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Theorem vitalilem4 25646
Description: Lemma for vitali 25648. (Contributed by Mario Carneiro, 16-Jun-2014.)
Hypotheses
Ref Expression
vitali.1 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]1)) ∧ (𝑥𝑦) ∈ ℚ)}
vitali.2 𝑆 = ((0[,]1) / )
vitali.3 (𝜑𝐹 Fn 𝑆)
vitali.4 (𝜑 → ∀𝑧𝑆 (𝑧 ≠ ∅ → (𝐹𝑧) ∈ 𝑧))
vitali.5 (𝜑𝐺:ℕ–1-1-onto→(ℚ ∩ (-1[,]1)))
vitali.6 𝑇 = (𝑛 ∈ ℕ ↦ {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑛)) ∈ ran 𝐹})
vitali.7 (𝜑 → ¬ ran 𝐹 ∈ (𝒫 ℝ ∖ dom vol))
Assertion
Ref Expression
vitalilem4 ((𝜑𝑚 ∈ ℕ) → (vol*‘(𝑇𝑚)) = 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.
StepHypRef Expression
1 fveq2 6906 . . . . . . . . 9 (𝑛 = 𝑚 → (𝐺𝑛) = (𝐺𝑚))
21oveq2d 7447 . . . . . . . 8 (𝑛 = 𝑚 → (𝑠 − (𝐺𝑛)) = (𝑠 − (𝐺𝑚)))
32eleq1d 2826 . . . . . . 7 (𝑛 = 𝑚 → ((𝑠 − (𝐺𝑛)) ∈ ran 𝐹 ↔ (𝑠 − (𝐺𝑚)) ∈ ran 𝐹))
43rabbidv 3444 . . . . . 6 (𝑛 = 𝑚 → {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑛)) ∈ ran 𝐹} = {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑚)) ∈ ran 𝐹})
5 vitali.6 . . . . . 6 𝑇 = (𝑛 ∈ ℕ ↦ {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑛)) ∈ ran 𝐹})
6 reex 11246 . . . . . . 7 ℝ ∈ V
76rabex 5339 . . . . . 6 {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑚)) ∈ ran 𝐹} ∈ V
84, 5, 7fvmpt 7016 . . . . 5 (𝑚 ∈ ℕ → (𝑇𝑚) = {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑚)) ∈ ran 𝐹})
98adantl 481 . . . 4 ((𝜑𝑚 ∈ ℕ) → (𝑇𝑚) = {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑚)) ∈ ran 𝐹})
109fveq2d 6910 . . 3 ((𝜑𝑚 ∈ ℕ) → (vol*‘(𝑇𝑚)) = (vol*‘{𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑚)) ∈ ran 𝐹}))
11 vitali.1 . . . . . . . 8 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]1)) ∧ (𝑥𝑦) ∈ ℚ)}
12 vitali.2 . . . . . . . 8 𝑆 = ((0[,]1) / )
13 vitali.3 . . . . . . . 8 (𝜑𝐹 Fn 𝑆)
14 vitali.4 . . . . . . . 8 (𝜑 → ∀𝑧𝑆 (𝑧 ≠ ∅ → (𝐹𝑧) ∈ 𝑧))
15 vitali.5 . . . . . . . 8 (𝜑𝐺:ℕ–1-1-onto→(ℚ ∩ (-1[,]1)))
16 vitali.7 . . . . . . . 8 (𝜑 → ¬ ran 𝐹 ∈ (𝒫 ℝ ∖ dom vol))
1711, 12, 13, 14, 15, 5, 16vitalilem2 25644 . . . . . . 7 (𝜑 → (ran 𝐹 ⊆ (0[,]1) ∧ (0[,]1) ⊆ 𝑚 ∈ ℕ (𝑇𝑚) ∧ 𝑚 ∈ ℕ (𝑇𝑚) ⊆ (-1[,]2)))
1817simp1d 1143 . . . . . 6 (𝜑 → ran 𝐹 ⊆ (0[,]1))
19 unitssre 13539 . . . . . 6 (0[,]1) ⊆ ℝ
2018, 19sstrdi 3996 . . . . 5 (𝜑 → ran 𝐹 ⊆ ℝ)
2120adantr 480 . . . 4 ((𝜑𝑚 ∈ ℕ) → ran 𝐹 ⊆ ℝ)
22 neg1rr 12381 . . . . . 6 -1 ∈ ℝ
23 1re 11261 . . . . . 6 1 ∈ ℝ
