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Theorem vitalilem4 23425
Description: Lemma for vitali 23427. (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 6229 . . . . . . . . 9 (𝑛 = 𝑚 → (𝐺𝑛) = (𝐺𝑚))
21oveq2d 6706 . . . . . . . 8 (𝑛 = 𝑚 → (𝑠 − (𝐺𝑛)) = (𝑠 − (𝐺𝑚)))
32eleq1d 2715 . . . . . . 7 (𝑛 = 𝑚 → ((𝑠 − (𝐺𝑛)) ∈ ran 𝐹 ↔ (𝑠 − (𝐺𝑚)) ∈ ran 𝐹))
43rabbidv 3220 . . . . . 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 10065 . . . . . . 7 ℝ ∈ V
76rabex 4845 . . . . . 6 {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑚)) ∈ ran 𝐹} ∈ V
84, 5, 7fvmpt 6321 . . . . 5 (𝑚 ∈ ℕ → (𝑇𝑚) = {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑚)) ∈ ran 𝐹})
98adantl 481 . . . 4 ((𝜑𝑚 ∈ ℕ) → (𝑇𝑚) = {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑚)) ∈ ran 𝐹})
109fveq2d 6233 . . 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 23423 . . . . . . 7 (𝜑 → (ran 𝐹 ⊆ (0[,]1) ∧ (0[,]1) ⊆ 𝑚 ∈ ℕ (𝑇𝑚) ∧ 𝑚 ∈ ℕ (𝑇𝑚) ⊆ (-1[,]2)))
1817simp1d 1093 . . . . . 6 (𝜑 → ran 𝐹 ⊆ (0[,]1))
19 unitssre 12357 . . . . . 6 (0[,]1) ⊆ ℝ
2018, 19syl6ss 3648 . . . . 5 (𝜑 → ran 𝐹 ⊆ ℝ)
2120adantr 480 . . . 4 ((𝜑𝑚 ∈ ℕ) → ran 𝐹 ⊆ ℝ)
22 neg1rr 11163 . . . . . 6 -1 ∈ ℝ
23 1re 10077 . . . . . 6 1 ∈ ℝ
24 iccssre 12293 . . . . . 6 ((-1 ∈ ℝ ∧ 1 ∈ ℝ) → (-1[,]1) ⊆ ℝ)
2522, 23, 24mp2an 708 . . . . 5 (-1[,]1) ⊆ ℝ
26 inss2 3867 . . . . . 6 (ℚ ∩ (-1[,]1)) ⊆ (-1[,]1)
27 f1of 6175 . . . . . . . 8 (𝐺:ℕ–1-1-onto→(ℚ ∩ (-1[,]1)) → 𝐺:ℕ⟶(ℚ ∩ (-1[,]1)))
2815, 27syl 17 . . . . . . 7 (𝜑𝐺:ℕ⟶(ℚ ∩ (-1[,]1)))
2928ffvelrnda 6399 . . . . . 6 ((𝜑𝑚 ∈ ℕ) → (𝐺𝑚) ∈ (ℚ ∩ (-1[,]1)))
3026, 29sseldi 3634 . . . . 5 ((𝜑𝑚 ∈ ℕ) → (𝐺𝑚) ∈ (-1[,]1))
3125, 30sseldi 3634 . . . 4 ((𝜑𝑚 ∈ ℕ) → (𝐺𝑚) ∈ ℝ)
32 eqidd 2652 . . . 4 ((𝜑𝑚 ∈ ℕ) → {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑚)) ∈ ran 𝐹} = {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑚)) ∈ ran 𝐹})
3321, 31, 32ovolshft 23325 . . 3 ((𝜑𝑚 ∈ ℕ) → (vol*‘ran 𝐹) = (vol*‘{𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑚)) ∈ ran 𝐹}))
3410, 33eqtr4d 2688 . 2 ((𝜑𝑚 ∈ ℕ) → (vol*‘(𝑇𝑚)) = (vol*‘ran 𝐹))
35 3re 11132 . . . . . . . 8 3 ∈ ℝ
3635rexri 10135 . . . . . . 7 3 ∈ ℝ*
3736a1i 11 . . . . . 6 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → 3 ∈ ℝ*)
38 3nn 11224 . . . . . . . . . . . . . 14 3 ∈ ℕ
39 nnrp 11880 . . . . . . . . . . . . . 14 (3 ∈ ℕ → 3 ∈ ℝ+)
4038, 39ax-mp 5 . . . . . . . . . . . . 13 3 ∈ ℝ+
41 0re 10078 . . . . . . . . . . . . . . . . . . . 20 0 ∈ ℝ
42 0le1 10589 . . . . . . . . . . . . . . . . . . . 20 0 ≤ 1
