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Theorem ovolfsval 25590

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 6872	. . 3 ⊢ (𝐺‘𝑁) = (((abs ∘ − ) ∘ 𝐹)‘𝑁)
3		fvco3 6971	. . 3 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐹)‘𝑁) = ((abs ∘ − )‘(𝐹‘𝑁)))
4	2, 3	eqtrid 2812	. 2 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (𝐺‘𝑁) = ((abs ∘ − )‘(𝐹‘𝑁)))
5		ffvelcdm 7066	. . . . . . 7 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (𝐹‘𝑁) ∈ ( ≤ ∩ (ℝ × ℝ)))
6	5	elin2d 4160	. . . . . 6 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (𝐹‘𝑁) ∈ (ℝ × ℝ))
7		1st2nd2 8013	. . . . . 6 ⊢ ((𝐹‘𝑁) ∈ (ℝ × ℝ) → (𝐹‘𝑁) = ⟨(1^st ‘(𝐹‘𝑁)), (2^nd ‘(𝐹‘𝑁))⟩)
8	6, 7	syl 18	. . . . 5 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (𝐹‘𝑁) = ⟨(1^st ‘(𝐹‘𝑁)), (2^nd ‘(𝐹‘𝑁))⟩)
9	8	fveq2d 6875	. . . 4 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → ((abs ∘ − )‘(𝐹‘𝑁)) = ((abs ∘ − )‘⟨(1^st ‘(𝐹‘𝑁)), (2^nd ‘(𝐹‘𝑁))⟩))
10		df-ov 7403	. . . 4 ⊢ ((1^st ‘(𝐹‘𝑁))(abs ∘ − )(2^nd ‘(𝐹‘𝑁))) = ((abs ∘ − )‘⟨(1^st ‘(𝐹‘𝑁)), (2^nd ‘(𝐹‘𝑁))⟩)
11	9, 10	eqtr4di 2818	. . 3 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → ((abs ∘ − )‘(𝐹‘𝑁)) = ((1^st ‘(𝐹‘𝑁))(abs ∘ − )(2^nd ‘(𝐹‘𝑁))))
12		ovolfcl 25586	. . . . . . 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))))
13	12	simp1d 1158	. . . . . 6 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (1^st ‘(𝐹‘𝑁)) ∈ ℝ)
14	13	recnd 11225	. . . . 5 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (1^st ‘(𝐹‘𝑁)) ∈ ℂ)
15	12	simp2d 1159	. . . . . 6 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (2^nd ‘(𝐹‘𝑁)) ∈ ℝ)
16	15	recnd 11225	. . . . 5 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (2^nd ‘(𝐹‘𝑁)) ∈ ℂ)
17		eqid 2765	. . . . . 6 ⊢ (abs ∘ − ) = (abs ∘ − )
18	17	cnmetdval 24888	. . . . 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)))))
19	14, 16, 18	syl2anc 595	. . . 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)))))
20		abssuble0 15370	. . . . 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))))
21	12, 20	syl 18	. . . 4 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (abs‘((1^st ‘(𝐹‘𝑁)) − (2^nd ‘(𝐹‘𝑁)))) = ((2^nd ‘(𝐹‘𝑁)) − (1^st ‘(𝐹‘𝑁))))
22	19, 21	eqtrd 2800	. . 3 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → ((1^st ‘(𝐹‘𝑁))(abs ∘ − )(2^nd ‘(𝐹‘𝑁))) = ((2^nd ‘(𝐹‘𝑁)) − (1^st ‘(𝐹‘𝑁))))
23	11, 22	eqtrd 2800	. 2 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → ((abs ∘ − )‘(𝐹‘𝑁)) = ((2^nd ‘(𝐹‘𝑁)) − (1^st ‘(𝐹‘𝑁))))
24	4, 23	eqtrd 2800	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 400 ∧ w3a 1101 = wceq 1563 ∈ wcel 2145 ∩ cin 3906 ⟨cop 4591 class class class wbr 5105 × cxp 5650 ∘ ccom 5656 ⟶wf 6521 ‘cfv 6525 (class class class)co 7400 1^st c1st 7972 2^nd c2nd 7973 ℂcc 11086 ℝcr 11087 ≤ cle 11232 − cmin 11429 ℕcn 12224 abscabs 15275

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1818 ax-4 1832 ax-5 1933 ax-6 1990 ax-7 2031 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2737 ax-sep 5251 ax-nul 5261 ax-pow 5327 ax-pr 5395 ax-un 7722 ax-cnex 11144 ax-resscn 11145 ax-1cn 11146 ax-icn 11147 ax-addcl 11148 ax-addrcl 11149 ax-mulcl 11150 ax-mulrcl 11151 ax-mulcom 11152 ax-addass 11153 ax-mulass 11154 ax-distr 11155 ax-i2m1 11156 ax-1ne0 11157 ax-1rid 11158 ax-rnegex 11159 ax-rrecex 11160 ax-cnre 11161 ax-pre-lttri 11162 ax-pre-lttrn 11163 ax-pre-ltadd 11164 ax-pre-mulgt0 11165 ax-pre-sup 11166

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1566 df-fal 1576 df-ex 1803 df-nf 1807 df-sb 2094 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3370 df-reu 3371 df-rab 3418 df-v 3459 df-sbc 3748 df-csb 3856 df-dif 3910 df-un 3912 df-in 3914 df-ss 3924 df-pss 3927 df-nul 4289 df-if 4484 df-pw 4560 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4869 df-iun 4954 df-br 5106 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5547 df-eprel 5552 df-po 5560 df-so 5561 df-fr 5605 df-we 5607 df-xp 5658 df-rel 5659 df-cnv 5660 df-co 5661 df-dm 5662 df-rn 5663 df-res 5664 df-ima 5665 df-pred 6292 df-ord 6353 df-on 6354 df-lim 6355 df-suc 6356 df-iota 6481 df-fun 6527 df-fn 6528 df-f 6529 df-f1 6530 df-fo 6531 df-f1o 6532 df-fv 6533 df-riota 7357 df-ov 7403 df-oprab 7404 df-mpo 7405 df-om 7851 df-1st 7974 df-2nd 7975 df-frecs 8266 df-wrecs 8297 df-recs 8346 df-rdg 8385 df-er 8682 df-en 8932 df-dom 8933 df-sdom 8934 df-sup 9390 df-pnf 11233 df-mnf 11234 df-xr 11235 df-ltxr 11236 df-le 11237 df-sub 11431 df-neg 11432 df-div 11860 df-nn 12225 df-2 12294 df-3 12295 df-n0 12496 df-z 12583 df-uz 12854 df-rp 13008 df-seq 14029 df-exp 14089 df-cj 15140 df-re 15141 df-im 15142 df-sqrt 15276 df-abs 15277

This theorem is referenced by: ovolfsf 25591 ovollb2lem 25608 ovolunlem1a 25616 ovoliunlem1 25622 ovolshftlem1 25629 ovolscalem1 25633 ovolicc1 25636 ovolicc2lem4 25640 ioombl1lem3 25680 ovolfs2 25691 uniioovol 25699 uniioombllem3 25705

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