Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  vitalilem4 Structured version   Visualization version   GIF version

Theorem vitalilem4 23186
 Description: Lemma for vitali 23188. (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 6103 . . . . . . . . 9 (𝑛 = 𝑚 → (𝐺𝑛) = (𝐺𝑚))
21oveq2d 6565 . . . . . . . 8 (𝑛 = 𝑚 → (𝑠 − (𝐺𝑛)) = (𝑠 − (𝐺𝑚)))
32eleq1d 2672 . . . . . . 7 (𝑛 = 𝑚 → ((𝑠 − (𝐺𝑛)) ∈ ran 𝐹 ↔ (𝑠 − (𝐺𝑚)) ∈ ran 𝐹))
43rabbidv 3164 . . . . . 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 9906 . . . . . . 7 ℝ ∈ V
76rabex 4740 . . . . . 6 {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑚)) ∈ ran 𝐹} ∈ V
84, 5, 7fvmpt 6191 . . . . 5 (𝑚 ∈ ℕ → (𝑇𝑚) = {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑚)) ∈ ran 𝐹})
98adantl 481 . . . 4 ((𝜑𝑚 ∈ ℕ) → (𝑇𝑚) = {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑚)) ∈ ran 𝐹})
109fveq2d 6107 . . 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 23184 . . . . . . 7 (𝜑 → (ran 𝐹 ⊆ (0[,]1) ∧ (0[,]1) ⊆ 𝑚 ∈ ℕ (𝑇𝑚) ∧ 𝑚 ∈ ℕ (𝑇𝑚) ⊆ (-1[,]2)))
1817simp1d 1066 . . . . . 6 (𝜑 → ran 𝐹 ⊆ (0[,]1))
19 unitssre 12190 . . . . . 6 (0[,]1) ⊆ ℝ
2018, 19syl6ss 3580 . . . . 5 (𝜑 → ran 𝐹 ⊆ ℝ)
2120adantr 480 . . . 4 ((𝜑𝑚 ∈ ℕ) → ran 𝐹 ⊆ ℝ)
22 neg1rr 11002 . . . . . 6 -1 ∈ ℝ
23 1re 9918 . . . . . 6 1 ∈ ℝ
24 iccssre 12126 . . . . . 6 ((-1 ∈ ℝ ∧ 1 ∈ ℝ) → (-1[,]1) ⊆ ℝ)
2522, 23, 24mp2an 704 . . . . 5 (-1[,]1) ⊆ ℝ
26 inss2 3796 . . . . . 6 (ℚ ∩ (-1[,]1)) ⊆ (-1[,]1)
27 f1of 6050 . . . . . . . 8 (𝐺:ℕ–1-1-onto→(ℚ ∩ (-1[,]1)) → 𝐺:ℕ⟶(ℚ ∩ (-1[,]1)))
2815, 27syl 17 . . . . . . 7 (𝜑𝐺:ℕ⟶(ℚ ∩ (-1[,]1)))
2928ffvelrnda 6267 . . . . . 6 ((𝜑𝑚 ∈ ℕ) → (𝐺𝑚) ∈ (ℚ ∩ (-1[,]1)))
3026, 29sseldi 3566 . . . . 5 ((𝜑𝑚 ∈ ℕ) → (𝐺𝑚) ∈ (-1[,]1))
3125, 30sseldi 3566 . . . 4 ((𝜑𝑚 ∈ ℕ) → (𝐺𝑚) ∈ ℝ)
32 eqidd 2611 . . . 4 ((𝜑𝑚 ∈ ℕ) → {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑚)) ∈ ran 𝐹} = {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑚)) ∈ ran 𝐹})
3321, 31, 32ovolshft 23086 . . 3 ((𝜑𝑚 ∈ ℕ) → (vol*‘ran 𝐹) = (vol*‘{𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑚)) ∈ ran 𝐹}))
3410, 33eqtr4d 2647 . 2 ((𝜑𝑚 ∈ ℕ) → (vol*‘(𝑇𝑚)) = (vol*‘ran 𝐹))
35 3re 10971 . . . . . . . 8 3 ∈ ℝ
3635rexri 9976 . . . . . . 7 3 ∈ ℝ*
3736a1i 11 . . . . . 6 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → 3 ∈ ℝ*)
38 3nn 11063 . . . . . . . . . . . . . 14 3 ∈ ℕ
39 nnrp 11718 . . . . . . . . . . . . . 14 (3 ∈ ℕ → 3 ∈ ℝ+)
4038, 39ax-mp 5 . . . . . . . . . . . . 13 3 ∈ ℝ+
41 0re 9919 . . . . . . . . . . . . . . . . . . . 20 0 ∈ ℝ
42 0le1 10430 . . . . . . . . . . . . . . . . . . . 20 0 ≤ 1
43 ovolicc 23098 . . . . . . . . . . . . . . . . . . . 20 ((0 ∈ ℝ ∧ 1 ∈ ℝ ∧ 0 ≤ 1) → (vol*‘(0[,]1)) = (1 − 0))
4441, 23, 42, 43mp3an 1416 . . . . . . . . . . . . . . . . . . 19 (vol*‘(0[,]1)) = (1 − 0)