24 iccssre 13469 . . . . . 6 ((-1 ∈ ℝ ∧ 1 ∈ ℝ) → (-1[,]1) ⊆ ℝ)
2522, 23, 24mp2an 692 . . . . 5 (-1[,]1) ⊆ ℝ
26 f1of 6848 . . . . . . . 8 (𝐺:ℕ–1-1-onto→(ℚ ∩ (-1[,]1)) → 𝐺:ℕ⟶(ℚ ∩ (-1[,]1)))
2715, 26syl 17 . . . . . . 7 (𝜑𝐺:ℕ⟶(ℚ ∩ (-1[,]1)))
2827ffvelcdmda 7104 . . . . . 6 ((𝜑𝑚 ∈ ℕ) → (𝐺𝑚) ∈ (ℚ ∩ (-1[,]1)))
2928elin2d 4205 . . . . 5 ((𝜑𝑚 ∈ ℕ) → (𝐺𝑚) ∈ (-1[,]1))
3025, 29sselid 3981 . . . 4 ((𝜑𝑚 ∈ ℕ) → (𝐺𝑚) ∈ ℝ)
31 eqidd 2738 . . . 4 ((𝜑𝑚 ∈ ℕ) → {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑚)) ∈ ran 𝐹} = {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑚)) ∈ ran 𝐹})
3221, 30, 31ovolshft 25546 . . 3 ((𝜑𝑚 ∈ ℕ) → (vol*‘ran 𝐹) = (vol*‘{𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑚)) ∈ ran 𝐹}))
3310, 32eqtr4d 2780 . 2 ((𝜑𝑚 ∈ ℕ) → (vol*‘(𝑇𝑚)) = (vol*‘ran 𝐹))
34 3re 12346 . . . . . . . 8 3 ∈ ℝ
3534rexri 11319 . . . . . . 7 3 ∈ ℝ*
3635a1i 11 . . . . . 6 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → 3 ∈ ℝ*)
37 3rp 13040 . . . . . . . . . . . . 13 3 ∈ ℝ+
38 0re 11263 . . . . . . . . . . . . . . . . . . . 20 0 ∈ ℝ
39 0le1 11786 . . . . . . . . . . . . . . . . . . . 20 0 ≤ 1
40 ovolicc 25558 . . . . . . . . . . . . . . . . . . . 20 ((0 ∈ ℝ ∧ 1 ∈ ℝ ∧ 0 ≤ 1) → (vol*‘(0[,]1)) = (1 − 0))
4138, 23, 39, 40mp3an 1463 . . . . . . . . . . . . . . . . . . 19 (vol*‘(0[,]1)) = (1 − 0)
42 1m0e1 12387 . . . . . . . . . . . . . . . . . . 19 (1 − 0) = 1
4341, 42eqtri 2765 . . . . . . . . . . . . . . . . . 18 (vol*‘(0[,]1)) = 1
4443, 23eqeltri 2837 . . . . . . . . . . . . . . . . 17 (vol*‘(0[,]1)) ∈ ℝ
45 ovolsscl 25521 . . . . . . . . . . . . . . . . 17 ((ran 𝐹 ⊆ (0[,]1) ∧ (0[,]1) ⊆ ℝ ∧ (vol*‘(0[,]1)) ∈ ℝ) → (vol*‘ran 𝐹) ∈ ℝ)
4619, 44, 45mp3an23 1455 . . . . . . . . . . . . . . . 16 (ran 𝐹 ⊆ (0[,]1) → (vol*‘ran 𝐹) ∈ ℝ)
4718, 46syl 17 . . . . . . . . . . . . . . 15 (𝜑 → (vol*‘ran 𝐹) ∈ ℝ)
4847adantr 480 . . . . . . . . . . . . . 14 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → (vol*‘ran 𝐹) ∈ ℝ)
49 simpr 484 . . . . . . . . . . . . . 14 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → 0 < (vol*‘ran 𝐹))
5048, 49elrpd 13074 . . . . . . . . . . . . 13 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → (vol*‘ran 𝐹) ∈ ℝ+)
51 rpdivcl 13060 . . . . . . . . . . . . 13 ((3 ∈ ℝ+ ∧ (vol*‘ran 𝐹) ∈ ℝ+) → (3 / (vol*‘ran 𝐹)) ∈ ℝ+)
5237, 50, 51sylancr 587 . . . . . . . . . . . 12 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → (3 / (vol*‘ran 𝐹)) ∈ ℝ+)
