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Theorem vitalilem4 24991
Description: Lemma for vitali 24993. (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 6847 . . . . . . . . 9 (𝑛 = π‘š β†’ (πΊβ€˜π‘›) = (πΊβ€˜π‘š))
21oveq2d 7378 . . . . . . . 8 (𝑛 = π‘š β†’ (𝑠 βˆ’ (πΊβ€˜π‘›)) = (𝑠 βˆ’ (πΊβ€˜π‘š)))
32eleq1d 2823 . . . . . . 7 (𝑛 = π‘š β†’ ((𝑠 βˆ’ (πΊβ€˜π‘›)) ∈ ran 𝐹 ↔ (𝑠 βˆ’ (πΊβ€˜π‘š)) ∈ ran 𝐹))
43rabbidv 3418 . . . . . 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 11149 . . . . . . 7 ℝ ∈ V
76rabex 5294 . . . . . 6 {𝑠 ∈ ℝ ∣ (𝑠 βˆ’ (πΊβ€˜π‘š)) ∈ ran 𝐹} ∈ V
84, 5, 7fvmpt 6953 . . . . 5 (π‘š ∈ β„• β†’ (π‘‡β€˜π‘š) = {𝑠 ∈ ℝ ∣ (𝑠 βˆ’ (πΊβ€˜π‘š)) ∈ ran 𝐹})
98adantl 483 . . . 4 ((πœ‘ ∧ π‘š ∈ β„•) β†’ (π‘‡β€˜π‘š) = {𝑠 ∈ ℝ ∣ (𝑠 βˆ’ (πΊβ€˜π‘š)) ∈ ran 𝐹})
109fveq2d 6851 . . 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 24989 . . . . . . 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 1143 . . . . . 6 (πœ‘ β†’ ran 𝐹 βŠ† (0[,]1))
19 unitssre 13423 . . . . . 6 (0[,]1) βŠ† ℝ
2018, 19sstrdi 3961 . . . . 5 (πœ‘ β†’ ran 𝐹 βŠ† ℝ)
2120adantr 482 . . . 4 ((πœ‘ ∧ π‘š ∈ β„•) β†’ ran 𝐹 βŠ† ℝ)
22 neg1rr 12275 . . . . . 6 -1 ∈ ℝ
23 1re 11162 . . . . . 6 1 ∈ ℝ
24 iccssre 13353 . . . . . 6 ((-1 ∈ ℝ ∧ 1 ∈ ℝ) β†’ (-1[,]1) βŠ† ℝ)
2522, 23, 24mp2an 691 . . . . 5 (-1[,]1) βŠ† ℝ
26 f1of 6789 . . . . . . . 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 7040 . . . . . 6 ((πœ‘ ∧ π‘š ∈ β„•) β†’ (πΊβ€˜π‘š) ∈ (β„š ∩ (-1[,]1)))
2928elin2d 4164 . . . . 5 ((πœ‘ ∧ π‘š ∈ β„•) β†’ (πΊβ€˜π‘š) ∈ (-1[,]1))
3025, 29sselid 3947 . . . 4 ((πœ‘ ∧ π‘š ∈ β„•) β†’ (πΊβ€˜π‘š) ∈ ℝ)
31 eqidd 2738 . . . 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 24891 . . 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 2780 . 2 ((πœ‘ ∧ π‘š ∈ β„•) β†’ (vol*β€˜(π‘‡β€˜π‘š)) = (vol*β€˜ran 𝐹))
34 3re 12240 . . . . . . . 8 3 ∈ ℝ
3534rexri 11220 . . . . . . 7 3 ∈ ℝ*
3635a1i 11 . . . . . 6 ((πœ‘ ∧ 0 < (vol*β€˜ran 𝐹)) β†’ 3 ∈ ℝ*)
37 3rp 12928 . . . . . . . . . . . . 13 3 ∈ ℝ+
38 0re 11164 . . . . . . . . . . . . . . . . . . . 20 0 ∈ ℝ
39 0le1 11685 . . . . . . . . . . . . . . . . . . . 20 0 ≀ 1
40 ovolicc 24903 . . . . . . . . . . . . . . . . . . . 20 ((0 ∈ ℝ ∧ 1 ∈ ℝ ∧ 0 ≀ 1) β†’ (vol*β€˜(0[,]1)) = (1 βˆ’ 0))
4138, 23, 39, 40mp3an 1462 . . . . . . . . . . . . . . . . . . 19 (vol*β€˜(0[,]1)) = (1 βˆ’ 0)
42 1m0e1 12281 . . . . . . . . . . . . . . . . . . 19 (1 βˆ’ 0) = 1