45 1m0e1 11008 . . . . . . . . . . . . . . . . . . 19 (1 − 0) = 1
4644, 45eqtri 2632 . . . . . . . . . . . . . . . . . 18 (vol*‘(0[,]1)) = 1
4746, 23eqeltri 2684 . . . . . . . . . . . . . . . . 17 (vol*‘(0[,]1)) ∈ ℝ
48 ovolsscl 23061 . . . . . . . . . . . . . . . . 17 ((ran 𝐹 ⊆ (0[,]1) ∧ (0[,]1) ⊆ ℝ ∧ (vol*‘(0[,]1)) ∈ ℝ) → (vol*‘ran 𝐹) ∈ ℝ)
4919, 47, 48mp3an23 1408 . . . . . . . . . . . . . . . 16 (ran 𝐹 ⊆ (0[,]1) → (vol*‘ran 𝐹) ∈ ℝ)
5018, 49syl 17 . . . . . . . . . . . . . . 15 (𝜑 → (vol*‘ran 𝐹) ∈ ℝ)
5150adantr 480 . . . . . . . . . . . . . 14 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → (vol*‘ran 𝐹) ∈ ℝ)
52 simpr 476 . . . . . . . . . . . . . 14 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → 0 < (vol*‘ran 𝐹))
5351, 52elrpd 11745 . . . . . . . . . . . . 13 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → (vol*‘ran 𝐹) ∈ ℝ+)
54 rpdivcl 11732 . . . . . . . . . . . . 13 ((3 ∈ ℝ+ ∧ (vol*‘ran 𝐹) ∈ ℝ+) → (3 / (vol*‘ran 𝐹)) ∈ ℝ+)
5540, 53, 54sylancr 694 . . . . . . . . . . . 12 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → (3 / (vol*‘ran 𝐹)) ∈ ℝ+)
5655rpred 11748 . . . . . . . . . . 11 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → (3 / (vol*‘ran 𝐹)) ∈ ℝ)
5755rpge0d 11752 . . . . . . . . . . 11 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → 0 ≤ (3 / (vol*‘ran 𝐹)))
58 flge0nn0 12483 . . . . . . . . . . 11 (((3 / (vol*‘ran 𝐹)) ∈ ℝ ∧ 0 ≤ (3 / (vol*‘ran 𝐹))) → (⌊‘(3 / (vol*‘ran 𝐹))) ∈ ℕ0)
5956, 57, 58syl2anc 691 . . . . . . . . . 10 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → (⌊‘(3 / (vol*‘ran 𝐹))) ∈ ℕ0)
60 nn0p1nn 11209 . . . . . . . . . 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 10912 . . . . . . . 8 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → ((⌊‘(3 / (vol*‘ran 𝐹))) + 1) ∈ ℝ)
6362, 51remulcld 9949 . . . . . . 7 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → (((⌊‘(3 / (vol*‘ran 𝐹))) + 1) · (vol*‘ran 𝐹)) ∈ ℝ)
6463rexrd 9968 . . . . . 6 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → (((⌊‘(3 / (vol*‘ran 𝐹))) + 1) · (vol*‘ran 𝐹)) ∈ ℝ*)
656elpw2 4755 . . . . . . . . . . . . . . . . . . 19 (ran 𝐹 ∈ 𝒫 ℝ ↔ ran 𝐹 ⊆ ℝ)
6620, 65sylibr 223 . . . . . . . . . . . . . . . . . 18 (𝜑 → ran 𝐹 ∈ 𝒫 ℝ)
6766anim1i 590 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ¬ ran 𝐹 ∈ dom vol) → (ran 𝐹 ∈ 𝒫 ℝ ∧ ¬ ran 𝐹 ∈ dom vol))
68 eldif 3550 . . . . . . . . . . . . . . . . 17 (ran 𝐹 ∈ (𝒫 ℝ ∖ dom vol) ↔ (ran 𝐹 ∈ 𝒫 ℝ ∧ ¬ ran 𝐹 ∈ dom vol))
6967, 68sylibr 223 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ¬ ran 𝐹 ∈ dom vol) → ran 𝐹 ∈ (𝒫 ℝ ∖ dom vol))
7069ex 449 . . . . . . . . . . . . . . 15 (𝜑 → (¬ ran 𝐹 ∈ dom vol → ran 𝐹 ∈ (𝒫 ℝ ∖ dom vol)))