43 ovolicc 23337 . . . . . . . . . . . . . . . . . . . 20 ((0 ∈ ℝ ∧ 1 ∈ ℝ ∧ 0 ≤ 1) → (vol*‘(0[,]1)) = (1 − 0))
4441, 23, 42, 43mp3an 1464 . . . . . . . . . . . . . . . . . . 19 (vol*‘(0[,]1)) = (1 − 0)
45 1m0e1 11169 . . . . . . . . . . . . . . . . . . 19 (1 − 0) = 1
4644, 45eqtri 2673 . . . . . . . . . . . . . . . . . 18 (vol*‘(0[,]1)) = 1
4746, 23eqeltri 2726 . . . . . . . . . . . . . . . . 17 (vol*‘(0[,]1)) ∈ ℝ
48 ovolsscl 23300 . . . . . . . . . . . . . . . . 17 ((ran 𝐹 ⊆ (0[,]1) ∧ (0[,]1) ⊆ ℝ ∧ (vol*‘(0[,]1)) ∈ ℝ) → (vol*‘ran 𝐹) ∈ ℝ)
4919, 47, 48mp3an23 1456 . . . . . . . . . . . . . . . 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 11907 . . . . . . . . . . . . 13 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → (vol*‘ran 𝐹) ∈ ℝ+)
54 rpdivcl 11894 . . . . . . . . . . . . 13 ((3 ∈ ℝ+ ∧ (vol*‘ran 𝐹) ∈ ℝ+) → (3 / (vol*‘ran 𝐹)) ∈ ℝ+)
5540, 53, 54sylancr 696 . . . . . . . . . . . 12 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → (3 / (vol*‘ran 𝐹)) ∈ ℝ+)
5655rpred 11910 . . . . . . . . . . 11 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → (3 / (vol*‘ran 𝐹)) ∈ ℝ)
5755rpge0d 11914 . . . . . . . . . . 11 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → 0 ≤ (3 / (vol*‘ran 𝐹)))
58 flge0nn0 12661 . . . . . . . . . . 11 (((3 / (vol*‘ran 𝐹)) ∈ ℝ ∧ 0 ≤ (3 / (vol*‘ran 𝐹))) → (⌊‘(3 / (vol*‘ran 𝐹))) ∈ ℕ0)
5956, 57, 58syl2anc 694 . . . . . . . . . 10 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → (⌊‘(3 / (vol*‘ran 𝐹))) ∈ ℕ0)
60 nn0p1nn 11370 . . . . . . . . . 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 11073 . . . . . . . 8 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → ((⌊‘(3 / (vol*‘ran 𝐹))) + 1) ∈ ℝ)
6362, 51remulcld 10108 . . . . . . 7 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → (((⌊‘(3 / (vol*‘ran 𝐹))) + 1) · (vol*‘ran 𝐹)) ∈ ℝ)
6463rexrd 10127 . . . . . 6 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → (((⌊‘(3 / (vol*‘ran 𝐹))) + 1) · (vol*‘ran 𝐹)) ∈ ℝ*)
656elpw2 4858 . . . . . . . . . . . . . . . . . . 19 (ran 𝐹 ∈ 𝒫 ℝ ↔ ran 𝐹 ⊆ ℝ)
6620, 65sylibr 224 . . . . . . . . . . . . . . . . . 18 (𝜑 → ran 𝐹 ∈ 𝒫 ℝ)
6766anim1i 591 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ¬ ran 𝐹 ∈ dom vol) → (ran 𝐹 ∈ 𝒫 ℝ ∧ ¬ ran 𝐹 ∈ dom vol))
68 eldif 3617 . . . . . . . . . . . . . . . . 17 (ran 𝐹 ∈ (𝒫 ℝ ∖ dom vol) ↔ (ran 𝐹 ∈ 𝒫 ℝ ∧ ¬ ran 𝐹 ∈ dom vol))
6967, 68sylibr 224 . . . . . . . . . . . . . . . 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 140 . . . . . . . . . . . . . 14 (𝜑 → ran 𝐹 ∈ dom vol)
7271adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → ran 𝐹 ∈ dom vol)
