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| Mirrors > Home > MPE Home > Th. List > ioombl1lem2 | Structured version Visualization version GIF version | ||
| Description: Lemma for ioombl1 25593. (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 2752 | . . . . . . 7 ⊢ ((abs ∘ − ) ∘ 𝐹) = ((abs ∘ − ) ∘ 𝐹) | |
| 3 | ioombl1.s | . . . . . . 7 ⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹)) | |
| 4 | 2, 3 | ovolsf 25503 | . . . . . 6 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑆:ℕ⟶(0[,)+∞)) |
| 5 | 1, 4 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑆:ℕ⟶(0[,)+∞)) |
| 6 | 5 | frnd 6685 | . . . 4 ⊢ (𝜑 → ran 𝑆 ⊆ (0[,)+∞)) |
| 7 | icossxr 13422 | . . . 4 ⊢ (0[,)+∞) ⊆ ℝ* | |
| 8 | 6, 7 | sstrdi 3939 | . . 3 ⊢ (𝜑 → ran 𝑆 ⊆ ℝ*) |
| 9 | supxrcl 13304 | . . 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 13023 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℝ) |
| 14 | 11, 13 | readdcld 11197 | . 2 ⊢ (𝜑 → ((vol*‘𝐸) + 𝐶) ∈ ℝ) |
| 15 | mnfxr 11225 | . . . 4 ⊢ -∞ ∈ ℝ* | |
| 16 | 15 | a1i 11 | . . 3 ⊢ (𝜑 → -∞ ∈ ℝ*) |
| 17 | 5 | ffnd 6677 | . . . . 5 ⊢ (𝜑 → 𝑆 Fn ℕ) |
| 18 | 1nn 12207 | . . . . 5 ⊢ 1 ∈ ℕ | |
| 19 | fnfvelrn 7046 | . . . . 5 ⊢ ((𝑆 Fn ℕ ∧ 1 ∈ ℕ) → (𝑆‘1) ∈ ran 𝑆) | |
| 20 | 17, 18, 19 | sylancl 594 | . . . 4 ⊢ (𝜑 → (𝑆‘1) ∈ ran 𝑆) |
| 21 | 8, 20 | sseldd 3928 | . . 3 ⊢ (𝜑 → (𝑆‘1) ∈ ℝ*) |
| 22 | rge0ssre 13446 | . . . . 5 ⊢ (0[,)+∞) ⊆ ℝ | |
| 23 | ffvelcdm 7047 | . . . . . 6 ⊢ ((𝑆:ℕ⟶(0[,)+∞) ∧ 1 ∈ ℕ) → (𝑆‘1) ∈ (0[,)+∞)) | |
| 24 | 5, 18, 23 | sylancl 594 | . . . . 5 ⊢ (𝜑 → (𝑆‘1) ∈ (0[,)+∞)) |
| 25 | 22, 24 | sselid 3925 | . . . 4 ⊢ (𝜑 → (𝑆‘1) ∈ ℝ) |
| 26 | 25 | mnfltd 13112 | . . 3 ⊢ (𝜑 → -∞ < (𝑆‘1)) |
| 27 | supxrub 13313 | . . . 4 ⊢ ((ran 𝑆 ⊆ ℝ* ∧ (𝑆‘1) ∈ ran 𝑆) → (𝑆‘1) ≤ sup(ran 𝑆, ℝ*, < )) | |
| 28 | 8, 20, 27 | syl2anc 592 | . . 3 ⊢ (𝜑 → (𝑆‘1) ≤ sup(ran 𝑆, ℝ*, < )) |
| 29 | 16, 21, 10, 26, 28 | xrltletrd 13149 | . 2 ⊢ (𝜑 → -∞ < sup(ran 𝑆, ℝ*, < )) |
| 30 | ioombl1.f3 | . 2 ⊢ (𝜑 → sup(ran 𝑆, ℝ*, < ) ≤ ((vol*‘𝐸) + 𝐶)) | |
| 31 | xrre 13158 | . 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 847 | 1 ⊢ (𝜑 → sup(ran 𝑆, ℝ*, < ) ∈ ℝ) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1550 ∈ wcel 2132 ∩ cin 3894 ⊆ wss 3895 ifcif 4470 〈cop 4578 ∪ cuni 4855 class class class wbr 5090 ↦ cmpt 5171 × cxp 5634 ran crn 5637 ∘ ccom 5640 Fn wfn 6501 ⟶wf 6502 ‘cfv 6506 (class class class)co 7381 1st c1st 7953 2nd c2nd 7954 supcsup 9372 ℝcr 11058 0cc0 11059 1c1 11060 + caddc 11062 +∞cpnf 11199 -∞cmnf 11200 ℝ*cxr 11201 < clt 11202 ≤ cle 11203 − cmin 11400 ℕcn 12196 ℝ+crp 12979 (,)cioo 13335 [,)cico 13337 seqcseq 14000 abscabs 15233 vol*covol 25493 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1805 ax-4 1819 ax-5 1920 ax-6 1977 ax-7 2018 ax-8 2134 ax-9 2142 ax-10 2165 ax-11 2181 ax-12 2202 ax-ext 2724 ax-sep 5236 ax-nul 5246 ax-pow 5312 ax-pr 5380 ax-un 7703 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 ax-pre-sup 11137 |
| This theorem depends on definitions: df-bi 209 df-an 399 df-or 857 df-3or 1096 df-3an 1097 df-tru 1553 df-fal 1563 df-ex 1790 df-nf 1794 df-sb 2081 df-mo 2556 df-eu 2586 df-clab 2731 df-cleq 2744 df-clel 2827 df-nfc 2901 df-ne 2948 df-nel 3052 df-ral 3067 df-rex 3077 df-rmo 3357 df-reu 3358 df-rab 3405 df-v 3446 df-sbc 3736 df-csb 3844 df-dif 3898 df-un 3900 df-in 3902 df-ss 3912 df-pss 3915 df-nul 4277 df-if 4471 df-pw 4547 df-sn 4573 df-pr 4575 df-op 4579 df-uni 4856 df-iun 4941 df-br 5091 df-opab 5153 df-mpt 5172 df-tr 5198 df-id 5531 df-eprel 5536 df-po 5544 df-so 5545 df-fr 5589 df-we 5591 df-xp 5642 df-rel 5643 df-cnv 5644 df-co 5645 df-dm 5646 df-rn 5647 df-res 5648 df-ima 5649 df-pred 6273 df-ord 6334 df-on 6335 df-lim 6336 df-suc 6337 df-iota 6462 df-fun 6508 df-fn 6509 df-f 6510 df-f1 6511 df-fo 6512 df-f1o 6513 df-fv 6514 df-riota 7338 df-ov 7384 df-oprab 7385 df-mpo 7386 df-om 7832 df-1st 7955 df-2nd 7956 df-frecs 8246 df-wrecs 8277 df-recs 8326 df-rdg 8365 df-er 8662 df-en 8913 df-dom 8914 df-sdom 8915 df-sup 9374 df-pnf 11204 df-mnf 11205 df-xr 11206 df-ltxr 11207 df-le 11208 df-sub 11402 df-neg 11403 df-div 11831 df-nn 12197 df-2 12266 df-3 12267 df-n0 12468 df-z 12555 df-uz 12826 df-rp 12980 df-ico 13341 df-fz 13499 df-seq 14001 df-exp 14061 df-cj 15098 df-re 15099 df-im 15100 df-sqrt 15234 df-abs 15235 |
| This theorem is referenced by: ioombl1lem4 25592 |
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