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Theorem vitalilem4 23103
Description: Lemma for vitali 23105. (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 6088 . . . . . . . . 9 (𝑛 = 𝑚 → (𝐺𝑛) = (𝐺𝑚))
21oveq2d 6543 . . . . . . . 8 (𝑛 = 𝑚 → (𝑠 − (𝐺𝑛)) = (𝑠 − (𝐺𝑚)))
32eleq1d 2671 . . . . . . 7 (𝑛 = 𝑚 → ((𝑠 − (𝐺𝑛)) ∈ ran 𝐹 ↔ (𝑠 − (𝐺𝑚)) ∈ ran 𝐹))
43rabbidv 3163 . . . . . 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 9883 . . . . . . 7 ℝ ∈ V
76rabex 4735 . . . . . 6 {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑚)) ∈ ran 𝐹} ∈ V
84, 5, 7fvmpt 6176 . . . . 5 (𝑚 ∈ ℕ → (𝑇𝑚) = {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑚)) ∈ ran 𝐹})
98adantl 480 . . . 4 ((𝜑𝑚 ∈ ℕ) → (𝑇𝑚) = {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑚)) ∈ ran 𝐹})
109fveq2d 6092 . . 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 23101 . . . . . . 7 (𝜑 → (ran 𝐹 ⊆ (0[,]1) ∧ (0[,]1) ⊆ 𝑚 ∈ ℕ (𝑇𝑚) ∧ 𝑚 ∈ ℕ (𝑇𝑚) ⊆ (-1[,]2)))
1817simp1d 1065 . . . . . 6 (𝜑 → ran 𝐹 ⊆ (0[,]1))
19 unitssre 12146 . . . . . 6 (0[,]1) ⊆ ℝ
2018, 19syl6ss 3579 . . . . 5 (𝜑 → ran 𝐹 ⊆ ℝ)
2120adantr 479 . . . 4 ((𝜑𝑚 ∈ ℕ) → ran 𝐹 ⊆ ℝ)
22 neg1rr 10972 . . . . . 6 -1 ∈ ℝ
23 1re 9895 . . . . . 6 1 ∈ ℝ
24 iccssre 12082 . . . . . 6 ((-1 ∈ ℝ ∧ 1 ∈ ℝ) → (-1[,]1) ⊆ ℝ)
2522, 23, 24mp2an 703 . . . . 5 (-1[,]1) ⊆ ℝ
26 inss2 3795 . . . . . 6 (ℚ ∩ (-1[,]1)) ⊆ (-1[,]1)
27 f1of 6035 . . . . . . . 8 (𝐺:ℕ–1-1-onto→(ℚ ∩ (-1[,]1)) → 𝐺:ℕ⟶(ℚ ∩ (-1[,]1)))
2815, 27syl 17 . . . . . . 7 (𝜑𝐺:ℕ⟶(ℚ ∩ (-1[,]1)))
2928ffvelrnda 6252 . . . . . 6 ((𝜑𝑚 ∈ ℕ) → (𝐺𝑚) ∈ (ℚ ∩ (-1[,]1)))
3026, 29sseldi 3565 . . . . 5 ((𝜑𝑚 ∈ ℕ) → (𝐺𝑚) ∈ (-1[,]1))
3125, 30sseldi 3565 . . . 4 ((𝜑𝑚 ∈ ℕ) → (𝐺𝑚) ∈ ℝ)
32 eqidd 2610 . . . 4 ((𝜑𝑚 ∈ ℕ) → {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑚)) ∈ ran 𝐹} = {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑚)) ∈ ran 𝐹})
3321, 31, 32ovolshft 23003 . . 3 ((𝜑𝑚 ∈ ℕ) → (vol*‘ran 𝐹) = (vol*‘{𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑚)) ∈ ran 𝐹}))
3410, 33eqtr4d 2646 . 2 ((𝜑𝑚 ∈ ℕ) → (vol*‘(𝑇𝑚)) = (vol*‘ran 𝐹))
35 3re 10941 . . . . . . . 8 3 ∈ ℝ
3635rexri 9948 . . . . . . 7 3 ∈ ℝ*
3736a1i 11 . . . . . 6 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → 3 ∈ ℝ*)
38 3nn 11033 . . . . . . . . . . . . . 14 3 ∈ ℕ
39 nnrp 11674 . . . . . . . . . . . . . 14 (3 ∈ ℕ → 3 ∈ ℝ+)
4038, 39ax-mp 5 . . . . . . . . . . . . 13 3 ∈ ℝ+
41 0re 9896 . . . . . . . . . . . . . . . . . . . 20 0 ∈ ℝ
42 0le1 10400 . . . . . . . . . . . . . . . . . . . 20 0 ≤ 1
43 ovolicc 23015 . . . . . . . . . . . . . . . . . . . 20 ((0 ∈ ℝ ∧ 1 ∈ ℝ ∧ 0 ≤ 1) → (vol*‘(0[,]1)) = (1 − 0))
4441, 23, 42, 43mp3an 1415 . . . . . . . . . . . . . . . . . . 19 (vol*‘(0[,]1)) = (1 − 0)
