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Theorem vitalilem4 25512

Description: Lemma for vitali 25514. (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 6858	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → (𝐺‘𝑛) = (𝐺‘𝑚))
2	1	oveq2d 7403	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → (𝑠 − (𝐺‘𝑛)) = (𝑠 − (𝐺‘𝑚)))
3	2	eleq1d 2813	. . . . . . 7 ⊢ (𝑛 = 𝑚 → ((𝑠 − (𝐺‘𝑛)) ∈ ran 𝐹 ↔ (𝑠 − (𝐺‘𝑚)) ∈ ran 𝐹))
4	3	rabbidv 3413	. . . . . 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 11159	. . . . . . 7 ⊢ ℝ ∈ V
7	6	rabex 5294	. . . . . 6 ⊢ {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑚)) ∈ ran 𝐹} ∈ V
8	4, 5, 7	fvmpt 6968	. . . . 5 ⊢ (𝑚 ∈ ℕ → (𝑇‘𝑚) = {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑚)) ∈ ran 𝐹})
9	8	adantl 481	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑇‘𝑚) = {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑚)) ∈ ran 𝐹})
10	9	fveq2d 6862	. . 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 25510	. . . . . . 7 ⊢ (𝜑 → (ran 𝐹 ⊆ (0[,]1) ∧ (0[,]1) ⊆ ∪ 𝑚 ∈ ℕ (𝑇‘𝑚) ∧ ∪ 𝑚 ∈ ℕ (𝑇‘𝑚) ⊆ (-1[,]2)))
18	17	simp1d 1142	. . . . . 6 ⊢ (𝜑 → ran 𝐹 ⊆ (0[,]1))
19		unitssre 13460	. . . . . 6 ⊢ (0[,]1) ⊆ ℝ
20	18, 19	sstrdi 3959	. . . . 5 ⊢ (𝜑 → ran 𝐹 ⊆ ℝ)
21	20	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ran 𝐹 ⊆ ℝ)
22		neg1rr 12172	. . . . . 6 ⊢ -1 ∈ ℝ
23		1re 11174	. . . . . 6 ⊢ 1 ∈ ℝ
24		iccssre 13390	. . . . . 6 ⊢ ((-1 ∈ ℝ ∧ 1 ∈ ℝ) → (-1[,]1) ⊆ ℝ)
25	22, 23, 24	mp2an 692	. . . . 5 ⊢ (-1[,]1) ⊆ ℝ
26		f1of 6800	. . . . . . . 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	ffvelcdmda 7056	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐺‘𝑚) ∈ (ℚ ∩ (-1[,]1)))
29	28	elin2d 4168	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐺‘𝑚) ∈ (-1[,]1))
30	25, 29	sselid 3944	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐺‘𝑚) ∈ ℝ)
31		eqidd 2730	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑚)) ∈ ran 𝐹} = {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑚)) ∈ ran 𝐹})
32	21, 30, 31	ovolshft 25412	. . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (vol‘ran 𝐹) = (vol‘{𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑚)) ∈ ran 𝐹}))
33	10, 32	eqtr4d 2767	. 2 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (vol‘(𝑇‘𝑚)) = (vol‘ran 𝐹))
34		3re 12266	. . . . . . . 8 ⊢ 3 ∈ ℝ
35	34	rexri 11232	. . . . . . 7 ⊢ 3 ∈ ℝ^*
36	35	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → 3 ∈ ℝ^)
37		3rp 12957	. . . . . . . . . . . . 13 ⊢ 3 ∈ ℝ⁺
38		0re 11176	. . . . . . . . . . . . . . . . . . . 20 ⊢ 0 ∈ ℝ
39		0le1 11701	. . . . . . . . . . . . . . . . . . . 20 ⊢ 0 ≤ 1
40		ovolicc 25424	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ ∧ 0 ≤ 1) → (vol*‘(0[,]1)) = (1 − 0))
