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Mirrors > Home > MPE Home > Th. List > ioombl1lem2 | Structured version Visualization version GIF version |
Description: Lemma for ioombl1 24431. (Contributed by Mario Carneiro, 18-Aug-2014.) |
Ref | Expression |
---|---|
ioombl1.b | ⊢ 𝐵 = (𝐴(,)+∞) |
ioombl1.a | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
ioombl1.e | ⊢ (𝜑 → 𝐸 ⊆ ℝ) |
ioombl1.v | ⊢ (𝜑 → (vol*‘𝐸) ∈ ℝ) |
ioombl1.c | ⊢ (𝜑 → 𝐶 ∈ ℝ+) |
ioombl1.s | ⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹)) |
ioombl1.t | ⊢ 𝑇 = seq1( + , ((abs ∘ − ) ∘ 𝐺)) |
ioombl1.u | ⊢ 𝑈 = seq1( + , ((abs ∘ − ) ∘ 𝐻)) |
ioombl1.f1 | ⊢ (𝜑 → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) |
ioombl1.f2 | ⊢ (𝜑 → 𝐸 ⊆ ∪ ran ((,) ∘ 𝐹)) |
ioombl1.f3 | ⊢ (𝜑 → sup(ran 𝑆, ℝ*, < ) ≤ ((vol*‘𝐸) + 𝐶)) |
ioombl1.p | ⊢ 𝑃 = (1st ‘(𝐹‘𝑛)) |
ioombl1.q | ⊢ 𝑄 = (2nd ‘(𝐹‘𝑛)) |
ioombl1.g | ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ 〈if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄), 𝑄〉) |
ioombl1.h | ⊢ 𝐻 = (𝑛 ∈ ℕ ↦ 〈𝑃, if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄)〉) |
Ref | Expression |
---|---|
ioombl1lem2 | ⊢ (𝜑 → sup(ran 𝑆, ℝ*, < ) ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ioombl1.f1 | . . . . . 6 ⊢ (𝜑 → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) | |
2 | eqid 2734 | . . . . . . 7 ⊢ ((abs ∘ − ) ∘ 𝐹) = ((abs ∘ − ) ∘ 𝐹) | |
3 | ioombl1.s | . . . . . . 7 ⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹)) | |
4 | 2, 3 | ovolsf 24341 | . . . . . 6 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑆:ℕ⟶(0[,)+∞)) |
5 | 1, 4 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑆:ℕ⟶(0[,)+∞)) |
6 | 5 | frnd 6542 | . . . 4 ⊢ (𝜑 → ran 𝑆 ⊆ (0[,)+∞)) |
7 | icossxr 13003 | . . . 4 ⊢ (0[,)+∞) ⊆ ℝ* | |
8 | 6, 7 | sstrdi 3903 | . . 3 ⊢ (𝜑 → ran 𝑆 ⊆ ℝ*) |
9 | supxrcl 12888 | . . 3 ⊢ (ran 𝑆 ⊆ ℝ* → sup(ran 𝑆, ℝ*, < ) ∈ ℝ*) | |
10 | 8, 9 | syl 17 | . 2 ⊢ (𝜑 → sup(ran 𝑆, ℝ*, < ) ∈ ℝ*) |
11 | ioombl1.v | . . 3 ⊢ (𝜑 → (vol*‘𝐸) ∈ ℝ) | |
12 | ioombl1.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ ℝ+) | |
13 | 12 | rpred 12611 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℝ) |
14 | 11, 13 | readdcld 10845 | . 2 ⊢ (𝜑 → ((vol*‘𝐸) + 𝐶) ∈ ℝ) |
15 | mnfxr 10873 | . . . 4 ⊢ -∞ ∈ ℝ* | |
16 | 15 | a1i 11 | . . 3 ⊢ (𝜑 → -∞ ∈ ℝ*) |
