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Theorem vitalilem4 25932

Description: Lemma for vitali 25934. (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 6885	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → (𝐺‘𝑛) = (𝐺‘𝑚))
2	1	oveq2d 7436	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → (𝑠 − (𝐺‘𝑛)) = (𝑠 − (𝐺‘𝑚)))
3	2	eleq1d 2845	. . . . . . 7 ⊢ (𝑛 = 𝑚 → ((𝑠 − (𝐺‘𝑛)) ∈ ran 𝐹 ↔ (𝑠 − (𝐺‘𝑚)) ∈ ran 𝐹))
4	3	rabbidv 3419	. . . . . 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 11291	. . . . . . 7 ⊢ ℝ ∈ V
7	6	rabex 5299	. . . . . 6 ⊢ {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑚)) ∈ ran 𝐹} ∈ V
8	4, 5, 7	fvmpt 6993	. . . . 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 6889	. . 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 25930	. . . . . . 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 13630	. . . . . 6 ⊢ (0[,]1) ⊆ ℝ
20	18, 19	sstrdi 3920	. . . . 5 ⊢ (𝜑 → ran 𝐹 ⊆ ℝ)
21	20	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ran 𝐹 ⊆ ℝ)
22		neg1rr 12306	. . . . . 6 ⊢ -1 ∈ ℝ
23		1re 11308	. . . . . 6 ⊢ 1 ∈ ℝ
24		iccssre 13560	. . . . . 6 ⊢ (( -1 ∈ ℝ ∧ 1 ∈ ℝ) → ( -1[,]1) ⊆ ℝ)
25	22, 23, 24	mp2an 705	. . . . 5 ⊢ ( -1[,]1) ⊆ ℝ
26		f1of 6824	. . . . . . . 8 ⊢ (𝐺:ℕ–1-1-onto→(ℚ ∩ ( -1[,]1)) → 𝐺:ℕ⟶(ℚ ∩ ( -1[,]1)))
27	15, 26	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝐺:ℕ⟶(ℚ ∩ ( -1[,]1)))
28	27	ffvelcdmda 7084	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐺‘𝑚) ∈ (ℚ ∩ ( -1[,]1)))
29	28	elin2d 4102	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐺‘𝑚) ∈ ( -1[,]1))
30	25, 29	sselid 3906	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐺‘𝑚) ∈ ℝ)
31		eqidd 2761	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑚)) ∈ ran 𝐹} = {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑚)) ∈ ran 𝐹})
32	21, 30, 31	ovolshft 25832	. . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (vol‘ran 𝐹) = (vol‘{𝑠 ∈ ℝ ∣ (𝑠 − (𝐺‘𝑚)) ∈ ran 𝐹}))
33	10, 32	eqtr4d 2798	. 2 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (vol‘(𝑇‘𝑚)) = (vol‘ran 𝐹))
34		3re 12423	. . . . . . . 8 ⊢ 3 ∈ ℝ
35	34	rexri 11367	. . . . . . 7 ⊢ 3 ∈ ℝ^*
36	35	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → 3 ∈ ℝ^)
37		3rp 13126	. . . . . . . . . . . . 13 ⊢ 3 ∈ ℝ⁺
38		0re 11310	. . . . . . . . . . . . . . . . . . . 20 ⊢ 0 ∈ ℝ
39		0le1 11839	. . . . . . . . . . . . . . . . . . . 20 ⊢ 0 ≤ 1
40		ovolicc 25844	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ ∧ 0 ≤ 1) → (vol*‘(0[,]1)) = (1 − 0))
41	38, 23, 39, 40	mp3an 1490	. . . . . . . . . . . . . . . . . . 19 ⊢ (vol*‘(0[,]1)) = (1 − 0)
42		1m0e1 12462	. . . . . . . . . . . . . . . . . . 19 ⊢ (1 − 0) = 1
43	41, 42	eqtri 2783	. . . . . . . . . . . . . . . . . 18 ⊢ (vol*‘(0[,]1)) = 1
44	43, 23	eqeltri 2856	. . . . . . . . . . . . . . . . 17 ⊢ (vol*‘(0[,]1)) ∈ ℝ
45		ovolsscl 25807	. . . . . . . . . . . . . . . . 17 ⊢ ((ran 𝐹 ⊆ (0[,]1) ∧ (0[,]1) ⊆ ℝ ∧ (vol‘(0[,]1)) ∈ ℝ) → (vol‘ran 𝐹) ∈ ℝ)
46	19, 44, 45	mp3an23 1482	. . . . . . . . . . . . . . . 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 13161	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → (vol‘ran 𝐹) ∈ ℝ⁺)