45 1m0e1 10978 . . . . . . . . . . . . . . . . . . 19 (1 − 0) = 1
4644, 45eqtri 2631 . . . . . . . . . . . . . . . . . 18 (vol*‘(0[,]1)) = 1
4746, 23eqeltri 2683 . . . . . . . . . . . . . . . . 17 (vol*‘(0[,]1)) ∈ ℝ
48 ovolsscl 22978 . . . . . . . . . . . . . . . . 17 ((ran 𝐹 ⊆ (0[,]1) ∧ (0[,]1) ⊆ ℝ ∧ (vol*‘(0[,]1)) ∈ ℝ) → (vol*‘ran 𝐹) ∈ ℝ)
4919, 47, 48mp3an23 1407 . . . . . . . . . . . . . . . 16 (ran 𝐹 ⊆ (0[,]1) → (vol*‘ran 𝐹) ∈ ℝ)
5018, 49syl 17 . . . . . . . . . . . . . . 15 (𝜑 → (vol*‘ran 𝐹) ∈ ℝ)
5150adantr 479 . . . . . . . . . . . . . 14 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → (vol*‘ran 𝐹) ∈ ℝ)
52 simpr 475 . . . . . . . . . . . . . 14 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → 0 < (vol*‘ran 𝐹))
5351, 52elrpd 11701 . . . . . . . . . . . . 13 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → (vol*‘ran 𝐹) ∈ ℝ+)
54 rpdivcl 11688 . . . . . . . . . . . . 13 ((3 ∈ ℝ+ ∧ (vol*‘ran 𝐹) ∈ ℝ+) → (3 / (vol*‘ran 𝐹)) ∈ ℝ+)
5540, 53, 54sylancr 693 . . . . . . . . . . . 12 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → (3 / (vol*‘ran 𝐹)) ∈ ℝ+)
5655rpred 11704 . . . . . . . . . . 11 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → (3 / (vol*‘ran 𝐹)) ∈ ℝ)
5755rpge0d 11708 . . . . . . . . . . 11 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → 0 ≤ (3 / (vol*‘ran 𝐹)))
58 flge0nn0 12438 . . . . . . . . . . 11 (((3 / (vol*‘ran 𝐹)) ∈ ℝ ∧ 0 ≤ (3 / (vol*‘ran 𝐹))) → (⌊‘(3 / (vol*‘ran 𝐹))) ∈ ℕ0)
5956, 57, 58syl2anc 690 . . . . . . . . . 10 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → (⌊‘(3 / (vol*‘ran 𝐹))) ∈ ℕ0)
60 nn0p1nn 11179 . . . . . . . . . 10 ((⌊‘(3 / (vol*‘ran 𝐹))) ∈ ℕ0 → ((⌊‘(3 / (vol*‘ran 𝐹))) + 1) ∈ ℕ)
6159, 60syl 17 . . . . . . . . 9 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → ((⌊‘(3 / (vol*‘ran 𝐹))) + 1) ∈ ℕ)
6261nnred 10882 . . . . . . . 8 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → ((⌊‘(3 / (vol*‘ran 𝐹))) + 1) ∈ ℝ)
6362, 51remulcld 9926 . . . . . . 7 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → (((⌊‘(3 / (vol*‘ran 𝐹))) + 1) · (vol*‘ran 𝐹)) ∈ ℝ)
6463rexrd 9945 . . . . . 6 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → (((⌊‘(3 / (vol*‘ran 𝐹))) + 1) · (vol*‘ran 𝐹)) ∈ ℝ*)
656elpw2 4750 . . . . . . . . . . . . . . . . . . 19 (ran 𝐹 ∈ 𝒫 ℝ ↔ ran 𝐹 ⊆ ℝ)
6620, 65sylibr 222 . . . . . . . . . . . . . . . . . 18 (𝜑 → ran 𝐹 ∈ 𝒫 ℝ)
6766anim1i 589 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ¬ ran 𝐹 ∈ dom vol) → (ran 𝐹 ∈ 𝒫 ℝ ∧ ¬ ran 𝐹 ∈ dom vol))
68 eldif 3549 . . . . . . . . . . . . . . . . 17 (ran 𝐹 ∈ (𝒫 ℝ ∖ dom vol) ↔ (ran 𝐹 ∈ 𝒫 ℝ ∧ ¬ ran 𝐹 ∈ dom vol))
