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Theorem vitalilem4 24215

Description: Lemma for vitali 24217. (Contributed by Mario Carneiro, 16-Jun-2014.)

**Hypotheses**
Ref	Expression
vitali.1	⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]1)) ∧ (𝑥 − 𝑦) ∈ ℚ)}
vitali.2	⊢ 𝑆 = ((0[,]1) / ∼ )
vitali.3	⊢ (𝜑 → 𝐹 Fn 𝑆)
vitali.4	⊢ (𝜑 → ∀𝑧 ∈ 𝑆 (𝑧 ≠ ∅ → (𝐹‘𝑧) ∈ 𝑧))
vitali.5	⊢ (𝜑 → 𝐺:ℕ–1-1-onto→(ℚ ∩ (-1[,]1)))
vitali.6	⊢ 𝑇 = (𝑛 ∈ ℕ ↦ {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑛)) ∈ ran 𝐹})
vitali.7	⊢ (𝜑 → ¬ ran 𝐹 ∈ (𝒫 ℝ ∖ dom vol))

**Assertion**
Ref	Expression
vitalilem4	⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (vol*‘(𝑇‘𝑚)) = 0)

Distinct variable groups: 𝑚,𝑛,𝑠,𝑥,𝑦,𝑧,𝐺 𝜑,𝑚,𝑛,𝑥,𝑧 𝑧,𝑆 𝑇,𝑚,𝑥 𝑚,𝐹,𝑛,𝑠,𝑥,𝑦,𝑧 ∼ ,𝑚,𝑛,𝑠,𝑥,𝑦,𝑧 Allowed substitution hints: 𝜑(𝑦,𝑠) 𝑆(𝑥,𝑦,𝑚,𝑛,𝑠) 𝑇(𝑦,𝑧,𝑛,𝑠)

Proof of Theorem vitalilem4 Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6645	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → (𝐺‘𝑛) = (𝐺‘𝑚))
2	1	oveq2d 7151	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → (𝑠 − (𝐺‘𝑛)) = (𝑠 − (𝐺‘𝑚)))
3	2	eleq1d 2874	. . . . . . 7 ⊢ (𝑛 = 𝑚 → ((𝑠 − (𝐺‘𝑛)) ∈ ran 𝐹 ↔ (𝑠 − (𝐺‘𝑚)) ∈ ran 𝐹))
4	3	rabbidv 3427	. . . . . 6 ⊢ (𝑛 = 𝑚 → {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑛)) ∈ ran 𝐹} = {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑚)) ∈ ran 𝐹})
5		vitali.6	. . . . . 6 ⊢ 𝑇 = (𝑛 ∈ ℕ ↦ {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑛)) ∈ ran 𝐹})
6		reex 10617	. . . . . . 7 ⊢ ℝ ∈ V
7	6	rabex 5199	. . . . . 6 ⊢ {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑚)) ∈ ran 𝐹} ∈ V
8	4, 5, 7	fvmpt 6745	. . . . 5 ⊢ (𝑚 ∈ ℕ → (𝑇‘𝑚) = {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑚)) ∈ ran 𝐹})
9	8	adantl 485	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑇‘𝑚) = {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑚)) ∈ ran 𝐹})
10	9	fveq2d 6649	. . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (vol‘(𝑇‘𝑚)) = (vol‘{𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑚)) ∈ ran 𝐹}))
11		vitali.1	. . . . . . . 8 ⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]1)) ∧ (𝑥 − 𝑦) ∈ ℚ)}
12		vitali.2	. . . . . . . 8 ⊢ 𝑆 = ((0[,]1) / ∼ )
13		vitali.3	. . . . . . . 8 ⊢ (𝜑 → 𝐹 Fn 𝑆)
14		vitali.4	. . . . . . . 8 ⊢ (𝜑 → ∀𝑧 ∈ 𝑆 (𝑧 ≠ ∅ → (𝐹‘𝑧) ∈ 𝑧))
15		vitali.5	. . . . . . . 8 ⊢ (𝜑 → 𝐺:ℕ–1-1-onto→(ℚ ∩ (-1[,]1)))
16		vitali.7	. . . . . . . 8 ⊢ (𝜑 → ¬ ran 𝐹 ∈ (𝒫 ℝ ∖ dom vol))
17	11, 12, 13, 14, 15, 5, 16	vitalilem2 24213	. . . . . . 7 ⊢ (𝜑 → (ran 𝐹 ⊆ (0[,]1) ∧ (0[,]1) ⊆ ∪ 𝑚 ∈ ℕ (𝑇‘𝑚) ∧ ∪ 𝑚 ∈ ℕ (𝑇‘𝑚) ⊆ (-1[,]2)))