51		rpdivcl 13147	. . . . . . . . . . . . 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 13164	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → (3 / (vol‘ran 𝐹)) ∈ ℝ)
54	52	rpge0d 13168	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → 0 ≤ (3 / (vol‘ran 𝐹)))
55		flge0nn0 13960	. . . . . . . . . . 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 12645	. . . . . . . . . 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 12350	. . . . . . . 8 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → ((⌊‘(3 / (vol‘ran 𝐹))) + 1) ∈ ℝ)
60	59, 48	remulcld 11339	. . . . . . 7 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → (((⌊‘(3 / (vol‘ran 𝐹))) + 1) · (vol*‘ran 𝐹)) ∈ ℝ)
61	60	rexrd 11359	. . . . . 6 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → (((⌊‘(3 / (vol‘ran 𝐹))) + 1) · (vol‘ran 𝐹)) ∈ ℝ^)
62	6	elpw2 5295	. . . . . . . . . . . . . . . . . 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 4049	. . . . . . . . . . . . . . . 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 4133	. . . . . . . . . . . . . . . 16 ⊢ (ℚ ∩ ( -1[,]1)) ⊆ ℚ
70		qssre 13086	. . . . . . . . . . . . . . . 16 ⊢ ℚ ⊆ ℝ
71	69, 70	sstri 3917	. . . . . . . . . . . . . . 15 ⊢ (ℚ ∩ ( -1[,]1)) ⊆ ℝ
72		fss 6726	. . . . . . . . . . . . . . 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 7084	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐺‘𝑛) ∈ ℝ)
75		shftmbl 25859	. . . . . . . . . . . . 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 7114	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑇:ℕ⟶dom vol)
78	77	ffvelcdmda 7084	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑇‘𝑚) ∈ dom vol)
79	78	ralrimiva 3154	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑚 ∈ ℕ (𝑇‘𝑚) ∈ dom vol)
80		iunmbl 25874	. . . . . . . . 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 25852	. . . . . . . 8 ⊢ (∪ 𝑚 ∈ ℕ (𝑇‘𝑚) ∈ dom vol → ∪ 𝑚 ∈ ℕ (𝑇‘𝑚) ⊆ ℝ)
83		ovolcl 25799	. . . . . . . 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 13940	. . . . . . . 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 13216	. . . . . . 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 13004	. . . . . . . . . . 11 ⊢ ℕ = (ℤ_≥‘1)
92		1zzd 12727	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → 1 ∈ ℤ)
93		mblvol 25851	. . . . . . . . . . . . . . . . 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 2795	. . . . . . . . . . . . . . 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 2860	. . . . . . . . . . . . . 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 2760	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℕ ↦ (vol‘(𝑇‘𝑚))) = (𝑚 ∈ ℕ ↦ (vol‘(𝑇‘𝑚)))
100	98, 99	fmptd 7114	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → (𝑚 ∈ ℕ ↦ (vol‘(𝑇‘𝑚))):ℕ⟶ℝ)
101	100	ffvelcdmda 7084	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 0 < (vol*‘ran 𝐹)) ∧ 𝑘 ∈ ℕ) → ((𝑚 ∈ ℕ ↦ (vol‘(𝑇‘𝑚)))‘𝑘) ∈ ℝ)
102	91, 92, 101	serfre 14174	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇‘𝑚)))):ℕ⟶ℝ)
103	102	frnd 6718	. . . . . . . . 9 ⊢ ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → ran seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇‘𝑚)))) ⊆ ℝ)
