ovolfsval - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > ovolfsval		Structured version Visualization version GIF version

Theorem ovolfsval 23458

Description: The value of the interval length function. (Contributed by Mario Carneiro, 16-Mar-2014.)

**Hypothesis**
Ref	Expression
ovolfs.1	⊢ 𝐺 = ((abs ∘ − ) ∘ 𝐹)

**Assertion**
Ref	Expression
ovolfsval	⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (𝐺‘𝑁) = ((2^nd ‘(𝐹‘𝑁)) − (1^st ‘(𝐹‘𝑁))))

**Proof of Theorem ovolfsval**
Step	Hyp	Ref	Expression
1		ovolfs.1	. . . 4 ⊢ 𝐺 = ((abs ∘ − ) ∘ 𝐹)
2	1	fveq1i 6334	. . 3 ⊢ (𝐺‘𝑁) = (((abs ∘ − ) ∘ 𝐹)‘𝑁)
3		fvco3 6419	. . 3 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐹)‘𝑁) = ((abs ∘ − )‘(𝐹‘𝑁)))
4	2, 3	syl5eq 2817	. 2 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (𝐺‘𝑁) = ((abs ∘ − )‘(𝐹‘𝑁)))
5		inss2 3982	. . . . . . 7 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
6		ffvelrn 6502	. . . . . . 7 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (𝐹‘𝑁) ∈ ( ≤ ∩ (ℝ × ℝ)))
7	5, 6	sseldi 3750	. . . . . 6 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (𝐹‘𝑁) ∈ (ℝ × ℝ))
8		1st2nd2 7358	. . . . . 6 ⊢ ((𝐹‘𝑁) ∈ (ℝ × ℝ) → (𝐹‘𝑁) = ⟨(1^st ‘(𝐹‘𝑁)), (2^nd ‘(𝐹‘𝑁))⟩)
9	7, 8	syl 17	. . . . 5 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (𝐹‘𝑁) = ⟨(1^st ‘(𝐹‘𝑁)), (2^nd ‘(𝐹‘𝑁))⟩)
10	9	fveq2d 6337	. . . 4 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → ((abs ∘ − )‘(𝐹‘𝑁)) = ((abs ∘ − )‘⟨(1^st ‘(𝐹‘𝑁)), (2^nd ‘(𝐹‘𝑁))⟩))
11		df-ov 6799	. . . 4 ⊢ ((1^st ‘(𝐹‘𝑁))(abs ∘ − )(2^nd ‘(𝐹‘𝑁))) = ((abs ∘ − )‘⟨(1^st ‘(𝐹‘𝑁)), (2^nd ‘(𝐹‘𝑁))⟩)
12	10, 11	syl6eqr 2823	. . 3 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → ((abs ∘ − )‘(𝐹‘𝑁)) = ((1^st ‘(𝐹‘𝑁))(abs ∘ − )(2^nd ‘(𝐹‘𝑁))))
13		ovolfcl 23454	. . . . . . 7 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → ((1^st ‘(𝐹‘𝑁)) ∈ ℝ ∧ (2^nd ‘(𝐹‘𝑁)) ∈ ℝ ∧ (1^st ‘(𝐹‘𝑁)) ≤ (2^nd ‘(𝐹‘𝑁))))
14	13	simp1d 1136	. . . . . 6 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (1^st ‘(𝐹‘𝑁)) ∈ ℝ)
15	14	recnd 10274	. . . . 5 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (1^st ‘(𝐹‘𝑁)) ∈ ℂ)
16	13	simp2d 1137	. . . . . 6 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (2^nd ‘(𝐹‘𝑁)) ∈ ℝ)
17	16	recnd 10274	. . . . 5 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (2^nd ‘(𝐹‘𝑁)) ∈ ℂ)
18		eqid 2771	. . . . . 6 ⊢ (abs ∘ − ) = (abs ∘ − )
19	18	cnmetdval 22794	. . . . 5 ⊢ (((1^st ‘(𝐹‘𝑁)) ∈ ℂ ∧ (2^nd ‘(𝐹‘𝑁)) ∈ ℂ) → ((1^st ‘(𝐹‘𝑁))(abs ∘ − )(2^nd ‘(𝐹‘𝑁))) = (abs‘((1^st ‘(𝐹‘𝑁)) − (2^nd ‘(𝐹‘𝑁)))))
20	15, 17, 19	syl2anc 573	. . . 4 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → ((1^st ‘(𝐹‘𝑁))(abs ∘ − )(2^nd ‘(𝐹‘𝑁))) = (abs‘((1^st ‘(𝐹‘𝑁)) − (2^nd ‘(𝐹‘𝑁)))))
21		abssuble0 14276	. . . . 5 ⊢ (((1^st ‘(𝐹‘𝑁)) ∈ ℝ ∧ (2^nd ‘(𝐹‘𝑁)) ∈ ℝ ∧ (1^st ‘(𝐹‘𝑁)) ≤ (2^nd ‘(𝐹‘𝑁))) → (abs‘((1^st ‘(𝐹‘𝑁)) − (2^nd ‘(𝐹‘𝑁)))) = ((2^nd ‘(𝐹‘𝑁)) − (1^st ‘(𝐹‘𝑁))))
22	13, 21	syl 17	. . . 4 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (abs‘((1^st ‘(𝐹‘𝑁)) − (2^nd ‘(𝐹‘𝑁)))) = ((2^nd ‘(𝐹‘𝑁)) − (1^st ‘(𝐹‘𝑁))))
23	20, 22	eqtrd 2805	. . 3 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → ((1^st ‘(𝐹‘𝑁))(abs ∘ − )(2^nd ‘(𝐹‘𝑁))) = ((2^nd ‘(𝐹‘𝑁)) − (1^st ‘(𝐹‘𝑁))))
24	12, 23	eqtrd 2805	. 2 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → ((abs ∘ − )‘(𝐹‘𝑁)) = ((2^nd ‘(𝐹‘𝑁)) − (1^st ‘(𝐹‘𝑁))))
25	4, 24	eqtrd 2805	1 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (𝐺‘𝑁) = ((2^nd ‘(𝐹‘𝑁)) − (1^st ‘(𝐹‘𝑁))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 382 ∧ w3a 1071 = wceq 1631 ∈ wcel 2145 ∩ cin 3722 ⟨cop 4323 class class class wbr 4787 × cxp 5248 ∘ ccom 5254 ⟶wf 6026 ‘cfv 6030 (class class class)co 6796 1^st c1st 7317 2^nd c2nd 7318 ℂcc 10140 ℝcr 10141 ≤ cle 10281 − cmin 10472 ℕcn 11226 abscabs 14182

