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Theorem vitalilem4 25545

Description: Lemma for vitali 25547. (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 6840	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → (𝐺‘𝑛) = (𝐺‘𝑚))
2	1	oveq2d 7385	. . . . . . . 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 3410	. . . . . 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 11135	. . . . . . 7 ⊢ ℝ ∈ V
7	6	rabex 5289	. . . . . 6 ⊢ {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑚)) ∈ ran 𝐹} ∈ V
8	4, 5, 7	fvmpt 6950	. . . . 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 6844	. . 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 25543	. . . . . . 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 13436	. . . . . 6 ⊢ (0[,]1) ⊆ ℝ
20	18, 19	sstrdi 3956	. . . . 5 ⊢ (𝜑 → ran 𝐹 ⊆ ℝ)
21	20	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ran 𝐹 ⊆ ℝ)
22		neg1rr 12148	. . . . . 6 ⊢ -1 ∈ ℝ
23		1re 11150	. . . . . 6 ⊢ 1 ∈ ℝ
24		iccssre 13366	. . . . . 6 ⊢ ((-1 ∈ ℝ ∧ 1 ∈ ℝ) → (-1[,]1) ⊆ ℝ)
25	22, 23, 24	mp2an 692	. . . . 5 ⊢ (-1[,]1) ⊆ ℝ
26		f1of 6782	. . . . . . . 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 7038	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐺‘𝑚) ∈ (ℚ ∩ (-1[,]1)))
29	28	elin2d 4164	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐺‘𝑚) ∈ (-1[,]1))
30	25, 29	sselid 3941	. . . 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 25445	. . 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 12242	. . . . . . . 8 ⊢ 3 ∈ ℝ
35	34	rexri 11208	. . . . . . 7 ⊢ 3 ∈ ℝ^*
36	35	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → 3 ∈ ℝ^)
37		3rp 12933	. . . . . . . . . . . . 13 ⊢ 3 ∈ ℝ⁺
38		0re 11152	. . . . . . . . . . . . . . . . . . . 20 ⊢ 0 ∈ ℝ
39		0le1 11677	. . . . . . . . . . . . . . . . . . . 20 ⊢ 0 ≤ 1
40		ovolicc 25457	. . . . . . . . . . . . . . . . . . . 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 12278	. . . . . . . . . . . . . . . . . . 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 25420	. . . . . . . . . . . . . . . . 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 12968	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → (vol‘ran 𝐹) ∈ ℝ⁺)
51		rpdivcl 12954	. . . . . . . . . . . . 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 12971	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → (3 / (vol‘ran 𝐹)) ∈ ℝ)
54	52	rpge0d 12975	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → 0 ≤ (3 / (vol‘ran 𝐹)))
55		flge0nn0 13758	. . . . . . . . . . 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 12457	. . . . . . . . . 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 12177	. . . . . . . 8 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → ((⌊‘(3 / (vol‘ran 𝐹))) + 1) ∈ ℝ)
60	59, 48	remulcld 11180	. . . . . . 7 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → (((⌊‘(3 / (vol‘ran 𝐹))) + 1) · (vol*‘ran 𝐹)) ∈ ℝ)
61	60	rexrd 11200	. . . . . 6 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → (((⌊‘(3 / (vol‘ran 𝐹))) + 1) · (vol‘ran 𝐹)) ∈ ℝ^)
62	6	elpw2 5284	. . . . . . . . . . . . . . . . . 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 3921	. . . . . . . . . . . . . . . 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 4196	. . . . . . . . . . . . . . . 16 ⊢ (ℚ ∩ (-1[,]1)) ⊆ ℚ
70		qssre 12894	. . . . . . . . . . . . . . . 16 ⊢ ℚ ⊆ ℝ
71	69, 70	sstri 3953	. . . . . . . . . . . . . . 15 ⊢ (ℚ ∩ (-1[,]1)) ⊆ ℝ
72		fss 6686	. . . . . . . . . . . . . . 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 7038	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐺‘𝑛) ∈ ℝ)
75		shftmbl 25472	. . . . . . . . . . . . 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 7068	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑇:ℕ⟶dom vol)
78	77	ffvelcdmda 7038	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑇‘𝑚) ∈ dom vol)
79	78	ralrimiva 3125	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑚 ∈ ℕ (𝑇‘𝑚) ∈ dom vol)
80		iunmbl 25487	. . . . . . . . 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 25465	. . . . . . . 8 ⊢ (∪ 𝑚 ∈ ℕ (𝑇‘𝑚) ∈ dom vol → ∪ 𝑚 ∈ ℕ (𝑇‘𝑚) ⊆ ℝ)