5352rpred 13077 . . . . . . . . . . 11 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → (3 / (vol*‘ran 𝐹)) ∈ ℝ)
5452rpge0d 13081 . . . . . . . . . . 11 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → 0 ≤ (3 / (vol*‘ran 𝐹)))
55 flge0nn0 13860 . . . . . . . . . . 11 (((3 / (vol*‘ran 𝐹)) ∈ ℝ ∧ 0 ≤ (3 / (vol*‘ran 𝐹))) → (⌊‘(3 / (vol*‘ran 𝐹))) ∈ ℕ0)
5653, 54, 55syl2anc 584 . . . . . . . . . 10 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → (⌊‘(3 / (vol*‘ran 𝐹))) ∈ ℕ0)
57 nn0p1nn 12565 . . . . . . . . . 10 ((⌊‘(3 / (vol*‘ran 𝐹))) ∈ ℕ0 → ((⌊‘(3 / (vol*‘ran 𝐹))) + 1) ∈ ℕ)
5856, 57syl 17 . . . . . . . . 9 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → ((⌊‘(3 / (vol*‘ran 𝐹))) + 1) ∈ ℕ)
5958nnred 12281 . . . . . . . 8 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → ((⌊‘(3 / (vol*‘ran 𝐹))) + 1) ∈ ℝ)
6059, 48remulcld 11291 . . . . . . 7 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → (((⌊‘(3 / (vol*‘ran 𝐹))) + 1) · (vol*‘ran 𝐹)) ∈ ℝ)
6160rexrd 11311 . . . . . 6 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → (((⌊‘(3 / (vol*‘ran 𝐹))) + 1) · (vol*‘ran 𝐹)) ∈ ℝ*)
626elpw2 5334 . . . . . . . . . . . . . . . . . 18 (ran 𝐹 ∈ 𝒫 ℝ ↔ ran 𝐹 ⊆ ℝ)
6320, 62sylibr 234 . . . . . . . . . . . . . . . . 17 (𝜑 → ran 𝐹 ∈ 𝒫 ℝ)
6463anim1i 615 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ¬ ran 𝐹 ∈ dom vol) → (ran 𝐹 ∈ 𝒫 ℝ ∧ ¬ ran 𝐹 ∈ dom vol))
65 eldif 3961 . . . . . . . . . . . . . . . 16 (ran 𝐹 ∈ (𝒫 ℝ ∖ dom vol) ↔ (ran 𝐹 ∈ 𝒫 ℝ ∧ ¬ ran 𝐹 ∈ dom vol))
6664, 65sylibr 234 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ¬ ran 𝐹 ∈ dom vol) → ran 𝐹 ∈ (𝒫 ℝ ∖ dom vol))
6766ex 412 . . . . . . . . . . . . . 14 (𝜑 → (¬ ran 𝐹 ∈ dom vol → ran 𝐹 ∈ (𝒫 ℝ ∖ dom vol)))
6816, 67mt3d 148 . . . . . . . . . . . . 13 (𝜑 → ran 𝐹 ∈ dom vol)
69 inss1 4237 . . . . . . . . . . . . . . . 16 (ℚ ∩ (-1[,]1)) ⊆ ℚ
70 qssre 13001 . . . . . . . . . . . . . . . 16 ℚ ⊆ ℝ
7169, 70sstri 3993 . . . . . . . . . . . . . . 15 (ℚ ∩ (-1[,]1)) ⊆ ℝ
72 fss 6752 . . . . . . . . . . . . . . 15 ((𝐺:ℕ⟶(ℚ ∩ (-1[,]1)) ∧ (ℚ ∩ (-1[,]1)) ⊆ ℝ) → 𝐺:ℕ⟶ℝ)
7327, 71, 72sylancl 586 . . . . . . . . . . . . . 14 (𝜑𝐺:ℕ⟶ℝ)
7473ffvelcdmda 7104 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → (𝐺𝑛) ∈ ℝ)
75 shftmbl 25573 . . . . . . . . . . . . 13 ((ran 𝐹 ∈ dom vol ∧ (𝐺𝑛) ∈ ℝ) → {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑛)) ∈ ran 𝐹} ∈ dom vol)