6967, 68sylibr 222 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ¬ ran 𝐹 ∈ dom vol) → ran 𝐹 ∈ (𝒫 ℝ ∖ dom vol))
7069ex 448 . . . . . . . . . . . . . . 15 (𝜑 → (¬ ran 𝐹 ∈ dom vol → ran 𝐹 ∈ (𝒫 ℝ ∖ dom vol)))
7116, 70mt3d 138 . . . . . . . . . . . . . 14 (𝜑 → ran 𝐹 ∈ dom vol)
7271adantr 479 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → ran 𝐹 ∈ dom vol)
73 inss1 3794 . . . . . . . . . . . . . . . 16 (ℚ ∩ (-1[,]1)) ⊆ ℚ
74 qssre 11630 . . . . . . . . . . . . . . . 16 ℚ ⊆ ℝ
7573, 74sstri 3576 . . . . . . . . . . . . . . 15 (ℚ ∩ (-1[,]1)) ⊆ ℝ
76 fss 5955 . . . . . . . . . . . . . . 15 ((𝐺:ℕ⟶(ℚ ∩ (-1[,]1)) ∧ (ℚ ∩ (-1[,]1)) ⊆ ℝ) → 𝐺:ℕ⟶ℝ)
7728, 75, 76sylancl 692 . . . . . . . . . . . . . 14 (𝜑𝐺:ℕ⟶ℝ)
7877ffvelrnda 6252 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → (𝐺𝑛) ∈ ℝ)
79 shftmbl 23030 . . . . . . . . . . . . 13 ((ran 𝐹 ∈ dom vol ∧ (𝐺𝑛) ∈ ℝ) → {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑛)) ∈ ran 𝐹} ∈ dom vol)
8072, 78, 79syl2anc 690 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑛)) ∈ ran 𝐹} ∈ dom vol)
8180, 5fmptd 6277 . . . . . . . . . . 11 (𝜑𝑇:ℕ⟶dom vol)
8281ffvelrnda 6252 . . . . . . . . . 10 ((𝜑𝑚 ∈ ℕ) → (𝑇𝑚) ∈ dom vol)
8382ralrimiva 2948 . . . . . . . . 9 (𝜑 → ∀𝑚 ∈ ℕ (𝑇𝑚) ∈ dom vol)
84 iunmbl 23045 . . . . . . . . 9 (∀𝑚 ∈ ℕ (𝑇𝑚) ∈ dom vol → 𝑚 ∈ ℕ (𝑇𝑚) ∈ dom vol)
8583, 84syl 17 . . . . . . . 8 (𝜑 𝑚 ∈ ℕ (𝑇𝑚) ∈ dom vol)
86 mblss 23023 . . . . . . . 8 ( 𝑚 ∈ ℕ (𝑇𝑚) ∈ dom vol → 𝑚 ∈ ℕ (𝑇𝑚) ⊆ ℝ)
87 ovolcl 22970 . . . . . . . 8 ( 𝑚 ∈ ℕ (𝑇𝑚) ⊆ ℝ → (vol*‘ 𝑚 ∈ ℕ (𝑇𝑚)) ∈ ℝ*)
8885, 86, 873syl 18 . . . . . . 7 (𝜑 → (vol*‘ 𝑚 ∈ ℕ (𝑇𝑚)) ∈ ℝ*)
8988adantr 479 . . . . . 6 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → (vol*‘ 𝑚 ∈ ℕ (𝑇𝑚)) ∈ ℝ*)
90 flltp1 12418 . . . . . . . 8 ((3 / (vol*‘ran 𝐹)) ∈ ℝ → (3 / (vol*‘ran 𝐹)) < ((⌊‘(3 / (vol*‘ran 𝐹))) + 1))
9156, 90syl 17 . . . . . . 7 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → (3 / (vol*‘ran 𝐹)) < ((⌊‘(3 / (vol*‘ran 𝐹))) + 1))
9235a1i 11 . . . . . . . 8 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → 3 ∈ ℝ)
9392, 62, 53ltdivmul2d 11756 . . . . . . 7 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → ((3 / (vol*‘ran 𝐹)) < ((⌊‘(3 / (vol*‘ran 𝐹))) + 1) ↔ 3 < (((⌊‘(3 / (vol*‘ran 𝐹))) + 1) · (vol*‘ran 𝐹))))
9491, 93mpbid 220 . . . . . 6 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → 3 < (((⌊‘(3 / (vol*‘ran 𝐹))) + 1) · (vol*‘ran 𝐹)))
95 nnuz 11555 . . . . . . . . . . 11 ℕ = (ℤ‘1)
96 1zzd 11241 . . . . . . . . . . 11 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → 1 ∈ ℤ)
97 mblvol 23022 . . . . . . . . . . . . . . . . 17 ((𝑇𝑚) ∈ dom vol → (vol‘(𝑇𝑚)) = (vol*‘(𝑇𝑚)))
9882, 97syl 17 . . . . . . . . . . . . . . . 16 ((𝜑𝑚 ∈ ℕ) → (vol‘(𝑇𝑚)) = (vol*‘(𝑇𝑚)))