7116, 70mt3d 139 . . . . . . . . . . . . . 14 (𝜑 → ran 𝐹 ∈ dom vol)
7271adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → ran 𝐹 ∈ dom vol)
73 inss1 3795 . . . . . . . . . . . . . . . 16 (ℚ ∩ (-1[,]1)) ⊆ ℚ
74 qssre 11674 . . . . . . . . . . . . . . . 16 ℚ ⊆ ℝ
7573, 74sstri 3577 . . . . . . . . . . . . . . 15 (ℚ ∩ (-1[,]1)) ⊆ ℝ
76 fss 5969 . . . . . . . . . . . . . . 15 ((𝐺:ℕ⟶(ℚ ∩ (-1[,]1)) ∧ (ℚ ∩ (-1[,]1)) ⊆ ℝ) → 𝐺:ℕ⟶ℝ)
7728, 75, 76sylancl 693 . . . . . . . . . . . . . 14 (𝜑𝐺:ℕ⟶ℝ)
7877ffvelrnda 6267 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → (𝐺𝑛) ∈ ℝ)
79 shftmbl 23113 . . . . . . . . . . . . 13 ((ran 𝐹 ∈ dom vol ∧ (𝐺𝑛) ∈ ℝ) → {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑛)) ∈ ran 𝐹} ∈ dom vol)
8072, 78, 79syl2anc 691 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑛)) ∈ ran 𝐹} ∈ dom vol)
8180, 5fmptd 6292 . . . . . . . . . . 11 (𝜑𝑇:ℕ⟶dom vol)
8281ffvelrnda 6267 . . . . . . . . . 10 ((𝜑𝑚 ∈ ℕ) → (𝑇𝑚) ∈ dom vol)
8382ralrimiva 2949 . . . . . . . . 9 (𝜑 → ∀𝑚 ∈ ℕ (𝑇𝑚) ∈ dom vol)
84 iunmbl 23128 . . . . . . . . 9 (∀𝑚 ∈ ℕ (𝑇𝑚) ∈ dom vol → 𝑚 ∈ ℕ (𝑇𝑚) ∈ dom vol)
8583, 84syl 17 . . . . . . . 8 (𝜑 𝑚 ∈ ℕ (𝑇𝑚) ∈ dom vol)
86 mblss 23106 . . . . . . . 8 ( 𝑚 ∈ ℕ (𝑇𝑚) ∈ dom vol → 𝑚 ∈ ℕ (𝑇𝑚) ⊆ ℝ)
87 ovolcl 23053 . . . . . . . 8 ( 𝑚 ∈ ℕ (𝑇𝑚) ⊆ ℝ → (vol*‘ 𝑚 ∈ ℕ (𝑇𝑚)) ∈ ℝ*)
8885, 86, 873syl 18 . . . . . . 7 (𝜑 → (vol*‘ 𝑚 ∈ ℕ (𝑇𝑚)) ∈ ℝ*)
8988adantr 480 . . . . . 6 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → (vol*‘ 𝑚 ∈ ℕ (𝑇𝑚)) ∈ ℝ*)
90 flltp1 12463 . . . . . . . 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 11800 . . . . . . 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 221 . . . . . 6 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → 3 < (((⌊‘(3 / (vol*‘ran 𝐹))) + 1) · (vol*‘ran 𝐹)))
95 nnuz 11599 . . . . . . . . . . 11 ℕ = (ℤ‘1)
96 1zzd 11285 . . . . . . . . . . 11 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → 1 ∈ ℤ)
97 mblvol 23105 . . . . . . . . . . . . . . . . 17 ((𝑇𝑚) ∈ dom vol → (vol‘(𝑇𝑚)) = (vol*‘(𝑇𝑚)))
9882, 97syl 17 . . . . . . . . . . . . . . . 16 ((𝜑𝑚 ∈ ℕ) → (vol‘(𝑇𝑚)) = (vol*‘(𝑇𝑚)))
9998, 34eqtrd 2644 . . . . . . . . . . . . . . 15 ((𝜑𝑚 ∈ ℕ) → (vol‘(𝑇𝑚)) = (vol*‘ran 𝐹))
10050adantr 480 . . . . . . . . . . . . . . 15 ((𝜑𝑚 ∈ ℕ) → (vol*‘ran 𝐹) ∈ ℝ)
10199, 100eqeltrd 2688 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℕ) → (vol‘(𝑇𝑚)) ∈ ℝ)
102101adantlr 747 . . . . . . . . . . . . 13 (((𝜑 ∧ 0 < (vol*‘ran 𝐹)) ∧ 𝑚 ∈ ℕ) → (vol‘(𝑇𝑚)) ∈ ℝ)
103 eqid 2610 . . . . . . . . . . . . 13 (𝑚 ∈ ℕ ↦ (vol‘(𝑇𝑚))) = (𝑚 ∈ ℕ ↦ (vol‘(𝑇𝑚)))
104102, 103fmptd 6292 . . . . . . . . . . . 12 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → (𝑚 ∈ ℕ ↦ (vol‘(𝑇𝑚))):ℕ⟶ℝ)
105104ffvelrnda 6267 . . . . . . . . . . 11 (((𝜑 ∧ 0 < (vol*‘ran 𝐹)) ∧ 𝑘 ∈ ℕ) → ((𝑚 ∈ ℕ ↦ (vol‘(𝑇𝑚)))‘𝑘) ∈ ℝ)