83		ovolcl 25412	. . . . . . . 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 13738	. . . . . . . 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 13023	. . . . . . 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 12812	. . . . . . . . . . 11 ⊢ ℕ = (ℤ_≥‘1)
92		1zzd 12540	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → 1 ∈ ℤ)
93		mblvol 25464	. . . . . . . . . . . . . . . . 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 7068	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → (𝑚 ∈ ℕ ↦ (vol‘(𝑇‘𝑚))):ℕ⟶ℝ)
101	100	ffvelcdmda 7038	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 0 < (vol*‘ran 𝐹)) ∧ 𝑘 ∈ ℕ) → ((𝑚 ∈ ℕ ↦ (vol‘(𝑇‘𝑚)))‘𝑘) ∈ ℝ)
102	91, 92, 101	serfre 13972	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇‘𝑚)))):ℕ⟶ℝ)
103	102	frnd 6678	. . . . . . . . 9 ⊢ ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → ran seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇‘𝑚)))) ⊆ ℝ)
104		ressxr 11194	. . . . . . . . 9 ⊢ ℝ ⊆ ℝ^*
105	103, 104	sstrdi 3956	. . . . . . . 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 5195	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → (𝑚 ∈ ℕ ↦ (vol‘(𝑇‘𝑚))) = (𝑚 ∈ ℕ ↦ (vol‘ran 𝐹)))
108		fconstmpt 5693	. . . . . . . . . . . . 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 13950	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇‘𝑚)))) = seq1( + , (ℕ × {(vol‘ran 𝐹)})))
111	110	fveq1d 6842	. . . . . . . . . 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 11178	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → (vol‘ran 𝐹) ∈ ℂ)
113		ser1const 13999	. . . . . . . . . . 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 6671	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇‘𝑚)))) Fn ℕ)
117		fnfvelrn 7034	. . . . . . . . . 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 13260	. . . . . . . 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 25464	. . . . . . . . 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 25544	. . . . . . . . . 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 25488	. . . . . . . . 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 5130	. . . . . 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 13097	. . . . 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 12236	. . . . . . . . 9 ⊢ 2 ∈ ℝ
138		iccssre 13366	. . . . . . . . 9 ⊢ ((-1 ∈ ℝ ∧ 2 ∈ ℝ) → (-1[,]2) ⊆ ℝ)
139	22, 137, 138	mp2an 692	. . . . . . . 8 ⊢ (-1[,]2) ⊆ ℝ
140		ovolss 25419	. . . . . . . 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 12237	. . . . . . . . 9 ⊢ 2 ∈ ℂ
143		ax-1cn 11102	. . . . . . . . 9 ⊢ 1 ∈ ℂ
144	142, 143	subnegi 11477	. . . . . . . 8 ⊢ (2 − -1) = (2 + 1)
145		neg1lt0 12150	. . . . . . . . . . 11 ⊢ -1 < 0
146		2pos 12265	. . . . . . . . . . 11 ⊢ 0 < 2
147	22, 38, 137	lttri 11276	. . . . . . . . . . 11 ⊢ ((-1 < 0 ∧ 0 < 2) → -1 < 2)
148	145, 146, 147	mp2an 692	. . . . . . . . . 10 ⊢ -1 < 2
149	22, 137, 148	ltleii 11273	. . . . . . . . 9 ⊢ -1 ≤ 2
150		ovolicc 25457	. . . . . . . . 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 12226	. . . . . . . 8 ⊢ 3 = (2 + 1)
153	144, 151, 152	3eqtr4i 2762	. . . . . . 7 ⊢ (vol*‘(-1[,]2)) = 3
154	141, 153	breqtrdi 5143	. . . . . 6 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → (vol‘∪ 𝑚 ∈ ℕ (𝑇‘𝑚)) ≤ 3)
155		xrlenlt 11215	. . . . . . 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 25415	. . . . . . 7 ⊢ (ran 𝐹 ⊆ ℝ → 0 ≤ (vol*‘ran 𝐹))
160	20, 159	syl 17	. . . . . 6 ⊢ (𝜑 → 0 ≤ (vol*‘ran 𝐹))
161		0xr 11197	. . . . . . 7 ⊢ 0 ∈ ℝ^*
162		ovolcl 25412	. . . . . . . 8 ⊢ (ran 𝐹 ⊆ ℝ → (vol‘ran 𝐹) ∈ ℝ^)
163	20, 162	syl 17	. . . . . . 7 ⊢ (𝜑 → (vol‘ran 𝐹) ∈ ℝ^)
164		xrleloe 13080	. . . . . . 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 3402 ∖ cdif 3908 ∩ cin 3910 ⊆ wss 3911 ∅c0 4292 𝒫 cpw 4559 {csn 4585 ∪ ciun 4951 Disj wdisj 5069 class class class wbr 5102 {copab 5164 ↦ cmpt 5183 × cxp 5629 dom cdm 5631 ran crn 5632 Fn wfn 6494 ⟶wf 6495 –1-1-onto→wf1o 6498 ‘cfv 6499 (class class class)co 7369 / cqs 8647 supcsup 9367 ℂcc 11042 ℝcr 11043 0cc0 11044 1c1 11045 + caddc 11047 · cmul 11049 ℝ^*cxr 11183 < clt 11184 ≤ cle 11185 − cmin 11381 -cneg 11382 / cdiv 11811 ℕcn 12162 2c2 12217 3c3 12218 ℕ₀cn0 12418 ℚcq 12883 ℝ⁺crp 12927 [,]cicc 13285 ⌊cfl 13728 seqcseq 13942 vol*covol 25396 volcvol 25397

