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Theorem vitalilem4 25959

Description: Lemma for vitali 25961. (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⟶1-1onto(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 6905	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → (𝐺‘𝑛) = (𝐺‘𝑚))
2	1	oveq2d 7456	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → (𝑠 − (𝐺‘𝑛)) = (𝑠 − (𝐺‘𝑚)))
3	2	eleq1d 2849	. . . . . . 7 ⊢ (𝑛 = 𝑚 → ((𝑠 − (𝐺‘𝑛)) ∈ ran 𝐹 ↔ (𝑠 − (𝐺‘𝑚)) ∈ ran 𝐹))
4	3	rabbidv 3423	. . . . . 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 11315	. . . . . . 7 ⊢ ℝ ∈ V
7	6	rabex 5304	. . . . . 6 ⊢ {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑚)) ∈ ran 𝐹} ∈ V
8	4, 5, 7	fvmpt 7013	. . . . 5 ⊢ (𝑚 ∈ ℕ → (𝑇‘𝑚) = {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑚)) ∈ ran 𝐹})
9	8	adantl 487	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑇‘𝑚) = {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑚)) ∈ ran 𝐹})
10	9	fveq2d 6909	. . 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⟶1-1onto(ℚ ∩ ( -1[,]1)))
16		vitali.7	. . . . . . . 8 ⊢ (𝜑 → ¬ ran 𝐹 ∈ (𝒫 ℝ ∖ dom vol))
17	11, 12, 13, 14, 15, 5, 16	vitalilem2 25957	. . . . . . 7 ⊢ (𝜑 → (ran 𝐹 ⊆ (0[,]1) ∧ (0[,]1) ⊆ ∪ 𝑚 ∈ ℕ (𝑇‘𝑚) ∧ ∪ 𝑚 ∈ ℕ (𝑇‘𝑚) ⊆ ( -1[,]2)))
18	17	simp1d 1160	. . . . . 6 ⊢ (𝜑 → ran 𝐹 ⊆ (0[,]1))
19		unitssre 13654	. . . . . 6 ⊢ (0[,]1) ⊆ ℝ
20	18, 19	sstrdi 3925	. . . . 5 ⊢ (𝜑 → ran 𝐹 ⊆ ℝ)
21	20	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ran 𝐹 ⊆ ℝ)
22		neg1rr 12330	. . . . . 6 ⊢ -1 ∈ ℝ
23		1re 11332	. . . . . 6 ⊢ 1 ∈ ℝ
24		iccssre 13584	. . . . . 6 ⊢ (( -1 ∈ ℝ ∧ 1 ∈ ℝ) → ( -1[,]1) ⊆ ℝ)
25	22, 23, 24	mp2an 705	. . . . 5 ⊢ ( -1[,]1) ⊆ ℝ
26		f1of 6843	. . . . . . . 8 ⊢ (𝐺:ℕ⟶1-1onto(ℚ ∩ ( -1[,]1)) → 𝐺:ℕ⟶(ℚ ∩ ( -1[,]1)))
27	15, 26	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝐺:ℕ⟶(ℚ ∩ ( -1[,]1)))
28	27	ffvelcdmda 7105	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐺‘𝑚) ∈ (ℚ ∩ ( -1[,]1)))
29	28	elin2d 4105	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐺‘𝑚) ∈ ( -1[,]1))
30	25, 29	sselid 3911	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐺‘𝑚) ∈ ℝ)
31		eqidd 2764	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑚)) ∈ ran 𝐹} = {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑚)) ∈ ran 𝐹})
32	21, 30, 31	ovolshft 25859	. . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (vol‘ran 𝐹) = (vol‘{𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑚)) ∈ ran 𝐹}))
33	10, 32	eqtr4d 2801	. 2 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (vol‘(𝑇‘𝑚)) = (vol‘ran 𝐹))
34		3re 12447	. . . . . . . 8 ⊢ 3 ∈ ℝ
35	34	rexri 11391	. . . . . . 7 ⊢ 3 ∈ ℝ^*
36	35	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → 3 ∈ ℝ^)
37		3rp 13150	. . . . . . . . . . . . 13 ⊢ 3 ∈ ℝ⁺
38		0re 11334	. . . . . . . . . . . . . . . . . . . 20 ⊢ 0 ∈ ℝ
39		0le1 11863	. . . . . . . . . . . . . . . . . . . 20 ⊢ 0 ≤ 1