10695, 96, 105serfre 12692 . . . . . . . . . 10 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇𝑚)))):ℕ⟶ℝ)
107 frn 5966 . . . . . . . . . 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 9962 . . . . . . . . 9 ℝ ⊆ ℝ*
110108, 109syl6ss 3580 . . . . . . . 8 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → ran seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇𝑚)))) ⊆ ℝ*)
11199adantlr 747 . . . . . . . . . . . . . 14 (((𝜑 ∧ 0 < (vol*‘ran 𝐹)) ∧ 𝑚 ∈ ℕ) → (vol‘(𝑇𝑚)) = (vol*‘ran 𝐹))
112111mpteq2dva 4672 . . . . . . . . . . . . 13 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → (𝑚 ∈ ℕ ↦ (vol‘(𝑇𝑚))) = (𝑚 ∈ ℕ ↦ (vol*‘ran 𝐹)))
113 fconstmpt 5085 . . . . . . . . . . . . 13 (ℕ × {(vol*‘ran 𝐹)}) = (𝑚 ∈ ℕ ↦ (vol*‘ran 𝐹))
114112, 113syl6eqr 2662 . . . . . . . . . . . 12 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → (𝑚 ∈ ℕ ↦ (vol‘(𝑇𝑚))) = (ℕ × {(vol*‘ran 𝐹)}))
115114seqeq3d 12671 . . . . . . . . . . 11 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇𝑚)))) = seq1( + , (ℕ × {(vol*‘ran 𝐹)})))
116115fveq1d 6105 . . . . . . . . . 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 9947 . . . . . . . . . . 11 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → (vol*‘ran 𝐹) ∈ ℂ)
118 ser1const 12719 . . . . . . . . . . 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 691 . . . . . . . . . 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 2644 . . . . . . . . 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 5958 . . . . . . . . . . 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 6264 . . . . . . . . . 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 691 . . . . . . . . 9 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → (seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇𝑚))))‘((⌊‘(3 / (vol*‘ran 𝐹))) + 1)) ∈ ran seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇𝑚)))))
125120, 124eqeltrrd 2689 . . . . . . . 8 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → (((⌊‘(3 / (vol*‘ran 𝐹))) + 1) · (vol*‘ran 𝐹)) ∈ ran seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇𝑚)))))
126 supxrub 12026 . . . . . . . 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 691 . . . . . . 7 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → (((⌊‘(3 / (vol*‘ran 𝐹))) + 1) · (vol*‘ran 𝐹)) ≤ sup(ran seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇𝑚)))), ℝ*, < ))
12885adantr 480 . . . . . . . . 9 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → 𝑚 ∈ ℕ (𝑇𝑚) ∈ dom vol)
129 mblvol 23105 . . . . . . . . 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 553 . . . . . . . . . . 11 ((𝜑𝑚 ∈ ℕ) → ((𝑇𝑚) ∈ dom vol ∧ (vol‘(𝑇𝑚)) ∈ ℝ))
132131ralrimiva 2949 . . . . . . . . . 10 (𝜑 → ∀𝑚 ∈ ℕ ((𝑇𝑚) ∈ dom vol ∧ (vol‘(𝑇𝑚)) ∈ ℝ))