7668, 74, 75syl2an2r 685 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑛)) ∈ ran 𝐹} ∈ dom vol)
7776, 5fmptd 7134 . . . . . . . . . . 11 (𝜑𝑇:ℕ⟶dom vol)
7877ffvelcdmda 7104 . . . . . . . . . 10 ((𝜑𝑚 ∈ ℕ) → (𝑇𝑚) ∈ dom vol)
7978ralrimiva 3146 . . . . . . . . 9 (𝜑 → ∀𝑚 ∈ ℕ (𝑇𝑚) ∈ dom vol)
80 iunmbl 25588 . . . . . . . . 9 (∀𝑚 ∈ ℕ (𝑇𝑚) ∈ dom vol → 𝑚 ∈ ℕ (𝑇𝑚) ∈ dom vol)
8179, 80syl 17 . . . . . . . 8 (𝜑 𝑚 ∈ ℕ (𝑇𝑚) ∈ dom vol)
82 mblss 25566 . . . . . . . 8 ( 𝑚 ∈ ℕ (𝑇𝑚) ∈ dom vol → 𝑚 ∈ ℕ (𝑇𝑚) ⊆ ℝ)
83 ovolcl 25513 . . . . . . . 8 ( 𝑚 ∈ ℕ (𝑇𝑚) ⊆ ℝ → (vol*‘ 𝑚 ∈ ℕ (𝑇𝑚)) ∈ ℝ*)
8481, 82, 833syl 18 . . . . . . 7 (𝜑 → (vol*‘ 𝑚 ∈ ℕ (𝑇𝑚)) ∈ ℝ*)
8584adantr 480 . . . . . 6 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → (vol*‘ 𝑚 ∈ ℕ (𝑇𝑚)) ∈ ℝ*)
86 flltp1 13840 . . . . . . . 8 ((3 / (vol*‘ran 𝐹)) ∈ ℝ → (3 / (vol*‘ran 𝐹)) < ((⌊‘(3 / (vol*‘ran 𝐹))) + 1))
8753, 86syl 17 . . . . . . 7 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → (3 / (vol*‘ran 𝐹)) < ((⌊‘(3 / (vol*‘ran 𝐹))) + 1))
8834a1i 11 . . . . . . . 8 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → 3 ∈ ℝ)
8988, 59, 50ltdivmul2d 13129 . . . . . . 7 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → ((3 / (vol*‘ran 𝐹)) < ((⌊‘(3 / (vol*‘ran 𝐹))) + 1) ↔ 3 < (((⌊‘(3 / (vol*‘ran 𝐹))) + 1) · (vol*‘ran 𝐹))))
9087, 89mpbid 232 . . . . . 6 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → 3 < (((⌊‘(3 / (vol*‘ran 𝐹))) + 1) · (vol*‘ran 𝐹)))
91 nnuz 12921 . . . . . . . . . . 11 ℕ = (ℤ‘1)
92 1zzd 12648 . . . . . . . . . . 11 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → 1 ∈ ℤ)
93 mblvol 25565 . . . . . . . . . . . . . . . . 17 ((𝑇𝑚) ∈ dom vol → (vol‘(𝑇𝑚)) = (vol*‘(𝑇𝑚)))
9478, 93syl 17 . . . . . . . . . . . . . . . 16 ((𝜑𝑚 ∈ ℕ) → (vol‘(𝑇𝑚)) = (vol*‘(𝑇𝑚)))
9594, 33eqtrd 2777 . . . . . . . . . . . . . . 15 ((𝜑𝑚 ∈ ℕ) → (vol‘(𝑇𝑚)) = (vol*‘ran 𝐹))
9647adantr 480 . . . . . . . . . . . . . . 15 ((𝜑𝑚 ∈ ℕ) → (vol*‘ran 𝐹) ∈ ℝ)
9795, 96eqeltrd 2841 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℕ) → (vol‘(𝑇𝑚)) ∈ ℝ)
9897adantlr 715 . . . . . . . . . . . . 13 (((𝜑 ∧ 0 < (vol*‘ran 𝐹)) ∧ 𝑚 ∈ ℕ) → (vol‘(𝑇𝑚)) ∈ ℝ)
99 eqid 2737 . . . . . . . . . . . . 13 (𝑚 ∈ ℕ ↦ (vol‘(𝑇𝑚))) = (𝑚 ∈ ℕ ↦ (vol‘(𝑇𝑚)))
10098, 99fmptd 7134 . . . . . . . . . . . 12 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → (𝑚 ∈ ℕ ↦ (vol‘(𝑇𝑚))):ℕ⟶ℝ)
101100ffvelcdmda 7104 . . . . . . . . . . 11 (((𝜑 ∧ 0 < (vol*‘ran 𝐹)) ∧ 𝑘 ∈ ℕ) → ((𝑚 ∈ ℕ ↦ (vol‘(𝑇𝑚)))‘𝑘) ∈ ℝ)
