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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 6908 | . . 3 ⊢ (𝐺‘𝑁) = (((abs ∘ − ) ∘ 𝐹)‘𝑁) |
3 | fvco3 7008 | . . 3 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐹)‘𝑁) = ((abs ∘ − )‘(𝐹‘𝑁))) | |
4 | 2, 3 | eqtrid 2787 | . 2 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (𝐺‘𝑁) = ((abs ∘ − )‘(𝐹‘𝑁))) |
5 | ffvelcdm 7101 | . . . . . . 7 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (𝐹‘𝑁) ∈ ( ≤ ∩ (ℝ × ℝ))) | |
6 | 5 | elin2d 4215 | . . . . . 6 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (𝐹‘𝑁) ∈ (ℝ × ℝ)) |
7 | 1st2nd2 8052 | . . . . . 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 6911 | . . . 4 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → ((abs ∘ − )‘(𝐹‘𝑁)) = ((abs ∘ − )‘〈(1st ‘(𝐹‘𝑁)), (2nd ‘(𝐹‘𝑁))〉)) |
10 | df-ov 7434 | . . . 4 ⊢ ((1st ‘(𝐹‘𝑁))(abs ∘ − )(2nd ‘(𝐹‘𝑁))) = ((abs ∘ − )‘〈(1st ‘(𝐹‘𝑁)), (2nd ‘(𝐹‘𝑁))〉) | |
11 | 9, 10 | eqtr4di 2793 | . . 3 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → ((abs ∘ − )‘(𝐹‘𝑁)) = ((1st ‘(𝐹‘𝑁))(abs ∘ − )(2nd ‘(𝐹‘𝑁)))) |
12 | ovolfcl 25515 | . . . . . . 7 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → ((1st ‘(𝐹‘𝑁)) ∈ ℝ ∧ (2nd ‘(𝐹‘𝑁)) ∈ ℝ ∧ (1st ‘(𝐹‘𝑁)) ≤ (2nd ‘(𝐹‘𝑁)))) | |
13 | 12 | simp1d 1141 | . . . . . 6 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (1st ‘(𝐹‘𝑁)) ∈ ℝ) |
14 | 13 | recnd 11287 | . . . . 5 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (1st ‘(𝐹‘𝑁)) ∈ ℂ) |
15 | 12 | simp2d 1142 | . . . . . 6 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (2nd ‘(𝐹‘𝑁)) ∈ ℝ) |
16 | 15 | recnd 11287 | . . . . 5 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (2nd ‘(𝐹‘𝑁)) ∈ ℂ) |
17 | eqid 2735 | . . . . . 6 ⊢ (abs ∘ − ) = (abs ∘ − ) | |
18 | 17 | cnmetdval 24807 | . . . . 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 584 | . . . 4 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → ((1st ‘(𝐹‘𝑁))(abs ∘ − )(2nd ‘(𝐹‘𝑁))) = (abs‘((1st ‘(𝐹‘𝑁)) − (2nd ‘(𝐹‘𝑁))))) |
20 | abssuble0 15364 | . . . . 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 2775 | . . 3 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → ((1st ‘(𝐹‘𝑁))(abs ∘ − )(2nd ‘(𝐹‘𝑁))) = ((2nd ‘(𝐹‘𝑁)) − (1st ‘(𝐹‘𝑁)))) |
23 | 11, 22 | eqtrd 2775 | . 2 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → ((abs ∘ − )‘(𝐹‘𝑁)) = ((2nd ‘(𝐹‘𝑁)) − (1st ‘(𝐹‘𝑁)))) |
24 | 4, 23 | eqtrd 2775 | 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 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ∩ cin 3962 〈cop 4637 class class class wbr 5148 × cxp 5687 ∘ ccom 5693 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 1st c1st 8011 2nd c2nd 8012 ℂcc 11151 ℝcr 11152 ≤ cle 11294 − cmin 11490 ℕcn 12264 abscabs 15270 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-sup 9480 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-z 12612 df-uz 12877 df-rp 13033 df-seq 14040 df-exp 14100 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 |
This theorem is referenced by: ovolfsf 25520 ovollb2lem 25537 ovolunlem1a 25545 ovoliunlem1 25551 ovolshftlem1 25558 ovolscalem1 25562 ovolicc1 25565 ovolicc2lem4 25569 ioombl1lem3 25609 ovolfs2 25620 uniioovol 25628 uniioombllem3 25634 |
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