17 | 5 | ffnd 6535 | . . . . 5 ⊢ (𝜑 → 𝑆 Fn ℕ) |
18 | 1nn 11824 | . . . . 5 ⊢ 1 ∈ ℕ | |
19 | fnfvelrn 6890 | . . . . 5 ⊢ ((𝑆 Fn ℕ ∧ 1 ∈ ℕ) → (𝑆‘1) ∈ ran 𝑆) | |
20 | 17, 18, 19 | sylancl 589 | . . . 4 ⊢ (𝜑 → (𝑆‘1) ∈ ran 𝑆) |
21 | 8, 20 | sseldd 3892 | . . 3 ⊢ (𝜑 → (𝑆‘1) ∈ ℝ*) |
22 | rge0ssre 13027 | . . . . 5 ⊢ (0[,)+∞) ⊆ ℝ | |
23 | ffvelrn 6891 | . . . . . 6 ⊢ ((𝑆:ℕ⟶(0[,)+∞) ∧ 1 ∈ ℕ) → (𝑆‘1) ∈ (0[,)+∞)) | |
24 | 5, 18, 23 | sylancl 589 | . . . . 5 ⊢ (𝜑 → (𝑆‘1) ∈ (0[,)+∞)) |
25 | 22, 24 | sseldi 3889 | . . . 4 ⊢ (𝜑 → (𝑆‘1) ∈ ℝ) |
26 | 25 | mnfltd 12699 | . . 3 ⊢ (𝜑 → -∞ < (𝑆‘1)) |
27 | supxrub 12897 | . . . 4 ⊢ ((ran 𝑆 ⊆ ℝ* ∧ (𝑆‘1) ∈ ran 𝑆) → (𝑆‘1) ≤ sup(ran 𝑆, ℝ*, < )) | |
28 | 8, 20, 27 | syl2anc 587 | . . 3 ⊢ (𝜑 → (𝑆‘1) ≤ sup(ran 𝑆, ℝ*, < )) |
29 | 16, 21, 10, 26, 28 | xrltletrd 12734 | . 2 ⊢ (𝜑 → -∞ < sup(ran 𝑆, ℝ*, < )) |
30 | ioombl1.f3 | . 2 ⊢ (𝜑 → sup(ran 𝑆, ℝ*, < ) ≤ ((vol*‘𝐸) + 𝐶)) | |
31 | xrre 12742 | . 2 ⊢ (((sup(ran 𝑆, ℝ*, < ) ∈ ℝ* ∧ ((vol*‘𝐸) + 𝐶) ∈ ℝ) ∧ (-∞ < sup(ran 𝑆, ℝ*, < ) ∧ sup(ran 𝑆, ℝ*, < ) ≤ ((vol*‘𝐸) + 𝐶))) → sup(ran 𝑆, ℝ*, < ) ∈ ℝ) | |
32 | 10, 14, 29, 30, 31 | syl22anc 839 | 1 ⊢ (𝜑 → sup(ran 𝑆, ℝ*, < ) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2110 ∩ cin 3856 ⊆ wss 3857 ifcif 4429 〈cop 4537 ∪ cuni 4809 class class class wbr 5043 ↦ cmpt 5124 × cxp 5538 ran crn 5541 ∘ ccom 5544 Fn wfn 6364 ⟶wf 6365 ‘cfv 6369 (class class class)co 7202 1st c1st 7748 2nd c2nd 7749 supcsup 9045 ℝcr 10711 0cc0 10712 1c1 10713 + caddc 10715 +∞cpnf 10847 -∞cmnf 10848 ℝ*cxr 10849 < clt 10850 ≤ cle 10851 − cmin 11045 ℕcn 11813 ℝ+crp 12569 (,)cioo 12918 [,)cico 12920 seqcseq 13557 abscabs 14780 vol*covol 24331 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 ax-pre-sup 10790 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-om 7634 df-1st 7750 df-2nd 7751 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-er 8380 df-en 8616 df-dom 8617 df-sdom 8618 df-sup 9047 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-div 11473 df-nn 11814 df-2 11876 df-3 11877 df-n0 12074 df-z 12160 df-uz 12422 df-rp 12570 df-ico 12924 df-fz 13079 df-seq 13558 df-exp 13619 df-cj 14645 df-re 14646 df-im 14647 df-sqrt 14781 df-abs 14782 |
This theorem is referenced by: ioombl1lem4 24430 |
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