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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 6664 | . . 3 ⊢ (𝐺‘𝑁) = (((abs ∘ − ) ∘ 𝐹)‘𝑁) |
3 | fvco3 6753 | . . 3 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐹)‘𝑁) = ((abs ∘ − )‘(𝐹‘𝑁))) | |
4 | 2, 3 | syl5eq 2866 | . 2 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (𝐺‘𝑁) = ((abs ∘ − )‘(𝐹‘𝑁))) |
5 | ffvelrn 6842 | . . . . . . 7 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (𝐹‘𝑁) ∈ ( ≤ ∩ (ℝ × ℝ))) | |
6 | 5 | elin2d 4174 | . . . . . 6 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (𝐹‘𝑁) ∈ (ℝ × ℝ)) |
7 | 1st2nd2 7720 | . . . . . 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 6667 | . . . 4 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → ((abs ∘ − )‘(𝐹‘𝑁)) = ((abs ∘ − )‘〈(1st ‘(𝐹‘𝑁)), (2nd ‘(𝐹‘𝑁))〉)) |
10 | df-ov 7151 | . . . 4 ⊢ ((1st ‘(𝐹‘𝑁))(abs ∘ − )(2nd ‘(𝐹‘𝑁))) = ((abs ∘ − )‘〈(1st ‘(𝐹‘𝑁)), (2nd ‘(𝐹‘𝑁))〉) | |
11 | 9, 10 | syl6eqr 2872 | . . 3 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → ((abs ∘ − )‘(𝐹‘𝑁)) = ((1st ‘(𝐹‘𝑁))(abs ∘ − )(2nd ‘(𝐹‘𝑁)))) |
12 | ovolfcl 24059 | . . . . . . 7 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → ((1st ‘(𝐹‘𝑁)) ∈ ℝ ∧ (2nd ‘(𝐹‘𝑁)) ∈ ℝ ∧ (1st ‘(𝐹‘𝑁)) ≤ (2nd ‘(𝐹‘𝑁)))) | |
13 | 12 | simp1d 1137 | . . . . . 6 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (1st ‘(𝐹‘𝑁)) ∈ ℝ) |
14 | 13 | recnd 10661 | . . . . 5 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (1st ‘(𝐹‘𝑁)) ∈ ℂ) |
15 | 12 | simp2d 1138 | . . . . . 6 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (2nd ‘(𝐹‘𝑁)) ∈ ℝ) |
16 | 15 | recnd 10661 | . . . . 5 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (2nd ‘(𝐹‘𝑁)) ∈ ℂ) |
17 | eqid 2819 | . . . . . 6 ⊢ (abs ∘ − ) = (abs ∘ − ) | |
18 | 17 | cnmetdval 23371 | . . . . 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 586 | . . . 4 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → ((1st ‘(𝐹‘𝑁))(abs ∘ − )(2nd ‘(𝐹‘𝑁))) = (abs‘((1st ‘(𝐹‘𝑁)) − (2nd ‘(𝐹‘𝑁))))) |
20 | abssuble0 14680 | . . . . 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 2854 | . . 3 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → ((1st ‘(𝐹‘𝑁))(abs ∘ − )(2nd ‘(𝐹‘𝑁))) = ((2nd ‘(𝐹‘𝑁)) − (1st ‘(𝐹‘𝑁)))) |
23 | 11, 22 | eqtrd 2854 | . 2 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → ((abs ∘ − )‘(𝐹‘𝑁)) = ((2nd ‘(𝐹‘𝑁)) − (1st ‘(𝐹‘𝑁)))) |
24 | 4, 23 | eqtrd 2854 | 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 398 ∧ w3a 1082 = wceq 1531 ∈ wcel 2108 ∩ cin 3933 〈cop 4565 class class class wbr 5057 × cxp 5546 ∘ ccom 5552 ⟶wf 6344 ‘cfv 6348 (class class class)co 7148 1st c1st 7679 2nd c2nd 7680 ℂcc 10527 ℝcr 10528 ≤ cle 10668 − cmin 10862 ℕcn 11630 abscabs 14585 |
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 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-cnex 10585 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 ax-pre-mulgt0 10606 ax-pre-sup 10607 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rmo 3144 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-om 7573 df-1st 7681 df-2nd 7682 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-er 8281 df-en 8502 df-dom 8503 df-sdom 8504 df-sup 8898 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 df-div 11290 df-nn 11631 df-2 11692 df-3 11693 df-n0 11890 df-z 11974 df-uz 12236 df-rp 12382 df-seq 13362 df-exp 13422 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 |
This theorem is referenced by: ovolfsf 24064 ovollb2lem 24081 ovolunlem1a 24089 ovoliunlem1 24095 ovolshftlem1 24102 ovolscalem1 24106 ovolicc1 24109 ovolicc2lem4 24113 ioombl1lem3 24153 ovolfs2 24164 uniioovol 24172 uniioombllem3 24178 |
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