41	38, 23, 39, 40	mp3an 1463	. . . . . . . . . . . . . . . . . . 19 ⊢ (vol*‘(0[,]1)) = (1 − 0)
42		1m0e1 12302	. . . . . . . . . . . . . . . . . . 19 ⊢ (1 − 0) = 1
43	41, 42	eqtri 2752	. . . . . . . . . . . . . . . . . 18 ⊢ (vol*‘(0[,]1)) = 1
44	43, 23	eqeltri 2824	. . . . . . . . . . . . . . . . 17 ⊢ (vol*‘(0[,]1)) ∈ ℝ
45		ovolsscl 25387	. . . . . . . . . . . . . . . . 17 ⊢ ((ran 𝐹 ⊆ (0[,]1) ∧ (0[,]1) ⊆ ℝ ∧ (vol‘(0[,]1)) ∈ ℝ) → (vol‘ran 𝐹) ∈ ℝ)
46	19, 44, 45	mp3an23 1455	. . . . . . . . . . . . . . . 16 ⊢ (ran 𝐹 ⊆ (0[,]1) → (vol*‘ran 𝐹) ∈ ℝ)
47	18, 46	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (vol*‘ran 𝐹) ∈ ℝ)
48	47	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → (vol‘ran 𝐹) ∈ ℝ)
49		simpr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → 0 < (vol‘ran 𝐹))
50	48, 49	elrpd 12992	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → (vol‘ran 𝐹) ∈ ℝ⁺)
51		rpdivcl 12978	. . . . . . . . . . . . 13 ⊢ ((3 ∈ ℝ⁺ ∧ (vol‘ran 𝐹) ∈ ℝ⁺) → (3 / (vol‘ran 𝐹)) ∈ ℝ⁺)
52	37, 50, 51	sylancr 587	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → (3 / (vol‘ran 𝐹)) ∈ ℝ⁺)
53	52	rpred 12995	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → (3 / (vol‘ran 𝐹)) ∈ ℝ)
54	52	rpge0d 12999	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → 0 ≤ (3 / (vol‘ran 𝐹)))
55		flge0nn0 13782	. . . . . . . . . . 11 ⊢ (((3 / (vol‘ran 𝐹)) ∈ ℝ ∧ 0 ≤ (3 / (vol‘ran 𝐹))) → (⌊‘(3 / (vol*‘ran 𝐹))) ∈ ℕ₀)
56	53, 54, 55	syl2anc 584	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → (⌊‘(3 / (vol‘ran 𝐹))) ∈ ℕ₀)
57		nn0p1nn 12481	. . . . . . . . . 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 12201	. . . . . . . 8 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → ((⌊‘(3 / (vol‘ran 𝐹))) + 1) ∈ ℝ)
60	59, 48	remulcld 11204	. . . . . . 7 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → (((⌊‘(3 / (vol‘ran 𝐹))) + 1) · (vol*‘ran 𝐹)) ∈ ℝ)
61	60	rexrd 11224	. . . . . 6 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → (((⌊‘(3 / (vol‘ran 𝐹))) + 1) · (vol‘ran 𝐹)) ∈ ℝ^)
62	6	elpw2 5289	. . . . . . . . . . . . . . . . . 18 ⊢ (ran 𝐹 ∈ 𝒫 ℝ ↔ ran 𝐹 ⊆ ℝ)
63	20, 62	sylibr 234	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ran 𝐹 ∈ 𝒫 ℝ)
64	63	anim1i 615	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ¬ ran 𝐹 ∈ dom vol) → (ran 𝐹 ∈ 𝒫 ℝ ∧ ¬ ran 𝐹 ∈ dom vol))
65		eldif 3924	. . . . . . . . . . . . . . . 16 ⊢ (ran 𝐹 ∈ (𝒫 ℝ ∖ dom vol) ↔ (ran 𝐹 ∈ 𝒫 ℝ ∧ ¬ ran 𝐹 ∈ dom vol))
66	64, 65	sylibr 234	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ¬ ran 𝐹 ∈ dom vol) → ran 𝐹 ∈ (𝒫 ℝ ∖ dom vol))
67	66	ex 412	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (¬ ran 𝐹 ∈ dom vol → ran 𝐹 ∈ (𝒫 ℝ ∖ dom vol)))