18	17	simp1d 1139	. . . . . 6 ⊢ (𝜑 → ran 𝐹 ⊆ (0[,]1))
19		unitssre 12877	. . . . . 6 ⊢ (0[,]1) ⊆ ℝ
20	18, 19	sstrdi 3927	. . . . 5 ⊢ (𝜑 → ran 𝐹 ⊆ ℝ)
21	20	adantr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ran 𝐹 ⊆ ℝ)
22		neg1rr 11740	. . . . . 6 ⊢ -1 ∈ ℝ
23		1re 10630	. . . . . 6 ⊢ 1 ∈ ℝ
24		iccssre 12807	. . . . . 6 ⊢ ((-1 ∈ ℝ ∧ 1 ∈ ℝ) → (-1[,]1) ⊆ ℝ)
25	22, 23, 24	mp2an 691	. . . . 5 ⊢ (-1[,]1) ⊆ ℝ
26		f1of 6590	. . . . . . . 8 ⊢ (𝐺:ℕ–1-1-onto→(ℚ ∩ (-1[,]1)) → 𝐺:ℕ⟶(ℚ ∩ (-1[,]1)))
27	15, 26	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐺:ℕ⟶(ℚ ∩ (-1[,]1)))
28	27	ffvelrnda 6828	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐺‘𝑚) ∈ (ℚ ∩ (-1[,]1)))
29	28	elin2d 4126	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐺‘𝑚) ∈ (-1[,]1))
30	25, 29	sseldi 3913	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐺‘𝑚) ∈ ℝ)
31		eqidd 2799	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑚)) ∈ ran 𝐹} = {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑚)) ∈ ran 𝐹})
32	21, 30, 31	ovolshft 24115	. . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (vol‘ran 𝐹) = (vol‘{𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑚)) ∈ ran 𝐹}))
33	10, 32	eqtr4d 2836	. 2 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (vol‘(𝑇‘𝑚)) = (vol‘ran 𝐹))
34		3re 11705	. . . . . . . 8 ⊢ 3 ∈ ℝ
35	34	rexri 10688	. . . . . . 7 ⊢ 3 ∈ ℝ^*
36	35	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → 3 ∈ ℝ^)
37		3rp 12383	. . . . . . . . . . . . 13 ⊢ 3 ∈ ℝ⁺
38		0re 10632	. . . . . . . . . . . . . . . . . . . 20 ⊢ 0 ∈ ℝ
39		0le1 11152	. . . . . . . . . . . . . . . . . . . 20 ⊢ 0 ≤ 1
40		ovolicc 24127	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ ∧ 0 ≤ 1) → (vol*‘(0[,]1)) = (1 − 0))
41	38, 23, 39, 40	mp3an 1458	. . . . . . . . . . . . . . . . . . 19 ⊢ (vol*‘(0[,]1)) = (1 − 0)
42		1m0e1 11746	. . . . . . . . . . . . . . . . . . 19 ⊢ (1 − 0) = 1
43	41, 42	eqtri 2821	. . . . . . . . . . . . . . . . . 18 ⊢ (vol*‘(0[,]1)) = 1
44	43, 23	eqeltri 2886	. . . . . . . . . . . . . . . . 17 ⊢ (vol*‘(0[,]1)) ∈ ℝ
45		ovolsscl 24090	. . . . . . . . . . . . . . . . 17 ⊢ ((ran 𝐹 ⊆ (0[,]1) ∧ (0[,]1) ⊆ ℝ ∧ (vol‘(0[,]1)) ∈ ℝ) → (vol‘ran 𝐹) ∈ ℝ)
46	19, 44, 45	mp3an23 1450	. . . . . . . . . . . . . . . 16 ⊢ (ran 𝐹 ⊆ (0[,]1) → (vol*‘ran 𝐹) ∈ ℝ)
47	18, 46	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (vol*‘ran 𝐹) ∈ ℝ)
48	47	adantr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → (vol‘ran 𝐹) ∈ ℝ)