73 inss1 3866 . . . . . . . . . . . . . . . 16 (ℚ ∩ (-1[,]1)) ⊆ ℚ
74 qssre 11836 . . . . . . . . . . . . . . . 16 ℚ ⊆ ℝ
7573, 74sstri 3645 . . . . . . . . . . . . . . 15 (ℚ ∩ (-1[,]1)) ⊆ ℝ
76 fss 6094 . . . . . . . . . . . . . . 15 ((𝐺:ℕ⟶(ℚ ∩ (-1[,]1)) ∧ (ℚ ∩ (-1[,]1)) ⊆ ℝ) → 𝐺:ℕ⟶ℝ)
7728, 75, 76sylancl 695 . . . . . . . . . . . . . 14 (𝜑𝐺:ℕ⟶ℝ)
7877ffvelrnda 6399 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → (𝐺𝑛) ∈ ℝ)
79 shftmbl 23352 . . . . . . . . . . . . 13 ((ran 𝐹 ∈ dom vol ∧ (𝐺𝑛) ∈ ℝ) → {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑛)) ∈ ran 𝐹} ∈ dom vol)
8072, 78, 79syl2anc 694 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → {𝑠 ∈ ℝ ∣ (𝑠 − (𝐺𝑛)) ∈ ran 𝐹} ∈ dom vol)
8180, 5fmptd 6425 . . . . . . . . . . 11 (𝜑𝑇:ℕ⟶dom vol)
8281ffvelrnda 6399 . . . . . . . . . 10 ((𝜑𝑚 ∈ ℕ) → (𝑇𝑚) ∈ dom vol)
8382ralrimiva 2995 . . . . . . . . 9 (𝜑 → ∀𝑚 ∈ ℕ (𝑇𝑚) ∈ dom vol)
84 iunmbl 23367 . . . . . . . . 9 (∀𝑚 ∈ ℕ (𝑇𝑚) ∈ dom vol → 𝑚 ∈ ℕ (𝑇𝑚) ∈ dom vol)
8583, 84syl 17 . . . . . . . 8 (𝜑 𝑚 ∈ ℕ (𝑇𝑚) ∈ dom vol)
86 mblss 23345 . . . . . . . 8 ( 𝑚 ∈ ℕ (𝑇𝑚) ∈ dom vol → 𝑚 ∈ ℕ (𝑇𝑚) ⊆ ℝ)
87 ovolcl 23292 . . . . . . . 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 12641 . . . . . . . 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 11962 . . . . . . 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 222 . . . . . 6 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → 3 < (((⌊‘(3 / (vol*‘ran 𝐹))) + 1) · (vol*‘ran 𝐹)))
95 nnuz 11761 . . . . . . . . . . 11 ℕ = (ℤ‘1)
96 1zzd 11446 . . . . . . . . . . 11 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → 1 ∈ ℤ)
97 mblvol 23344 . . . . . . . . . . . . . . . . 17 ((𝑇𝑚) ∈ dom vol → (vol‘(𝑇𝑚)) = (vol*‘(𝑇𝑚)))
9882, 97syl 17 . . . . . . . . . . . . . . . 16 ((𝜑𝑚 ∈ ℕ) → (vol‘(𝑇𝑚)) = (vol*‘(𝑇𝑚)))
9998, 34eqtrd 2685 . . . . . . . . . . . . . . 15 ((𝜑𝑚 ∈ ℕ) → (vol‘(𝑇𝑚)) = (vol*‘ran 𝐹))
10050adantr 480 . . . . . . . . . . . . . . 15 ((𝜑𝑚 ∈ ℕ) → (vol*‘ran 𝐹) ∈ ℝ)
10199, 100eqeltrd 2730 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℕ) → (vol‘(𝑇𝑚)) ∈ ℝ)
102101adantlr 751 . . . . . . . . . . . . 13 (((𝜑 ∧ 0 < (vol*‘ran 𝐹)) ∧ 𝑚 ∈ ℕ) → (vol‘(𝑇𝑚)) ∈ ℝ)
103 eqid 2651 . . . . . . . . . . . . 13 (𝑚 ∈ ℕ ↦ (vol‘(𝑇𝑚))) = (𝑚 ∈ ℕ ↦ (vol‘(𝑇𝑚)))
104102, 103fmptd 6425 . . . . . . . . . . . 12 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → (𝑚 ∈ ℕ ↦ (vol‘(𝑇𝑚))):ℕ⟶ℝ)
105104ffvelrnda 6399 . . . . . . . . . . 11 (((𝜑 ∧ 0 < (vol*‘ran 𝐹)) ∧ 𝑘 ∈ ℕ) → ((𝑚 ∈ ℕ ↦ (vol‘(𝑇𝑚)))‘𝑘) ∈ ℝ)