68	16, 67	mt3d 148	. . . . . . . . . . . . 13 ⊢ (𝜑 → ran 𝐹 ∈ dom vol)
69		inss1 4200	. . . . . . . . . . . . . . . 16 ⊢ (ℚ ∩ (-1[,]1)) ⊆ ℚ
70		qssre 12918	. . . . . . . . . . . . . . . 16 ⊢ ℚ ⊆ ℝ
71	69, 70	sstri 3956	. . . . . . . . . . . . . . 15 ⊢ (ℚ ∩ (-1[,]1)) ⊆ ℝ
72		fss 6704	. . . . . . . . . . . . . . 15 ⊢ ((𝐺:ℕ⟶(ℚ ∩ (-1[,]1)) ∧ (ℚ ∩ (-1[,]1)) ⊆ ℝ) → 𝐺:ℕ⟶ℝ)
73	27, 71, 72	sylancl 586	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐺:ℕ⟶ℝ)
74	73	ffvelcdmda 7056	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐺‘𝑛) ∈ ℝ)
75		shftmbl 25439	. . . . . . . . . . . . 13 ⊢ ((ran 𝐹 ∈ dom vol ∧ (𝐺‘𝑛) ∈ ℝ) → {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑛)) ∈ ran 𝐹} ∈ dom vol)
76	68, 74, 75	syl2an2r 685	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑛)) ∈ ran 𝐹} ∈ dom vol)
77	76, 5	fmptd 7086	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑇:ℕ⟶dom vol)
78	77	ffvelcdmda 7056	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑇‘𝑚) ∈ dom vol)
79	78	ralrimiva 3125	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑚 ∈ ℕ (𝑇‘𝑚) ∈ dom vol)
80		iunmbl 25454	. . . . . . . . 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 25432	. . . . . . . 8 ⊢ (∪ 𝑚 ∈ ℕ (𝑇‘𝑚) ∈ dom vol → ∪ 𝑚 ∈ ℕ (𝑇‘𝑚) ⊆ ℝ)
83		ovolcl 25379	. . . . . . . 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 480	. . . . . 6 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → (vol‘∪ 𝑚 ∈ ℕ (𝑇‘𝑚)) ∈ ℝ^*)
86		flltp1 13762	. . . . . . . 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 13047	. . . . . . 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 232	. . . . . 6 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → 3 < (((⌊‘(3 / (vol‘ran 𝐹))) + 1) · (vol*‘ran 𝐹)))
91		nnuz 12836	. . . . . . . . . . 11 ⊢ ℕ = (ℤ_≥‘1)
92		1zzd 12564	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → 1 ∈ ℤ)
93		mblvol 25431	. . . . . . . . . . . . . . . . 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 2764	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (vol‘(𝑇‘𝑚)) = (vol*‘ran 𝐹))
96	47	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (vol*‘ran 𝐹) ∈ ℝ)
97	95, 96	eqeltrd 2828	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (vol‘(𝑇‘𝑚)) ∈ ℝ)
98	97	adantlr 715	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 0 < (vol*‘ran 𝐹)) ∧ 𝑚 ∈ ℕ) → (vol‘(𝑇‘𝑚)) ∈ ℝ)
99		eqid 2729	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℕ ↦ (vol‘(𝑇‘𝑚))) = (𝑚 ∈ ℕ ↦ (vol‘(𝑇‘𝑚)))
100	98, 99	fmptd 7086	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → (𝑚 ∈ ℕ ↦ (vol‘(𝑇‘𝑚))):ℕ⟶ℝ)
101	100	ffvelcdmda 7056	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 0 < (vol*‘ran 𝐹)) ∧ 𝑘 ∈ ℕ) → ((𝑚 ∈ ℕ ↦ (vol‘(𝑇‘𝑚)))‘𝑘) ∈ ℝ)