49		simpr 488	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → 0 < (vol‘ran 𝐹))
50	48, 49	elrpd 12416	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → (vol‘ran 𝐹) ∈ ℝ⁺)
51		rpdivcl 12402	. . . . . . . . . . . . 13 ⊢ ((3 ∈ ℝ⁺ ∧ (vol‘ran 𝐹) ∈ ℝ⁺) → (3 / (vol‘ran 𝐹)) ∈ ℝ⁺)
52	37, 50, 51	sylancr 590	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → (3 / (vol‘ran 𝐹)) ∈ ℝ⁺)
53	52	rpred 12419	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → (3 / (vol‘ran 𝐹)) ∈ ℝ)
54	52	rpge0d 12423	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → 0 ≤ (3 / (vol‘ran 𝐹)))
55		flge0nn0 13185	. . . . . . . . . . 11 ⊢ (((3 / (vol‘ran 𝐹)) ∈ ℝ ∧ 0 ≤ (3 / (vol‘ran 𝐹))) → (⌊‘(3 / (vol*‘ran 𝐹))) ∈ ℕ₀)
56	53, 54, 55	syl2anc 587	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → (⌊‘(3 / (vol‘ran 𝐹))) ∈ ℕ₀)
57		nn0p1nn 11924	. . . . . . . . . 10 ⊢ ((⌊‘(3 / (vol‘ran 𝐹))) ∈ ℕ₀ → ((⌊‘(3 / (vol‘ran 𝐹))) + 1) ∈ ℕ)
58	56, 57	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → ((⌊‘(3 / (vol‘ran 𝐹))) + 1) ∈ ℕ)
59	58	nnred 11640	. . . . . . . 8 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → ((⌊‘(3 / (vol‘ran 𝐹))) + 1) ∈ ℝ)
60	59, 48	remulcld 10660	. . . . . . 7 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → (((⌊‘(3 / (vol‘ran 𝐹))) + 1) · (vol*‘ran 𝐹)) ∈ ℝ)
61	60	rexrd 10680	. . . . . 6 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → (((⌊‘(3 / (vol‘ran 𝐹))) + 1) · (vol‘ran 𝐹)) ∈ ℝ^)
62	6	elpw2 5212	. . . . . . . . . . . . . . . . . 18 ⊢ (ran 𝐹 ∈ 𝒫 ℝ ↔ ran 𝐹 ⊆ ℝ)
63	20, 62	sylibr 237	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ran 𝐹 ∈ 𝒫 ℝ)
64	63	anim1i 617	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ¬ ran 𝐹 ∈ dom vol) → (ran 𝐹 ∈ 𝒫 ℝ ∧ ¬ ran 𝐹 ∈ dom vol))
65		eldif 3891	. . . . . . . . . . . . . . . 16 ⊢ (ran 𝐹 ∈ (𝒫 ℝ ∖ dom vol) ↔ (ran 𝐹 ∈ 𝒫 ℝ ∧ ¬ ran 𝐹 ∈ dom vol))
66	64, 65	sylibr 237	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ¬ ran 𝐹 ∈ dom vol) → ran 𝐹 ∈ (𝒫 ℝ ∖ dom vol))
67	66	ex 416	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (¬ ran 𝐹 ∈ dom vol → ran 𝐹 ∈ (𝒫 ℝ ∖ dom vol)))
68	16, 67	mt3d 150	. . . . . . . . . . . . 13 ⊢ (𝜑 → ran 𝐹 ∈ dom vol)
69		inss1 4155	. . . . . . . . . . . . . . . 16 ⊢ (ℚ ∩ (-1[,]1)) ⊆ ℚ
70		qssre 12346	. . . . . . . . . . . . . . . 16 ⊢ ℚ ⊆ ℝ
71	69, 70	sstri 3924	. . . . . . . . . . . . . . 15 ⊢ (ℚ ∩ (-1[,]1)) ⊆ ℝ
72		fss 6501	. . . . . . . . . . . . . . 15 ⊢ ((𝐺:ℕ⟶(ℚ ∩ (-1[,]1)) ∧ (ℚ ∩ (-1[,]1)) ⊆ ℝ) → 𝐺:ℕ⟶ℝ)
73	27, 71, 72	sylancl 589	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐺:ℕ⟶ℝ)