10695, 96, 105serfre 12870 . . . . . . . . . 10 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇𝑚)))):ℕ⟶ℝ)
107 frn 6091 . . . . . . . . . 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 10121 . . . . . . . . 9 ℝ ⊆ ℝ*
110108, 109syl6ss 3648 . . . . . . . 8 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → ran seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇𝑚)))) ⊆ ℝ*)
11199adantlr 751 . . . . . . . . . . . . . 14 (((𝜑 ∧ 0 < (vol*‘ran 𝐹)) ∧ 𝑚 ∈ ℕ) → (vol‘(𝑇𝑚)) = (vol*‘ran 𝐹))
112111mpteq2dva 4777 . . . . . . . . . . . . 13 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → (𝑚 ∈ ℕ ↦ (vol‘(𝑇𝑚))) = (𝑚 ∈ ℕ ↦ (vol*‘ran 𝐹)))
113 fconstmpt 5197 . . . . . . . . . . . . 13 (ℕ × {(vol*‘ran 𝐹)}) = (𝑚 ∈ ℕ ↦ (vol*‘ran 𝐹))
114112, 113syl6eqr 2703 . . . . . . . . . . . 12 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → (𝑚 ∈ ℕ ↦ (vol‘(𝑇𝑚))) = (ℕ × {(vol*‘ran 𝐹)}))
115114seqeq3d 12849 . . . . . . . . . . 11 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇𝑚)))) = seq1( + , (ℕ × {(vol*‘ran 𝐹)})))
116115fveq1d 6231 . . . . . . . . . 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 10106 . . . . . . . . . . 11 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → (vol*‘ran 𝐹) ∈ ℂ)
118 ser1const 12897 . . . . . . . . . . 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 694 . . . . . . . . . 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 2685 . . . . . . . . 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 6083 . . . . . . . . . . 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 6396 . . . . . . . . . 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 694 . . . . . . . . 9 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → (seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇𝑚))))‘((⌊‘(3 / (vol*‘ran 𝐹))) + 1)) ∈ ran seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇𝑚)))))
125120, 124eqeltrrd 2731 . . . . . . . 8 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → (((⌊‘(3 / (vol*‘ran 𝐹))) + 1) · (vol*‘ran 𝐹)) ∈ ran seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇𝑚)))))
126 supxrub 12192 . . . . . . . 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 694 . . . . . . 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 23344 . . . . . . . . 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 2995 . . . . . . . . . 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 23424 . . . . . . . . . 10 (𝜑Disj 𝑚 ∈ ℕ (𝑇𝑚))
135134adantr 480 . . . . . . . . 9 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → Disj 𝑚 ∈ ℕ (𝑇𝑚))
