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Mirrors > Home > MPE Home > Th. List > ovolfs2 | Structured version Visualization version GIF version |
Description: Alternative expression for the interval length function. (Contributed by Mario Carneiro, 26-Mar-2015.) |
Ref | Expression |
---|---|
ovolfs2.1 | ⊢ 𝐺 = ((abs ∘ − ) ∘ 𝐹) |
Ref | Expression |
---|---|
ovolfs2 | ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝐺 = ((vol* ∘ (,)) ∘ 𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ovolfcl 24630 | . . . . 5 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐹‘𝑛)) ∈ ℝ ∧ (2nd ‘(𝐹‘𝑛)) ∈ ℝ ∧ (1st ‘(𝐹‘𝑛)) ≤ (2nd ‘(𝐹‘𝑛)))) | |
2 | ovolioo 24732 | . . . . 5 ⊢ (((1st ‘(𝐹‘𝑛)) ∈ ℝ ∧ (2nd ‘(𝐹‘𝑛)) ∈ ℝ ∧ (1st ‘(𝐹‘𝑛)) ≤ (2nd ‘(𝐹‘𝑛))) → (vol*‘((1st ‘(𝐹‘𝑛))(,)(2nd ‘(𝐹‘𝑛)))) = ((2nd ‘(𝐹‘𝑛)) − (1st ‘(𝐹‘𝑛)))) | |
3 | 1, 2 | syl 17 | . . . 4 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (vol*‘((1st ‘(𝐹‘𝑛))(,)(2nd ‘(𝐹‘𝑛)))) = ((2nd ‘(𝐹‘𝑛)) − (1st ‘(𝐹‘𝑛)))) |
4 | inss2 4163 | . . . . . . . . . 10 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ) | |
5 | rexpssxrxp 11020 | . . . . . . . . . 10 ⊢ (ℝ × ℝ) ⊆ (ℝ* × ℝ*) | |
6 | 4, 5 | sstri 3930 | . . . . . . . . 9 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*) |
7 | ffvelrn 6959 | . . . . . . . . 9 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ ( ≤ ∩ (ℝ × ℝ))) | |
8 | 6, 7 | sselid 3919 | . . . . . . . 8 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ (ℝ* × ℝ*)) |
9 | 1st2nd2 7870 | . . . . . . . 8 ⊢ ((𝐹‘𝑛) ∈ (ℝ* × ℝ*) → (𝐹‘𝑛) = 〈(1st ‘(𝐹‘𝑛)), (2nd ‘(𝐹‘𝑛))〉) | |
10 | 8, 9 | syl 17 | . . . . . . 7 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) = 〈(1st ‘(𝐹‘𝑛)), (2nd ‘(𝐹‘𝑛))〉) |
11 | 10 | fveq2d 6778 | . . . . . 6 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → ((,)‘(𝐹‘𝑛)) = ((,)‘〈(1st ‘(𝐹‘𝑛)), (2nd ‘(𝐹‘𝑛))〉)) |
12 | df-ov 7278 | . . . . . 6 ⊢ ((1st ‘(𝐹‘𝑛))(,)(2nd ‘(𝐹‘𝑛))) = ((,)‘〈(1st ‘(𝐹‘𝑛)), (2nd ‘(𝐹‘𝑛))〉) | |
13 | 11, 12 | eqtr4di 2796 | . . . . 5 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → ((,)‘(𝐹‘𝑛)) = ((1st ‘(𝐹‘𝑛))(,)(2nd ‘(𝐹‘𝑛)))) |
14 | 13 | fveq2d 6778 | . . . 4 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (vol*‘((,)‘(𝐹‘𝑛))) = (vol*‘((1st ‘(𝐹‘𝑛))(,)(2nd ‘(𝐹‘𝑛))))) |
15 | ovolfs2.1 | . . . . 5 ⊢ 𝐺 = ((abs ∘ − ) ∘ 𝐹) | |
16 | 15 | ovolfsval 24634 | . . . 4 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (𝐺‘𝑛) = ((2nd ‘(𝐹‘𝑛)) − (1st ‘(𝐹‘𝑛)))) |
17 | 3, 14, 16 | 3eqtr4rd 2789 | . . 3 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (𝐺‘𝑛) = (vol*‘((,)‘(𝐹‘𝑛)))) |
18 | 17 | mpteq2dva 5174 | . 2 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → (𝑛 ∈ ℕ ↦ (𝐺‘𝑛)) = (𝑛 ∈ ℕ ↦ (vol*‘((,)‘(𝐹‘𝑛))))) |
19 | 15 | ovolfsf 24635 | . . 3 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝐺:ℕ⟶(0[,)+∞)) |