74	73	ffvelrnda 6828	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐺‘𝑛) ∈ ℝ)
75		shftmbl 24142	. . . . . . . . . . . . 13 ⊢ ((ran 𝐹 ∈ dom vol ∧ (𝐺‘𝑛) ∈ ℝ) → {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑛)) ∈ ran 𝐹} ∈ dom vol)
76	68, 74, 75	syl2an2r 684	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑛)) ∈ ran 𝐹} ∈ dom vol)
77	76, 5	fmptd 6855	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑇:ℕ⟶dom vol)
78	77	ffvelrnda 6828	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑇‘𝑚) ∈ dom vol)
79	78	ralrimiva 3149	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑚 ∈ ℕ (𝑇‘𝑚) ∈ dom vol)
80		iunmbl 24157	. . . . . . . . 9 ⊢ (∀𝑚 ∈ ℕ (𝑇‘𝑚) ∈ dom vol → ∪ 𝑚 ∈ ℕ (𝑇‘𝑚) ∈ dom vol)
81	79, 80	syl 17	. . . . . . . 8 ⊢ (𝜑 → ∪ 𝑚 ∈ ℕ (𝑇‘𝑚) ∈ dom vol)
82		mblss 24135	. . . . . . . 8 ⊢ (∪ 𝑚 ∈ ℕ (𝑇‘𝑚) ∈ dom vol → ∪ 𝑚 ∈ ℕ (𝑇‘𝑚) ⊆ ℝ)
83		ovolcl 24082	. . . . . . . 8 ⊢ (∪ 𝑚 ∈ ℕ (𝑇‘𝑚) ⊆ ℝ → (vol‘∪ 𝑚 ∈ ℕ (𝑇‘𝑚)) ∈ ℝ^)
84	81, 82, 83	3syl 18	. . . . . . 7 ⊢ (𝜑 → (vol‘∪ 𝑚 ∈ ℕ (𝑇‘𝑚)) ∈ ℝ^)
85	84	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → (vol‘∪ 𝑚 ∈ ℕ (𝑇‘𝑚)) ∈ ℝ^*)
86		flltp1 13165	. . . . . . . 8 ⊢ ((3 / (vol‘ran 𝐹)) ∈ ℝ → (3 / (vol‘ran 𝐹)) < ((⌊‘(3 / (vol*‘ran 𝐹))) + 1))
87	53, 86	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → (3 / (vol‘ran 𝐹)) < ((⌊‘(3 / (vol*‘ran 𝐹))) + 1))
88	34	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → 3 ∈ ℝ)
89	88, 59, 50	ltdivmul2d 12471	. . . . . . 7 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → ((3 / (vol‘ran 𝐹)) < ((⌊‘(3 / (vol‘ran 𝐹))) + 1) ↔ 3 < (((⌊‘(3 / (vol‘ran 𝐹))) + 1) · (vol*‘ran 𝐹))))
90	87, 89	mpbid 235	. . . . . 6 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → 3 < (((⌊‘(3 / (vol‘ran 𝐹))) + 1) · (vol*‘ran 𝐹)))
91		nnuz 12269	. . . . . . . . . . 11 ⊢ ℕ = (ℤ_≥‘1)
92		1zzd 12001	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → 1 ∈ ℤ)
93		mblvol 24134	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑇‘𝑚) ∈ dom vol → (vol‘(𝑇‘𝑚)) = (vol*‘(𝑇‘𝑚)))
94	78, 93	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (vol‘(𝑇‘𝑚)) = (vol*‘(𝑇‘𝑚)))
95	94, 33	eqtrd 2833	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (vol‘(𝑇‘𝑚)) = (vol*‘ran 𝐹))
96	47	adantr 484	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (vol*‘ran 𝐹) ∈ ℝ)
97	95, 96	eqeltrd 2890	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (vol‘(𝑇‘𝑚)) ∈ ℝ)
98	97	adantlr 714	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 0 < (vol*‘ran 𝐹)) ∧ 𝑚 ∈ ℕ) → (vol‘(𝑇‘𝑚)) ∈ ℝ)
