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Theorem vitalilem4 25119
Description: Lemma for vitali 25121. (Contributed by Mario Carneiro, 16-Jun-2014.)
Hypotheses
Ref Expression
vitali.1 ∼ = {⟨π‘₯, π‘¦βŸ© ∣ ((π‘₯ ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]1)) ∧ (π‘₯ βˆ’ 𝑦) ∈ β„š)}
vitali.2 𝑆 = ((0[,]1) / ∼ )
vitali.3 (πœ‘ β†’ 𝐹 Fn 𝑆)
vitali.4 (πœ‘ β†’ βˆ€π‘§ ∈ 𝑆 (𝑧 β‰  βˆ… β†’ (πΉβ€˜π‘§) ∈ 𝑧))
vitali.5 (πœ‘ β†’ 𝐺:ℕ–1-1-ontoβ†’(β„š ∩ (-1[,]1)))
vitali.6 𝑇 = (𝑛 ∈ β„• ↦ {𝑠 ∈ ℝ ∣ (𝑠 βˆ’ (πΊβ€˜π‘›)) ∈ ran 𝐹})
vitali.7 (πœ‘ β†’ Β¬ ran 𝐹 ∈ (𝒫 ℝ βˆ– dom vol))
Assertion
Ref Expression
vitalilem4 ((πœ‘ ∧ π‘š ∈ β„•) β†’ (vol*β€˜(π‘‡β€˜π‘š)) = 0)
Distinct variable groups:   π‘š,𝑛,𝑠,π‘₯,𝑦,𝑧,𝐺   πœ‘,π‘š,𝑛,π‘₯,𝑧   𝑧,𝑆   𝑇,π‘š,π‘₯   π‘š,𝐹,𝑛,𝑠,π‘₯,𝑦,𝑧   ∼ ,π‘š,𝑛,𝑠,π‘₯,𝑦,𝑧
Allowed substitution hints:   πœ‘(𝑦,𝑠)   𝑆(π‘₯,𝑦,π‘š,𝑛,𝑠)   𝑇(𝑦,𝑧,𝑛,𝑠)
Proof of Theorem vitalilem4
Dummy variable π‘˜ is distinct from all other variables.
StepHypRef Expression
1 fveq2 6888 . . . . . . . . 9 (𝑛 = π‘š β†’ (πΊβ€˜π‘›) = (πΊβ€˜π‘š))
21oveq2d 7421 . . . . . . . 8 (𝑛 = π‘š β†’ (𝑠 βˆ’ (πΊβ€˜π‘›)) = (𝑠 βˆ’ (πΊβ€˜π‘š)))
32eleq1d 2818 . . . . . . 7 (𝑛 = π‘š β†’ ((𝑠 βˆ’ (πΊβ€˜π‘›)) ∈ ran 𝐹 ↔ (𝑠 βˆ’ (πΊβ€˜π‘š)) ∈ ran 𝐹))
43rabbidv 3440 . . . . . 6 (𝑛 = π‘š β†’ {𝑠 ∈ ℝ ∣ (𝑠 βˆ’ (πΊβ€˜π‘›)) ∈ ran 𝐹} = {𝑠 ∈ ℝ ∣ (𝑠 βˆ’ (πΊβ€˜π‘š)) ∈ ran 𝐹})
5 vitali.6 . . . . . 6 𝑇 = (𝑛 ∈ β„• ↦ {𝑠 ∈ ℝ ∣ (𝑠 βˆ’ (πΊβ€˜π‘›)) ∈ ran 𝐹})
6 reex 11197 . . . . . . 7 ℝ ∈ V
76rabex 5331 . . . . . 6 {𝑠 ∈ ℝ ∣ (𝑠 βˆ’ (πΊβ€˜π‘š)) ∈ ran 𝐹} ∈ V
84, 5, 7fvmpt 6995 . . . . 5 (π‘š ∈ β„• β†’ (π‘‡β€˜π‘š) = {𝑠 ∈ ℝ ∣ (𝑠 βˆ’ (πΊβ€˜π‘š)) ∈ ran 𝐹})
98adantl 482 . . . 4 ((πœ‘ ∧ π‘š ∈ β„•) β†’ (π‘‡β€˜π‘š) = {𝑠 ∈ ℝ ∣ (𝑠 βˆ’ (πΊβ€˜π‘š)) ∈ ran 𝐹})
109fveq2d 6892 . . 3 ((πœ‘ ∧ π‘š ∈ β„•) β†’ (vol*β€˜(π‘‡β€˜π‘š)) = (vol*β€˜{𝑠 ∈ ℝ ∣ (𝑠 βˆ’ (πΊβ€˜π‘š)) ∈ ran 𝐹}))
11 vitali.1 . . . . . . . 8 ∼ = {⟨π‘₯, π‘¦βŸ© ∣ ((π‘₯ ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]1)) ∧ (π‘₯ βˆ’ 𝑦) ∈ β„š)}
12 vitali.2 . . . . . . . 8 𝑆 = ((0[,]1) / ∼ )
13 vitali.3 . . . . . . . 8 (πœ‘ β†’ 𝐹 Fn 𝑆)
14 vitali.4 . . . . . . . 8 (πœ‘ β†’ βˆ€π‘§ ∈ 𝑆 (𝑧 β‰  βˆ… β†’ (πΉβ€˜π‘§) ∈ 𝑧))
15 vitali.5 . . . . . . . 8 (πœ‘ β†’ 𝐺:ℕ–1-1-ontoβ†’(β„š ∩ (-1[,]1)))
16 vitali.7 . . . . . . . 8 (πœ‘ β†’ Β¬ ran 𝐹 ∈ (𝒫 ℝ βˆ– dom vol))
1711, 12, 13, 14, 15, 5, 16vitalilem2 25117 . . . . . . 7 (πœ‘ β†’ (ran 𝐹 βŠ† (0[,]1) ∧ (0[,]1) βŠ† βˆͺ π‘š ∈ β„• (π‘‡β€˜π‘š) ∧ βˆͺ π‘š ∈ β„• (π‘‡β€˜π‘š) βŠ† (-1[,]2)))
1817simp1d 1142 . . . . . 6 (πœ‘ β†’ ran 𝐹 βŠ† (0[,]1))
19 unitssre 13472 . . . . . 6 (0[,]1) βŠ† ℝ
2018, 19sstrdi 3993 . . . . 5 (πœ‘ β†’ ran 𝐹 βŠ† ℝ)
