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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 24218 | . . . . 5 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐹‘𝑛)) ∈ ℝ ∧ (2nd ‘(𝐹‘𝑛)) ∈ ℝ ∧ (1st ‘(𝐹‘𝑛)) ≤ (2nd ‘(𝐹‘𝑛)))) | |
2 | ovolioo 24320 | . . . . 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 4120 | . . . . . . . . . 10 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ) | |
5 | rexpssxrxp 10764 | . . . . . . . . . 10 ⊢ (ℝ × ℝ) ⊆ (ℝ* × ℝ*) | |
6 | 4, 5 | sstri 3886 | . . . . . . . . 9 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*) |
7 | ffvelrn 6859 | . . . . . . . . 9 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ ( ≤ ∩ (ℝ × ℝ))) | |
8 | 6, 7 | sseldi 3875 | . . . . . . . 8 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ (ℝ* × ℝ*)) |
9 | 1st2nd2 7753 | . . . . . . . 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 6678 | . . . . . 6 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → ((,)‘(𝐹‘𝑛)) = ((,)‘〈(1st ‘(𝐹‘𝑛)), (2nd ‘(𝐹‘𝑛))〉)) |
12 | df-ov 7173 | . . . . . 6 ⊢ ((1st ‘(𝐹‘𝑛))(,)(2nd ‘(𝐹‘𝑛))) = ((,)‘〈(1st ‘(𝐹‘𝑛)), (2nd ‘(𝐹‘𝑛))〉) | |
13 | 11, 12 | eqtr4di 2791 | . . . . 5 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → ((,)‘(𝐹‘𝑛)) = ((1st ‘(𝐹‘𝑛))(,)(2nd ‘(𝐹‘𝑛)))) |
14 | 13 | fveq2d 6678 | . . . 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 24222 | . . . 4 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (𝐺‘𝑛) = ((2nd ‘(𝐹‘𝑛)) − (1st ‘(𝐹‘𝑛)))) |
17 | 3, 14, 16 | 3eqtr4rd 2784 | . . 3 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (𝐺‘𝑛) = (vol*‘((,)‘(𝐹‘𝑛)))) |
18 | 17 | mpteq2dva 5125 | . 2 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → (𝑛 ∈ ℕ ↦ (𝐺‘𝑛)) = (𝑛 ∈ ℕ ↦ (vol*‘((,)‘(𝐹‘𝑛))))) |
19 | 15 | ovolfsf 24223 | . . 3 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝐺:ℕ⟶(0[,)+∞)) |
20 | 19 | feqmptd 6737 | . 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 6737 | . . 3 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝐹 = (𝑛 ∈ ℕ ↦ (𝐹‘𝑛))) |
23 | ioof 12921 | . . . . . 6 ⊢ (,):(ℝ* × ℝ*)⟶𝒫 ℝ | |
24 | 23 | a1i 11 | . . . . 5 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → (,):(ℝ* × ℝ*)⟶𝒫 ℝ) |
25 | 24 | ffvelrnda 6861 | . . . 4 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ (ℝ* × ℝ*)) → ((,)‘𝑥) ∈ 𝒫 ℝ) |
26 | 24 | feqmptd 6737 | . . . 4 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → (,) = (𝑥 ∈ (ℝ* × ℝ*) ↦ ((,)‘𝑥))) |
27 | ovolf 24234 | . . . . . 6 ⊢ vol*:𝒫 ℝ⟶(0[,]+∞) | |
28 | 27 | a1i 11 | . . . . 5 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → vol*:𝒫 ℝ⟶(0[,]+∞)) |
29 | 28 | feqmptd 6737 | . . . 4 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → vol* = (𝑦 ∈ 𝒫 ℝ ↦ (vol*‘𝑦))) |
30 | fveq2 6674 | . . . 4 ⊢ (𝑦 = ((,)‘𝑥) → (vol*‘𝑦) = (vol*‘((,)‘𝑥))) | |
31 | 25, 26, 29, 30 | fmptco 6901 | . . 3 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → (vol* ∘ (,)) = (𝑥 ∈ (ℝ* × ℝ*) ↦ (vol*‘((,)‘𝑥)))) |
32 | 2fveq3 6679 | . . 3 ⊢ (𝑥 = (𝐹‘𝑛) → (vol*‘((,)‘𝑥)) = (vol*‘((,)‘(𝐹‘𝑛)))) | |
33 | 8, 22, 31, 32 | fmptco 6901 | . 2 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ((vol* ∘ (,)) ∘ 𝐹) = (𝑛 ∈ ℕ ↦ (vol*‘((,)‘(𝐹‘𝑛))))) |
34 | 18, 20, 33 | 3eqtr4d 2783 | 1 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝐺 = ((vol* ∘ (,)) ∘ 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1088 = wceq 1542 ∈ wcel 2114 ∩ cin 3842 𝒫 cpw 4488 〈cop 4522 class class class wbr 5030 ↦ cmpt 5110 × cxp 5523 ∘ ccom 5529 ⟶wf 6335 ‘cfv 6339 (class class class)co 7170 1st c1st 7712 2nd c2nd 7713 ℝcr 10614 0cc0 10615 +∞cpnf 10750 ℝ*cxr 10752 ≤ cle 10754 − cmin 10948 ℕcn 11716 (,)cioo 12821 [,)cico 12823 [,]cicc 12824 abscabs 14683 vol*covol 24214 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-inf2 9177 ax-cnex 10671 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692 ax-pre-sup 10693 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-int 4837 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-se 5484 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-isom 6348 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-of 7425 df-om 7600 df-1st 7714 df-2nd 7715 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-1o 8131 df-2o 8132 df-er 8320 df-map 8439 df-pm 8440 df-en 8556 df-dom 8557 df-sdom 8558 df-fin 8559 df-fi 8948 df-sup 8979 df-inf 8980 df-oi 9047 df-dju 9403 df-card 9441 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-div 11376 df-nn 11717 df-2 11779 df-3 11780 df-n0 11977 df-z 12063 df-uz 12325 df-q 12431 df-rp 12473 df-xneg 12590 df-xadd 12591 df-xmul 12592 df-ioo 12825 df-ico 12827 df-icc 12828 df-fz 12982 df-fzo 13125 df-fl 13253 df-seq 13461 df-exp 13522 df-hash 13783 df-cj 14548 df-re 14549 df-im 14550 df-sqrt 14684 df-abs 14685 df-clim 14935 df-rlim 14936 df-sum 15136 df-rest 16799 df-topgen 16820 df-psmet 20209 df-xmet 20210 df-met 20211 df-bl 20212 df-mopn 20213 df-top 21645 df-topon 21662 df-bases 21697 df-cmp 22138 df-ovol 24216 df-vol 24217 |
This theorem is referenced by: uniioombllem2 24335 |
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