99		eqid 2798	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℕ ↦ (vol‘(𝑇‘𝑚))) = (𝑚 ∈ ℕ ↦ (vol‘(𝑇‘𝑚)))
100	98, 99	fmptd 6855	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → (𝑚 ∈ ℕ ↦ (vol‘(𝑇‘𝑚))):ℕ⟶ℝ)
101	100	ffvelrnda 6828	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 0 < (vol*‘ran 𝐹)) ∧ 𝑘 ∈ ℕ) → ((𝑚 ∈ ℕ ↦ (vol‘(𝑇‘𝑚)))‘𝑘) ∈ ℝ)
102	91, 92, 101	serfre 13395	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇‘𝑚)))):ℕ⟶ℝ)
103	102	frnd 6494	. . . . . . . . 9 ⊢ ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → ran seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇‘𝑚)))) ⊆ ℝ)
104		ressxr 10674	. . . . . . . . 9 ⊢ ℝ ⊆ ℝ^*
105	103, 104	sstrdi 3927	. . . . . . . 8 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → ran seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇‘𝑚)))) ⊆ ℝ^)
106	95	adantlr 714	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 0 < (vol‘ran 𝐹)) ∧ 𝑚 ∈ ℕ) → (vol‘(𝑇‘𝑚)) = (vol‘ran 𝐹))
107	106	mpteq2dva 5125	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → (𝑚 ∈ ℕ ↦ (vol‘(𝑇‘𝑚))) = (𝑚 ∈ ℕ ↦ (vol‘ran 𝐹)))
108		fconstmpt 5578	. . . . . . . . . . . . 13 ⊢ (ℕ × {(vol‘ran 𝐹)}) = (𝑚 ∈ ℕ ↦ (vol‘ran 𝐹))
109	107, 108	eqtr4di 2851	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → (𝑚 ∈ ℕ ↦ (vol‘(𝑇‘𝑚))) = (ℕ × {(vol‘ran 𝐹)}))
110	109	seqeq3d 13372	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇‘𝑚)))) = seq1( + , (ℕ × {(vol‘ran 𝐹)})))
111	110	fveq1d 6647	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → (seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇‘𝑚))))‘((⌊‘(3 / (vol‘ran 𝐹))) + 1)) = (seq1( + , (ℕ × {(vol‘ran 𝐹)}))‘((⌊‘(3 / (vol‘ran 𝐹))) + 1)))
112	48	recnd 10658	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → (vol‘ran 𝐹) ∈ ℂ)
113		ser1const 13422	. . . . . . . . . . 11 ⊢ (((vol‘ran 𝐹) ∈ ℂ ∧ ((⌊‘(3 / (vol‘ran 𝐹))) + 1) ∈ ℕ) → (seq1( + , (ℕ × {(vol‘ran 𝐹)}))‘((⌊‘(3 / (vol‘ran 𝐹))) + 1)) = (((⌊‘(3 / (vol‘ran 𝐹))) + 1) · (vol‘ran 𝐹)))
114	112, 58, 113	syl2anc 587	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → (seq1( + , (ℕ × {(vol‘ran 𝐹)}))‘((⌊‘(3 / (vol‘ran 𝐹))) + 1)) = (((⌊‘(3 / (vol‘ran 𝐹))) + 1) · (vol*‘ran 𝐹)))
115	111, 114	eqtrd 2833	. . . . . . . . 9 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → (seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇‘𝑚))))‘((⌊‘(3 / (vol‘ran 𝐹))) + 1)) = (((⌊‘(3 / (vol‘ran 𝐹))) + 1) · (vol‘ran 𝐹)))
116	102	ffnd 6488	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇‘𝑚)))) Fn ℕ)