102	91, 92, 101	serfre 13996	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇‘𝑚)))):ℕ⟶ℝ)
103	102	frnd 6696	. . . . . . . . 9 ⊢ ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → ran seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇‘𝑚)))) ⊆ ℝ)
104		ressxr 11218	. . . . . . . . 9 ⊢ ℝ ⊆ ℝ^*
105	103, 104	sstrdi 3959	. . . . . . . 8 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → ran seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇‘𝑚)))) ⊆ ℝ^)
106	95	adantlr 715	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 0 < (vol‘ran 𝐹)) ∧ 𝑚 ∈ ℕ) → (vol‘(𝑇‘𝑚)) = (vol‘ran 𝐹))
107	106	mpteq2dva 5200	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → (𝑚 ∈ ℕ ↦ (vol‘(𝑇‘𝑚))) = (𝑚 ∈ ℕ ↦ (vol‘ran 𝐹)))
108		fconstmpt 5700	. . . . . . . . . . . . 13 ⊢ (ℕ × {(vol‘ran 𝐹)}) = (𝑚 ∈ ℕ ↦ (vol‘ran 𝐹))
109	107, 108	eqtr4di 2782	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → (𝑚 ∈ ℕ ↦ (vol‘(𝑇‘𝑚))) = (ℕ × {(vol‘ran 𝐹)}))
110	109	seqeq3d 13974	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇‘𝑚)))) = seq1( + , (ℕ × {(vol‘ran 𝐹)})))
111	110	fveq1d 6860	. . . . . . . . . 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 11202	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → (vol‘ran 𝐹) ∈ ℂ)
113		ser1const 14023	. . . . . . . . . . 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 584	. . . . . . . . . 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 2764	. . . . . . . . 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 6689	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇‘𝑚)))) Fn ℕ)
117		fnfvelrn 7052	. . . . . . . . . 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 584	. . . . . . . . 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 2829	. . . . . . . 8 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → (((⌊‘(3 / (vol‘ran 𝐹))) + 1) · (vol*‘ran 𝐹)) ∈ ran seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇‘𝑚)))))
120		supxrub 13284	. . . . . . . 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 584	. . . . . . 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 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → ∪ 𝑚 ∈ ℕ (𝑇‘𝑚) ∈ dom vol)
123		mblvol 25431	. . . . . . . . 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 511	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑇‘𝑚) ∈ dom vol ∧ (vol‘(𝑇‘𝑚)) ∈ ℝ))
126	125	ralrimiva 3125	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑚 ∈ ℕ ((𝑇‘𝑚) ∈ dom vol ∧ (vol‘(𝑇‘𝑚)) ∈ ℝ))
127	11, 12, 13, 14, 15, 5, 16	vitalilem3 25511	. . . . . . . . . 10 ⊢ (𝜑 → Disj 𝑚 ∈ ℕ (𝑇‘𝑚))
128	127	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → Disj 𝑚 ∈ ℕ (𝑇‘𝑚))