136 eqid 2651 . . . . . . . . . 10 seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇𝑚)))) = seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇𝑚))))
137136, 103voliun 23368 . . . . . . . . 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 694 . . . . . . . 8 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → (vol‘ 𝑚 ∈ ℕ (𝑇𝑚)) = sup(ran seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇𝑚)))), ℝ*, < ))
139130, 138eqtr3d 2687 . . . . . . 7 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → (vol*‘ 𝑚 ∈ ℕ (𝑇𝑚)) = sup(ran seq1( + , (𝑚 ∈ ℕ ↦ (vol‘(𝑇𝑚)))), ℝ*, < ))
140127, 139breqtrrd 4713 . . . . . 6 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → (((⌊‘(3 / (vol*‘ran 𝐹))) + 1) · (vol*‘ran 𝐹)) ≤ (vol*‘ 𝑚 ∈ ℕ (𝑇𝑚)))
14137, 64, 89, 94, 140xrltletrd 12030 . . . . 5 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → 3 < (vol*‘ 𝑚 ∈ ℕ (𝑇𝑚)))
14217simp3d 1095 . . . . . . . . 9 (𝜑 𝑚 ∈ ℕ (𝑇𝑚) ⊆ (-1[,]2))
143142adantr 480 . . . . . . . 8 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → 𝑚 ∈ ℕ (𝑇𝑚) ⊆ (-1[,]2))
144 2re 11128 . . . . . . . . 9 2 ∈ ℝ
145 iccssre 12293 . . . . . . . . 9 ((-1 ∈ ℝ ∧ 2 ∈ ℝ) → (-1[,]2) ⊆ ℝ)
14622, 144, 145mp2an 708 . . . . . . . 8 (-1[,]2) ⊆ ℝ
147 ovolss 23299 . . . . . . . 8 (( 𝑚 ∈ ℕ (𝑇𝑚) ⊆ (-1[,]2) ∧ (-1[,]2) ⊆ ℝ) → (vol*‘ 𝑚 ∈ ℕ (𝑇𝑚)) ≤ (vol*‘(-1[,]2)))
148143, 146, 147sylancl 695 . . . . . . 7 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → (vol*‘ 𝑚 ∈ ℕ (𝑇𝑚)) ≤ (vol*‘(-1[,]2)))
149 2cn 11129 . . . . . . . . 9 2 ∈ ℂ
150 ax-1cn 10032 . . . . . . . . 9 1 ∈ ℂ
151149, 150subnegi 10398 . . . . . . . 8 (2 − -1) = (2 + 1)
152 neg1lt0 11165 . . . . . . . . . . 11 -1 < 0
153 2pos 11150 . . . . . . . . . . 11 0 < 2
15422, 41, 144lttri 10201 . . . . . . . . . . 11 ((-1 < 0 ∧ 0 < 2) → -1 < 2)
155152, 153, 154mp2an 708 . . . . . . . . . 10 -1 < 2
15622, 144, 155ltleii 10198 . . . . . . . . 9 -1 ≤ 2
157 ovolicc 23337 . . . . . . . . 9 ((-1 ∈ ℝ ∧ 2 ∈ ℝ ∧ -1 ≤ 2) → (vol*‘(-1[,]2)) = (2 − -1))
15822, 144, 156, 157mp3an 1464 . . . . . . . 8 (vol*‘(-1[,]2)) = (2 − -1)
159 df-3 11118 . . . . . . . 8 3 = (2 + 1)
160151, 158, 1593eqtr4i 2683 . . . . . . 7 (vol*‘(-1[,]2)) = 3
161148, 160syl6breq 4726 . . . . . 6 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → (vol*‘ 𝑚 ∈ ℕ (𝑇𝑚)) ≤ 3)
162 xrlenlt 10141 . . . . . . 7 (((vol*‘ 𝑚 ∈ ℕ (𝑇𝑚)) ∈ ℝ* ∧ 3 ∈ ℝ*) → ((vol*‘ 𝑚 ∈ ℕ (𝑇𝑚)) ≤ 3 ↔ ¬ 3 < (vol*‘ 𝑚 ∈ ℕ (𝑇𝑚))))
16389, 36, 162sylancl 695 . . . . . 6 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → ((vol*‘ 𝑚 ∈ ℕ (𝑇𝑚)) ≤ 3 ↔ ¬ 3 < (vol*‘ 𝑚 ∈ ℕ (𝑇𝑚))))