117		fnfvelrn 6825	. . . . . . . . . 10 ⊢ ((seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇‘𝑚)))) Fn ℕ ∧ ((⌊‘(3 / (vol‘ran 𝐹))) + 1) ∈ ℕ) → (seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇‘𝑚))))‘((⌊‘(3 / (vol‘ran 𝐹))) + 1)) ∈ ran seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇‘𝑚)))))
118	116, 58, 117	syl2anc 587	. . . . . . . . 9 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → (seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇‘𝑚))))‘((⌊‘(3 / (vol‘ran 𝐹))) + 1)) ∈ ran seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇‘𝑚)))))
119	115, 118	eqeltrrd 2891	. . . . . . . 8 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → (((⌊‘(3 / (vol‘ran 𝐹))) + 1) · (vol*‘ran 𝐹)) ∈ ran seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇‘𝑚)))))
120		supxrub 12705	. . . . . . . 8 ⊢ ((ran seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇‘𝑚)))) ⊆ ℝ^* ∧ (((⌊‘(3 / (vol‘ran 𝐹))) + 1) · (vol‘ran 𝐹)) ∈ ran seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇‘𝑚))))) → (((⌊‘(3 / (vol‘ran 𝐹))) + 1) · (vol‘ran 𝐹)) ≤ sup(ran seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇‘𝑚)))), ℝ^*, < ))
121	105, 119, 120	syl2anc 587	. . . . . . 7 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → (((⌊‘(3 / (vol‘ran 𝐹))) + 1) · (vol‘ran 𝐹)) ≤ sup(ran seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇‘𝑚)))), ℝ^, < ))
122	81	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → ∪ 𝑚 ∈ ℕ (𝑇‘𝑚) ∈ dom vol)
123		mblvol 24134	. . . . . . . . 9 ⊢ (∪ 𝑚 ∈ ℕ (𝑇‘𝑚) ∈ dom vol → (vol‘∪ 𝑚 ∈ ℕ (𝑇‘𝑚)) = (vol*‘∪ 𝑚 ∈ ℕ (𝑇‘𝑚)))
124	122, 123	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → (vol‘∪ 𝑚 ∈ ℕ (𝑇‘𝑚)) = (vol‘∪ 𝑚 ∈ ℕ (𝑇‘𝑚)))
125	78, 97	jca 515	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑇‘𝑚) ∈ dom vol ∧ (vol‘(𝑇‘𝑚)) ∈ ℝ))
126	125	ralrimiva 3149	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑚 ∈ ℕ ((𝑇‘𝑚) ∈ dom vol ∧ (vol‘(𝑇‘𝑚)) ∈ ℝ))
127	11, 12, 13, 14, 15, 5, 16	vitalilem3 24214	. . . . . . . . . 10 ⊢ (𝜑 → Disj 𝑚 ∈ ℕ (𝑇‘𝑚))
128	127	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → Disj 𝑚 ∈ ℕ (𝑇‘𝑚))
129		eqid 2798	. . . . . . . . . 10 ⊢ seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇‘𝑚)))) = seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇‘𝑚))))
130	129, 99	voliun 24158	. . . . . . . . 9 ⊢ ((∀𝑚 ∈ ℕ ((𝑇‘𝑚) ∈ dom vol ∧ (vol‘(𝑇‘𝑚)) ∈ ℝ) ∧ Disj 𝑚 ∈ ℕ (𝑇‘𝑚)) → (vol‘∪ 𝑚 ∈ ℕ (𝑇‘𝑚)) = sup(ran seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇‘𝑚)))), ℝ^*, < ))