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 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-inf2 9570 ax-cc 10364 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121 ax-pre-sup 11122

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 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4868 df-int 4907 df-iun 4953 df-disj 5070 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-isom 6508 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-of 7633 df-om 7823 df-1st 7947 df-2nd 7948 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-1o 8411 df-2o 8412 df-er 8648 df-ec 8650 df-qs 8654 df-map 8778 df-pm 8779 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fi 9338 df-sup 9369 df-inf 9370 df-oi 9439 df-dju 9830 df-card 9868 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-div 11812 df-nn 12163 df-2 12225 df-3 12226 df-n0 12419 df-z 12506 df-uz 12770 df-q 12884 df-rp 12928 df-xneg 13048 df-xadd 13049 df-xmul 13050 df-ioo 13286 df-ico 13288 df-icc 13289 df-fz 13445 df-fzo 13592 df-fl 13730 df-seq 13943 df-exp 14003 df-hash 14272 df-cj 15041 df-re 15042 df-im 15043 df-sqrt 15177 df-abs 15178 df-clim 15430 df-rlim 15431 df-sum 15629 df-rest 17361 df-topgen 17382 df-psmet 21288 df-xmet 21289 df-met 21290 df-bl 21291 df-mopn 21292 df-top 22814 df-topon 22831 df-bases 22866 df-cmp 23307 df-ovol 25398 df-vol 25399

This theorem is referenced by: vitalilem5 25546

W3C validator