10291, 92, 101serfre 14072 . . . . . . . . . 10 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇𝑚)))):ℕ⟶ℝ)
103102frnd 6744 . . . . . . . . 9 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → ran seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇𝑚)))) ⊆ ℝ)
104 ressxr 11305 . . . . . . . . 9 ℝ ⊆ ℝ*
105103, 104sstrdi 3996 . . . . . . . 8 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → ran seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇𝑚)))) ⊆ ℝ*)
10695adantlr 715 . . . . . . . . . . . . . 14 (((𝜑 ∧ 0 < (vol*‘ran 𝐹)) ∧ 𝑚 ∈ ℕ) → (vol‘(𝑇𝑚)) = (vol*‘ran 𝐹))
107106mpteq2dva 5242 . . . . . . . . . . . . 13 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → (𝑚 ∈ ℕ ↦ (vol‘(𝑇𝑚))) = (𝑚 ∈ ℕ ↦ (vol*‘ran 𝐹)))
108 fconstmpt 5747 . . . . . . . . . . . . 13 (ℕ × {(vol*‘ran 𝐹)}) = (𝑚 ∈ ℕ ↦ (vol*‘ran 𝐹))
109107, 108eqtr4di 2795 . . . . . . . . . . . 12 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → (𝑚 ∈ ℕ ↦ (vol‘(𝑇𝑚))) = (ℕ × {(vol*‘ran 𝐹)}))
110109seqeq3d 14050 . . . . . . . . . . 11 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇𝑚)))) = seq1( + , (ℕ × {(vol*‘ran 𝐹)})))
111110fveq1d 6908 . . . . . . . . . 10 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → (seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇𝑚))))‘((⌊‘(3 / (vol*‘ran 𝐹))) + 1)) = (seq1( + , (ℕ × {(vol*‘ran 𝐹)}))‘((⌊‘(3 / (vol*‘ran 𝐹))) + 1)))
11248recnd 11289 . . . . . . . . . . 11 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → (vol*‘ran 𝐹) ∈ ℂ)
113 ser1const 14099 . . . . . . . . . . 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)))
114112, 58, 113syl2anc 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)))
115111, 114eqtrd 2777 . . . . . . . . 9 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → (seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇𝑚))))‘((⌊‘(3 / (vol*‘ran 𝐹))) + 1)) = (((⌊‘(3 / (vol*‘ran 𝐹))) + 1) · (vol*‘ran 𝐹)))
116102ffnd 6737 . . . . . . . . . 10 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇𝑚)))) Fn ℕ)
117 fnfvelrn 7100 . . . . . . . . . 10 ((seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇𝑚)))) Fn ℕ ∧ ((⌊‘(3 / (vol*‘ran 𝐹))) + 1) ∈ ℕ) → (seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇𝑚))))‘((⌊‘(3 / (vol*‘ran 𝐹))) + 1)) ∈ ran seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇𝑚)))))