131	126, 128, 130	syl2an2r 684	. . . . . . . 8 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → (vol‘∪ 𝑚 ∈ ℕ (𝑇‘𝑚)) = sup(ran seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇‘𝑚)))), ℝ^, < ))
132	124, 131	eqtr3d 2835	. . . . . . 7 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → (vol‘∪ 𝑚 ∈ ℕ (𝑇‘𝑚)) = sup(ran seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇‘𝑚)))), ℝ^*, < ))
133	121, 132	breqtrrd 5058	. . . . . 6 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → (((⌊‘(3 / (vol‘ran 𝐹))) + 1) · (vol‘ran 𝐹)) ≤ (vol‘∪ 𝑚 ∈ ℕ (𝑇‘𝑚)))
134	36, 61, 85, 90, 133	xrltletrd 12542	. . . . 5 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → 3 < (vol‘∪ 𝑚 ∈ ℕ (𝑇‘𝑚)))
135	17	simp3d 1141	. . . . . . . . 9 ⊢ (𝜑 → ∪ 𝑚 ∈ ℕ (𝑇‘𝑚) ⊆ (-1[,]2))
136	135	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → ∪ 𝑚 ∈ ℕ (𝑇‘𝑚) ⊆ (-1[,]2))
137		2re 11699	. . . . . . . . 9 ⊢ 2 ∈ ℝ
138		iccssre 12807	. . . . . . . . 9 ⊢ ((-1 ∈ ℝ ∧ 2 ∈ ℝ) → (-1[,]2) ⊆ ℝ)
139	22, 137, 138	mp2an 691	. . . . . . . 8 ⊢ (-1[,]2) ⊆ ℝ
140		ovolss 24089	. . . . . . . 8 ⊢ ((∪ 𝑚 ∈ ℕ (𝑇‘𝑚) ⊆ (-1[,]2) ∧ (-1[,]2) ⊆ ℝ) → (vol‘∪ 𝑚 ∈ ℕ (𝑇‘𝑚)) ≤ (vol‘(-1[,]2)))
141	136, 139, 140	sylancl 589	. . . . . . 7 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → (vol‘∪ 𝑚 ∈ ℕ (𝑇‘𝑚)) ≤ (vol*‘(-1[,]2)))
142		2cn 11700	. . . . . . . . 9 ⊢ 2 ∈ ℂ
143		ax-1cn 10584	. . . . . . . . 9 ⊢ 1 ∈ ℂ
144	142, 143	subnegi 10954	. . . . . . . 8 ⊢ (2 − -1) = (2 + 1)
145		neg1lt0 11742	. . . . . . . . . . 11 ⊢ -1 < 0
146		2pos 11728	. . . . . . . . . . 11 ⊢ 0 < 2
147	22, 38, 137	lttri 10755	. . . . . . . . . . 11 ⊢ ((-1 < 0 ∧ 0 < 2) → -1 < 2)
148	145, 146, 147	mp2an 691	. . . . . . . . . 10 ⊢ -1 < 2
149	22, 137, 148	ltleii 10752	. . . . . . . . 9 ⊢ -1 ≤ 2
150		ovolicc 24127	. . . . . . . . 9 ⊢ ((-1 ∈ ℝ ∧ 2 ∈ ℝ ∧ -1 ≤ 2) → (vol*‘(-1[,]2)) = (2 − -1))
151	22, 137, 149, 150	mp3an 1458	. . . . . . . 8 ⊢ (vol*‘(-1[,]2)) = (2 − -1)
152		df-3 11689	. . . . . . . 8 ⊢ 3 = (2 + 1)
153	144, 151, 152	3eqtr4i 2831	. . . . . . 7 ⊢ (vol*‘(-1[,]2)) = 3
154	141, 153	breqtrdi 5071	. . . . . 6 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → (vol‘∪ 𝑚 ∈ ℕ (𝑇‘𝑚)) ≤ 3)
155		xrlenlt 10695	. . . . . . 7 ⊢ (((vol‘∪ 𝑚 ∈ ℕ (𝑇‘𝑚)) ∈ ℝ^ ∧ 3 ∈ ℝ^) → ((vol‘∪ 𝑚 ∈ ℕ (𝑇‘𝑚)) ≤ 3 ↔ ¬ 3 < (vol*‘∪ 𝑚 ∈ ℕ (𝑇‘𝑚))))
156	85, 35, 155	sylancl 589	. . . . . 6 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → ((vol‘∪ 𝑚 ∈ ℕ (𝑇‘𝑚)) ≤ 3 ↔ ¬ 3 < (vol*‘∪ 𝑚 ∈ ℕ (𝑇‘𝑚))))
157	154, 156	mpbid 235	. . . . 5 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → ¬ 3 < (vol‘∪ 𝑚 ∈ ℕ (𝑇‘𝑚)))