4341, 42eqtri 2765 . . . . . . . . . . . . . . . . . 18 (vol*β€˜(0[,]1)) = 1
4443, 23eqeltri 2834 . . . . . . . . . . . . . . . . 17 (vol*β€˜(0[,]1)) ∈ ℝ
45 ovolsscl 24866 . . . . . . . . . . . . . . . . 17 ((ran 𝐹 βŠ† (0[,]1) ∧ (0[,]1) βŠ† ℝ ∧ (vol*β€˜(0[,]1)) ∈ ℝ) β†’ (vol*β€˜ran 𝐹) ∈ ℝ)
4619, 44, 45mp3an23 1454 . . . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . . 14 ((πœ‘ ∧ 0 < (vol*β€˜ran 𝐹)) β†’ (vol*β€˜ran 𝐹) ∈ ℝ)
49 simpr 486 . . . . . . . . . . . . . 14 ((πœ‘ ∧ 0 < (vol*β€˜ran 𝐹)) β†’ 0 < (vol*β€˜ran 𝐹))
5048, 49elrpd 12961 . . . . . . . . . . . . 13 ((πœ‘ ∧ 0 < (vol*β€˜ran 𝐹)) β†’ (vol*β€˜ran 𝐹) ∈ ℝ+)
51 rpdivcl 12947 . . . . . . . . . . . . 13 ((3 ∈ ℝ+ ∧ (vol*β€˜ran 𝐹) ∈ ℝ+) β†’ (3 / (vol*β€˜ran 𝐹)) ∈ ℝ+)
5237, 50, 51sylancr 588 . . . . . . . . . . . 12 ((πœ‘ ∧ 0 < (vol*β€˜ran 𝐹)) β†’ (3 / (vol*β€˜ran 𝐹)) ∈ ℝ+)
5352rpred 12964 . . . . . . . . . . 11 ((πœ‘ ∧ 0 < (vol*β€˜ran 𝐹)) β†’ (3 / (vol*β€˜ran 𝐹)) ∈ ℝ)
5452rpge0d 12968 . . . . . . . . . . 11 ((πœ‘ ∧ 0 < (vol*β€˜ran 𝐹)) β†’ 0 ≀ (3 / (vol*β€˜ran 𝐹)))
55 flge0nn0 13732 . . . . . . . . . . 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 585 . . . . . . . . . 10 ((πœ‘ ∧ 0 < (vol*β€˜ran 𝐹)) β†’ (βŒŠβ€˜(3 / (vol*β€˜ran 𝐹))) ∈ β„•0)
57 nn0p1nn 12459 . . . . . . . . . 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 12175 . . . . . . . 8 ((πœ‘ ∧ 0 < (vol*β€˜ran 𝐹)) β†’ ((βŒŠβ€˜(3 / (vol*β€˜ran 𝐹))) + 1) ∈ ℝ)
6059, 48remulcld 11192 . . . . . . 7 ((πœ‘ ∧ 0 < (vol*β€˜ran 𝐹)) β†’ (((βŒŠβ€˜(3 / (vol*β€˜ran 𝐹))) + 1) Β· (vol*β€˜ran 𝐹)) ∈ ℝ)
6160rexrd 11212 . . . . . 6 ((πœ‘ ∧ 0 < (vol*β€˜ran 𝐹)) β†’ (((βŒŠβ€˜(3 / (vol*β€˜ran 𝐹))) + 1) Β· (vol*β€˜ran 𝐹)) ∈ ℝ*)
626elpw2 5307 . . . . . . . . . . . . . . . . . 18 (ran 𝐹 ∈ 𝒫 ℝ ↔ ran 𝐹 βŠ† ℝ)
6320, 62sylibr 233 . . . . . . . . . . . . . . . . 17 (πœ‘ β†’ ran 𝐹 ∈ 𝒫 ℝ)
6463anim1i 616 . . . . . . . . . . . . . . . 16 ((πœ‘ ∧ Β¬ ran 𝐹 ∈ dom vol) β†’ (ran 𝐹 ∈ 𝒫 ℝ ∧ Β¬ ran 𝐹 ∈ dom vol))
65 eldif 3925 . . . . . . . . . . . . . . . 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 414 . . . . . . . . . . . . . 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 4193 . . . . . . . . . . . . . . . 16 (β„š ∩ (-1[,]1)) βŠ† β„š
70 qssre 12891 . . . . . . . . . . . . . . . 16 β„š βŠ† ℝ
7169, 70sstri 3958 . . . . . . . . . . . . . . 15 (β„š ∩ (-1[,]1)) βŠ† ℝ
72 fss 6690 . . . . . . . . . . . . . . 15 ((𝐺:β„•βŸΆ(β„š ∩ (-1[,]1)) ∧ (β„š ∩ (-1[,]1)) βŠ† ℝ) β†’ 𝐺:β„•βŸΆβ„)
7327, 71, 72sylancl 587 . . . . . . . . . . . . . 14 (πœ‘ β†’ 𝐺:β„•βŸΆβ„)