2120adantr 481 . . . 4 ((πœ‘ ∧ π‘š ∈ β„•) β†’ ran 𝐹 βŠ† ℝ)
22 neg1rr 12323 . . . . . 6 -1 ∈ ℝ
23 1re 11210 . . . . . 6 1 ∈ ℝ
24 iccssre 13402 . . . . . 6 ((-1 ∈ ℝ ∧ 1 ∈ ℝ) β†’ (-1[,]1) βŠ† ℝ)
2522, 23, 24mp2an 690 . . . . 5 (-1[,]1) βŠ† ℝ
26 f1of 6830 . . . . . . . 8 (𝐺:ℕ–1-1-ontoβ†’(β„š ∩ (-1[,]1)) β†’ 𝐺:β„•βŸΆ(β„š ∩ (-1[,]1)))
2715, 26syl 17 . . . . . . 7 (πœ‘ β†’ 𝐺:β„•βŸΆ(β„š ∩ (-1[,]1)))
2827ffvelcdmda 7083 . . . . . 6 ((πœ‘ ∧ π‘š ∈ β„•) β†’ (πΊβ€˜π‘š) ∈ (β„š ∩ (-1[,]1)))
2928elin2d 4198 . . . . 5 ((πœ‘ ∧ π‘š ∈ β„•) β†’ (πΊβ€˜π‘š) ∈ (-1[,]1))
3025, 29sselid 3979 . . . 4 ((πœ‘ ∧ π‘š ∈ β„•) β†’ (πΊβ€˜π‘š) ∈ ℝ)
31 eqidd 2733 . . . 4 ((πœ‘ ∧ π‘š ∈ β„•) β†’ {𝑠 ∈ ℝ ∣ (𝑠 βˆ’ (πΊβ€˜π‘š)) ∈ ran 𝐹} = {𝑠 ∈ ℝ ∣ (𝑠 βˆ’ (πΊβ€˜π‘š)) ∈ ran 𝐹})
3221, 30, 31ovolshft 25019 . . 3 ((πœ‘ ∧ π‘š ∈ β„•) β†’ (vol*β€˜ran 𝐹) = (vol*β€˜{𝑠 ∈ ℝ ∣ (𝑠 βˆ’ (πΊβ€˜π‘š)) ∈ ran 𝐹}))
3310, 32eqtr4d 2775 . 2 ((πœ‘ ∧ π‘š ∈ β„•) β†’ (vol*β€˜(π‘‡β€˜π‘š)) = (vol*β€˜ran 𝐹))
34 3re 12288 . . . . . . . 8 3 ∈ ℝ
3534rexri 11268 . . . . . . 7 3 ∈ ℝ*
3635a1i 11 . . . . . 6 ((πœ‘ ∧ 0 < (vol*β€˜ran 𝐹)) β†’ 3 ∈ ℝ*)
37 3rp 12976 . . . . . . . . . . . . 13 3 ∈ ℝ+
38 0re 11212 . . . . . . . . . . . . . . . . . . . 20 0 ∈ ℝ
39 0le1 11733 . . . . . . . . . . . . . . . . . . . 20 0 ≀ 1
40 ovolicc 25031 . . . . . . . . . . . . . . . . . . . 20 ((0 ∈ ℝ ∧ 1 ∈ ℝ ∧ 0 ≀ 1) β†’ (vol*β€˜(0[,]1)) = (1 βˆ’ 0))
4138, 23, 39, 40mp3an 1461 . . . . . . . . . . . . . . . . . . 19 (vol*β€˜(0[,]1)) = (1 βˆ’ 0)
42 1m0e1 12329 . . . . . . . . . . . . . . . . . . 19 (1 βˆ’ 0) = 1
4341, 42eqtri 2760 . . . . . . . . . . . . . . . . . 18 (vol*β€˜(0[,]1)) = 1
4443, 23eqeltri 2829 . . . . . . . . . . . . . . . . 17 (vol*β€˜(0[,]1)) ∈ ℝ
45 ovolsscl 24994 . . . . . . . . . . . . . . . . 17 ((ran 𝐹 βŠ† (0[,]1) ∧ (0[,]1) βŠ† ℝ ∧ (vol*β€˜(0[,]1)) ∈ ℝ) β†’ (vol*β€˜ran 𝐹) ∈ ℝ)
4619, 44, 45mp3an23 1453 . . . . . . . . . . . . . . . 16 (ran 𝐹 βŠ† (0[,]1) β†’ (vol*β€˜ran 𝐹) ∈ ℝ)
4718, 46syl 17 . . . . . . . . . . . . . . 15 (πœ‘ β†’ (vol*β€˜ran 𝐹) ∈ ℝ)
4847adantr 481 . . . . . . . . . . . . . 14 ((πœ‘ ∧ 0 < (vol*β€˜ran 𝐹)) β†’ (vol*β€˜ran 𝐹) ∈ ℝ)
49 simpr 485 . . . . . . . . . . . . . 14 ((πœ‘ ∧ 0 < (vol*β€˜ran 𝐹)) β†’ 0 < (vol*β€˜ran 𝐹))
5048, 49elrpd 13009 . . . . . . . . . . . . 13 ((πœ‘ ∧ 0 < (vol*β€˜ran 𝐹)) β†’ (vol*β€˜ran 𝐹) ∈ ℝ+)
51 rpdivcl 12995 . . . . . . . . . . . . 13 ((3 ∈ ℝ+ ∧ (vol*β€˜ran 𝐹) ∈ ℝ+) β†’ (3 / (vol*β€˜ran 𝐹)) ∈ ℝ+)
5237, 50, 51sylancr 587 . . . . . . . . . . . 12 ((πœ‘ ∧ 0 < (vol*β€˜ran 𝐹)) β†’ (3 / (vol*β€˜ran 𝐹)) ∈ ℝ+)