133132adantr 480 . . . . . . . . 9 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → ∀𝑚 ∈ ℕ ((𝑇𝑚) ∈ dom vol ∧ (vol‘(𝑇𝑚)) ∈ ℝ))
13411, 12, 13, 14, 15, 5, 16vitalilem3 23185 . . . . . . . . . 10 (𝜑Disj 𝑚 ∈ ℕ (𝑇𝑚))
135134adantr 480 . . . . . . . . 9 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → Disj 𝑚 ∈ ℕ (𝑇𝑚))
136 eqid 2610 . . . . . . . . . 10 seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇𝑚)))) = seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇𝑚))))
137136, 103voliun 23129 . . . . . . . . 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 691 . . . . . . . 8 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → (vol‘ 𝑚 ∈ ℕ (𝑇𝑚)) = sup(ran seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇𝑚)))), ℝ*, < ))
139130, 138eqtr3d 2646 . . . . . . 7 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → (vol*‘ 𝑚 ∈ ℕ (𝑇𝑚)) = sup(ran seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇𝑚)))), ℝ*, < ))
140127, 139breqtrrd 4611 . . . . . 6 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → (((⌊‘(3 / (vol*‘ran 𝐹))) + 1) · (vol*‘ran 𝐹)) ≤ (vol*‘ 𝑚 ∈ ℕ (𝑇𝑚)))
14137, 64, 89, 94, 140xrltletrd 11868 . . . . 5 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → 3 < (vol*‘ 𝑚 ∈ ℕ (𝑇𝑚)))
14217simp3d 1068 . . . . . . . . 9 (𝜑 𝑚 ∈ ℕ (𝑇𝑚) ⊆ (-1[,]2))
143142adantr 480 . . . . . . . 8 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → 𝑚 ∈ ℕ (𝑇𝑚) ⊆ (-1[,]2))
144 2re 10967 . . . . . . . . 9 2 ∈ ℝ
145 iccssre 12126 . . . . . . . . 9 ((-1 ∈ ℝ ∧ 2 ∈ ℝ) → (-1[,]2) ⊆ ℝ)
14622, 144, 145mp2an 704 . . . . . . . 8 (-1[,]2) ⊆ ℝ
147 ovolss 23060 . . . . . . . 8 (( 𝑚 ∈ ℕ (𝑇𝑚) ⊆ (-1[,]2) ∧ (-1[,]2) ⊆ ℝ) → (vol*‘ 𝑚 ∈ ℕ (𝑇𝑚)) ≤ (vol*‘(-1[,]2)))
148143, 146, 147sylancl 693 . . . . . . 7 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → (vol*‘ 𝑚 ∈ ℕ (𝑇𝑚)) ≤ (vol*‘(-1[,]2)))
149 2cn 10968 . . . . . . . . 9 2 ∈ ℂ
150 ax-1cn 9873 . . . . . . . . 9 1 ∈ ℂ
151149, 150subnegi 10239 . . . . . . . 8 (2 − -1) = (2 + 1)
152 neg1lt0 11004 . . . . . . . . . . 11 -1 < 0
153 2pos 10989 . . . . . . . . . . 11 0 < 2
15422, 41, 144lttri 10042 . . . . . . . . . . 11 ((-1 < 0 ∧ 0 < 2) → -1 < 2)
155152, 153, 154mp2an 704 . . . . . . . . . 10 -1 < 2
15622, 144, 155ltleii 10039 . . . . . . . . 9 -1 ≤ 2
157 ovolicc 23098 . . . . . . . . 9 ((-1 ∈ ℝ ∧ 2 ∈ ℝ ∧ -1 ≤ 2) → (vol*‘(-1[,]2)) = (2 − -1))
15822, 144, 156, 157mp3an 1416 . . . . . . . 8 (vol*‘(-1[,]2)) = (2 − -1)
159 df-3 10957 . . . . . . . 8 3 = (2 + 1)
160151, 158, 1593eqtr4i 2642 . . . . . . 7 (vol*‘(-1[,]2)) = 3
161148, 160syl6breq 4624 . . . . . 6 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → (vol*‘ 𝑚 ∈ ℕ (𝑇𝑚)) ≤ 3)