9998, 34eqtrd 2643 . . . . . . . . . . . . . . 15 ((𝜑𝑚 ∈ ℕ) → (vol‘(𝑇𝑚)) = (vol*‘ran 𝐹))
10050adantr 479 . . . . . . . . . . . . . . 15 ((𝜑𝑚 ∈ ℕ) → (vol*‘ran 𝐹) ∈ ℝ)
10199, 100eqeltrd 2687 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℕ) → (vol‘(𝑇𝑚)) ∈ ℝ)
102101adantlr 746 . . . . . . . . . . . . 13 (((𝜑 ∧ 0 < (vol*‘ran 𝐹)) ∧ 𝑚 ∈ ℕ) → (vol‘(𝑇𝑚)) ∈ ℝ)
103 eqid 2609 . . . . . . . . . . . . 13 (𝑚 ∈ ℕ ↦ (vol‘(𝑇𝑚))) = (𝑚 ∈ ℕ ↦ (vol‘(𝑇𝑚)))
104102, 103fmptd 6277 . . . . . . . . . . . 12 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → (𝑚 ∈ ℕ ↦ (vol‘(𝑇𝑚))):ℕ⟶ℝ)
105104ffvelrnda 6252 . . . . . . . . . . 11 (((𝜑 ∧ 0 < (vol*‘ran 𝐹)) ∧ 𝑘 ∈ ℕ) → ((𝑚 ∈ ℕ ↦ (vol‘(𝑇𝑚)))‘𝑘) ∈ ℝ)
10695, 96, 105serfre 12647 . . . . . . . . . 10 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇𝑚)))):ℕ⟶ℝ)
107 frn 5952 . . . . . . . . . 10 (seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇𝑚)))):ℕ⟶ℝ → ran seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇𝑚)))) ⊆ ℝ)
108106, 107syl 17 . . . . . . . . 9 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → ran seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇𝑚)))) ⊆ ℝ)
109 ressxr 9939 . . . . . . . . 9 ℝ ⊆ ℝ*
110108, 109syl6ss 3579 . . . . . . . 8 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → ran seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇𝑚)))) ⊆ ℝ*)
11199adantlr 746 . . . . . . . . . . . . . 14 (((𝜑 ∧ 0 < (vol*‘ran 𝐹)) ∧ 𝑚 ∈ ℕ) → (vol‘(𝑇𝑚)) = (vol*‘ran 𝐹))
112111mpteq2dva 4666 . . . . . . . . . . . . 13 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → (𝑚 ∈ ℕ ↦ (vol‘(𝑇𝑚))) = (𝑚 ∈ ℕ ↦ (vol*‘ran 𝐹)))
113 fconstmpt 5075 . . . . . . . . . . . . 13 (ℕ × {(vol*‘ran 𝐹)}) = (𝑚 ∈ ℕ ↦ (vol*‘ran 𝐹))
114112, 113syl6eqr 2661 . . . . . . . . . . . 12 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → (𝑚 ∈ ℕ ↦ (vol‘(𝑇𝑚))) = (ℕ × {(vol*‘ran 𝐹)}))
115114seqeq3d 12626 . . . . . . . . . . 11 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇𝑚)))) = seq1( + , (ℕ × {(vol*‘ran 𝐹)})))
116115fveq1d 6090 . . . . . . . . . 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)))
11751recnd 9924 . . . . . . . . . . 11 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → (vol*‘ran 𝐹) ∈ ℂ)
118 ser1const 12674 . . . . . . . . . . 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)))
119117, 61, 118syl2anc 690 . . . . . . . . . 10 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → (seq1( + , (ℕ × {(vol*‘ran 𝐹)}))‘((⌊‘(3 / (vol*‘ran 𝐹))) + 1)) = (((⌊‘(3 / (vol*‘ran 𝐹))) + 1) · (vol*‘ran 𝐹)))
120116, 119eqtrd 2643 . . . . . . . . 9 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → (seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇𝑚))))‘((⌊‘(3 / (vol*‘ran 𝐹))) + 1)) = (((⌊‘(3 / (vol*‘ran 𝐹))) + 1) · (vol*‘ran 𝐹)))
