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Mirrors > Home > MPE Home > Th. List > ovolfsval | Structured version Visualization version GIF version |
Description: The value of the interval length function. (Contributed by Mario Carneiro, 16-Mar-2014.) |
Ref | Expression |
---|---|
ovolfs.1 | ⊢ 𝐺 = ((abs ∘ − ) ∘ 𝐹) |
Ref | Expression |
---|---|
ovolfsval | ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (𝐺‘𝑁) = ((2nd ‘(𝐹‘𝑁)) − (1st ‘(𝐹‘𝑁)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ovolfs.1 | . . . 4 ⊢ 𝐺 = ((abs ∘ − ) ∘ 𝐹) | |
2 | 1 | fveq1i 6505 | . . 3 ⊢ (𝐺‘𝑁) = (((abs ∘ − ) ∘ 𝐹)‘𝑁) |
3 | fvco3 6594 | . . 3 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐹)‘𝑁) = ((abs ∘ − )‘(𝐹‘𝑁))) | |
4 | 2, 3 | syl5eq 2828 | . 2 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (𝐺‘𝑁) = ((abs ∘ − )‘(𝐹‘𝑁))) |
5 | ffvelrn 6680 | . . . . . . 7 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (𝐹‘𝑁) ∈ ( ≤ ∩ (ℝ × ℝ))) | |
6 | 5 | elin2d 4067 | . . . . . 6 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (𝐹‘𝑁) ∈ (ℝ × ℝ)) |
7 | 1st2nd2 7546 | . . . . . 6 ⊢ ((𝐹‘𝑁) ∈ (ℝ × ℝ) → (𝐹‘𝑁) = 〈(1st ‘(𝐹‘𝑁)), (2nd ‘(𝐹‘𝑁))〉) | |
8 | 6, 7 | syl 17 | . . . . 5 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (𝐹‘𝑁) = 〈(1st ‘(𝐹‘𝑁)), (2nd ‘(𝐹‘𝑁))〉) |
9 | 8 | fveq2d 6508 | . . . 4 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → ((abs ∘ − )‘(𝐹‘𝑁)) = ((abs ∘ − )‘〈(1st ‘(𝐹‘𝑁)), (2nd ‘(𝐹‘𝑁))〉)) |
10 | df-ov 6985 | . . . 4 ⊢ ((1st ‘(𝐹‘𝑁))(abs ∘ − )(2nd ‘(𝐹‘𝑁))) = ((abs ∘ − )‘〈(1st ‘(𝐹‘𝑁)), (2nd ‘(𝐹‘𝑁))〉) | |
11 | 9, 10 | syl6eqr 2834 | . . 3 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → ((abs ∘ − )‘(𝐹‘𝑁)) = ((1st ‘(𝐹‘𝑁))(abs ∘ − )(2nd ‘(𝐹‘𝑁)))) |
12 | ovolfcl 23785 | . . . . . . 7 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → ((1st ‘(𝐹‘𝑁)) ∈ ℝ ∧ (2nd ‘(𝐹‘𝑁)) ∈ ℝ ∧ (1st ‘(𝐹‘𝑁)) ≤ (2nd ‘(𝐹‘𝑁)))) | |
13 | 12 | simp1d 1123 | . . . . . 6 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (1st ‘(𝐹‘𝑁)) ∈ ℝ) |
14 | 13 | recnd 10474 | . . . . 5 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (1st ‘(𝐹‘𝑁)) ∈ ℂ) |
15 | 12 | simp2d 1124 | . . . . . 6 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (2nd ‘(𝐹‘𝑁)) ∈ ℝ) |
16 | 15 | recnd 10474 | . . . . 5 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (2nd ‘(𝐹‘𝑁)) ∈ ℂ) |
17 | eqid 2780 | . . . . . 6 ⊢ (abs ∘ − ) = (abs ∘ − ) | |
18 | 17 | cnmetdval 23097 | . . . . 5 ⊢ (((1st ‘(𝐹‘𝑁)) ∈ ℂ ∧ (2nd ‘(𝐹‘𝑁)) ∈ ℂ) → ((1st ‘(𝐹‘𝑁))(abs ∘ − )(2nd ‘(𝐹‘𝑁))) = (abs‘((1st ‘(𝐹‘𝑁)) − (2nd ‘(𝐹‘𝑁))))) |
19 | 14, 16, 18 | syl2anc 576 | . . . 4 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → ((1st ‘(𝐹‘𝑁))(abs ∘ − )(2nd ‘(𝐹‘𝑁))) = (abs‘((1st ‘(𝐹‘𝑁)) − (2nd ‘(𝐹‘𝑁))))) |
20 | abssuble0 14555 | . . . . 5 ⊢ (((1st ‘(𝐹‘𝑁)) ∈ ℝ ∧ (2nd ‘(𝐹‘𝑁)) ∈ ℝ ∧ (1st ‘(𝐹‘𝑁)) ≤ (2nd ‘(𝐹‘𝑁))) → (abs‘((1st ‘(𝐹‘𝑁)) − (2nd ‘(𝐹‘𝑁)))) = ((2nd ‘(𝐹‘𝑁)) − (1st ‘(𝐹‘𝑁)))) | |