7473ffvelcdmda 7040 . . . . . . . . . . . . 13 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (πΊβ€˜π‘›) ∈ ℝ)
75 shftmbl 24918 . . . . . . . . . . . . 13 ((ran 𝐹 ∈ dom vol ∧ (πΊβ€˜π‘›) ∈ ℝ) β†’ {𝑠 ∈ ℝ ∣ (𝑠 βˆ’ (πΊβ€˜π‘›)) ∈ ran 𝐹} ∈ dom vol)
7668, 74, 75syl2an2r 684 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ {𝑠 ∈ ℝ ∣ (𝑠 βˆ’ (πΊβ€˜π‘›)) ∈ ran 𝐹} ∈ dom vol)
7776, 5fmptd 7067 . . . . . . . . . . 11 (πœ‘ β†’ 𝑇:β„•βŸΆdom vol)
7877ffvelcdmda 7040 . . . . . . . . . 10 ((πœ‘ ∧ π‘š ∈ β„•) β†’ (π‘‡β€˜π‘š) ∈ dom vol)
7978ralrimiva 3144 . . . . . . . . 9 (πœ‘ β†’ βˆ€π‘š ∈ β„• (π‘‡β€˜π‘š) ∈ dom vol)
80 iunmbl 24933 . . . . . . . . 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 24911 . . . . . . . 8 (βˆͺ π‘š ∈ β„• (π‘‡β€˜π‘š) ∈ dom vol β†’ βˆͺ π‘š ∈ β„• (π‘‡β€˜π‘š) βŠ† ℝ)
83 ovolcl 24858 . . . . . . . 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 482 . . . . . 6 ((πœ‘ ∧ 0 < (vol*β€˜ran 𝐹)) β†’ (vol*β€˜βˆͺ π‘š ∈ β„• (π‘‡β€˜π‘š)) ∈ ℝ*)
86 flltp1 13712 . . . . . . . 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 13016 . . . . . . 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 12813 . . . . . . . . . . 11 β„• = (β„€β‰₯β€˜1)
92 1zzd 12541 . . . . . . . . . . 11 ((πœ‘ ∧ 0 < (vol*β€˜ran 𝐹)) β†’ 1 ∈ β„€)
93 mblvol 24910 . . . . . . . . . . . . . . . . 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 2777 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ π‘š ∈ β„•) β†’ (volβ€˜(π‘‡β€˜π‘š)) = (vol*β€˜ran 𝐹))
9647adantr 482 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ π‘š ∈ β„•) β†’ (vol*β€˜ran 𝐹) ∈ ℝ)
9795, 96eqeltrd 2838 . . . . . . . . . . . . . 14 ((πœ‘ ∧ π‘š ∈ β„•) β†’ (volβ€˜(π‘‡β€˜π‘š)) ∈ ℝ)
9897adantlr 714 . . . . . . . . . . . . 13 (((πœ‘ ∧ 0 < (vol*β€˜ran 𝐹)) ∧ π‘š ∈ β„•) β†’ (volβ€˜(π‘‡β€˜π‘š)) ∈ ℝ)
99 eqid 2737 . . . . . . . . . . . . 13 (π‘š ∈ β„• ↦ (volβ€˜(π‘‡β€˜π‘š))) = (π‘š ∈ β„• ↦ (volβ€˜(π‘‡β€˜π‘š)))
10098, 99fmptd 7067 . . . . . . . . . . . 12 ((πœ‘ ∧ 0 < (vol*β€˜ran 𝐹)) β†’ (π‘š ∈ β„• ↦ (volβ€˜(π‘‡β€˜π‘š))):β„•βŸΆβ„)
101100ffvelcdmda 7040 . . . . . . . . . . 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 13944 . . . . . . . . . 10 ((πœ‘ ∧ 0 < (vol*β€˜ran 𝐹)) β†’ seq1( + , (π‘š ∈ β„• ↦ (volβ€˜(π‘‡β€˜π‘š)))):β„•βŸΆβ„)
103102frnd 6681 . . . . . . . . 9 ((πœ‘ ∧ 0 < (vol*β€˜ran 𝐹)) β†’ ran seq1( + , (π‘š ∈ β„• ↦ (volβ€˜(π‘‡β€˜π‘š)))) βŠ† ℝ)
104 ressxr 11206 . . . . . . . . 9 ℝ βŠ† ℝ*