5352rpred 13012 . . . . . . . . . . 11 ((πœ‘ ∧ 0 < (vol*β€˜ran 𝐹)) β†’ (3 / (vol*β€˜ran 𝐹)) ∈ ℝ)
5452rpge0d 13016 . . . . . . . . . . 11 ((πœ‘ ∧ 0 < (vol*β€˜ran 𝐹)) β†’ 0 ≀ (3 / (vol*β€˜ran 𝐹)))
55 flge0nn0 13781 . . . . . . . . . . 11 (((3 / (vol*β€˜ran 𝐹)) ∈ ℝ ∧ 0 ≀ (3 / (vol*β€˜ran 𝐹))) β†’ (βŒŠβ€˜(3 / (vol*β€˜ran 𝐹))) ∈ β„•0)
5653, 54, 55syl2anc 584 . . . . . . . . . 10 ((πœ‘ ∧ 0 < (vol*β€˜ran 𝐹)) β†’ (βŒŠβ€˜(3 / (vol*β€˜ran 𝐹))) ∈ β„•0)
57 nn0p1nn 12507 . . . . . . . . . 10 ((βŒŠβ€˜(3 / (vol*β€˜ran 𝐹))) ∈ β„•0 β†’ ((βŒŠβ€˜(3 / (vol*β€˜ran 𝐹))) + 1) ∈ β„•)
5856, 57syl 17 . . . . . . . . 9 ((πœ‘ ∧ 0 < (vol*β€˜ran 𝐹)) β†’ ((βŒŠβ€˜(3 / (vol*β€˜ran 𝐹))) + 1) ∈ β„•)
5958nnred 12223 . . . . . . . 8 ((πœ‘ ∧ 0 < (vol*β€˜ran 𝐹)) β†’ ((βŒŠβ€˜(3 / (vol*β€˜ran 𝐹))) + 1) ∈ ℝ)
6059, 48remulcld 11240 . . . . . . 7 ((πœ‘ ∧ 0 < (vol*β€˜ran 𝐹)) β†’ (((βŒŠβ€˜(3 / (vol*β€˜ran 𝐹))) + 1) Β· (vol*β€˜ran 𝐹)) ∈ ℝ)
6160rexrd 11260 . . . . . 6 ((πœ‘ ∧ 0 < (vol*β€˜ran 𝐹)) β†’ (((βŒŠβ€˜(3 / (vol*β€˜ran 𝐹))) + 1) Β· (vol*β€˜ran 𝐹)) ∈ ℝ*)
626elpw2 5344 . . . . . . . . . . . . . . . . . 18 (ran 𝐹 ∈ 𝒫 ℝ ↔ ran 𝐹 βŠ† ℝ)
6320, 62sylibr 233 . . . . . . . . . . . . . . . . 17 (πœ‘ β†’ ran 𝐹 ∈ 𝒫 ℝ)
6463anim1i 615 . . . . . . . . . . . . . . . 16 ((πœ‘ ∧ Β¬ ran 𝐹 ∈ dom vol) β†’ (ran 𝐹 ∈ 𝒫 ℝ ∧ Β¬ ran 𝐹 ∈ dom vol))
65 eldif 3957 . . . . . . . . . . . . . . . 16 (ran 𝐹 ∈ (𝒫 ℝ βˆ– dom vol) ↔ (ran 𝐹 ∈ 𝒫 ℝ ∧ Β¬ ran 𝐹 ∈ dom vol))
6664, 65sylibr 233 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ Β¬ ran 𝐹 ∈ dom vol) β†’ ran 𝐹 ∈ (𝒫 ℝ βˆ– dom vol))
6766ex 413 . . . . . . . . . . . . . 14 (πœ‘ β†’ (Β¬ ran 𝐹 ∈ dom vol β†’ ran 𝐹 ∈ (𝒫 ℝ βˆ– dom vol)))
6816, 67mt3d 148 . . . . . . . . . . . . 13 (πœ‘ β†’ ran 𝐹 ∈ dom vol)
69 inss1 4227 . . . . . . . . . . . . . . . 16 (β„š ∩ (-1[,]1)) βŠ† β„š
70 qssre 12939 . . . . . . . . . . . . . . . 16 β„š βŠ† ℝ
7169, 70sstri 3990 . . . . . . . . . . . . . . 15 (β„š ∩ (-1[,]1)) βŠ† ℝ
72 fss 6731 . . . . . . . . . . . . . . 15 ((𝐺:β„•βŸΆ(β„š ∩ (-1[,]1)) ∧ (β„š ∩ (-1[,]1)) βŠ† ℝ) β†’ 𝐺:β„•βŸΆβ„)
7327, 71, 72sylancl 586 . . . . . . . . . . . . . 14 (πœ‘ β†’ 𝐺:β„•βŸΆβ„)
7473ffvelcdmda 7083 . . . . . . . . . . . . 13 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (πΊβ€˜π‘›) ∈ ℝ)
75 shftmbl 25046 . . . . . . . . . . . . 13 ((ran 𝐹 ∈ dom vol ∧ (πΊβ€˜π‘›) ∈ ℝ) β†’ {𝑠 ∈ ℝ ∣ (𝑠 βˆ’ (πΊβ€˜π‘›)) ∈ ran 𝐹} ∈ dom vol)
7668, 74, 75syl2an2r 683 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ {𝑠 ∈ ℝ ∣ (𝑠 βˆ’ (πΊβ€˜π‘›)) ∈ ran 𝐹} ∈ dom vol)
7776, 5fmptd 7110 . . . . . . . . . . 11 (πœ‘ β†’ 𝑇:β„•βŸΆdom vol)
7877ffvelcdmda 7083 . . . . . . . . . 10 ((πœ‘ ∧ π‘š ∈ β„•) β†’ (π‘‡β€˜π‘š) ∈ dom vol)