121 ffn 5944 . . . . . . . . . . 11 (seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇𝑚)))):ℕ⟶ℝ → seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇𝑚)))) Fn ℕ)
122106, 121syl 17 . . . . . . . . . 10 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇𝑚)))) Fn ℕ)
123 fnfvelrn 6249 . . . . . . . . . 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)))))
124122, 61, 123syl2anc 690 . . . . . . . . 9 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → (seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇𝑚))))‘((⌊‘(3 / (vol*‘ran 𝐹))) + 1)) ∈ ran seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇𝑚)))))
125120, 124eqeltrrd 2688 . . . . . . . 8 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → (((⌊‘(3 / (vol*‘ran 𝐹))) + 1) · (vol*‘ran 𝐹)) ∈ ran seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇𝑚)))))
126 supxrub 11982 . . . . . . . 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*, < ))
127110, 125, 126syl2anc 690 . . . . . . 7 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → (((⌊‘(3 / (vol*‘ran 𝐹))) + 1) · (vol*‘ran 𝐹)) ≤ sup(ran seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇𝑚)))), ℝ*, < ))
12885adantr 479 . . . . . . . . 9 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → 𝑚 ∈ ℕ (𝑇𝑚) ∈ dom vol)
129 mblvol 23022 . . . . . . . . 9 ( 𝑚 ∈ ℕ (𝑇𝑚) ∈ dom vol → (vol‘ 𝑚 ∈ ℕ (𝑇𝑚)) = (vol*‘ 𝑚 ∈ ℕ (𝑇𝑚)))
130128, 129syl 17 . . . . . . . 8 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → (vol‘ 𝑚 ∈ ℕ (𝑇𝑚)) = (vol*‘ 𝑚 ∈ ℕ (𝑇𝑚)))
13182, 101jca 552 . . . . . . . . . . 11 ((𝜑𝑚 ∈ ℕ) → ((𝑇𝑚) ∈ dom vol ∧ (vol‘(𝑇𝑚)) ∈ ℝ))
132131ralrimiva 2948 . . . . . . . . . 10 (𝜑 → ∀𝑚 ∈ ℕ ((𝑇𝑚) ∈ dom vol ∧ (vol‘(𝑇𝑚)) ∈ ℝ))
133132adantr 479 . . . . . . . . 9 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → ∀𝑚 ∈ ℕ ((𝑇𝑚) ∈ dom vol ∧ (vol‘(𝑇𝑚)) ∈ ℝ))
13411, 12, 13, 14, 15, 5, 16vitalilem3 23102 . . . . . . . . . 10 (𝜑Disj 𝑚 ∈ ℕ (𝑇𝑚))
135134adantr 479 . . . . . . . . 9 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → Disj 𝑚 ∈ ℕ (𝑇𝑚))
136 eqid 2609 . . . . . . . . . 10 seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇𝑚)))) = seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇𝑚))))
137136, 103voliun 23046 . . . . . . . . 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*, < ))
138133, 135, 137syl2anc 690 . . . . . . . 8 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → (vol‘ 𝑚 ∈ ℕ (𝑇𝑚)) = sup(ran seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇𝑚)))), ℝ*, < ))
