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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 6902 | . . 3 ⊢ (𝐺‘𝑁) = (((abs ∘ − ) ∘ 𝐹)‘𝑁) |
3 | fvco3 7001 | . . 3 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐹)‘𝑁) = ((abs ∘ − )‘(𝐹‘𝑁))) | |
4 | 2, 3 | eqtrid 2778 | . 2 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (𝐺‘𝑁) = ((abs ∘ − )‘(𝐹‘𝑁))) |
5 | ffvelcdm 7095 | . . . . . . 7 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (𝐹‘𝑁) ∈ ( ≤ ∩ (ℝ × ℝ))) | |
6 | 5 | elin2d 4200 | . . . . . 6 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (𝐹‘𝑁) ∈ (ℝ × ℝ)) |
7 | 1st2nd2 8042 | . . . . . 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 6905 | . . . 4 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → ((abs ∘ − )‘(𝐹‘𝑁)) = ((abs ∘ − )‘〈(1st ‘(𝐹‘𝑁)), (2nd ‘(𝐹‘𝑁))〉)) |
10 | df-ov 7427 | . . . 4 ⊢ ((1st ‘(𝐹‘𝑁))(abs ∘ − )(2nd ‘(𝐹‘𝑁))) = ((abs ∘ − )‘〈(1st ‘(𝐹‘𝑁)), (2nd ‘(𝐹‘𝑁))〉) | |
11 | 9, 10 | eqtr4di 2784 | . . 3 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → ((abs ∘ − )‘(𝐹‘𝑁)) = ((1st ‘(𝐹‘𝑁))(abs ∘ − )(2nd ‘(𝐹‘𝑁)))) |
12 | ovolfcl 25486 | . . . . . . 7 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → ((1st ‘(𝐹‘𝑁)) ∈ ℝ ∧ (2nd ‘(𝐹‘𝑁)) ∈ ℝ ∧ (1st ‘(𝐹‘𝑁)) ≤ (2nd ‘(𝐹‘𝑁)))) | |
13 | 12 | simp1d 1139 | . . . . . 6 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (1st ‘(𝐹‘𝑁)) ∈ ℝ) |
14 | 13 | recnd 11292 | . . . . 5 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (1st ‘(𝐹‘𝑁)) ∈ ℂ) |
15 | 12 | simp2d 1140 | . . . . . 6 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (2nd ‘(𝐹‘𝑁)) ∈ ℝ) |
16 | 15 | recnd 11292 | . . . . 5 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (2nd ‘(𝐹‘𝑁)) ∈ ℂ) |
17 | eqid 2726 | . . . . . 6 ⊢ (abs ∘ − ) = (abs ∘ − ) | |
18 | 17 | cnmetdval 24778 | . . . . 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 582 | . . . 4 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → ((1st ‘(𝐹‘𝑁))(abs ∘ − )(2nd ‘(𝐹‘𝑁))) = (abs‘((1st ‘(𝐹‘𝑁)) − (2nd ‘(𝐹‘𝑁))))) |
20 | abssuble0 15333 | . . . . 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 2766 | . . 3 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → ((1st ‘(𝐹‘𝑁))(abs ∘ − )(2nd ‘(𝐹‘𝑁))) = ((2nd ‘(𝐹‘𝑁)) − (1st ‘(𝐹‘𝑁)))) |
23 | 11, 22 | eqtrd 2766 | . 2 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → ((abs ∘ − )‘(𝐹‘𝑁)) = ((2nd ‘(𝐹‘𝑁)) − (1st ‘(𝐹‘𝑁)))) |
24 | 4, 23 | eqtrd 2766 | 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 394 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 ∩ cin 3946 〈cop 4639 class class class wbr 5153 × cxp 5680 ∘ ccom 5686 ⟶wf 6550 ‘cfv 6554 (class class class)co 7424 1st c1st 8001 2nd c2nd 8002 ℂcc 11156 ℝcr 11157 ≤ cle 11299 − cmin 11494 ℕcn 12264 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-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-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: ovolfsf 25491 ovollb2lem 25508 ovolunlem1a 25516 ovoliunlem1 25522 ovolshftlem1 25529 ovolscalem1 25533 ovolicc1 25536 ovolicc2lem4 25540 ioombl1lem3 25580 ovolfs2 25591 uniioovol 25599 uniioombllem3 25605 |
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