7978ralrimiva 3146 . . . . . . . . 9 (πœ‘ β†’ βˆ€π‘š ∈ β„• (π‘‡β€˜π‘š) ∈ dom vol)
80 iunmbl 25061 . . . . . . . . 9 (βˆ€π‘š ∈ β„• (π‘‡β€˜π‘š) ∈ dom vol β†’ βˆͺ π‘š ∈ β„• (π‘‡β€˜π‘š) ∈ dom vol)
8179, 80syl 17 . . . . . . . 8 (πœ‘ β†’ βˆͺ π‘š ∈ β„• (π‘‡β€˜π‘š) ∈ dom vol)
82 mblss 25039 . . . . . . . 8 (βˆͺ π‘š ∈ β„• (π‘‡β€˜π‘š) ∈ dom vol β†’ βˆͺ π‘š ∈ β„• (π‘‡β€˜π‘š) βŠ† ℝ)
83 ovolcl 24986 . . . . . . . 8 (βˆͺ π‘š ∈ β„• (π‘‡β€˜π‘š) βŠ† ℝ β†’ (vol*β€˜βˆͺ π‘š ∈ β„• (π‘‡β€˜π‘š)) ∈ ℝ*)
8481, 82, 833syl 18 . . . . . . 7 (πœ‘ β†’ (vol*β€˜βˆͺ π‘š ∈ β„• (π‘‡β€˜π‘š)) ∈ ℝ*)
8584adantr 481 . . . . . 6 ((πœ‘ ∧ 0 < (vol*β€˜ran 𝐹)) β†’ (vol*β€˜βˆͺ π‘š ∈ β„• (π‘‡β€˜π‘š)) ∈ ℝ*)
86 flltp1 13761 . . . . . . . 8 ((3 / (vol*β€˜ran 𝐹)) ∈ ℝ β†’ (3 / (vol*β€˜ran 𝐹)) < ((βŒŠβ€˜(3 / (vol*β€˜ran 𝐹))) + 1))
8753, 86syl 17 . . . . . . 7 ((πœ‘ ∧ 0 < (vol*β€˜ran 𝐹)) β†’ (3 / (vol*β€˜ran 𝐹)) < ((βŒŠβ€˜(3 / (vol*β€˜ran 𝐹))) + 1))
8834a1i 11 . . . . . . . 8 ((πœ‘ ∧ 0 < (vol*β€˜ran 𝐹)) β†’ 3 ∈ ℝ)
8988, 59, 50ltdivmul2d 13064 . . . . . . 7 ((πœ‘ ∧ 0 < (vol*β€˜ran 𝐹)) β†’ ((3 / (vol*β€˜ran 𝐹)) < ((βŒŠβ€˜(3 / (vol*β€˜ran 𝐹))) + 1) ↔ 3 < (((βŒŠβ€˜(3 / (vol*β€˜ran 𝐹))) + 1) Β· (vol*β€˜ran 𝐹))))
9087, 89mpbid 231 . . . . . 6 ((πœ‘ ∧ 0 < (vol*β€˜ran 𝐹)) β†’ 3 < (((βŒŠβ€˜(3 / (vol*β€˜ran 𝐹))) + 1) Β· (vol*β€˜ran 𝐹)))
91 nnuz 12861 . . . . . . . . . . 11 β„• = (β„€β‰₯β€˜1)
92 1zzd 12589 . . . . . . . . . . 11 ((πœ‘ ∧ 0 < (vol*β€˜ran 𝐹)) β†’ 1 ∈ β„€)
93 mblvol 25038 . . . . . . . . . . . . . . . . 17 ((π‘‡β€˜π‘š) ∈ dom vol β†’ (volβ€˜(π‘‡β€˜π‘š)) = (vol*β€˜(π‘‡β€˜π‘š)))
9478, 93syl 17 . . . . . . . . . . . . . . . 16 ((πœ‘ ∧ π‘š ∈ β„•) β†’ (volβ€˜(π‘‡β€˜π‘š)) = (vol*β€˜(π‘‡β€˜π‘š)))
9594, 33eqtrd 2772 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ π‘š ∈ β„•) β†’ (volβ€˜(π‘‡β€˜π‘š)) = (vol*β€˜ran 𝐹))
9647adantr 481 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ π‘š ∈ β„•) β†’ (vol*β€˜ran 𝐹) ∈ ℝ)
9795, 96eqeltrd 2833 . . . . . . . . . . . . . 14 ((πœ‘ ∧ π‘š ∈ β„•) β†’ (volβ€˜(π‘‡β€˜π‘š)) ∈ ℝ)
9897adantlr 713 . . . . . . . . . . . . 13 (((πœ‘ ∧ 0 < (vol*β€˜ran 𝐹)) ∧ π‘š ∈ β„•) β†’ (volβ€˜(π‘‡β€˜π‘š)) ∈ ℝ)
99 eqid 2732 . . . . . . . . . . . . 13 (π‘š ∈ β„• ↦ (volβ€˜(π‘‡β€˜π‘š))) = (π‘š ∈ β„• ↦ (volβ€˜(π‘‡β€˜π‘š)))
10098, 99fmptd 7110 . . . . . . . . . . . 12 ((πœ‘ ∧ 0 < (vol*β€˜ran 𝐹)) β†’ (π‘š ∈ β„• ↦ (volβ€˜(π‘‡β€˜π‘š))):β„•βŸΆβ„)
101100ffvelcdmda 7083 . . . . . . . . . . 11 (((πœ‘ ∧ 0 < (vol*β€˜ran 𝐹)) ∧ π‘˜ ∈ β„•) β†’ ((π‘š ∈ β„• ↦ (volβ€˜(π‘‡β€˜π‘š)))β€˜π‘˜) ∈ ℝ)