118116, 58, 117syl2anc 584 . . . . . . . . 9 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → (seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇𝑚))))‘((⌊‘(3 / (vol*‘ran 𝐹))) + 1)) ∈ ran seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇𝑚)))))
119115, 118eqeltrrd 2842 . . . . . . . 8 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → (((⌊‘(3 / (vol*‘ran 𝐹))) + 1) · (vol*‘ran 𝐹)) ∈ ran seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇𝑚)))))
120 supxrub 13366 . . . . . . . 8 ((ran seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇𝑚)))) ⊆ ℝ* ∧ (((⌊‘(3 / (vol*‘ran 𝐹))) + 1) · (vol*‘ran 𝐹)) ∈ ran seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇𝑚))))) → (((⌊‘(3 / (vol*‘ran 𝐹))) + 1) · (vol*‘ran 𝐹)) ≤ sup(ran seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇𝑚)))), ℝ*, < ))
121105, 119, 120syl2anc 584 . . . . . . 7 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → (((⌊‘(3 / (vol*‘ran 𝐹))) + 1) · (vol*‘ran 𝐹)) ≤ sup(ran seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇𝑚)))), ℝ*, < ))
12281adantr 480 . . . . . . . . 9 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → 𝑚 ∈ ℕ (𝑇𝑚) ∈ dom vol)
123 mblvol 25565 . . . . . . . . 9 ( 𝑚 ∈ ℕ (𝑇𝑚) ∈ dom vol → (vol‘ 𝑚 ∈ ℕ (𝑇𝑚)) = (vol*‘ 𝑚 ∈ ℕ (𝑇𝑚)))
124122, 123syl 17 . . . . . . . 8 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → (vol‘ 𝑚 ∈ ℕ (𝑇𝑚)) = (vol*‘ 𝑚 ∈ ℕ (𝑇𝑚)))
12578, 97jca 511 . . . . . . . . . 10 ((𝜑𝑚 ∈ ℕ) → ((𝑇𝑚) ∈ dom vol ∧ (vol‘(𝑇𝑚)) ∈ ℝ))
126125ralrimiva 3146 . . . . . . . . 9 (𝜑 → ∀𝑚 ∈ ℕ ((𝑇𝑚) ∈ dom vol ∧ (vol‘(𝑇𝑚)) ∈ ℝ))
12711, 12, 13, 14, 15, 5, 16vitalilem3 25645 . . . . . . . . . 10 (𝜑Disj 𝑚 ∈ ℕ (𝑇𝑚))
128127adantr 480 . . . . . . . . 9 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → Disj 𝑚 ∈ ℕ (𝑇𝑚))
129 eqid 2737 . . . . . . . . . 10 seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇𝑚)))) = seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇𝑚))))
130129, 99voliun 25589 . . . . . . . . 9 ((∀𝑚 ∈ ℕ ((𝑇𝑚) ∈ dom vol ∧ (vol‘(𝑇𝑚)) ∈ ℝ) ∧ Disj 𝑚 ∈ ℕ (𝑇𝑚)) → (vol‘ 𝑚 ∈ ℕ (𝑇𝑚)) = sup(ran seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇𝑚)))), ℝ*, < ))
131126, 128, 130syl2an2r 685 . . . . . . . 8 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → (vol‘ 𝑚 ∈ ℕ (𝑇𝑚)) = sup(ran seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇𝑚)))), ℝ*, < ))