40		ovolicc 25871	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ ∧ 0 ≤ 1) → (vol*‘(0[,]1)) = (1 − 0))
41	38, 23, 39, 40	mp3an 1491	. . . . . . . . . . . . . . . . . . 19 ⊢ (vol*‘(0[,]1)) = (1 − 0)
42		1m0e1 12486	. . . . . . . . . . . . . . . . . . 19 ⊢ (1 − 0) = 1
43	41, 42	eqtri 2786	. . . . . . . . . . . . . . . . . 18 ⊢ (vol*‘(0[,]1)) = 1
44	43, 23	eqeltri 2860	. . . . . . . . . . . . . . . . 17 ⊢ (vol*‘(0[,]1)) ∈ ℝ
45		ovolsscl 25834	. . . . . . . . . . . . . . . . 17 ⊢ ((ran 𝐹 ⊆ (0[,]1) ∧ (0[,]1) ⊆ ℝ ∧ (vol‘(0[,]1)) ∈ ℝ) → (vol‘ran 𝐹) ∈ ℝ)
46	19, 44, 45	mp3an23 1483	. . . . . . . . . . . . . . . 16 ⊢ (ran 𝐹 ⊆ (0[,]1) → (vol*‘ran 𝐹) ∈ ℝ)
47	18, 46	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (vol*‘ran 𝐹) ∈ ℝ)
48	47	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → (vol‘ran 𝐹) ∈ ℝ)
49		simpr 490	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → 0 < (vol‘ran 𝐹))
50	48, 49	elrpd 13185	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → (vol‘ran 𝐹) ∈ ℝ⁺)
51		rpdivcl 13171	. . . . . . . . . . . . 13 ⊢ ((3 ∈ ℝ⁺ ∧ (vol‘ran 𝐹) ∈ ℝ⁺) → (3 / (vol‘ran 𝐹)) ∈ ℝ⁺)
52	37, 50, 51	sylancr 599	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → (3 / (vol‘ran 𝐹)) ∈ ℝ⁺)
53	52	rpred 13188	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → (3 / (vol‘ran 𝐹)) ∈ ℝ)
54	52	rpge0d 13192	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → 0 ≤ (3 / (vol‘ran 𝐹)))
55		flge0nn0 13984	. . . . . . . . . . 11 ⊢ (((3 / (vol‘ran 𝐹)) ∈ ℝ ∧ 0 ≤ (3 / (vol‘ran 𝐹))) → (⌊‘(3 / (vol*‘ran 𝐹))) ∈ ℕ₀)
56	53, 54, 55	syl2anc 596	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → (⌊‘(3 / (vol‘ran 𝐹))) ∈ ℕ₀)
57		nn0p1nn 12669	. . . . . . . . . 10 ⊢ ((⌊‘(3 / (vol‘ran 𝐹))) ∈ ℕ₀ → ((⌊‘(3 / (vol‘ran 𝐹))) + 1) ∈ ℕ)
58	56, 57	syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → ((⌊‘(3 / (vol‘ran 𝐹))) + 1) ∈ ℕ)
59	58	nnred 12374	. . . . . . . 8 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → ((⌊‘(3 / (vol‘ran 𝐹))) + 1) ∈ ℝ)
60	59, 48	remulcld 11363	. . . . . . 7 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → (((⌊‘(3 / (vol‘ran 𝐹))) + 1) · (vol*‘ran 𝐹)) ∈ ℝ)
61	60	rexrd 11383	. . . . . 6 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → (((⌊‘(3 / (vol‘ran 𝐹))) + 1) · (vol‘ran 𝐹)) ∈ ℝ^)
62	6	elpw2 5300	. . . . . . . . . . . . . . . . . 18 ⊢ (ran 𝐹 ∈ 𝒫 ℝ ↔ ran 𝐹 ⊆ ℝ)
63	20, 62	sylibr 237	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ran 𝐹 ∈ 𝒫 ℝ)
64	63	anim1i 627	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ¬ ran 𝐹 ∈ dom vol) → (ran 𝐹 ∈ 𝒫 ℝ ∧ ¬ ran 𝐹 ∈ dom vol))
65		eldif 4052	. . . . . . . . . . . . . . . 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 418	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (¬ ran 𝐹 ∈ dom vol → ran 𝐹 ∈ (𝒫 ℝ ∖ dom vol)))
68	16, 67	mt3d 149	. . . . . . . . . . . . 13 ⊢ (𝜑 → ran 𝐹 ∈ dom vol)
69		inss1 4136	. . . . . . . . . . . . . . . 16 ⊢ (ℚ ∩ ( -1[,]1)) ⊆ ℚ