10291, 92, 101serfre 13993 . . . . . . . . . 10 ((πœ‘ ∧ 0 < (vol*β€˜ran 𝐹)) β†’ seq1( + , (π‘š ∈ β„• ↦ (volβ€˜(π‘‡β€˜π‘š)))):β„•βŸΆβ„)
103102frnd 6722 . . . . . . . . 9 ((πœ‘ ∧ 0 < (vol*β€˜ran 𝐹)) β†’ ran seq1( + , (π‘š ∈ β„• ↦ (volβ€˜(π‘‡β€˜π‘š)))) βŠ† ℝ)
104 ressxr 11254 . . . . . . . . 9 ℝ βŠ† ℝ*
105103, 104sstrdi 3993 . . . . . . . 8 ((πœ‘ ∧ 0 < (vol*β€˜ran 𝐹)) β†’ ran seq1( + , (π‘š ∈ β„• ↦ (volβ€˜(π‘‡β€˜π‘š)))) βŠ† ℝ*)
10695adantlr 713 . . . . . . . . . . . . . 14 (((πœ‘ ∧ 0 < (vol*β€˜ran 𝐹)) ∧ π‘š ∈ β„•) β†’ (volβ€˜(π‘‡β€˜π‘š)) = (vol*β€˜ran 𝐹))
107106mpteq2dva 5247 . . . . . . . . . . . . 13 ((πœ‘ ∧ 0 < (vol*β€˜ran 𝐹)) β†’ (π‘š ∈ β„• ↦ (volβ€˜(π‘‡β€˜π‘š))) = (π‘š ∈ β„• ↦ (vol*β€˜ran 𝐹)))
108 fconstmpt 5736 . . . . . . . . . . . . 13 (β„• Γ— {(vol*β€˜ran 𝐹)}) = (π‘š ∈ β„• ↦ (vol*β€˜ran 𝐹))
109107, 108eqtr4di 2790 . . . . . . . . . . . 12 ((πœ‘ ∧ 0 < (vol*β€˜ran 𝐹)) β†’ (π‘š ∈ β„• ↦ (volβ€˜(π‘‡β€˜π‘š))) = (β„• Γ— {(vol*β€˜ran 𝐹)}))
110109seqeq3d 13970 . . . . . . . . . . 11 ((πœ‘ ∧ 0 < (vol*β€˜ran 𝐹)) β†’ seq1( + , (π‘š ∈ β„• ↦ (volβ€˜(π‘‡β€˜π‘š)))) = seq1( + , (β„• Γ— {(vol*β€˜ran 𝐹)})))
111110fveq1d 6890 . . . . . . . . . 10 ((πœ‘ ∧ 0 < (vol*β€˜ran 𝐹)) β†’ (seq1( + , (π‘š ∈ β„• ↦ (volβ€˜(π‘‡β€˜π‘š))))β€˜((βŒŠβ€˜(3 / (vol*β€˜ran 𝐹))) + 1)) = (seq1( + , (β„• Γ— {(vol*β€˜ran 𝐹)}))β€˜((βŒŠβ€˜(3 / (vol*β€˜ran 𝐹))) + 1)))
11248recnd 11238 . . . . . . . . . . 11 ((πœ‘ ∧ 0 < (vol*β€˜ran 𝐹)) β†’ (vol*β€˜ran 𝐹) ∈ β„‚)
113 ser1const 14020 . . . . . . . . . . 11 (((vol*β€˜ran 𝐹) ∈ β„‚ ∧ ((βŒŠβ€˜(3 / (vol*β€˜ran 𝐹))) + 1) ∈ β„•) β†’ (seq1( + , (β„• Γ— {(vol*β€˜ran 𝐹)}))β€˜((βŒŠβ€˜(3 / (vol*β€˜ran 𝐹))) + 1)) = (((βŒŠβ€˜(3 / (vol*β€˜ran 𝐹))) + 1) Β· (vol*β€˜ran 𝐹)))
114112, 58, 113syl2anc 584 . . . . . . . . . 10 ((πœ‘ ∧ 0 < (vol*β€˜ran 𝐹)) β†’ (seq1( + , (β„• Γ— {(vol*β€˜ran 𝐹)}))β€˜((βŒŠβ€˜(3 / (vol*β€˜ran 𝐹))) + 1)) = (((βŒŠβ€˜(3 / (vol*β€˜ran 𝐹))) + 1) Β· (vol*β€˜ran 𝐹)))
115111, 114eqtrd 2772 . . . . . . . . 9 ((πœ‘ ∧ 0 < (vol*β€˜ran 𝐹)) β†’ (seq1( + , (π‘š ∈ β„• ↦ (volβ€˜(π‘‡β€˜π‘š))))β€˜((βŒŠβ€˜(3 / (vol*β€˜ran 𝐹))) + 1)) = (((βŒŠβ€˜(3 / (vol*β€˜ran 𝐹))) + 1) Β· (vol*β€˜ran 𝐹)))
116102ffnd 6715 . . . . . . . . . 10 ((πœ‘ ∧ 0 < (vol*β€˜ran 𝐹)) β†’ seq1( + , (π‘š ∈ β„• ↦ (volβ€˜(π‘‡β€˜π‘š)))) Fn β„•)
117 fnfvelrn 7079 . . . . . . . . . 10 ((seq1( + , (π‘š ∈ β„• ↦ (volβ€˜(π‘‡β€˜π‘š)))) Fn β„• ∧ ((βŒŠβ€˜(3 / (vol*β€˜ran 𝐹))) + 1) ∈ β„•) β†’ (seq1( + , (π‘š ∈ β„• ↦ (volβ€˜(π‘‡β€˜π‘š))))β€˜((βŒŠβ€˜(3 / (vol*β€˜ran 𝐹))) + 1)) ∈ ran seq1( + , (π‘š ∈ β„• ↦ (volβ€˜(π‘‡β€˜π‘š)))))