105103, 104sstrdi 3961 . . . . . . . 8 ((πœ‘ ∧ 0 < (vol*β€˜ran 𝐹)) β†’ ran seq1( + , (π‘š ∈ β„• ↦ (volβ€˜(π‘‡β€˜π‘š)))) βŠ† ℝ*)
10695adantlr 714 . . . . . . . . . . . . . 14 (((πœ‘ ∧ 0 < (vol*β€˜ran 𝐹)) ∧ π‘š ∈ β„•) β†’ (volβ€˜(π‘‡β€˜π‘š)) = (vol*β€˜ran 𝐹))
107106mpteq2dva 5210 . . . . . . . . . . . . 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 5699 . . . . . . . . . . . . 13 (β„• Γ— {(vol*β€˜ran 𝐹)}) = (π‘š ∈ β„• ↦ (vol*β€˜ran 𝐹))
109107, 108eqtr4di 2795 . . . . . . . . . . . 12 ((πœ‘ ∧ 0 < (vol*β€˜ran 𝐹)) β†’ (π‘š ∈ β„• ↦ (volβ€˜(π‘‡β€˜π‘š))) = (β„• Γ— {(vol*β€˜ran 𝐹)}))
110109seqeq3d 13921 . . . . . . . . . . 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 6849 . . . . . . . . . 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 11190 . . . . . . . . . . 11 ((πœ‘ ∧ 0 < (vol*β€˜ran 𝐹)) β†’ (vol*β€˜ran 𝐹) ∈ β„‚)
113 ser1const 13971 . . . . . . . . . . 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 585 . . . . . . . . . 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 2777 . . . . . . . . 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 6674 . . . . . . . . . 10 ((πœ‘ ∧ 0 < (vol*β€˜ran 𝐹)) β†’ seq1( + , (π‘š ∈ β„• ↦ (volβ€˜(π‘‡β€˜π‘š)))) Fn β„•)
117 fnfvelrn 7036 . . . . . . . . . 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 585 . . . . . . . . 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 2839 . . . . . . . 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 13250 . . . . . . . 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 585 . . . . . . 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 482 . . . . . . . . 9 ((πœ‘ ∧ 0 < (vol*β€˜ran 𝐹)) β†’ βˆͺ π‘š ∈ β„• (π‘‡β€˜π‘š) ∈ dom vol)
123 mblvol 24910 . . . . . . . . 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 513 . . . . . . . . . 10 ((πœ‘ ∧ π‘š ∈ β„•) β†’ ((π‘‡β€˜π‘š) ∈ dom vol ∧ (volβ€˜(π‘‡β€˜π‘š)) ∈ ℝ))
126125ralrimiva 3144 . . . . . . . . 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 24990 . . . . . . . . . 10 (πœ‘ β†’ Disj π‘š ∈ β„• (π‘‡β€˜π‘š))
128127adantr 482 . . . . . . . . 9 ((πœ‘ ∧ 0 < (vol*β€˜ran 𝐹)) β†’ Disj π‘š ∈ β„• (π‘‡β€˜π‘š))
129 eqid 2737 . . . . . . . . . 10 seq1( + , (π‘š ∈ β„• ↦ (volβ€˜(π‘‡β€˜π‘š)))) = seq1( + , (π‘š ∈ β„• ↦ (volβ€˜(π‘‡β€˜π‘š))))