70		qssre 13110	. . . . . . . . . . . . . . . 16 ⊢ ℚ ⊆ ℝ
71	69, 70	sstri 3922	. . . . . . . . . . . . . . 15 ⊢ (ℚ ∩ ( -1[,]1)) ⊆ ℝ
72		fss 6745	. . . . . . . . . . . . . . 15 ⊢ ((𝐺:ℕ⟶(ℚ ∩ ( -1[,]1)) ∧ (ℚ ∩ ( -1[,]1)) ⊆ ℝ) → 𝐺:ℕ⟶ℝ)
73	27, 71, 72	sylancl 598	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐺:ℕ⟶ℝ)
74	73	ffvelcdmda 7105	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐺‘𝑛) ∈ ℝ)
75		shftmbl 25886	. . . . . . . . . . . . 13 ⊢ ((ran 𝐹 ∈ dom vol ∧ (𝐺‘𝑛) ∈ ℝ) → {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑛)) ∈ ran 𝐹} ∈ dom vol)
76	68, 74, 75	syl2an2r 698	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑛)) ∈ ran 𝐹} ∈ dom vol)
77	76, 5	fmptd 7135	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑇:ℕ⟶dom vol)
78	77	ffvelcdmda 7105	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑇‘𝑚) ∈ dom vol)
79	78	ralrimiva 3158	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑚 ∈ ℕ (𝑇‘𝑚) ∈ dom vol)
80		iunmbl 25901	. . . . . . . . 9 ⊢ (∀𝑚 ∈ ℕ (𝑇‘𝑚) ∈ dom vol → ∪ 𝑚 ∈ ℕ (𝑇‘𝑚) ∈ dom vol)
81	79, 80	syl 18	. . . . . . . 8 ⊢ (𝜑 → ∪ 𝑚 ∈ ℕ (𝑇‘𝑚) ∈ dom vol)
82		mblss 25879	. . . . . . . 8 ⊢ (∪ 𝑚 ∈ ℕ (𝑇‘𝑚) ∈ dom vol → ∪ 𝑚 ∈ ℕ (𝑇‘𝑚) ⊆ ℝ)
83		ovolcl 25826	. . . . . . . 8 ⊢ (∪ 𝑚 ∈ ℕ (𝑇‘𝑚) ⊆ ℝ → (vol‘∪ 𝑚 ∈ ℕ (𝑇‘𝑚)) ∈ ℝ^)
84	81, 82, 83	3syl 19	. . . . . . 7 ⊢ (𝜑 → (vol‘∪ 𝑚 ∈ ℕ (𝑇‘𝑚)) ∈ ℝ^)
85	84	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → (vol‘∪ 𝑚 ∈ ℕ (𝑇‘𝑚)) ∈ ℝ^*)
86		flltp1 13964	. . . . . . . 8 ⊢ ((3 / (vol‘ran 𝐹)) ∈ ℝ → (3 / (vol‘ran 𝐹)) < ((⌊‘(3 / (vol*‘ran 𝐹))) + 1))
87	53, 86	syl 18	. . . . . . 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 13240	. . . . . . 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 13028	. . . . . . . . . . 11 ⊢ ℕ = (ℤ_≥‘1)
92		1zzd 12751	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → 1 ∈ ℤ)
93		mblvol 25878	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑇‘𝑚) ∈ dom vol → (vol‘(𝑇‘𝑚)) = (vol*‘(𝑇‘𝑚)))
94	78, 93	syl 18	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (vol‘(𝑇‘𝑚)) = (vol*‘(𝑇‘𝑚)))
95	94, 33	eqtrd 2798	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (vol‘(𝑇‘𝑚)) = (vol*‘ran 𝐹))
96	47	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (vol*‘ran 𝐹) ∈ ℝ)
97	95, 96	eqeltrd 2864	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (vol‘(𝑇‘𝑚)) ∈ ℝ)
98	97	adantlr 728	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 0 < (vol*‘ran 𝐹)) ∧ 𝑚 ∈ ℕ) → (vol‘(𝑇‘𝑚)) ∈ ℝ)
99		eqid 2763	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℕ ↦ (vol‘(𝑇‘𝑚))) = (𝑚 ∈ ℕ ↦ (vol‘(𝑇‘𝑚)))
100	98, 99	fmptd 7135	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → (𝑚 ∈ ℕ ↦ (vol‘(𝑇‘𝑚))):ℕ⟶ℝ)
101	100	ffvelcdmda 7105	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 0 < (vol*‘ran 𝐹)) ∧ 𝑘 ∈ ℕ) → ((𝑚 ∈ ℕ ↦ (vol‘(𝑇‘𝑚)))‘𝑘) ∈ ℝ)
102	91, 92, 101	serfre 14198	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇‘𝑚)))):ℕ⟶ℝ)