118116, 58, 117syl2anc 584 . . . . . . . . 9 ((πœ‘ ∧ 0 < (vol*β€˜ran 𝐹)) β†’ (seq1( + , (π‘š ∈ β„• ↦ (volβ€˜(π‘‡β€˜π‘š))))β€˜((βŒŠβ€˜(3 / (vol*β€˜ran 𝐹))) + 1)) ∈ ran seq1( + , (π‘š ∈ β„• ↦ (volβ€˜(π‘‡β€˜π‘š)))))
119115, 118eqeltrrd 2834 . . . . . . . 8 ((πœ‘ ∧ 0 < (vol*β€˜ran 𝐹)) β†’ (((βŒŠβ€˜(3 / (vol*β€˜ran 𝐹))) + 1) Β· (vol*β€˜ran 𝐹)) ∈ ran seq1( + , (π‘š ∈ β„• ↦ (volβ€˜(π‘‡β€˜π‘š)))))
120 supxrub 13299 . . . . . . . 8 ((ran seq1( + , (π‘š ∈ β„• ↦ (volβ€˜(π‘‡β€˜π‘š)))) βŠ† ℝ* ∧ (((βŒŠβ€˜(3 / (vol*β€˜ran 𝐹))) + 1) Β· (vol*β€˜ran 𝐹)) ∈ ran seq1( + , (π‘š ∈ β„• ↦ (volβ€˜(π‘‡β€˜π‘š))))) β†’ (((βŒŠβ€˜(3 / (vol*β€˜ran 𝐹))) + 1) Β· (vol*β€˜ran 𝐹)) ≀ sup(ran seq1( + , (π‘š ∈ β„• ↦ (volβ€˜(π‘‡β€˜π‘š)))), ℝ*, < ))
121105, 119, 120syl2anc 584 . . . . . . 7 ((πœ‘ ∧ 0 < (vol*β€˜ran 𝐹)) β†’ (((βŒŠβ€˜(3 / (vol*β€˜ran 𝐹))) + 1) Β· (vol*β€˜ran 𝐹)) ≀ sup(ran seq1( + , (π‘š ∈ β„• ↦ (volβ€˜(π‘‡β€˜π‘š)))), ℝ*, < ))
12281adantr 481 . . . . . . . . 9 ((πœ‘ ∧ 0 < (vol*β€˜ran 𝐹)) β†’ βˆͺ π‘š ∈ β„• (π‘‡β€˜π‘š) ∈ dom vol)
123 mblvol 25038 . . . . . . . . 9 (βˆͺ π‘š ∈ β„• (π‘‡β€˜π‘š) ∈ dom vol β†’ (volβ€˜βˆͺ π‘š ∈ β„• (π‘‡β€˜π‘š)) = (vol*β€˜βˆͺ π‘š ∈ β„• (π‘‡β€˜π‘š)))
124122, 123syl 17 . . . . . . . 8 ((πœ‘ ∧ 0 < (vol*β€˜ran 𝐹)) β†’ (volβ€˜βˆͺ π‘š ∈ β„• (π‘‡β€˜π‘š)) = (vol*β€˜βˆͺ π‘š ∈ β„• (π‘‡β€˜π‘š)))
12578, 97jca 512 . . . . . . . . . 10 ((πœ‘ ∧ π‘š ∈ β„•) β†’ ((π‘‡β€˜π‘š) ∈ dom vol ∧ (volβ€˜(π‘‡β€˜π‘š)) ∈ ℝ))
126125ralrimiva 3146 . . . . . . . . 9 (πœ‘ β†’ βˆ€π‘š ∈ β„• ((π‘‡β€˜π‘š) ∈ dom vol ∧ (volβ€˜(π‘‡β€˜π‘š)) ∈ ℝ))
12711, 12, 13, 14, 15, 5, 16vitalilem3 25118 . . . . . . . . . 10 (πœ‘ β†’ Disj π‘š ∈ β„• (π‘‡β€˜π‘š))
128127adantr 481 . . . . . . . . 9 ((πœ‘ ∧ 0 < (vol*β€˜ran 𝐹)) β†’ Disj π‘š ∈ β„• (π‘‡β€˜π‘š))
129 eqid 2732 . . . . . . . . . 10 seq1( + , (π‘š ∈ β„• ↦ (volβ€˜(π‘‡β€˜π‘š)))) = seq1( + , (π‘š ∈ β„• ↦ (volβ€˜(π‘‡β€˜π‘š))))
130129, 99voliun 25062 . . . . . . . . 9 ((βˆ€π‘š ∈ β„• ((π‘‡β€˜π‘š) ∈ dom vol ∧ (volβ€˜(π‘‡β€˜π‘š)) ∈ ℝ) ∧ Disj π‘š ∈ β„• (π‘‡β€˜π‘š)) β†’ (volβ€˜βˆͺ π‘š ∈ β„• (π‘‡β€˜π‘š)) = sup(ran seq1( + , (π‘š ∈ β„• ↦ (volβ€˜(π‘‡β€˜π‘š)))), ℝ*, < ))
131126, 128, 130syl2an2r 683 . . . . . . . 8 ((πœ‘ ∧ 0 < (vol*β€˜ran 𝐹)) β†’ (volβ€˜βˆͺ π‘š ∈ β„• (π‘‡β€˜π‘š)) = sup(ran seq1( + , (π‘š ∈ β„• ↦ (volβ€˜(π‘‡β€˜π‘š)))), ℝ*, < ))
132124, 131eqtr3d 2774 . . . . . . 7 ((πœ‘ ∧ 0 < (vol*β€˜ran 𝐹)) β†’ (vol*β€˜βˆͺ π‘š ∈ β„• (π‘‡β€˜π‘š)) = sup(ran seq1( + , (π‘š ∈ β„• ↦ (volβ€˜(π‘‡β€˜π‘š)))), ℝ*, < ))
133121, 132breqtrrd 5175 . . . . . 6 ((πœ‘ ∧ 0 < (vol*β€˜ran 𝐹)) β†’ (((βŒŠβ€˜(3 / (vol*β€˜ran 𝐹))) + 1) Β· (vol*β€˜ran 𝐹)) ≀ (vol*β€˜βˆͺ π‘š ∈ β„• (π‘‡β€˜π‘š)))
