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Mirrors > Home > MPE Home > Th. List > ioombl1lem2 | Structured version Visualization version GIF version |
Description: Lemma for ioombl1 25507. (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 2725 | . . . . . . 7 ⊢ ((abs ∘ − ) ∘ 𝐹) = ((abs ∘ − ) ∘ 𝐹) | |
3 | ioombl1.s | . . . . . . 7 ⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹)) | |
4 | 2, 3 | ovolsf 25417 | . . . . . 6 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑆:ℕ⟶(0[,)+∞)) |
5 | 1, 4 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑆:ℕ⟶(0[,)+∞)) |
6 | 5 | frnd 6724 | . . . 4 ⊢ (𝜑 → ran 𝑆 ⊆ (0[,)+∞)) |
7 | icossxr 13439 | . . . 4 ⊢ (0[,)+∞) ⊆ ℝ* | |
8 | 6, 7 | sstrdi 3985 | . . 3 ⊢ (𝜑 → ran 𝑆 ⊆ ℝ*) |
9 | supxrcl 13324 | . . 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 13046 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℝ) |
14 | 11, 13 | readdcld 11271 | . 2 ⊢ (𝜑 → ((vol*‘𝐸) + 𝐶) ∈ ℝ) |
15 | mnfxr 11299 | . . . 4 ⊢ -∞ ∈ ℝ* | |
16 | 15 | a1i 11 | . . 3 ⊢ (𝜑 → -∞ ∈ ℝ*) |
17 | 5 | ffnd 6717 | . . . . 5 ⊢ (𝜑 → 𝑆 Fn ℕ) |
18 | 1nn 12251 | . . . . 5 ⊢ 1 ∈ ℕ | |
19 | fnfvelrn 7084 | . . . . 5 ⊢ ((𝑆 Fn ℕ ∧ 1 ∈ ℕ) → (𝑆‘1) ∈ ran 𝑆) | |
20 | 17, 18, 19 | sylancl 584 | . . . 4 ⊢ (𝜑 → (𝑆‘1) ∈ ran 𝑆) |
21 | 8, 20 | sseldd 3973 | . . 3 ⊢ (𝜑 → (𝑆‘1) ∈ ℝ*) |
22 | rge0ssre 13463 | . . . . 5 ⊢ (0[,)+∞) ⊆ ℝ | |
23 | ffvelcdm 7085 | . . . . . 6 ⊢ ((𝑆:ℕ⟶(0[,)+∞) ∧ 1 ∈ ℕ) → (𝑆‘1) ∈ (0[,)+∞)) | |
24 | 5, 18, 23 | sylancl 584 | . . . . 5 ⊢ (𝜑 → (𝑆‘1) ∈ (0[,)+∞)) |
25 | 22, 24 | sselid 3970 | . . . 4 ⊢ (𝜑 → (𝑆‘1) ∈ ℝ) |
26 | 25 | mnfltd 13134 | . . 3 ⊢ (𝜑 → -∞ < (𝑆‘1)) |
27 | supxrub 13333 | . . . 4 ⊢ ((ran 𝑆 ⊆ ℝ* ∧ (𝑆‘1) ∈ ran 𝑆) → (𝑆‘1) ≤ sup(ran 𝑆, ℝ*, < )) | |
28 | 8, 20, 27 | syl2anc 582 | . . 3 ⊢ (𝜑 → (𝑆‘1) ≤ sup(ran 𝑆, ℝ*, < )) |
29 | 16, 21, 10, 26, 28 | xrltletrd 13170 | . 2 ⊢ (𝜑 → -∞ < sup(ran 𝑆, ℝ*, < )) |
30 | ioombl1.f3 | . 2 ⊢ (𝜑 → sup(ran 𝑆, ℝ*, < ) ≤ ((vol*‘𝐸) + 𝐶)) | |
31 | xrre 13178 | . 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 837 | 1 ⊢ (𝜑 → sup(ran 𝑆, ℝ*, < ) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ∩ cin 3939 ⊆ wss 3940 ifcif 4524 ⟨cop 4630 ∪ cuni 4903 class class class wbr 5143 ↦ cmpt 5226 × cxp 5670 ran crn 5673 ∘ ccom 5676 Fn wfn 6537 ⟶wf 6538 ‘cfv 6542 (class class class)co 7415 1st c1st 7987 2nd c2nd 7988 supcsup 9461 ℝcr 11135 0cc0 11136 1c1 11137 + caddc 11139 +∞cpnf 11273 -∞cmnf 11274 ℝ*cxr 11275 < clt 11276 ≤ cle 11277 − cmin 11472 ℕcn 12240 ℝ+crp 13004 (,)cioo 13354 [,)cico 13356 seqcseq 13996 abscabs 15211 vol*covol 25407 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213 ax-pre-sup 11214 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3960 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7868 df-1st 7989 df-2nd 7990 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-er 8721 df-en 8961 df-dom 8962 df-sdom 8963 df-sup 9463 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-div 11900 df-nn 12241 df-2 12303 df-3 12304 df-n0 12501 df-z 12587 df-uz 12851 df-rp 13005 df-ico 13360 df-fz 13515 df-seq 13997 df-exp 14057 df-cj 15076 df-re 15077 df-im 15078 df-sqrt 15212 df-abs 15213 |
This theorem is referenced by: ioombl1lem4 25506 |
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