104		ressxr 11353	. . . . . . . . 9 ⊢ ℝ ⊆ ℝ^*
105	103, 104	sstrdi 3920	. . . . . . . 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 5197	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → (𝑚 ∈ ℕ ↦ (vol‘(𝑇‘𝑚))) = (𝑚 ∈ ℕ ↦ (vol‘ran 𝐹)))
108		fconstmpt 5712	. . . . . . . . . . . . 13 ⊢ (ℕ × {(vol‘ran 𝐹)}) = (𝑚 ∈ ℕ ↦ (vol‘ran 𝐹))
109	107, 108	eqtr4di 2813	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → (𝑚 ∈ ℕ ↦ (vol‘(𝑇‘𝑚))) = (ℕ × {(vol‘ran 𝐹)}))
110	109	seqeq3d 14152	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇‘𝑚)))) = seq1( + , (ℕ × {(vol‘ran 𝐹)})))
111	110	fveq1d 6887	. . . . . . . . . 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 11337	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → (vol‘ran 𝐹) ∈ ℂ)
113		ser1const 14201	. . . . . . . . . . 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 2795	. . . . . . . . 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 6710	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇‘𝑚)))) Fn ℕ)
117		fnfvelrn 7080	. . . . . . . . . 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 2861	. . . . . . . 8 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → (((⌊‘(3 / (vol‘ran 𝐹))) + 1) · (vol*‘ran 𝐹)) ∈ ran seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇‘𝑚)))))
120		supxrub 13454	. . . . . . . 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 25851	. . . . . . . . 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 3154	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑚 ∈ ℕ ((𝑇‘𝑚) ∈ dom vol ∧ (vol‘(𝑇‘𝑚)) ∈ ℝ))
127	11, 12, 13, 14, 15, 5, 16	vitalilem3 25931	. . . . . . . . . 10 ⊢ (𝜑 → Disj 𝑚 ∈ ℕ (𝑇‘𝑚))
128	127	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → Disj 𝑚 ∈ ℕ (𝑇‘𝑚))
129		eqid 2760	. . . . . . . . . 10 ⊢ seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇‘𝑚)))) = seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇‘𝑚))))
130	129, 99	voliun 25875	. . . . . . . . 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 2797	. . . . . . 7 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → (vol‘∪ 𝑚 ∈ ℕ (𝑇‘𝑚)) = sup(ran seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇‘𝑚)))), ℝ^*, < ))
133	121, 132	breqtrrd 5132	. . . . . 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 13290	. . . . 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 12417	. . . . . . . . 9 ⊢ 2 ∈ ℝ
138		iccssre 13560	. . . . . . . . 9 ⊢ (( -1 ∈ ℝ ∧ 2 ∈ ℝ) → ( -1[,]2) ⊆ ℝ)
139	22, 137, 138	mp2an 705	. . . . . . . 8 ⊢ ( -1[,]2) ⊆ ℝ
140		ovolss 25806	. . . . . . . 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 12418	. . . . . . . . 9 ⊢ 2 ∈ ℂ
143		ax-1cn 11258	. . . . . . . . 9 ⊢ 1 ∈ ℂ
144	142, 143	subnegi 11637	. . . . . . . 8 ⊢ (2 − -1) = (2 + 1)
145		neg1lt0 12308	. . . . . . . . . . 11 ⊢ -1 < 0
146		2pos 12447	. . . . . . . . . . 11 ⊢ 0 < 2
147	22, 38, 137	lttri 11436	. . . . . . . . . . 11 ⊢ (( -1 < 0 ∧ 0 < 2) → -1 < 2)
148	145, 146, 147	mp2an 705	. . . . . . . . . 10 ⊢ -1 < 2
149	22, 137, 148	ltleii 11433	. . . . . . . . 9 ⊢ -1 ≤ 2
150		ovolicc 25844	. . . . . . . . 9 ⊢ (( -1 ∈ ℝ ∧ 2 ∈ ℝ ∧ -1 ≤ 2) → (vol*‘( -1[,]2)) = (2 − -1))