13436, 61, 85, 90, 133xrltletrd 13136 . . . . 5 ((πœ‘ ∧ 0 < (vol*β€˜ran 𝐹)) β†’ 3 < (vol*β€˜βˆͺ π‘š ∈ β„• (π‘‡β€˜π‘š)))
13517simp3d 1144 . . . . . . . . 9 (πœ‘ β†’ βˆͺ π‘š ∈ β„• (π‘‡β€˜π‘š) βŠ† (-1[,]2))
136135adantr 481 . . . . . . . 8 ((πœ‘ ∧ 0 < (vol*β€˜ran 𝐹)) β†’ βˆͺ π‘š ∈ β„• (π‘‡β€˜π‘š) βŠ† (-1[,]2))
137 2re 12282 . . . . . . . . 9 2 ∈ ℝ
138 iccssre 13402 . . . . . . . . 9 ((-1 ∈ ℝ ∧ 2 ∈ ℝ) β†’ (-1[,]2) βŠ† ℝ)
13922, 137, 138mp2an 690 . . . . . . . 8 (-1[,]2) βŠ† ℝ
140 ovolss 24993 . . . . . . . 8 ((βˆͺ π‘š ∈ β„• (π‘‡β€˜π‘š) βŠ† (-1[,]2) ∧ (-1[,]2) βŠ† ℝ) β†’ (vol*β€˜βˆͺ π‘š ∈ β„• (π‘‡β€˜π‘š)) ≀ (vol*β€˜(-1[,]2)))
141136, 139, 140sylancl 586 . . . . . . 7 ((πœ‘ ∧ 0 < (vol*β€˜ran 𝐹)) β†’ (vol*β€˜βˆͺ π‘š ∈ β„• (π‘‡β€˜π‘š)) ≀ (vol*β€˜(-1[,]2)))
142 2cn 12283 . . . . . . . . 9 2 ∈ β„‚
143 ax-1cn 11164 . . . . . . . . 9 1 ∈ β„‚
144142, 143subnegi 11535 . . . . . . . 8 (2 βˆ’ -1) = (2 + 1)
145 neg1lt0 12325 . . . . . . . . . . 11 -1 < 0
146 2pos 12311 . . . . . . . . . . 11 0 < 2
14722, 38, 137lttri 11336 . . . . . . . . . . 11 ((-1 < 0 ∧ 0 < 2) β†’ -1 < 2)
148145, 146, 147mp2an 690 . . . . . . . . . 10 -1 < 2
14922, 137, 148ltleii 11333 . . . . . . . . 9 -1 ≀ 2
150 ovolicc 25031 . . . . . . . . 9 ((-1 ∈ ℝ ∧ 2 ∈ ℝ ∧ -1 ≀ 2) β†’ (vol*β€˜(-1[,]2)) = (2 βˆ’ -1))
15122, 137, 149, 150mp3an 1461 . . . . . . . 8 (vol*β€˜(-1[,]2)) = (2 βˆ’ -1)
152 df-3 12272 . . . . . . . 8 3 = (2 + 1)
153144, 151, 1523eqtr4i 2770 . . . . . . 7 (vol*β€˜(-1[,]2)) = 3
154141, 153breqtrdi 5188 . . . . . 6 ((πœ‘ ∧ 0 < (vol*β€˜ran 𝐹)) β†’ (vol*β€˜βˆͺ π‘š ∈ β„• (π‘‡β€˜π‘š)) ≀ 3)
155 xrlenlt 11275 . . . . . . 7 (((vol*β€˜βˆͺ π‘š ∈ β„• (π‘‡β€˜π‘š)) ∈ ℝ* ∧ 3 ∈ ℝ*) β†’ ((vol*β€˜βˆͺ π‘š ∈ β„• (π‘‡β€˜π‘š)) ≀ 3 ↔ Β¬ 3 < (vol*β€˜βˆͺ π‘š ∈ β„• (π‘‡β€˜π‘š))))
15685, 35, 155sylancl 586 . . . . . 6 ((πœ‘ ∧ 0 < (vol*β€˜ran 𝐹)) β†’ ((vol*β€˜βˆͺ π‘š ∈ β„• (π‘‡β€˜π‘š)) ≀ 3 ↔ Β¬ 3 < (vol*β€˜βˆͺ π‘š ∈ β„• (π‘‡β€˜π‘š))))
157154, 156mpbid 231 . . . . 5 ((πœ‘ ∧ 0 < (vol*β€˜ran 𝐹)) β†’ Β¬ 3 < (vol*β€˜βˆͺ π‘š ∈ β„• (π‘‡β€˜π‘š)))
158134, 157pm2.65da 815 . . . 4 (πœ‘ β†’ Β¬ 0 < (vol*β€˜ran 𝐹))
159 ovolge0 24989 . . . . . . 7 (ran 𝐹 βŠ† ℝ β†’ 0 ≀ (vol*β€˜ran 𝐹))
16020, 159syl 17 . . . . . 6 (πœ‘ β†’ 0 ≀ (vol*β€˜ran 𝐹))
161 0xr 11257 . . . . . . 7 0 ∈ ℝ*
162 ovolcl 24986 . . . . . . . 8 (ran 𝐹 βŠ† ℝ β†’ (vol*β€˜ran 𝐹) ∈ ℝ*)
16320, 162syl 17 . . . . . . 7 (πœ‘ β†’ (vol*β€˜ran 𝐹) ∈ ℝ*)
164 xrleloe 13119 . . . . . . 7 ((0 ∈ ℝ* ∧ (vol*β€˜ran 𝐹) ∈ ℝ*) β†’ (0 ≀ (vol*β€˜ran 𝐹) ↔ (0 < (vol*β€˜ran 𝐹) ∨ 0 = (vol*β€˜ran 𝐹))))
165161, 163, 164sylancr 587 . . . . . 6 (πœ‘ β†’ (0 ≀ (vol*β€˜ran 𝐹) ↔ (0 < (vol*β€˜ran 𝐹) ∨ 0 = (vol*β€˜ran 𝐹))))