139130, 138eqtr3d 2645 . . . . . . 7 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → (vol*‘ 𝑚 ∈ ℕ (𝑇𝑚)) = sup(ran seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇𝑚)))), ℝ*, < ))
140127, 139breqtrrd 4605 . . . . . 6 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → (((⌊‘(3 / (vol*‘ran 𝐹))) + 1) · (vol*‘ran 𝐹)) ≤ (vol*‘ 𝑚 ∈ ℕ (𝑇𝑚)))
14137, 64, 89, 94, 140xrltletrd 11827 . . . . 5 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → 3 < (vol*‘ 𝑚 ∈ ℕ (𝑇𝑚)))
14217simp3d 1067 . . . . . . . . 9 (𝜑 𝑚 ∈ ℕ (𝑇𝑚) ⊆ (-1[,]2))
143142adantr 479 . . . . . . . 8 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → 𝑚 ∈ ℕ (𝑇𝑚) ⊆ (-1[,]2))
144 2re 10937 . . . . . . . . 9 2 ∈ ℝ
145 iccssre 12082 . . . . . . . . 9 ((-1 ∈ ℝ ∧ 2 ∈ ℝ) → (-1[,]2) ⊆ ℝ)
14622, 144, 145mp2an 703 . . . . . . . 8 (-1[,]2) ⊆ ℝ
147 ovolss 22977 . . . . . . . 8 (( 𝑚 ∈ ℕ (𝑇𝑚) ⊆ (-1[,]2) ∧ (-1[,]2) ⊆ ℝ) → (vol*‘ 𝑚 ∈ ℕ (𝑇𝑚)) ≤ (vol*‘(-1[,]2)))
148143, 146, 147sylancl 692 . . . . . . 7 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → (vol*‘ 𝑚 ∈ ℕ (𝑇𝑚)) ≤ (vol*‘(-1[,]2)))
149 2cn 10938 . . . . . . . . 9 2 ∈ ℂ
150 ax-1cn 9850 . . . . . . . . 9 1 ∈ ℂ
151149, 150subnegi 10211 . . . . . . . 8 (2 − -1) = (2 + 1)
152 neg1lt0 10974 . . . . . . . . . . 11 -1 < 0
153 2pos 10959 . . . . . . . . . . 11 0 < 2
15422, 41, 144lttri 10014 . . . . . . . . . . 11 ((-1 < 0 ∧ 0 < 2) → -1 < 2)
155152, 153, 154mp2an 703 . . . . . . . . . 10 -1 < 2
15622, 144, 155ltleii 10011 . . . . . . . . 9 -1 ≤ 2
157 ovolicc 23015 . . . . . . . . 9 ((-1 ∈ ℝ ∧ 2 ∈ ℝ ∧ -1 ≤ 2) → (vol*‘(-1[,]2)) = (2 − -1))
15822, 144, 156, 157mp3an 1415 . . . . . . . 8 (vol*‘(-1[,]2)) = (2 − -1)
159 df-3 10927 . . . . . . . 8 3 = (2 + 1)
160151, 158, 1593eqtr4i 2641 . . . . . . 7 (vol*‘(-1[,]2)) = 3
161148, 160syl6breq 4618 . . . . . 6 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → (vol*‘ 𝑚 ∈ ℕ (𝑇𝑚)) ≤ 3)
162 xrlenlt 9954 . . . . . . 7 (((vol*‘ 𝑚 ∈ ℕ (𝑇𝑚)) ∈ ℝ* ∧ 3 ∈ ℝ*) → ((vol*‘ 𝑚 ∈ ℕ (𝑇𝑚)) ≤ 3 ↔ ¬ 3 < (vol*‘ 𝑚 ∈ ℕ (𝑇𝑚))))
16389, 36, 162sylancl 692 . . . . . 6 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → ((vol*‘ 𝑚 ∈ ℕ (𝑇𝑚)) ≤ 3 ↔ ¬ 3 < (vol*‘ 𝑚 ∈ ℕ (𝑇𝑚))))
164161, 163mpbid 220 . . . . 5 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → ¬ 3 < (vol*‘ 𝑚 ∈ ℕ (𝑇𝑚)))
165141, 164pm2.65da 597 . . . 4 (𝜑 → ¬ 0 < (vol*‘ran 𝐹))
166 ovolge0 22973 . . . . . . 7 (ran 𝐹 ⊆ ℝ → 0 ≤ (vol*‘ran 𝐹))
16720, 166syl 17 . . . . . 6 (𝜑 → 0 ≤ (vol*‘ran 𝐹))
168 0xr 9942 . . . . . . 7 0 ∈ ℝ*
169 ovolcl 22970 . . . . . . . 8 (ran 𝐹 ⊆ ℝ → (vol*‘ran 𝐹) ∈ ℝ*)
17020, 169syl 17 . . . . . . 7 (𝜑 → (vol*‘ran 𝐹) ∈ ℝ*)
171 xrleloe 11812 . . . . . . 7 ((0 ∈ ℝ* ∧ (vol*‘ran 𝐹) ∈ ℝ*) → (0 ≤ (vol*‘ran 𝐹) ↔ (0 < (vol*‘ran 𝐹) ∨ 0 = (vol*‘ran 𝐹))))
