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| Mirrors > Home > MPE Home > Th. List > ioombl1lem2 | Structured version Visualization version GIF version | ||
| Description: Lemma for ioombl1 25612. (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 2761 | . . . . . . 7 ⊢ ((abs ∘ − ) ∘ 𝐹) = ((abs ∘ − ) ∘ 𝐹) | |
| 3 | ioombl1.s | . . . . . . 7 ⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹)) | |
| 4 | 2, 3 | ovolsf 25522 | . . . . . 6 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑆:ℕ⟶(0[,)+∞)) |
| 5 | 1, 4 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑆:ℕ⟶(0[,)+∞)) |
| 6 | 5 | frnd 6695 | . . . 4 ⊢ (𝜑 → ran 𝑆 ⊆ (0[,)+∞)) |
| 7 | icossxr 13430 | . . . 4 ⊢ (0[,)+∞) ⊆ ℝ* | |
| 8 | 6, 7 | sstrdi 3946 | . . 3 ⊢ (𝜑 → ran 𝑆 ⊆ ℝ*) |
| 9 | supxrcl 13312 | . . 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 13031 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℝ) |
| 14 | 11, 13 | readdcld 11205 | . 2 ⊢ (𝜑 → ((vol*‘𝐸) + 𝐶) ∈ ℝ) |
| 15 | mnfxr 11233 | . . . 4 ⊢ -∞ ∈ ℝ* | |
| 16 | 15 | a1i 11 | . . 3 ⊢ (𝜑 → -∞ ∈ ℝ*) |
| 17 | 5 | ffnd 6687 | . . . . 5 ⊢ (𝜑 → 𝑆 Fn ℕ) |
| 18 | 1nn 12215 | . . . . 5 ⊢ 1 ∈ ℕ | |
| 19 | fnfvelrn 7056 | . . . . 5 ⊢ ((𝑆 Fn ℕ ∧ 1 ∈ ℕ) → (𝑆‘1) ∈ ran 𝑆) | |
| 20 | 17, 18, 19 | sylancl 595 | . . . 4 ⊢ (𝜑 → (𝑆‘1) ∈ ran 𝑆) |
| 21 | 8, 20 | sseldd 3935 | . . 3 ⊢ (𝜑 → (𝑆‘1) ∈ ℝ*) |
| 22 | rge0ssre 13454 | . . . . 5 ⊢ (0[,)+∞) ⊆ ℝ | |
| 23 | ffvelcdm 7057 | . . . . . 6 ⊢ ((𝑆:ℕ⟶(0[,)+∞) ∧ 1 ∈ ℕ) → (𝑆‘1) ∈ (0[,)+∞)) | |
| 24 | 5, 18, 23 | sylancl 595 | . . . . 5 ⊢ (𝜑 → (𝑆‘1) ∈ (0[,)+∞)) |
| 25 | 22, 24 | sselid 3932 | . . . 4 ⊢ (𝜑 → (𝑆‘1) ∈ ℝ) |
| 26 | 25 | mnfltd 13120 | . . 3 ⊢ (𝜑 → -∞ < (𝑆‘1)) |
| 27 | supxrub 13321 | . . . 4 ⊢ ((ran 𝑆 ⊆ ℝ* ∧ (𝑆‘1) ∈ ran 𝑆) → (𝑆‘1) ≤ sup(ran 𝑆, ℝ*, < )) | |
| 28 | 8, 20, 27 | syl2anc 593 | . . 3 ⊢ (𝜑 → (𝑆‘1) ≤ sup(ran 𝑆, ℝ*, < )) |
| 29 | 16, 21, 10, 26, 28 | xrltletrd 13157 | . 2 ⊢ (𝜑 → -∞ < sup(ran 𝑆, ℝ*, < )) |
| 30 | ioombl1.f3 | . 2 ⊢ (𝜑 → sup(ran 𝑆, ℝ*, < ) ≤ ((vol*‘𝐸) + 𝐶)) | |
| 31 | xrre 13166 | . 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 849 | 1 ⊢ (𝜑 → sup(ran 𝑆, ℝ*, < ) ∈ ℝ) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1559 ∈ wcel 2141 ∩ cin 3901 ⊆ wss 3902 ifcif 4477 〈cop 4585 ∪ cuni 4862 class class class wbr 5097 ↦ cmpt 5178 × cxp 5641 ran crn 5644 ∘ ccom 5647 Fn wfn 6511 ⟶wf 6512 ‘cfv 6516 (class class class)co 7391 1st c1st 7963 2nd c2nd 7964 supcsup 9380 ℝcr 11066 0cc0 11067 1c1 11068 + caddc 11070 +∞cpnf 11207 -∞cmnf 11208 ℝ*cxr 11209 < clt 11210 ≤ cle 11211 − cmin 11408 ℕcn 12204 ℝ+crp 12987 (,)cioo 13343 [,)cico 13345 seqcseq 14008 abscabs 15252 vol*covol 25512 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-sep 5243 ax-nul 5253 ax-pow 5319 ax-pr 5387 ax-un 7713 ax-cnex 11123 ax-resscn 11124 ax-1cn 11125 ax-icn 11126 ax-addcl 11127 ax-addrcl 11128 ax-mulcl 11129 ax-mulrcl 11130 ax-mulcom 11131 ax-addass 11132 ax-mulass 11133 ax-distr 11134 ax-i2m1 11135 ax-1ne0 11136 ax-1rid 11137 ax-rnegex 11138 ax-rrecex 11139 ax-cnre 11140 ax-pre-lttri 11141 ax-pre-lttrn 11142 ax-pre-ltadd 11143 ax-pre-mulgt0 11144 ax-pre-sup 11145 |
| This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4863 df-iun 4948 df-br 5098 df-opab 5160 df-mpt 5179 df-tr 5205 df-id 5538 df-eprel 5543 df-po 5551 df-so 5552 df-fr 5596 df-we 5598 df-xp 5649 df-rel 5650 df-cnv 5651 df-co 5652 df-dm 5653 df-rn 5654 df-res 5655 df-ima 5656 df-pred 6283 df-ord 6344 df-on 6345 df-lim 6346 df-suc 6347 df-iota 6472 df-fun 6518 df-fn 6519 df-f 6520 df-f1 6521 df-fo 6522 df-f1o 6523 df-fv 6524 df-riota 7348 df-ov 7394 df-oprab 7395 df-mpo 7396 df-om 7842 df-1st 7965 df-2nd 7966 df-frecs 8256 df-wrecs 8287 df-recs 8336 df-rdg 8375 df-er 8672 df-en 8922 df-dom 8923 df-sdom 8924 df-sup 9382 df-pnf 11212 df-mnf 11213 df-xr 11214 df-ltxr 11215 df-le 11216 df-sub 11410 df-neg 11411 df-div 11839 df-nn 12205 df-2 12274 df-3 12275 df-n0 12476 df-z 12563 df-uz 12834 df-rp 12988 df-ico 13349 df-fz 13507 df-seq 14009 df-exp 14069 df-cj 15117 df-re 15118 df-im 15119 df-sqrt 15253 df-abs 15254 |
| This theorem is referenced by: ioombl1lem4 25611 |
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