162 xrlenlt 9982 . . . . . . 7 (((vol*‘ 𝑚 ∈ ℕ (𝑇𝑚)) ∈ ℝ* ∧ 3 ∈ ℝ*) → ((vol*‘ 𝑚 ∈ ℕ (𝑇𝑚)) ≤ 3 ↔ ¬ 3 < (vol*‘ 𝑚 ∈ ℕ (𝑇𝑚))))
16389, 36, 162sylancl 693 . . . . . 6 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → ((vol*‘ 𝑚 ∈ ℕ (𝑇𝑚)) ≤ 3 ↔ ¬ 3 < (vol*‘ 𝑚 ∈ ℕ (𝑇𝑚))))
164161, 163mpbid 221 . . . . 5 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → ¬ 3 < (vol*‘ 𝑚 ∈ ℕ (𝑇𝑚)))
165141, 164pm2.65da 598 . . . 4 (𝜑 → ¬ 0 < (vol*‘ran 𝐹))
166 ovolge0 23056 . . . . . . 7 (ran 𝐹 ⊆ ℝ → 0 ≤ (vol*‘ran 𝐹))
16720, 166syl 17 . . . . . 6 (𝜑 → 0 ≤ (vol*‘ran 𝐹))
168 0xr 9965 . . . . . . 7 0 ∈ ℝ*
169 ovolcl 23053 . . . . . . . 8 (ran 𝐹 ⊆ ℝ → (vol*‘ran 𝐹) ∈ ℝ*)
17020, 169syl 17 . . . . . . 7 (𝜑 → (vol*‘ran 𝐹) ∈ ℝ*)
171 xrleloe 11853 . . . . . . 7 ((0 ∈ ℝ* ∧ (vol*‘ran 𝐹) ∈ ℝ*) → (0 ≤ (vol*‘ran 𝐹) ↔ (0 < (vol*‘ran 𝐹) ∨ 0 = (vol*‘ran 𝐹))))
172168, 170, 171sylancr 694 . . . . . 6 (𝜑 → (0 ≤ (vol*‘ran 𝐹) ↔ (0 < (vol*‘ran 𝐹) ∨ 0 = (vol*‘ran 𝐹))))
173167, 172mpbid 221 . . . . 5 (𝜑 → (0 < (vol*‘ran 𝐹) ∨ 0 = (vol*‘ran 𝐹)))
174173ord 391 . . . 4 (𝜑 → (¬ 0 < (vol*‘ran 𝐹) → 0 = (vol*‘ran 𝐹)))
175165, 174mpd 15 . . 3 (𝜑 → 0 = (vol*‘ran 𝐹))
176175adantr 480 . 2 ((𝜑𝑚 ∈ ℕ) → 0 = (vol*‘ran 𝐹))
17734, 176eqtr4d 2647 1 ((𝜑𝑚 ∈ ℕ) → (vol*‘(𝑇𝑚)) = 0)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  {crab 2900   ∖ cdif 3537   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108  {csn 4125  ∪ ciun 4455  Disj wdisj 4553   class class class wbr 4583  {copab 4642   ↦ cmpt 4643   × cxp 5036  dom cdm 5038  ran crn 5039   Fn wfn 5799  ⟶wf 5800  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549   / cqs 7628  supcsup 8229  ℂcc 9813  ℝcr 9814  0cc0 9815  1c1 9816   + caddc 9818   · cmul 9820  ℝ*cxr 9952   < clt 9953   ≤ cle 9954   − cmin 10145  -cneg 10146   / cdiv 10563  ℕcn 10897  2c2 10947  3c3 10948  ℕ0cn0 11169  ℚcq 11664  ℝ+crp 11708  [,]cicc 12049  ⌊cfl 12453  seqcseq 12663  vol*covol 23038  volcvol 23039 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cc 9140  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-disj 4554  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-ec 7631  df-qs 7635  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fi 8200  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-q 11665  df-rp 11709  df-xneg 11822  df-xadd 11823  df-xmul 11824  df-ioo 12050  df-ico 12052  df-icc 12053  df-fz 12198  df-fzo 12335  df-fl 12455  df-seq 12664  df-exp 12723  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-rlim 14068  df-sum 14265  df-rest 15906  df-topgen 15927  df-psmet 19559  df-xmet 19560  df-met 19561  df-bl 19562  df-mopn 19563  df-top 20521  df-bases 20522  df-topon 20523  df-cmp 21000  df-ovol 23040  df-vol 23041 This theorem is referenced by:  vitalilem5  23187
 Copyright terms: Public domain W3C validator