre-ltadd 11193  ax-pre-mulgt0 11194  ax-pre-sup 11195

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-int 4915  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-se 5617  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-isom 6549  df-riota 7376  df-ov 7422  df-oprab 7423  df-mpo 7424  df-of 7684  df-om 7869  df-1st 7992  df-2nd 7993  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-1o 8459  df-2o 8460  df-er 8700  df-map 8832  df-en 8950  df-dom 8951  df-sdom 8952  df-fin 8953  df-sup 9409  df-inf 9410  df-oi 9479  df-dju 9903  df-card 9941  df-pnf 11262  df-mnf 11263  df-xr 11264  df-ltxr 11265  df-le 11266  df-sub 11460  df-neg 11461  df-div 11889  df-nn 12251  df-2 12320  df-3 12321  df-n0 12522  df-z 12609  df-uz 12881  df-q 12991  df-rp 13035  df-xadd 13156  df-ioo 13394  df-ico 13396  df-icc 13397  df-fz 13554  df-fzo 13702  df-fl 13845  df-seq 14058  df-exp 14118  df-hash 14387  df-cj 15176  df-re 15177  df-im 15178  df-sqrt 15312  df-abs 15313  df-clim 15565  df-sum 15764  df-prod 15983  df-xmet 21567  df-met 21568  df-ovol 25676  df-vol 25677  df-sumge0 47137

This theorem is used by:  ovn0  47340