130129, 99voliun 24934 . . . . . . . . 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 684 . . . . . . . 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 2779 . . . . . . 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 5138 . . . . . 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 13087 . . . . 5 ((πœ‘ ∧ 0 < (vol*β€˜ran 𝐹)) β†’ 3 < (vol*β€˜βˆͺ π‘š ∈ β„• (π‘‡β€˜π‘š)))
13517simp3d 1145 . . . . . . . . 9 (πœ‘ β†’ βˆͺ π‘š ∈ β„• (π‘‡β€˜π‘š) βŠ† (-1[,]2))
136135adantr 482 . . . . . . . 8 ((πœ‘ ∧ 0 < (vol*β€˜ran 𝐹)) β†’ βˆͺ π‘š ∈ β„• (π‘‡β€˜π‘š) βŠ† (-1[,]2))
137 2re 12234 . . . . . . . . 9 2 ∈ ℝ
138 iccssre 13353 . . . . . . . . 9 ((-1 ∈ ℝ ∧ 2 ∈ ℝ) β†’ (-1[,]2) βŠ† ℝ)
13922, 137, 138mp2an 691 . . . . . . . 8 (-1[,]2) βŠ† ℝ
140 ovolss 24865 . . . . . . . 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 587 . . . . . . 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 12235 . . . . . . . . 9 2 ∈ β„‚
143 ax-1cn 11116 . . . . . . . . 9 1 ∈ β„‚
144142, 143subnegi 11487 . . . . . . . 8 (2 βˆ’ -1) = (2 + 1)
145 neg1lt0 12277 . . . . . . . . . . 11 -1 < 0
146 2pos 12263 . . . . . . . . . . 11 0 < 2
14722, 38, 137lttri 11288 . . . . . . . . . . 11 ((-1 < 0 ∧ 0 < 2) β†’ -1 < 2)
148145, 146, 147mp2an 691 . . . . . . . . . 10 -1 < 2
14922, 137, 148ltleii 11285 . . . . . . . . 9 -1 ≀ 2
150 ovolicc 24903 . . . . . . . . 9 ((-1 ∈ ℝ ∧ 2 ∈ ℝ ∧ -1 ≀ 2) β†’ (vol*β€˜(-1[,]2)) = (2 βˆ’ -1))
15122, 137, 149, 150mp3an 1462 . . . . . . . 8 (vol*β€˜(-1[,]2)) = (2 βˆ’ -1)
152 df-3 12224 . . . . . . . 8 3 = (2 + 1)
153144, 151, 1523eqtr4i 2775 . . . . . . 7 (vol*β€˜(-1[,]2)) = 3
154141, 153breqtrdi 5151 . . . . . 6 ((πœ‘ ∧ 0 < (vol*β€˜ran 𝐹)) β†’ (vol*β€˜βˆͺ π‘š ∈ β„• (π‘‡β€˜π‘š)) ≀ 3)
155 xrlenlt 11227 . . . . . . 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 587 . . . . . 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 816 . . . 4 (πœ‘ β†’ Β¬ 0 < (vol*β€˜ran 𝐹))
159 ovolge0 24861 . . . . . . 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 11209 . . . . . . 7 0 ∈ ℝ*
162 ovolcl 24858 . . . . . . . 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 13070 . . . . . . 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 588 . . . . . 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 863 . . . 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 482 . 2 ((πœ‘ ∧ π‘š ∈ β„•) β†’ 0 = (vol*β€˜ran 𝐹))