21 | 12, 20 | syl 17 | . . . 4 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (abs‘((1st ‘(𝐹‘𝑁)) − (2nd ‘(𝐹‘𝑁)))) = ((2nd ‘(𝐹‘𝑁)) − (1st ‘(𝐹‘𝑁)))) |
22 | 19, 21 | eqtrd 2816 | . . 3 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → ((1st ‘(𝐹‘𝑁))(abs ∘ − )(2nd ‘(𝐹‘𝑁))) = ((2nd ‘(𝐹‘𝑁)) − (1st ‘(𝐹‘𝑁)))) |
23 | 11, 22 | eqtrd 2816 | . 2 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → ((abs ∘ − )‘(𝐹‘𝑁)) = ((2nd ‘(𝐹‘𝑁)) − (1st ‘(𝐹‘𝑁)))) |
24 | 4, 23 | eqtrd 2816 | 1 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (𝐺‘𝑁) = ((2nd ‘(𝐹‘𝑁)) − (1st ‘(𝐹‘𝑁)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 ∧ w3a 1069 = wceq 1508 ∈ wcel 2051 ∩ cin 3830 〈cop 4450 class class class wbr 4934 × cxp 5409 ∘ ccom 5415 ⟶wf 6189 ‘cfv 6193 (class class class)co 6982 1st c1st 7505 2nd c2nd 7506 ℂcc 10339 ℝcr 10340 ≤ cle 10481 − cmin 10676 ℕcn 11445 abscabs 14460 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2752 ax-sep 5064 ax-nul 5071 ax-pow 5123 ax-pr 5190 ax-un 7285 ax-cnex 10397 ax-resscn 10398 ax-1cn 10399 ax-icn 10400 ax-addcl 10401 ax-addrcl 10402 ax-mulcl 10403 ax-mulrcl 10404 ax-mulcom 10405 ax-addass 10406 ax-mulass 10407 ax-distr 10408 ax-i2m1 10409 ax-1ne0 10410 ax-1rid 10411 ax-rnegex 10412 ax-rrecex 10413 ax-cnre 10414 ax-pre-lttri 10415 ax-pre-lttrn 10416 ax-pre-ltadd 10417 ax-pre-mulgt0 10418 ax-pre-sup 10419 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2551 df-eu 2589 df-clab 2761 df-cleq 2773 df-clel 2848 df-nfc 2920 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3419 df-sbc 3684 df-csb 3789 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-pss 3847 df-nul 4182 df-if 4354 df-pw 4427 df-sn 4445 df-pr 4447 df-tp 4449 df-op 4451 df-uni 4718 df-iun 4799 df-br 4935 df-opab 4997 df-mpt 5014 df-tr 5036 df-id 5316 df-eprel 5321 df-po 5330 df-so 5331 df-fr 5370 df-we 5372 df-xp 5417 df-rel 5418 df-cnv 5419 df-co 5420 df-dm 5421 df-rn 5422 df-res 5423 df-ima 5424 df-pred 5991 df-ord 6037 df-on 6038 df-lim 6039 df-suc 6040 df-iota 6157 df-fun 6195 df-fn 6196 df-f 6197 df-f1 6198 df-fo 6199 df-f1o 6200 df-fv 6201 df-riota 6943 df-ov 6985 df-oprab 6986 df-mpo 6987 df-om 7403 df-1st 7507 df-2nd 7508 df-wrecs 7756 df-recs 7818 df-rdg 7856 df-er 8095 df-en 8313 df-dom 8314 df-sdom 8315 df-sup 8707 df-pnf 10482 df-mnf 10483 df-xr 10484 df-ltxr 10485 df-le 10486 df-sub 10678 df-neg 10679 df-div 11105 df-nn 11446 df-2 11509 df-3 11510 df-n0 11714 df-z 11800 df-uz 12065 df-rp 12211 df-seq 13191 df-exp 13251 df-cj 14325 df-re 14326 df-im 14327 df-sqrt 14461 df-abs 14462 |
This theorem is referenced by: ovolfsf 23790 ovollb2lem 23807 ovolunlem1a 23815 ovoliunlem1 23821 ovolshftlem1 23828 ovolscalem1 23832 ovolicc1 23835 ovolicc2lem4 23839 ioombl1lem3 23879 ovolfs2 23890 uniioovol 23898 uniioombllem3 23904 |
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