132124, 131eqtr3d 2779 . . . . . . 7 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → (vol*‘ 𝑚 ∈ ℕ (𝑇𝑚)) = sup(ran seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇𝑚)))), ℝ*, < ))
133121, 132breqtrrd 5171 . . . . . 6 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → (((⌊‘(3 / (vol*‘ran 𝐹))) + 1) · (vol*‘ran 𝐹)) ≤ (vol*‘ 𝑚 ∈ ℕ (𝑇𝑚)))
13436, 61, 85, 90, 133xrltletrd 13203 . . . . 5 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → 3 < (vol*‘ 𝑚 ∈ ℕ (𝑇𝑚)))
13517simp3d 1145 . . . . . . . . 9 (𝜑 𝑚 ∈ ℕ (𝑇𝑚) ⊆ (-1[,]2))
136135adantr 480 . . . . . . . 8 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → 𝑚 ∈ ℕ (𝑇𝑚) ⊆ (-1[,]2))
137 2re 12340 . . . . . . . . 9 2 ∈ ℝ
138 iccssre 13469 . . . . . . . . 9 ((-1 ∈ ℝ ∧ 2 ∈ ℝ) → (-1[,]2) ⊆ ℝ)
13922, 137, 138mp2an 692 . . . . . . . 8 (-1[,]2) ⊆ ℝ
140 ovolss 25520 . . . . . . . 8 (( 𝑚 ∈ ℕ (𝑇𝑚) ⊆ (-1[,]2) ∧ (-1[,]2) ⊆ ℝ) → (vol*‘ 𝑚 ∈ ℕ (𝑇𝑚)) ≤ (vol*‘(-1[,]2)))
141136, 139, 140sylancl 586 . . . . . . 7 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → (vol*‘ 𝑚 ∈ ℕ (𝑇𝑚)) ≤ (vol*‘(-1[,]2)))
142 2cn 12341 . . . . . . . . 9 2 ∈ ℂ
143 ax-1cn 11213 . . . . . . . . 9 1 ∈ ℂ
144142, 143subnegi 11588 . . . . . . . 8 (2 − -1) = (2 + 1)
145 neg1lt0 12383 . . . . . . . . . . 11 -1 < 0
146 2pos 12369 . . . . . . . . . . 11 0 < 2
14722, 38, 137lttri 11387 . . . . . . . . . . 11 ((-1 < 0 ∧ 0 < 2) → -1 < 2)
148145, 146, 147mp2an 692 . . . . . . . . . 10 -1 < 2
14922, 137, 148ltleii 11384 . . . . . . . . 9 -1 ≤ 2
150 ovolicc 25558 . . . . . . . . 9 ((-1 ∈ ℝ ∧ 2 ∈ ℝ ∧ -1 ≤ 2) → (vol*‘(-1[,]2)) = (2 − -1))
15122, 137, 149, 150mp3an 1463 . . . . . . . 8 (vol*‘(-1[,]2)) = (2 − -1)
152 df-3 12330 . . . . . . . 8 3 = (2 + 1)
153144, 151, 1523eqtr4i 2775 . . . . . . 7 (vol*‘(-1[,]2)) = 3
154141, 153breqtrdi 5184 . . . . . 6 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → (vol*‘ 𝑚 ∈ ℕ (𝑇𝑚)) ≤ 3)
155 xrlenlt 11326 . . . . . . 7 (((vol*‘ 𝑚 ∈ ℕ (𝑇𝑚)) ∈ ℝ* ∧ 3 ∈ ℝ*) → ((vol*‘ 𝑚 ∈ ℕ (𝑇𝑚)) ≤ 3 ↔ ¬ 3 < (vol*‘ 𝑚 ∈ ℕ (𝑇𝑚))))
15685, 35, 155sylancl 586 . . . . . 6 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → ((vol*‘ 𝑚 ∈ ℕ (𝑇𝑚)) ≤ 3 ↔ ¬ 3 < (vol*‘ 𝑚 ∈ ℕ (𝑇𝑚))))
157154, 156mpbid 232 . . . . 5 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → ¬ 3 < (vol*‘ 𝑚 ∈ ℕ (𝑇𝑚)))
158134, 157pm2.65da 817 . . . 4 (𝜑 → ¬ 0 < (vol*‘ran 𝐹))
159 ovolge0 25516 . . . . . . 7 (ran 𝐹 ⊆ ℝ → 0 ≤ (vol*‘ran 𝐹))
16020, 159syl 17 . . . . . 6 (𝜑 → 0 ≤ (vol*‘ran 𝐹))
161 0xr 11308 . . . . . . 7 0 ∈ ℝ*