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4916 ax-nul 4924 ax-pow 4975 ax-pr 5035 ax-un 7100 ax-cnex 10198 ax-resscn 10199 ax-1cn 10200 ax-icn 10201 ax-addcl 10202 ax-addrcl 10203 ax-mulcl 10204 ax-mulrcl 10205 ax-mulcom 10206 ax-addass 10207 ax-mulass 10208 ax-distr 10209 ax-i2m1 10210 ax-1ne0 10211 ax-1rid 10212 ax-rnegex 10213 ax-rrecex 10214 ax-cnre 10215 ax-pre-lttri 10216 ax-pre-lttrn 10217 ax-pre-ltadd 10218 ax-pre-mulgt0 10219 ax-pre-sup 10220

This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-tp 4322 df-op 4324 df-uni 4576 df-iun 4657 df-br 4788 df-opab 4848 df-mpt 4865 df-tr 4888 df-id 5158 df-eprel 5163 df-po 5171 df-so 5172 df-fr 5209 df-we 5211 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-rn 5261 df-res 5262 df-ima 5263 df-pred 5822 df-ord 5868 df-on 5869 df-lim 5870 df-suc 5871 df-iota 5993 df-fun 6032 df-fn 6033 df-f 6034 df-f1 6035 df-fo 6036 df-f1o 6037 df-fv 6038 df-riota 6757 df-ov 6799 df-oprab 6800 df-mpt2 6801 df-om 7217 df-1st 7319 df-2nd 7320 df-wrecs 7563 df-recs 7625 df-rdg 7663 df-er 7900 df-en 8114 df-dom 8115 df-sdom 8116 df-sup 8508 df-pnf 10282 df-mnf 10283 df-xr 10284 df-ltxr 10285 df-le 10286 df-sub 10474 df-neg 10475 df-div 10891 df-nn 11227 df-2 11285 df-3 11286 df-n0 11500 df-z 11585 df-uz 11894 df-rp 12036 df-seq 13009 df-exp 13068 df-cj 14047 df-re 14048 df-im 14049 df-sqrt 14183 df-abs 14184

This theorem is referenced by: ovolfsf 23459 ovollb2lem 23476 ovolunlem1a 23484 ovoliunlem1 23490 ovolshftlem1 23497 ovolscalem1 23501 ovolicc1 23504 ovolicc2lem4 23508 ioombl1lem3 23548 ovolfs2 23559 uniioovol 23567 uniioombllem3 23573

W3C validator