151	22, 137, 149, 150	mp3an 1490	. . . . . . . 8 ⊢ (vol*‘( -1[,]2)) = (2 − -1)
152		df-3 12406	. . . . . . . 8 ⊢ 3 = (2 + 1)
153	144, 151, 152	3eqtr4i 2793	. . . . . . 7 ⊢ (vol*‘( -1[,]2)) = 3
154	141, 153	breqtrdi 5145	. . . . . 6 ⊢ ((𝜑 ∧ 0 < (vol‘ran 𝐹)) → (vol‘∪ 𝑚 ∈ ℕ (𝑇‘𝑚)) ≤ 3)
155		xrlenlt 11374	. . . . . . 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 25802	. . . . . . 7 ⊢ (ran 𝐹 ⊆ ℝ → 0 ≤ (vol*‘ran 𝐹))
160	20, 159	syl 18	. . . . . 6 ⊢ (𝜑 → 0 ≤ (vol*‘ran 𝐹))
161		0xr 11356	. . . . . . 7 ⊢ 0 ∈ ℝ^*
162		ovolcl 25799	. . . . . . . 8 ⊢ (ran 𝐹 ⊆ ℝ → (vol‘ran 𝐹) ∈ ℝ^)
163	20, 162	syl 18	. . . . . . 7 ⊢ (𝜑 → (vol‘ran 𝐹) ∈ ℝ^)
164		xrleloe 13273	. . . . . . 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 2798	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 1569 ∈ wcel 2144 ≠ wne 2955 ∀wral 3076 {crab 3412 ⊆ wss 3895 ∖ cdif 4045 ∩ cin 4089 ∅c0 4278 𝒫 cpw 4556 {csn 4583 ∪ ciun 4950 Disj wdisj 5069 class class class wbr 5102 {copab 5166 ↦ cmpt 5185 × cxp 5648 dom cdm 5650 ran crn 5651 Fn wfn 6533 ⟶wf 6534 –1-1-onto→wf1o 6537 ‘cfv 6538 (class class class)co 7420 / cqs 8716 supcsup 9432 ℂcc 11198 ℝcr 11199 0cc0 11200 1c1 11201 + caddc 11203 · cmul 11205 ℝ^*cxr 11342 < clt 11343 ≤ cle 11344 − cmin 11541 -cneg 11542 / cdiv 11973 ℕcn 12335 2c2 12397 3c3 12398 ℕ₀cn0 12606 ℚcq 13075 ℝ⁺crp 13120 [,]cicc 13479 ⌊cfl 13930 seqcseq 14144 vol*covol 25783 volcvol 25784

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7751 ax-inf2 9642 ax-cc 10513 ax-cnex 11256 ax-resscn 11257 ax-1cn 11258 ax-icn 11259 ax-addcl 11260 ax-addrcl 11261 ax-mulcl 11262 ax-mulrcl 11263 ax-mulcom 11264 ax-addass 11265 ax-mulass 11266 ax-distr 11267 ax-i2m1 11268 ax-1ne0 11269 ax-1rid 11270 ax-rnegex 11271 ax-rrecex 11272 ax-cnre 11273 ax-pre-lttri 11274 ax-pre-lttrn 11275 ax-pre-ltadd 11276 ax-pre-mulgt0 11277 ax-pre-sup 11278

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-op 4590 df-uni 4867 df-int 4907 df-iun 4952 df-disj 5070 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-se 5604 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-isom 6547 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7693 df-om 7878 df-1st 8001 df-2nd 8002 df-frecs 8299 df-wrecs 8330 df-recs 8379 df-rdg 8418 df-1o 8476 df-2o 8477 df-er 8717 df-ec 8719 df-qs 8723 df-map 8849 df-pm 8850 df-en 8974 df-dom 8975 df-sdom 8976 df-fin 8977 df-fi 9403 df-sup 9434 df-inf 9435 df-oi 9504 df-dju 9982 df-card 10020 df-pnf 11345 df-mnf 11346 df-xr 11347 df-ltxr 11348 df-le 11349 df-sub 11543 df-neg 11544 df-div 11974 df-nn 12336 df-2 12405 df-3 12406 df-n0 12607 df-z 12694 df-uz 12966 df-q 13076 df-rp 13121 df-xneg 13241 df-xadd 13242 df-xmul 13243 df-ioo 13480 df-ico 13482 df-icc 13483 df-fz 13640 df-fzo 13789 df-fl 13932 df-seq 14145 df-exp 14205 df-hash 14475 df-cj 15266 df-re 15267 df-im 15268 df-sqrt 15402 df-abs 15403 df-clim 15655 df-rlim 15656 df-sum 15854 df-rest 17593 df-topgen 17614 df-psmet 21670 df-xmet 21671 df-met 21672 df-bl 21673 df-mopn 21674 df-top 23212 df-topon 23229 df-bases 23264 df-cmp 23705 df-ovol 25785 df-vol 25786

This theorem is used by: vitalilem5 25933

W3C validator