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Mirrors > Home > MPE Home > Th. List > ioombl1lem2 | Structured version Visualization version GIF version |
Description: Lemma for ioombl1 24165. (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 2823 | . . . . . . 7 ⊢ ((abs ∘ − ) ∘ 𝐹) = ((abs ∘ − ) ∘ 𝐹) | |
3 | ioombl1.s | . . . . . . 7 ⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹)) | |
4 | 2, 3 | ovolsf 24075 | . . . . . 6 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑆:ℕ⟶(0[,)+∞)) |
5 | 1, 4 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑆:ℕ⟶(0[,)+∞)) |
6 | 5 | frnd 6523 | . . . 4 ⊢ (𝜑 → ran 𝑆 ⊆ (0[,)+∞)) |
7 | icossxr 12824 | . . . 4 ⊢ (0[,)+∞) ⊆ ℝ* | |
8 | 6, 7 | sstrdi 3981 | . . 3 ⊢ (𝜑 → ran 𝑆 ⊆ ℝ*) |
9 | supxrcl 12711 | . . 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 12434 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℝ) |
14 | 11, 13 | readdcld 10672 | . 2 ⊢ (𝜑 → ((vol*‘𝐸) + 𝐶) ∈ ℝ) |
15 | mnfxr 10700 | . . . 4 ⊢ -∞ ∈ ℝ* | |
16 | 15 | a1i 11 | . . 3 ⊢ (𝜑 → -∞ ∈ ℝ*) |
17 | 5 | ffnd 6517 | . . . . 5 ⊢ (𝜑 → 𝑆 Fn ℕ) |
18 | 1nn 11651 | . . . . 5 ⊢ 1 ∈ ℕ | |
19 | fnfvelrn 6850 | . . . . 5 ⊢ ((𝑆 Fn ℕ ∧ 1 ∈ ℕ) → (𝑆‘1) ∈ ran 𝑆) | |
20 | 17, 18, 19 | sylancl 588 | . . . 4 ⊢ (𝜑 → (𝑆‘1) ∈ ran 𝑆) |
21 | 8, 20 | sseldd 3970 | . . 3 ⊢ (𝜑 → (𝑆‘1) ∈ ℝ*) |
22 | rge0ssre 12847 | . . . . 5 ⊢ (0[,)+∞) ⊆ ℝ | |
23 | ffvelrn 6851 | . . . . . 6 ⊢ ((𝑆:ℕ⟶(0[,)+∞) ∧ 1 ∈ ℕ) → (𝑆‘1) ∈ (0[,)+∞)) | |
24 | 5, 18, 23 | sylancl 588 | . . . . 5 ⊢ (𝜑 → (𝑆‘1) ∈ (0[,)+∞)) |
25 | 22, 24 | sseldi 3967 | . . . 4 ⊢ (𝜑 → (𝑆‘1) ∈ ℝ) |
26 | 25 | mnfltd 12522 | . . 3 ⊢ (𝜑 → -∞ < (𝑆‘1)) |
27 | supxrub 12720 | . . . 4 ⊢ ((ran 𝑆 ⊆ ℝ* ∧ (𝑆‘1) ∈ ran 𝑆) → (𝑆‘1) ≤ sup(ran 𝑆, ℝ*, < )) | |
28 | 8, 20, 27 | syl2anc 586 | . . 3 ⊢ (𝜑 → (𝑆‘1) ≤ sup(ran 𝑆, ℝ*, < )) |
29 | 16, 21, 10, 26, 28 | xrltletrd 12557 | . 2 ⊢ (𝜑 → -∞ < sup(ran 𝑆, ℝ*, < )) |
30 | ioombl1.f3 | . 2 ⊢ (𝜑 → sup(ran 𝑆, ℝ*, < ) ≤ ((vol*‘𝐸) + 𝐶)) | |
31 | xrre 12565 | . 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 1537 ∈ wcel 2114 ∩ cin 3937 ⊆ wss 3938 ifcif 4469 〈cop 4575 ∪ cuni 4840 class class class wbr 5068 ↦ cmpt 5148 × cxp 5555 ran crn 5558 ∘ ccom 5561 Fn wfn 6352 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 1st c1st 7689 2nd c2nd 7690 supcsup 8906 ℝcr 10538 0cc0 10539 1c1 10540 + caddc 10542 +∞cpnf 10674 -∞cmnf 10675 ℝ*cxr 10676 < clt 10677 ≤ cle 10678 − cmin 10872 ℕcn 11640 ℝ+crp 12392 (,)cioo 12741 [,)cico 12743 seqcseq 13372 abscabs 14595 vol*covol 24065 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-sup 8908 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-n0 11901 df-z 11985 df-uz 12247 df-rp 12393 df-ico 12747 df-fz 12896 df-seq 13373 df-exp 13433 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 |
This theorem is referenced by: ioombl1lem4 24164 |
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