162 ovolcl 25513 . . . . . . . 8 (ran 𝐹 ⊆ ℝ → (vol*‘ran 𝐹) ∈ ℝ*)
16320, 162syl 17 . . . . . . 7 (𝜑 → (vol*‘ran 𝐹) ∈ ℝ*)
164 xrleloe 13186 . . . . . . 7 ((0 ∈ ℝ* ∧ (vol*‘ran 𝐹) ∈ ℝ*) → (0 ≤ (vol*‘ran 𝐹) ↔ (0 < (vol*‘ran 𝐹) ∨ 0 = (vol*‘ran 𝐹))))
165161, 163, 164sylancr 587 . . . . . 6 (𝜑 → (0 ≤ (vol*‘ran 𝐹) ↔ (0 < (vol*‘ran 𝐹) ∨ 0 = (vol*‘ran 𝐹))))
166160, 165mpbid 232 . . . . 5 (𝜑 → (0 < (vol*‘ran 𝐹) ∨ 0 = (vol*‘ran 𝐹)))
167166ord 865 . . . 4 (𝜑 → (¬ 0 < (vol*‘ran 𝐹) → 0 = (vol*‘ran 𝐹)))
168158, 167mpd 15 . . 3 (𝜑 → 0 = (vol*‘ran 𝐹))
169168adantr 480 . 2 ((𝜑𝑚 ∈ ℕ) → 0 = (vol*‘ran 𝐹))
17033, 169eqtr4d 2780 1 ((𝜑𝑚 ∈ ℕ) → (vol*‘(𝑇𝑚)) = 0)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 848   = wceq 1540  wcel 2108  wne 2940  wral 3061  {crab 3436  cdif 3948  cin 3950  wss 3951  c0 4333  𝒫 cpw 4600  {csn 4626   ciun 4991  Disj wdisj 5110   class class class wbr 5143  {copab 5205  cmpt 5225   × cxp 5683  dom cdm 5685  ran crn 5686   Fn wfn 6556  wf 6557  1-1-ontowf1o 6560  cfv 6561  (class class class)co 7431   / cqs 8744  supcsup 9480  cc 11153  cr 11154  0cc0 11155  1c1 11156   + caddc 11158   · cmul 11160  *cxr 11294   < clt 11295  cle 11296  cmin 11492  -cneg 11493   / cdiv 11920  cn 12266  2c2 12321  3c3 12322  0cn0 12526  cq 12990  +crp 13034  [,]cicc 13390  cfl 13830  seqcseq 14042  vol*covol 25497  volcvol 25498
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-cc 10475  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-disj 5111  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-er 8745  df-ec 8747  df-qs 8751  df-map 8868  df-pm 8869  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fi 9451  df-sup 9482  df-inf 9483  df-oi 9550  df-dju 9941  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-n0 12527  df-z 12614  df-uz 12879  df-q 12991  df-rp 13035  df-xneg 13154  df-xadd 13155  df-xmul 13156  df-ioo 13391  df-ico 13393  df-icc 13394  df-fz 13548  df-fzo 13695  df-fl 13832  df-seq 14043  df-exp 14103  df-hash 14370  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-clim 15524  df-rlim 15525  df-sum 15723  df-rest 17467  df-topgen 17488  df-psmet 21356  df-xmet 21357  df-met 21358  df-bl 21359  df-mopn 21360  df-top 22900  df-topon 22917  df-bases 22953  df-cmp 23395  df-ovol 25499  df-vol 25500
This theorem is referenced by:  vitalilem5  25647
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