103	102	frnd 6737	. . . . . . . . 9 ⊢ ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → ran seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇‘𝑚)))) ⊆ ℝ)
104		ressxr 11377	. . . . . . . . 9 ⊢ ℝ ⊆ ℝ^*
105	103, 104	sstrdi 3925	. . . . . . . 8 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → ran seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇‘𝑚)))) ⊆ ℝ^)
106	95	adantlr 728	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 0 < (vol‘ran 𝐹)) ∧ 𝑚 ∈ ℕ) → (vol‘(𝑇‘𝑚)) = (vol‘ran 𝐹))
107	106	mpteq2dva 5201	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → (𝑚 ∈ ℕ ↦ (vol‘(𝑇‘𝑚))) = (𝑚 ∈ ℕ ↦ (vol‘ran 𝐹)))
108		fconstmpt 5717	. . . . . . . . . . . . 13 ⊢ (ℕ × {(vol‘ran 𝐹)}) = (𝑚 ∈ ℕ ↦ (vol‘ran 𝐹))
109	107, 108	eqtr4di 2816	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → (𝑚 ∈ ℕ ↦ (vol‘(𝑇‘𝑚))) = (ℕ × {(vol‘ran 𝐹)}))
110	109	seqeq3d 14176	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇‘𝑚)))) = seq1( + , (ℕ × {(vol‘ran 𝐹)})))
111	110	fveq1d 6907	. . . . . . . . . 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 11361	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → (vol‘ran 𝐹) ∈ ℂ)
113		ser1const 14225	. . . . . . . . . . 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 596	. . . . . . . . . 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 2798	. . . . . . . . 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 6729	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇‘𝑚)))) Fn ℕ)
117		fnfvelrn 7101	. . . . . . . . . 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 596	. . . . . . . . 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 2865	. . . . . . . 8 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → (((⌊‘(3 / (vol‘ran 𝐹))) + 1) · (vol*‘ran 𝐹)) ∈ ran seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇‘𝑚)))))
120		supxrub 13478	. . . . . . . 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 596	. . . . . . 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 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → ∪ 𝑚 ∈ ℕ (𝑇‘𝑚) ∈ dom vol)
123		mblvol 25878	. . . . . . . . 9 ⊢ (∪ 𝑚 ∈ ℕ (𝑇‘𝑚) ∈ dom vol → (vol‘∪ 𝑚 ∈ ℕ (𝑇‘𝑚)) = (vol*‘∪ 𝑚 ∈ ℕ (𝑇‘𝑚)))
124	122, 123	syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → (vol‘∪ 𝑚 ∈ ℕ (𝑇‘𝑚)) = (vol‘∪ 𝑚 ∈ ℕ (𝑇‘𝑚)))
125	78, 97	jca 521	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑇‘𝑚) ∈ dom vol ∧ (vol‘(𝑇‘𝑚)) ∈ ℝ))
126	125	ralrimiva 3158	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑚 ∈ ℕ ((𝑇‘𝑚) ∈ dom vol ∧ (vol‘(𝑇‘𝑚)) ∈ ℝ))
127	11, 12, 13, 14, 15, 5, 16	vitalilem3 25958	. . . . . . . . . 10 ⊢ (𝜑 → Disj 𝑚 ∈ ℕ (𝑇‘𝑚))
128	127	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → Disj 𝑚 ∈ ℕ (𝑇‘𝑚))
129		eqid 2763	. . . . . . . . . 10 ⊢ seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇‘𝑚)))) = seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇‘𝑚))))