129		eqid 2729	. . . . . . . . . 10 ⊢ seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇‘𝑚)))) = seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇‘𝑚))))
130	129, 99	voliun 25455	. . . . . . . . 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 685	. . . . . . . 8 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → (vol‘∪ 𝑚 ∈ ℕ (𝑇‘𝑚)) = sup(ran seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇‘𝑚)))), ℝ^, < ))
132	124, 131	eqtr3d 2766	. . . . . . 7 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → (vol‘∪ 𝑚 ∈ ℕ (𝑇‘𝑚)) = sup(ran seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇‘𝑚)))), ℝ^*, < ))
133	121, 132	breqtrrd 5135	. . . . . 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 13121	. . . . 5 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → 3 < (vol‘∪ 𝑚 ∈ ℕ (𝑇‘𝑚)))
135	17	simp3d 1144	. . . . . . . . 9 ⊢ (𝜑 → ∪ 𝑚 ∈ ℕ (𝑇‘𝑚) ⊆ (-1[,]2))
136	135	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → ∪ 𝑚 ∈ ℕ (𝑇‘𝑚) ⊆ (-1[,]2))
137		2re 12260	. . . . . . . . 9 ⊢ 2 ∈ ℝ
138		iccssre 13390	. . . . . . . . 9 ⊢ ((-1 ∈ ℝ ∧ 2 ∈ ℝ) → (-1[,]2) ⊆ ℝ)
139	22, 137, 138	mp2an 692	. . . . . . . 8 ⊢ (-1[,]2) ⊆ ℝ
140		ovolss 25386	. . . . . . . 8 ⊢ ((∪ 𝑚 ∈ ℕ (𝑇‘𝑚) ⊆ (-1[,]2) ∧ (-1[,]2) ⊆ ℝ) → (vol‘∪ 𝑚 ∈ ℕ (𝑇‘𝑚)) ≤ (vol‘(-1[,]2)))
141	136, 139, 140	sylancl 586	. . . . . . 7 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → (vol‘∪ 𝑚 ∈ ℕ (𝑇‘𝑚)) ≤ (vol*‘(-1[,]2)))
142		2cn 12261	. . . . . . . . 9 ⊢ 2 ∈ ℂ
143		ax-1cn 11126	. . . . . . . . 9 ⊢ 1 ∈ ℂ
144	142, 143	subnegi 11501	. . . . . . . 8 ⊢ (2 − -1) = (2 + 1)
145		neg1lt0 12174	. . . . . . . . . . 11 ⊢ -1 < 0
146		2pos 12289	. . . . . . . . . . 11 ⊢ 0 < 2
147	22, 38, 137	lttri 11300	. . . . . . . . . . 11 ⊢ ((-1 < 0 ∧ 0 < 2) → -1 < 2)
148	145, 146, 147	mp2an 692	. . . . . . . . . 10 ⊢ -1 < 2
149	22, 137, 148	ltleii 11297	. . . . . . . . 9 ⊢ -1 ≤ 2
150		ovolicc 25424	. . . . . . . . 9 ⊢ ((-1 ∈ ℝ ∧ 2 ∈ ℝ ∧ -1 ≤ 2) → (vol*‘(-1[,]2)) = (2 − -1))
151	22, 137, 149, 150	mp3an 1463	. . . . . . . 8 ⊢ (vol*‘(-1[,]2)) = (2 − -1)
152		df-3 12250	. . . . . . . 8 ⊢ 3 = (2 + 1)
153	144, 151, 152	3eqtr4i 2762	. . . . . . 7 ⊢ (vol*‘(-1[,]2)) = 3
154	141, 153	breqtrdi 5148	. . . . . 6 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → (vol‘∪ 𝑚 ∈ ℕ (𝑇‘𝑚)) ≤ 3)
155		xrlenlt 11239	. . . . . . 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 586	. . . . . 6 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → ((vol‘∪ 𝑚 ∈ ℕ (𝑇‘𝑚)) ≤ 3 ↔ ¬ 3 < (vol*‘∪ 𝑚 ∈ ℕ (𝑇‘𝑚))))
157	154, 156	mpbid 232	. . . . 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 25382	. . . . . . 7 ⊢ (ran 𝐹 ⊆ ℝ → 0 ≤ (vol*‘ran 𝐹))
160	20, 159	syl 17	. . . . . 6 ⊢ (𝜑 → 0 ≤ (vol*‘ran 𝐹))
161		0xr 11221	. . . . . . 7 ⊢ 0 ∈ ℝ^*
162		ovolcl 25379	. . . . . . . 8 ⊢ (ran 𝐹 ⊆ ℝ → (vol‘ran 𝐹) ∈ ℝ^)