20 | 19 | feqmptd 6837 | . 2 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝐺 = (𝑛 ∈ ℕ ↦ (𝐺‘𝑛))) |
21 | id 22 | . . . 4 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) | |
22 | 21 | feqmptd 6837 | . . 3 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝐹 = (𝑛 ∈ ℕ ↦ (𝐹‘𝑛))) |
23 | ioof 13179 | . . . . . 6 ⊢ (,):(ℝ* × ℝ*)⟶𝒫 ℝ | |
24 | 23 | a1i 11 | . . . . 5 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → (,):(ℝ* × ℝ*)⟶𝒫 ℝ) |
25 | 24 | ffvelrnda 6961 | . . . 4 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ (ℝ* × ℝ*)) → ((,)‘𝑥) ∈ 𝒫 ℝ) |
26 | 24 | feqmptd 6837 | . . . 4 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → (,) = (𝑥 ∈ (ℝ* × ℝ*) ↦ ((,)‘𝑥))) |
27 | ovolf 24646 | . . . . . 6 ⊢ vol*:𝒫 ℝ⟶(0[,]+∞) | |
28 | 27 | a1i 11 | . . . . 5 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → vol*:𝒫 ℝ⟶(0[,]+∞)) |
29 | 28 | feqmptd 6837 | . . . 4 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → vol* = (𝑦 ∈ 𝒫 ℝ ↦ (vol*‘𝑦))) |
30 | fveq2 6774 | . . . 4 ⊢ (𝑦 = ((,)‘𝑥) → (vol*‘𝑦) = (vol*‘((,)‘𝑥))) | |
31 | 25, 26, 29, 30 | fmptco 7001 | . . 3 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → (vol* ∘ (,)) = (𝑥 ∈ (ℝ* × ℝ*) ↦ (vol*‘((,)‘𝑥)))) |
32 | 2fveq3 6779 | . . 3 ⊢ (𝑥 = (𝐹‘𝑛) → (vol*‘((,)‘𝑥)) = (vol*‘((,)‘(𝐹‘𝑛)))) | |
33 | 8, 22, 31, 32 | fmptco 7001 | . 2 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ((vol* ∘ (,)) ∘ 𝐹) = (𝑛 ∈ ℕ ↦ (vol*‘((,)‘(𝐹‘𝑛))))) |
34 | 18, 20, 33 | 3eqtr4d 2788 | 1 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝐺 = ((vol* ∘ (,)) ∘ 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ∩ cin 3886 𝒫 cpw 4533 〈cop 4567 class class class wbr 5074 ↦ cmpt 5157 × cxp 5587 ∘ ccom 5593 ⟶wf 6429 ‘cfv 6433 (class class class)co 7275 1st c1st 7829 2nd c2nd 7830 ℝcr 10870 0cc0 10871 +∞cpnf 11006 ℝ*cxr 11008 ≤ cle 11010 − cmin 11205 ℕcn 11973 (,)cioo 13079 [,)cico 13081 [,]cicc 13082 abscabs 14945 vol*covol 24626 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-inf2 9399 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-2o 8298 df-er 8498 df-map 8617 df-pm 8618 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fi 9170 df-sup 9201 df-inf 9202 df-oi 9269 df-dju 9659 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-n0 12234 df-z 12320 df-uz 12583 df-q 12689 df-rp 12731 df-xneg 12848 df-xadd 12849 df-xmul 12850 df-ioo 13083 df-ico 13085 df-icc 13086 df-fz 13240 df-fzo 13383 df-fl 13512 df-seq 13722 df-exp 13783 df-hash 14045 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-clim 15197 df-rlim 15198 df-sum 15398 df-rest 17133 df-topgen 17154 df-psmet 20589 df-xmet 20590 df-met 20591 df-bl 20592 df-mopn 20593 df-top 22043 df-topon 22060 df-bases 22096 df-cmp 22538 df-ovol 24628 df-vol 24629 |
This theorem is referenced by: uniioombllem2 24747 |
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