130	129, 99	voliun 25902	. . . . . . . . 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 698	. . . . . . . 8 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → (vol‘∪ 𝑚 ∈ ℕ (𝑇‘𝑚)) = sup(ran seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇‘𝑚)))), ℝ^, < ))
132	124, 131	eqtr3d 2800	. . . . . . 7 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → (vol‘∪ 𝑚 ∈ ℕ (𝑇‘𝑚)) = sup(ran seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇‘𝑚)))), ℝ^*, < ))
133	121, 132	breqtrrd 5136	. . . . . 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 13314	. . . . 5 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → 3 < (vol‘∪ 𝑚 ∈ ℕ (𝑇‘𝑚)))
135	17	simp3d 1162	. . . . . . . . 9 ⊢ (𝜑 → ∪ 𝑚 ∈ ℕ (𝑇‘𝑚) ⊆ ( -1[,]2))
136	135	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → ∪ 𝑚 ∈ ℕ (𝑇‘𝑚) ⊆ ( -1[,]2))
137		2re 12441	. . . . . . . . 9 ⊢ 2 ∈ ℝ
138		iccssre 13584	. . . . . . . . 9 ⊢ (( -1 ∈ ℝ ∧ 2 ∈ ℝ) → ( -1[,]2) ⊆ ℝ)
139	22, 137, 138	mp2an 705	. . . . . . . 8 ⊢ ( -1[,]2) ⊆ ℝ
140		ovolss 25833	. . . . . . . 8 ⊢ ((∪ 𝑚 ∈ ℕ (𝑇‘𝑚) ⊆ ( -1[,]2) ∧ ( -1[,]2) ⊆ ℝ) → (vol‘∪ 𝑚 ∈ ℕ (𝑇‘𝑚)) ≤ (vol‘( -1[,]2)))
141	136, 139, 140	sylancl 598	. . . . . . 7 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → (vol‘∪ 𝑚 ∈ ℕ (𝑇‘𝑚)) ≤ (vol*‘( -1[,]2)))
142		2cn 12442	. . . . . . . . 9 ⊢ 2 ∈ ℂ
143		ax-1cn 11282	. . . . . . . . 9 ⊢ 1 ∈ ℂ
144	142, 143	subnegi 11661	. . . . . . . 8 ⊢ (2 − -1) = (2 + 1)
145		neg1lt0 12332	. . . . . . . . . . 11 ⊢ -1 < 0
146		2pos 12471	. . . . . . . . . . 11 ⊢ 0 < 2
147	22, 38, 137	lttri 11460	. . . . . . . . . . 11 ⊢ (( -1 < 0 ∧ 0 < 2) → -1 < 2)
148	145, 146, 147	mp2an 705	. . . . . . . . . 10 ⊢ -1 < 2
149	22, 137, 148	ltleii 11457	. . . . . . . . 9 ⊢ -1 ≤ 2
150		ovolicc 25871	. . . . . . . . 9 ⊢ (( -1 ∈ ℝ ∧ 2 ∈ ℝ ∧ -1 ≤ 2) → (vol*‘( -1[,]2)) = (2 − -1))
151	22, 137, 149, 150	mp3an 1491	. . . . . . . 8 ⊢ (vol*‘( -1[,]2)) = (2 − -1)
152		df-3 12430	. . . . . . . 8 ⊢ 3 = (2 + 1)
153	144, 151, 152	3eqtr4i 2796	. . . . . . 7 ⊢ (vol*‘( -1[,]2)) = 3
154	141, 153	breqtrdi 5149	. . . . . 6 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → (vol‘∪ 𝑚 ∈ ℕ (𝑇‘𝑚)) ≤ 3)
155		xrlenlt 11398	. . . . . . 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 598	. . . . . 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 829	. . . 4 ⊢ (𝜑 → ¬ 0 < (vol*‘ran 𝐹))
159		ovolge0 25829	. . . . . . 7 ⊢ (ran 𝐹 ⊆ ℝ → 0 ≤ (vol*‘ran 𝐹))
160	20, 159	syl 18	. . . . . 6 ⊢ (𝜑 → 0 ≤ (vol*‘ran 𝐹))
161		0xr 11380	. . . . . . 7 ⊢ 0 ∈ ℝ^*
162		ovolcl 25826	. . . . . . . 8 ⊢ (ran 𝐹 ⊆ ℝ → (vol‘ran 𝐹) ∈ ℝ^)
163	20, 162	syl 18	. . . . . . 7 ⊢ (𝜑 → (vol‘ran 𝐹) ∈ ℝ^)
164		xrleloe 13297	. . . . . . 7 ⊢ ((0 ∈ ℝ^* ∧ (vol‘ran 𝐹) ∈ ℝ^) → (0 ≤ (vol‘ran 𝐹) ↔ (0 < (vol‘ran 𝐹) ∨ 0 = (vol*‘ran 𝐹))))
165	161, 163, 164	sylancr 599	. . . . . 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 878	. . . 4 ⊢ (𝜑 → (¬ 0 < (vol‘ran 𝐹) → 0 = (vol‘ran 𝐹)))
168	158, 167	mpd 16	. . 3 ⊢ (𝜑 → 0 = (vol*‘ran 𝐹))