166160, 165mpbid 231 . . . . 5 (πœ‘ β†’ (0 < (vol*β€˜ran 𝐹) ∨ 0 = (vol*β€˜ran 𝐹)))
167166ord 862 . . . 4 (πœ‘ β†’ (Β¬ 0 < (vol*β€˜ran 𝐹) β†’ 0 = (vol*β€˜ran 𝐹)))
168158, 167mpd 15 . . 3 (πœ‘ β†’ 0 = (vol*β€˜ran 𝐹))
169168adantr 481 . 2 ((πœ‘ ∧ π‘š ∈ β„•) β†’ 0 = (vol*β€˜ran 𝐹))
17033, 169eqtr4d 2775 1 ((πœ‘ ∧ π‘š ∈ β„•) β†’ (vol*β€˜(π‘‡β€˜π‘š)) = 0)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 845   = wceq 1541   ∈ wcel 2106   β‰  wne 2940  βˆ€wral 3061  {crab 3432   βˆ– cdif 3944   ∩ cin 3946   βŠ† wss 3947  βˆ…c0 4321  π’« cpw 4601  {csn 4627  βˆͺ ciun 4996  Disj wdisj 5112   class class class wbr 5147  {copab 5209   ↦ cmpt 5230   Γ— cxp 5673  dom cdm 5675  ran crn 5676   Fn wfn 6535  βŸΆwf 6536  β€“1-1-ontoβ†’wf1o 6539  β€˜cfv 6540  (class class class)co 7405   / cqs 8698  supcsup 9431  β„‚cc 11104  β„cr 11105  0cc0 11106  1c1 11107   + caddc 11109   Β· cmul 11111  β„*cxr 11243   < clt 11244   ≀ cle 11245   βˆ’ cmin 11440  -cneg 11441   / cdiv 11867  β„•cn 12208  2c2 12263  3c3 12264  β„•0cn0 12468  β„šcq 12928  β„+crp 12970  [,]cicc 13323  βŒŠcfl 13751  seqcseq 13962  vol*covol 24970  volcvol 24971
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-inf2 9632  ax-cc 10426  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183  ax-pre-sup 11184
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-disj 5113  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-se 5631  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-isom 6549  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-of 7666  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-2o 8463  df-er 8699  df-ec 8701  df-qs 8705  df-map 8818  df-pm 8819  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-fi 9402  df-sup 9433  df-inf 9434  df-oi 9501  df-dju 9892  df-card 9930  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-div 11868  df-nn 12209  df-2 12271  df-3 12272  df-n0 12469  df-z 12555  df-uz 12819  df-q 12929  df-rp 12971  df-xneg 13088  df-xadd 13089  df-xmul 13090  df-ioo 13324  df-ico 13326  df-icc 13327  df-fz 13481  df-fzo 13624  df-fl 13753  df-seq 13963  df-exp 14024  df-hash 14287  df-cj 15042  df-re 15043  df-im 15044  df-sqrt 15178  df-abs 15179  df-clim 15428  df-rlim 15429  df-sum 15629  df-rest 17364  df-topgen 17385  df-psmet 20928  df-xmet 20929  df-met 20930  df-bl 20931  df-mopn 20932  df-top 22387  df-topon 22404  df-bases 22440  df-cmp 22882  df-ovol 24972  df-vol 24973
This theorem is referenced by:  vitalilem5  25120
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