172168, 170, 171sylancr 693 . . . . . 6 (𝜑 → (0 ≤ (vol*‘ran 𝐹) ↔ (0 < (vol*‘ran 𝐹) ∨ 0 = (vol*‘ran 𝐹))))
173167, 172mpbid 220 . . . . 5 (𝜑 → (0 < (vol*‘ran 𝐹) ∨ 0 = (vol*‘ran 𝐹)))
174173ord 390 . . . 4 (𝜑 → (¬ 0 < (vol*‘ran 𝐹) → 0 = (vol*‘ran 𝐹)))
175165, 174mpd 15 . . 3 (𝜑 → 0 = (vol*‘ran 𝐹))
176175adantr 479 . 2 ((𝜑𝑚 ∈ ℕ) → 0 = (vol*‘ran 𝐹))
17734, 176eqtr4d 2646 1 ((𝜑𝑚 ∈ ℕ) → (vol*‘(𝑇𝑚)) = 0)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wo 381  wa 382   = wceq 1474  wcel 1976  wne 2779  wral 2895  {crab 2899  cdif 3536  cin 3538  wss 3539  c0 3873  𝒫 cpw 4107  {csn 4124   ciun 4449  Disj wdisj 4547   class class class wbr 4577  {copab 4636  cmpt 4637   × cxp 5026  dom cdm 5028  ran crn 5029   Fn wfn 5785  wf 5786  1-1-ontowf1o 5789  cfv 5790  (class class class)co 6527   / cqs 7605  supcsup 8206  cc 9790  cr 9791  0cc0 9792  1c1 9793   + caddc 9795   · cmul 9797  *cxr 9929   < clt 9930  cle 9931  cmin 10117  -cneg 10118   / cdiv 10533  cn 10867  2c2 10917  3c3 10918  0cn0 11139  cq 11620  +crp 11664  [,]cicc 12005  cfl 12408  seqcseq 12618  vol*covol 22955  volcvol 22956
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-inf2 8398  ax-cc 9117  ax-cnex 9848  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868  ax-pre-mulgt0 9869  ax-pre-sup 9870
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-fal 1480  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-disj 4548  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-se 4988  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-isom 5799  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-of 6772  df-om 6935  df-1st 7036  df-2nd 7037  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-1o 7424  df-2o 7425  df-oadd 7428  df-er 7606  df-ec 7608  df-qs 7612  df-map 7723  df-pm 7724  df-en 7819  df-dom 7820  df-sdom 7821  df-fin 7822  df-fi 8177  df-sup 8208  df-inf 8209  df-oi 8275  df-card 8625  df-cda 8850  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-sub 10119  df-neg 10120  df-div 10534  df-nn 10868  df-2 10926  df-3 10927  df-n0 11140  df-z 11211  df-uz 11520  df-q 11621  df-rp 11665  df-xneg 11778  df-xadd 11779  df-xmul 11780  df-ioo 12006  df-ico 12008  df-icc 12009  df-fz 12153  df-fzo 12290  df-fl 12410  df-seq 12619  df-exp 12678  df-hash 12935  df-cj 13633  df-re 13634  df-im 13635  df-sqrt 13769  df-abs 13770  df-clim 14013  df-rlim 14014  df-sum 14211  df-rest 15852  df-topgen 15873  df-psmet 19505  df-xmet 19506  df-met 19507  df-bl 19508  df-mopn 19509  df-top 20463  df-bases 20464  df-topon 20465  df-cmp 20942  df-ovol 22957  df-vol 22958
This theorem is referenced by:  vitalilem5  23104
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