![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > ovolsf | Structured version Visualization version GIF version |
Description: Closure for the partial sums of the interval length function. (Contributed by Mario Carneiro, 16-Mar-2014.) |
Ref | Expression |
---|---|
ovolfs.1 | ⊢ 𝐺 = ((abs ∘ − ) ∘ 𝐹) |
ovolfs.2 | ⊢ 𝑆 = seq1( + , 𝐺) |
Ref | Expression |
---|---|
ovolsf | ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑆:ℕ⟶(0[,)+∞)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnuz 12919 | . . 3 ⊢ ℕ = (ℤ≥‘1) | |
2 | 1zzd 12647 | . . 3 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 1 ∈ ℤ) | |
3 | ovolfs.1 | . . . . 5 ⊢ 𝐺 = ((abs ∘ − ) ∘ 𝐹) | |
4 | 3 | ovolfsf 25494 | . . . 4 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝐺:ℕ⟶(0[,)+∞)) |
5 | 4 | ffvelcdmda 7100 | . . 3 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (𝐺‘𝑥) ∈ (0[,)+∞)) |
6 | ge0addcl 13493 | . . . 4 ⊢ ((𝑥 ∈ (0[,)+∞) ∧ 𝑦 ∈ (0[,)+∞)) → (𝑥 + 𝑦) ∈ (0[,)+∞)) | |
7 | 6 | adantl 480 | . . 3 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑦 ∈ (0[,)+∞))) → (𝑥 + 𝑦) ∈ (0[,)+∞)) |
8 | 1, 2, 5, 7 | seqf 14045 | . 2 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → seq1( + , 𝐺):ℕ⟶(0[,)+∞)) |
9 | ovolfs.2 | . . 3 ⊢ 𝑆 = seq1( + , 𝐺) | |
10 | 9 | feq1i 6721 | . 2 ⊢ (𝑆:ℕ⟶(0[,)+∞) ↔ seq1( + , 𝐺):ℕ⟶(0[,)+∞)) |
11 | 8, 10 | sylibr 233 | 1 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑆:ℕ⟶(0[,)+∞)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ∩ cin 3946 × cxp 5682 ∘ ccom 5688 ⟶wf 6552 (class class class)co 7426 ℝcr 11159 0cc0 11160 1c1 11161 + caddc 11163 +∞cpnf 11297 ≤ cle 11301 − cmin 11496 ℕcn 12266 [,)cico 13382 seqcseq 14023 abscabs 15241 |
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 5306 ax-nul 5313 ax-pow 5371 ax-pr 5435 ax-un 7748 ax-cnex 11216 ax-resscn 11217 ax-1cn 11218 ax-icn 11219 ax-addcl 11220 ax-addrcl 11221 ax-mulcl 11222 ax-mulrcl 11223 ax-mulcom 11224 ax-addass 11225 ax-mulass 11226 ax-distr 11227 ax-i2m1 11228 ax-1ne0 11229 ax-1rid 11230 ax-rnegex 11231 ax-rrecex 11232 ax-cnre 11233 ax-pre-lttri 11234 ax-pre-lttrn 11235 ax-pre-ltadd 11236 ax-pre-mulgt0 11237 ax-pre-sup 11238 |
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 4916 df-iun 5005 df-br 5156 df-opab 5218 df-mpt 5239 df-tr 5273 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5639 df-we 5641 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6314 df-ord 6381 df-on 6382 df-lim 6383 df-suc 6384 df-iota 6508 df-fun 6558 df-fn 6559 df-f 6560 df-f1 6561 df-fo 6562 df-f1o 6563 df-fv 6564 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7879 df-1st 8005 df-2nd 8006 df-frecs 8298 df-wrecs 8329 df-recs 8403 df-rdg 8442 df-er 8736 df-en 8977 df-dom 8978 df-sdom 8979 df-sup 9487 df-pnf 11302 df-mnf 11303 df-xr 11304 df-ltxr 11305 df-le 11306 df-sub 11498 df-neg 11499 df-div 11924 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-z 12613 df-uz 12877 df-rp 13031 df-ico 13386 df-fz 13541 df-seq 14024 df-exp 14084 df-cj 15106 df-re 15107 df-im 15108 df-sqrt 15242 df-abs 15243 |
This theorem is referenced by: elovolm 25498 ovolmge0 25500 ovolgelb 25503 ovollb2lem 25511 ovollb2 25512 ovolunlem1a 25519 ovolunlem1 25520 ovoliunlem1 25525 ovoliunlem2 25526 ovolscalem1 25536 ovolicc1 25539 ovolicc2lem4 25543 ioombl1lem2 25582 ioombl1lem4 25584 uniioovol 25602 uniiccvol 25603 uniioombllem1 25604 uniioombllem2 25606 uniioombllem3 25608 uniioombllem6 25611 mblfinlem3 37362 mblfinlem4 37363 ismblfin 37364 |
Copyright terms: Public domain | W3C validator |