163	20, 162	syl 17	. . . . . . 7 ⊢ (𝜑 → (vol‘ran 𝐹) ∈ ℝ^)
164		xrleloe 13104	. . . . . . 7 ⊢ ((0 ∈ ℝ^* ∧ (vol‘ran 𝐹) ∈ ℝ^) → (0 ≤ (vol‘ran 𝐹) ↔ (0 < (vol‘ran 𝐹) ∨ 0 = (vol*‘ran 𝐹))))
165	161, 163, 164	sylancr 587	. . . . . 6 ⊢ (𝜑 → (0 ≤ (vol‘ran 𝐹) ↔ (0 < (vol‘ran 𝐹) ∨ 0 = (vol*‘ran 𝐹))))
166	160, 165	mpbid 232	. . . . 5 ⊢ (𝜑 → (0 < (vol‘ran 𝐹) ∨ 0 = (vol‘ran 𝐹)))
167	166	ord 864	. . . 4 ⊢ (𝜑 → (¬ 0 < (vol‘ran 𝐹) → 0 = (vol‘ran 𝐹)))
168	158, 167	mpd 15	. . 3 ⊢ (𝜑 → 0 = (vol*‘ran 𝐹))
169	168	adantr 480	. 2 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 0 = (vol*‘ran 𝐹))
170	33, 169	eqtr4d 2767	1 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (vol*‘(𝑇‘𝑚)) = 0)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 ∀wral 3044 {crab 3405 ∖ cdif 3911 ∩ cin 3913 ⊆ wss 3914 ∅c0 4296 𝒫 cpw 4563 {csn 4589 ∪ ciun 4955 Disj wdisj 5074 class class class wbr 5107 {copab 5169 ↦ cmpt 5188 × cxp 5636 dom cdm 5638 ran crn 5639 Fn wfn 6506 ⟶wf 6507 –1-1-onto→wf1o 6510 ‘cfv 6511 (class class class)co 7387 / cqs 8670 supcsup 9391 ℂcc 11066 ℝcr 11067 0cc0 11068 1c1 11069 + caddc 11071 · cmul 11073 ℝ^*cxr 11207 < clt 11208 ≤ cle 11209 − cmin 11405 -cneg 11406 / cdiv 11835 ℕcn 12186 2c2 12241 3c3 12242 ℕ₀cn0 12442 ℚcq 12907 ℝ⁺crp 12951 [,]cicc 13309 ⌊cfl 13752 seqcseq 13966 vol*covol 25363 volcvol 25364

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-inf2 9594 ax-cc 10388 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-pre-sup 11146

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-disj 5075 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-se 5592 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-isom 6520 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-of 7653 df-om 7843 df-1st 7968 df-2nd 7969 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-2o 8435 df-er 8671 df-ec 8673 df-qs 8677 df-map 8801 df-pm 8802 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-fi 9362 df-sup 9393 df-inf 9394 df-oi 9463 df-dju 9854 df-card 9892 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-div 11836 df-nn 12187 df-2 12249 df-3 12250 df-n0 12443 df-z 12530 df-uz 12794 df-q 12908 df-rp 12952 df-xneg 13072 df-xadd 13073 df-xmul 13074 df-ioo 13310 df-ico 13312 df-icc 13313 df-fz 13469 df-fzo 13616 df-fl 13754 df-seq 13967 df-exp 14027 df-hash 14296 df-cj 15065 df-re 15066 df-im 15067 df-sqrt 15201 df-abs 15202 df-clim 15454 df-rlim 15455 df-sum 15653 df-rest 17385 df-topgen 17406 df-psmet 21256 df-xmet 21257 df-met 21258 df-bl 21259 df-mopn 21260 df-top 22781 df-topon 22798 df-bases 22833 df-cmp 23274 df-ovol 25365 df-vol 25366

This theorem is referenced by: vitalilem5 25513

W3C validator