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Mirrors > Home > MPE Home > Th. List > ioombl1lem2 | Structured version Visualization version GIF version |
Description: Lemma for ioombl1 25446. (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 2726 | . . . . . . 7 ⊢ ((abs ∘ − ) ∘ 𝐹) = ((abs ∘ − ) ∘ 𝐹) | |
3 | ioombl1.s | . . . . . . 7 ⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹)) | |
4 | 2, 3 | ovolsf 25356 | . . . . . 6 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑆:ℕ⟶(0[,)+∞)) |
5 | 1, 4 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑆:ℕ⟶(0[,)+∞)) |
6 | 5 | frnd 6719 | . . . 4 ⊢ (𝜑 → ran 𝑆 ⊆ (0[,)+∞)) |
7 | icossxr 13415 | . . . 4 ⊢ (0[,)+∞) ⊆ ℝ* | |
8 | 6, 7 | sstrdi 3989 | . . 3 ⊢ (𝜑 → ran 𝑆 ⊆ ℝ*) |
9 | supxrcl 13300 | . . 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 13022 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℝ) |
14 | 11, 13 | readdcld 11247 | . 2 ⊢ (𝜑 → ((vol*‘𝐸) + 𝐶) ∈ ℝ) |
15 | mnfxr 11275 | . . . 4 ⊢ -∞ ∈ ℝ* | |
16 | 15 | a1i 11 | . . 3 ⊢ (𝜑 → -∞ ∈ ℝ*) |
17 | 5 | ffnd 6712 | . . . . 5 ⊢ (𝜑 → 𝑆 Fn ℕ) |
18 | 1nn 12227 | . . . . 5 ⊢ 1 ∈ ℕ | |
19 | fnfvelrn 7076 | . . . . 5 ⊢ ((𝑆 Fn ℕ ∧ 1 ∈ ℕ) → (𝑆‘1) ∈ ran 𝑆) | |
20 | 17, 18, 19 | sylancl 585 | . . . 4 ⊢ (𝜑 → (𝑆‘1) ∈ ran 𝑆) |
21 | 8, 20 | sseldd 3978 | . . 3 ⊢ (𝜑 → (𝑆‘1) ∈ ℝ*) |
22 | rge0ssre 13439 | . . . . 5 ⊢ (0[,)+∞) ⊆ ℝ | |
23 | ffvelcdm 7077 | . . . . . 6 ⊢ ((𝑆:ℕ⟶(0[,)+∞) ∧ 1 ∈ ℕ) → (𝑆‘1) ∈ (0[,)+∞)) | |
24 | 5, 18, 23 | sylancl 585 | . . . . 5 ⊢ (𝜑 → (𝑆‘1) ∈ (0[,)+∞)) |
25 | 22, 24 | sselid 3975 | . . . 4 ⊢ (𝜑 → (𝑆‘1) ∈ ℝ) |
26 | 25 | mnfltd 13110 | . . 3 ⊢ (𝜑 → -∞ < (𝑆‘1)) |
27 | supxrub 13309 | . . . 4 ⊢ ((ran 𝑆 ⊆ ℝ* ∧ (𝑆‘1) ∈ ran 𝑆) → (𝑆‘1) ≤ sup(ran 𝑆, ℝ*, < )) | |
28 | 8, 20, 27 | syl2anc 583 | . . 3 ⊢ (𝜑 → (𝑆‘1) ≤ sup(ran 𝑆, ℝ*, < )) |
29 | 16, 21, 10, 26, 28 | xrltletrd 13146 | . 2 ⊢ (𝜑 → -∞ < sup(ran 𝑆, ℝ*, < )) |
30 | ioombl1.f3 | . 2 ⊢ (𝜑 → sup(ran 𝑆, ℝ*, < ) ≤ ((vol*‘𝐸) + 𝐶)) | |
31 | xrre 13154 | . 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 836 | 1 ⊢ (𝜑 → sup(ran 𝑆, ℝ*, < ) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ∩ cin 3942 ⊆ wss 3943 ifcif 4523 ⟨cop 4629 ∪ cuni 4902 class class class wbr 5141 ↦ cmpt 5224 × cxp 5667 ran crn 5670 ∘ ccom 5673 Fn wfn 6532 ⟶wf 6533 ‘cfv 6537 (class class class)co 7405 1st c1st 7972 2nd c2nd 7973 supcsup 9437 ℝcr 11111 0cc0 11112 1c1 11113 + caddc 11115 +∞cpnf 11249 -∞cmnf 11250 ℝ*cxr 11251 < clt 11252 ≤ cle 11253 − cmin 11448 ℕcn 12216 ℝ+crp 12980 (,)cioo 13330 [,)cico 13332 seqcseq 13972 abscabs 15187 vol*covol 25346 |
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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-sup 9439 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-n0 12477 df-z 12563 df-uz 12827 df-rp 12981 df-ico 13336 df-fz 13491 df-seq 13973 df-exp 14033 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 |
This theorem is referenced by: ioombl1lem4 25445 |
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