158	134, 157	pm2.65da 816	. . . 4 ⊢ (𝜑 → ¬ 0 < (vol*‘ran 𝐹))
159		ovolge0 24085	. . . . . . 7 ⊢ (ran 𝐹 ⊆ ℝ → 0 ≤ (vol*‘ran 𝐹))
160	20, 159	syl 17	. . . . . 6 ⊢ (𝜑 → 0 ≤ (vol*‘ran 𝐹))
161		0xr 10677	. . . . . . 7 ⊢ 0 ∈ ℝ^*
162		ovolcl 24082	. . . . . . . 8 ⊢ (ran 𝐹 ⊆ ℝ → (vol‘ran 𝐹) ∈ ℝ^)
163	20, 162	syl 17	. . . . . . 7 ⊢ (𝜑 → (vol‘ran 𝐹) ∈ ℝ^)
164		xrleloe 12525	. . . . . . 7 ⊢ ((0 ∈ ℝ^* ∧ (vol‘ran 𝐹) ∈ ℝ^) → (0 ≤ (vol‘ran 𝐹) ↔ (0 < (vol‘ran 𝐹) ∨ 0 = (vol*‘ran 𝐹))))
165	161, 163, 164	sylancr 590	. . . . . 6 ⊢ (𝜑 → (0 ≤ (vol‘ran 𝐹) ↔ (0 < (vol‘ran 𝐹) ∨ 0 = (vol*‘ran 𝐹))))
166	160, 165	mpbid 235	. . . . 5 ⊢ (𝜑 → (0 < (vol‘ran 𝐹) ∨ 0 = (vol‘ran 𝐹)))
167	166	ord 861	. . . 4 ⊢ (𝜑 → (¬ 0 < (vol‘ran 𝐹) → 0 = (vol‘ran 𝐹)))
168	158, 167	mpd 15	. . 3 ⊢ (𝜑 → 0 = (vol*‘ran 𝐹))
169	168	adantr 484	. 2 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 0 = (vol*‘ran 𝐹))
170	33, 169	eqtr4d 2836	1 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (vol*‘(𝑇‘𝑚)) = 0)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∨ wo 844 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∀wral 3106 {crab 3110 ∖ cdif 3878 ∩ cin 3880 ⊆ wss 3881 ∅c0 4243 𝒫 cpw 4497 {csn 4525 ∪ ciun 4881 Disj wdisj 4995 class class class wbr 5030 {copab 5092 ↦ cmpt 5110 × cxp 5517 dom cdm 5519 ran crn 5520 Fn wfn 6319 ⟶wf 6320 –1-1-onto→wf1o 6323 ‘cfv 6324 (class class class)co 7135 / cqs 8271 supcsup 8888 ℂcc 10524 ℝcr 10525 0cc0 10526 1c1 10527 + caddc 10529 · cmul 10531 ℝ^*cxr 10663 < clt 10664 ≤ cle 10665 − cmin 10859 -cneg 10860 / cdiv 11286 ℕcn 11625 2c2 11680 3c3 11681 ℕ₀cn0 11885 ℚcq 12336 ℝ⁺crp 12377 [,]cicc 12729 ⌊cfl 13155 seqcseq 13364 vol*covol 24066 volcvol 24067

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cc 9846 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-disj 4996 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-ec 8274 df-qs 8278 df-map 8391 df-pm 8392 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fi 8859 df-sup 8890 df-inf 8891 df-oi 8958 df-dju 9314 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-ioo 12730 df-ico 12732 df-icc 12733 df-fz 12886 df-fzo 13029 df-fl 13157 df-seq 13365 df-exp 13426 df-hash 13687 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-clim 14837 df-rlim 14838 df-sum 15035 df-rest 16688 df-topgen 16709 df-psmet 20083 df-xmet 20084 df-met 20085 df-bl 20086 df-mopn 20087 df-top 21499 df-topon 21516 df-bases 21551 df-cmp 21992 df-ovol 24068 df-vol 24069

This theorem is referenced by: vitalilem5 24216

W3C validator