164161, 163mpbid 222 . . . . 5 ((𝜑 ∧ 0 < (vol*‘ran 𝐹)) → ¬ 3 < (vol*‘ 𝑚 ∈ ℕ (𝑇𝑚)))
165141, 164pm2.65da 599 . . . 4 (𝜑 → ¬ 0 < (vol*‘ran 𝐹))
166 ovolge0 23295 . . . . . . 7 (ran 𝐹 ⊆ ℝ → 0 ≤ (vol*‘ran 𝐹))
16720, 166syl 17 . . . . . 6 (𝜑 → 0 ≤ (vol*‘ran 𝐹))
168 0xr 10124 . . . . . . 7 0 ∈ ℝ*
169 ovolcl 23292 . . . . . . . 8 (ran 𝐹 ⊆ ℝ → (vol*‘ran 𝐹) ∈ ℝ*)
17020, 169syl 17 . . . . . . 7 (𝜑 → (vol*‘ran 𝐹) ∈ ℝ*)
171 xrleloe 12015 . . . . . . 7 ((0 ∈ ℝ* ∧ (vol*‘ran 𝐹) ∈ ℝ*) → (0 ≤ (vol*‘ran 𝐹) ↔ (0 < (vol*‘ran 𝐹) ∨ 0 = (vol*‘ran 𝐹))))
172168, 170, 171sylancr 696 . . . . . 6 (𝜑 → (0 ≤ (vol*‘ran 𝐹) ↔ (0 < (vol*‘ran 𝐹) ∨ 0 = (vol*‘ran 𝐹))))
173167, 172mpbid 222 . . . . 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 2688 1 ((𝜑𝑚 ∈ ℕ) → (vol*‘(𝑇𝑚)) = 0)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382  wa 383   = wceq 1523  wcel 2030  wne 2823  wral 2941  {crab 2945  cdif 3604  cin 3606  wss 3607  c0 3948  𝒫 cpw 4191  {csn 4210   ciun 4552  Disj wdisj 4652   class class class wbr 4685  {copab 4745  cmpt 4762   × cxp 5141  dom cdm 5143  ran crn 5144   Fn wfn 5921  wf 5922  1-1-ontowf1o 5925  cfv 5926  (class class class)co 6690   / cqs 7786  supcsup 8387  cc 9972  cr 9973  0cc0 9974  1c1 9975   + caddc 9977   · cmul 9979  *cxr 10111   < clt 10112  cle 10113  cmin 10304  -cneg 10305   / cdiv 10722  cn 11058  2c2 11108  3c3 11109  0cn0 11330  cq 11826  +crp 11870  [,]cicc 12216  cfl 12631  seqcseq 12841  vol*covol 23277  volcvol 23278
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cc 9295  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-disj 4653  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-ec 7789  df-qs 7793  df-map 7901  df-pm 7902  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-fi 8358  df-sup 8389  df-inf 8390  df-oi 8456  df-card 8803  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-n0 11331  df-z 11416  df-uz 11726  df-q 11827  df-rp 11871  df-xneg 11984  df-xadd 11985  df-xmul 11986  df-ioo 12217  df-ico 12219  df-icc 12220  df-fz 12365  df-fzo 12505  df-fl 12633  df-seq 12842  df-exp 12901  df-hash 13158  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-clim 14263  df-rlim 14264  df-sum 14461  df-rest 16130  df-topgen 16151  df-psmet 19786  df-xmet 19787  df-met 19788  df-bl 19789  df-mopn 19790  df-top 20747  df-topon 20764  df-bases 20798  df-cmp 21238  df-ovol 23279  df-vol 23280
This theorem is referenced by:  vitalilem5  23426
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