169	168	adantr 486	. 2 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 0 = (vol*‘ran 𝐹))
170	33, 169	eqtr4d 2801	1 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (vol*‘(𝑇‘𝑚)) = 0)

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 = wceq 1570 ∈ wcel 2147 ≠ wne 2959 ∀wral 3080 {crab 3416 ⊆ wss 3900 ∖ cdif 4048 ∩ cin 4092 ∅c0 4281 𝒫 cpw 4560 {csn 4588 ∪ ciun 4954 Disj wdisj 5073 class class class wbr 5106 {copab 5170 ↦ cmpt 5189 × cxp 5653 dom cdm 5655 ran crn 5656 Fn wfn 6550 ⟶wf 6551 ⟶1-1ontowf1o 6554 ‘cfv 6555 (class class class)co 7440 / cqs 8741 supcsup 9456 ℂcc 11222 ℝcr 11223 0cc0 11224 1c1 11225 + caddc 11227 · cmul 11229 ℝ^*cxr 11366 < clt 11367 ≤ cle 11368 − cmin 11565 -cneg 11566 / cdiv 11997 ℕcn 12359 2c2 12421 3c3 12422 ℕ₀cn0 12630 ℚcq 13099 ℝ⁺crp 13144 [,]cicc 13503 ⌊cfl 13954 seqcseq 14168 vol*covol 25810 volcvol 25811

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 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5235 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7772 ax-inf2 9666 ax-cc 10537 ax-cnex 11280 ax-resscn 11281 ax-1cn 11282 ax-icn 11283 ax-addcl 11284 ax-addrcl 11285 ax-mulcl 11286 ax-mulrcl 11287 ax-mulcom 11288 ax-addass 11289 ax-mulass 11290 ax-distr 11291 ax-i2m1 11292 ax-1ne0 11293 ax-1rid 11294 ax-rnegex 11295 ax-rrecex 11296 ax-cnre 11297 ax-pre-lttri 11298 ax-pre-lttrn 11299 ax-pre-ltadd 11300 ax-pre-mulgt0 11301 ax-pre-sup 11302

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 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4839 df-uni 4871 df-int 4911 df-iun 4956 df-disj 5074 df-br 5107 df-opab 5171 df-mpt 5190 df-tr 5216 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6319 df-ord 6380 df-on 6381 df-lim 6382 df-suc 6383 df-iota 6509 df-fun 6557 df-fn 6558 df-f 6559 df-f1 6560 df-fo 6561 df-f1o 6562 df-fv 6563 df-isom 6564 df-riota 7397 df-ov 7443 df-oprab 7444 df-mpo 7445 df-of 7714 df-om 7899 df-1st 8022 df-2nd 8023 df-frecs 8324 df-wrecs 8355 df-recs 8404 df-rdg 8443 df-1o 8501 df-2o 8502 df-er 8742 df-ec 8744 df-qs 8748 df-map 8874 df-pm 8875 df-en 8999 df-dom 9000 df-sdom 9001 df-fin 9002 df-fi 9427 df-sup 9458 df-inf 9459 df-oi 9528 df-dju 10006 df-card 10044 df-pnf 11369 df-mnf 11370 df-xr 11371 df-ltxr 11372 df-le 11373 df-sub 11567 df-neg 11568 df-div 11998 df-nn 12360 df-2 12429 df-3 12430 df-n0 12631 df-z 12718 df-uz 12990 df-q 13100 df-rp 13145 df-xneg 13265 df-xadd 13266 df-xmul 13267 df-ioo 13504 df-ico 13506 df-icc 13507 df-fz 13664 df-fzo 13813 df-fl 13956 df-seq 14169 df-exp 14229 df-hash 14499 df-cj 15292 df-re 15293 df-im 15294 df-sqrt 15428 df-abs 15429 df-clim 15681 df-rlim 15682 df-sum 15880 df-rest 17619 df-topgen 17640 df-psmet 21697 df-xmet 21698 df-met 21699 df-bl 21700 df-mopn 21701 df-top 23239 df-topon 23256 df-bases 23291 df-cmp 23732 df-ovol 25812 df-vol 25813

This theorem is used by: vitalilem5 25960

W3C validator