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Mirrors > Home > MPE Home > Th. List > elovolm | Structured version Visualization version GIF version |
Description: Elementhood in the set 𝑀 of approximations to the outer measure. (Contributed by Mario Carneiro, 16-Mar-2014.) |
Ref | Expression |
---|---|
elovolm.1 | ⊢ 𝑀 = {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))} |
Ref | Expression |
---|---|
elovolm | ⊢ (𝐵 ∈ 𝑀 ↔ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝐵 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqeq1 2730 | . . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ↔ 𝐵 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))) | |
2 | 1 | anbi2d 628 | . . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < )) ↔ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝐵 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < )))) |
3 | 2 | rexbidv 3169 | . . 3 ⊢ (𝑦 = 𝐵 → (∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < )) ↔ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝐵 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < )))) |
4 | elovolm.1 | . . 3 ⊢ 𝑀 = {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))} | |
5 | 3, 4 | elrab2 3684 | . 2 ⊢ (𝐵 ∈ 𝑀 ↔ (𝐵 ∈ ℝ* ∧ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝐵 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < )))) |
6 | elovolmlem 25494 | . . . . . . . . . 10 ⊢ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ↔ 𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ))) | |
7 | eqid 2726 | . . . . . . . . . . 11 ⊢ ((abs ∘ − ) ∘ 𝑓) = ((abs ∘ − ) ∘ 𝑓) | |
8 | eqid 2726 | . . . . . . . . . . 11 ⊢ seq1( + , ((abs ∘ − ) ∘ 𝑓)) = seq1( + , ((abs ∘ − ) ∘ 𝑓)) | |
9 | 7, 8 | ovolsf 25492 | . . . . . . . . . 10 ⊢ (𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → seq1( + , ((abs ∘ − ) ∘ 𝑓)):ℕ⟶(0[,)+∞)) |
10 | 6, 9 | sylbi 216 | . . . . . . . . 9 ⊢ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) → seq1( + , ((abs ∘ − ) ∘ 𝑓)):ℕ⟶(0[,)+∞)) |
11 | icossxr 13463 | . . . . . . . . 9 ⊢ (0[,)+∞) ⊆ ℝ* | |
12 | fss 6744 | . . . . . . . . 9 ⊢ ((seq1( + , ((abs ∘ − ) ∘ 𝑓)):ℕ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ ℝ*) → seq1( + , ((abs ∘ − ) ∘ 𝑓)):ℕ⟶ℝ*) | |
13 | 10, 11, 12 | sylancl 584 | . . . . . . . 8 ⊢ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) → seq1( + , ((abs ∘ − ) ∘ 𝑓)):ℕ⟶ℝ*) |
14 | frn 6735 | . . . . . . . 8 ⊢ (seq1( + , ((abs ∘ − ) ∘ 𝑓)):ℕ⟶ℝ* → ran seq1( + , ((abs ∘ − ) ∘ 𝑓)) ⊆ ℝ*) | |
15 | supxrcl 13348 | . . . . . . . 8 ⊢ (ran seq1( + , ((abs ∘ − ) ∘ 𝑓)) ⊆ ℝ* → sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ∈ ℝ*) | |
16 | 13, 14, 15 | 3syl 18 | . . . . . . 7 ⊢ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) → sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ∈ ℝ*) |
17 | eleq1 2814 | . . . . . . 7 ⊢ (𝐵 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) → (𝐵 ∈ ℝ* ↔ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ∈ ℝ*)) | |
18 | 16, 17 | syl5ibrcom 246 | . . . . . 6 ⊢ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) → (𝐵 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) → 𝐵 ∈ ℝ*)) |
19 | 18 | imp 405 | . . . . 5 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ 𝐵 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < )) → 𝐵 ∈ ℝ*) |
20 | 19 | adantrl 714 | . . . 4 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝐵 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))) → 𝐵 ∈ ℝ*) |
21 | 20 | rexlimiva 3137 | . . 3 ⊢ (∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝐵 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < )) → 𝐵 ∈ ℝ*) |
22 | 21 | pm4.71ri 559 | . 2 ⊢ (∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝐵 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < )) ↔ (𝐵 ∈ ℝ* ∧ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝐵 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < )))) |
23 | 5, 22 | bitr4i 277 | 1 ⊢ (𝐵 ∈ 𝑀 ↔ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝐵 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ∃wrex 3060 {crab 3419 ∩ cin 3946 ⊆ wss 3947 ∪ cuni 4913 × cxp 5680 ran crn 5683 ∘ ccom 5686 ⟶wf 6550 (class class class)co 7424 ↑m cmap 8855 supcsup 9483 ℝcr 11157 0cc0 11158 1c1 11159 + caddc 11161 +∞cpnf 11295 ℝ*cxr 11297 < clt 11298 ≤ cle 11299 − cmin 11494 ℕcn 12264 (,)cioo 13378 [,)cico 13380 seqcseq 14021 abscabs 15239 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 ax-pre-sup 11236 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-1st 8003 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-er 8734 df-map 8857 df-en 8975 df-dom 8976 df-sdom 8977 df-sup 9485 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-div 11922 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-z 12611 df-uz 12875 df-rp 13029 df-ico 13384 df-fz 13539 df-seq 14022 df-exp 14082 df-cj 15104 df-re 15105 df-im 15106 df-sqrt 15240 df-abs 15241 |
This theorem is referenced by: elovolmr 25496 ovolmge0 25497 ovolicc2 25542 |
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