17033, 169eqtr4d 2780 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 397   ∨ wo 846   = wceq 1542   ∈ wcel 2107   β‰  wne 2944  βˆ€wral 3065  {crab 3410   βˆ– cdif 3912   ∩ cin 3914   βŠ† wss 3915  βˆ…c0 4287  π’« cpw 4565  {csn 4591  βˆͺ ciun 4959  Disj wdisj 5075   class class class wbr 5110  {copab 5172   ↦ cmpt 5193   Γ— cxp 5636  dom cdm 5638  ran crn 5639   Fn wfn 6496  βŸΆwf 6497  β€“1-1-ontoβ†’wf1o 6500  β€˜cfv 6501  (class class class)co 7362   / cqs 8654  supcsup 9383  β„‚cc 11056  β„cr 11057  0cc0 11058  1c1 11059   + caddc 11061   Β· cmul 11063  β„*cxr 11195   < clt 11196   ≀ cle 11197   βˆ’ cmin 11392  -cneg 11393   / cdiv 11819  β„•cn 12160  2c2 12215  3c3 12216  β„•0cn0 12420  β„šcq 12880  β„+crp 12922  [,]cicc 13274  βŒŠcfl 13702  seqcseq 13913  vol*covol 24842  volcvol 24843
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-inf2 9584  ax-cc 10378  ax-cnex 11114  ax-resscn 11115  ax-1cn 11116  ax-icn 11117  ax-addcl 11118  ax-addrcl 11119  ax-mulcl 11120  ax-mulrcl 11121  ax-mulcom 11122  ax-addass 11123  ax-mulass 11124  ax-distr 11125  ax-i2m1 11126  ax-1ne0 11127  ax-1rid 11128  ax-rnegex 11129  ax-rrecex 11130  ax-cnre 11131  ax-pre-lttri 11132  ax-pre-lttrn 11133  ax-pre-ltadd 11134  ax-pre-mulgt0 11135  ax-pre-sup 11136
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-rmo 3356  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-disj 5076  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-se 5594  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-isom 6510  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-of 7622  df-om 7808  df-1st 7926  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-2o 8418  df-er 8655  df-ec 8657  df-qs 8661  df-map 8774  df-pm 8775  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-fi 9354  df-sup 9385  df-inf 9386  df-oi 9453  df-dju 9844  df-card 9882  df-pnf 11198  df-mnf 11199  df-xr 11200  df-ltxr 11201  df-le 11202  df-sub 11394  df-neg 11395  df-div 11820  df-nn 12161  df-2 12223  df-3 12224  df-n0 12421  df-z 12507  df-uz 12771  df-q 12881  df-rp 12923  df-xneg 13040  df-xadd 13041  df-xmul 13042  df-ioo 13275  df-ico 13277  df-icc 13278  df-fz 13432  df-fzo 13575  df-fl 13704  df-seq 13914  df-exp 13975  df-hash 14238  df-cj 14991  df-re 14992  df-im 14993  df-sqrt 15127  df-abs 15128  df-clim 15377  df-rlim 15378  df-sum 15578  df-rest 17311  df-topgen 17332  df-psmet 20804  df-xmet 20805  df-met 20806  df-bl 20807  df-mopn 20808  df-top 22259  df-topon 22276  df-bases 22312  df-cmp 22754  df-ovol 24844  df-vol 24845
This theorem is referenced by:  vitalilem5  24992
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