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Theorem ovolfsval 24740

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 6826	. . 3 ⊢ (𝐺‘𝑁) = (((abs ∘ − ) ∘ 𝐹)‘𝑁)
3		fvco3 6923	. . 3 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐹)‘𝑁) = ((abs ∘ − )‘(𝐹‘𝑁)))
4	2, 3	eqtrid 2788	. 2 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (𝐺‘𝑁) = ((abs ∘ − )‘(𝐹‘𝑁)))
5		ffvelcdm 7015	. . . . . . 7 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (𝐹‘𝑁) ∈ ( ≤ ∩ (ℝ × ℝ)))
6	5	elin2d 4146	. . . . . 6 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (𝐹‘𝑁) ∈ (ℝ × ℝ))
7		1st2nd2 7938	. . . . . 6 ⊢ ((𝐹‘𝑁) ∈ (ℝ × ℝ) → (𝐹‘𝑁) = ⟨(1^st ‘(𝐹‘𝑁)), (2^nd ‘(𝐹‘𝑁))⟩)
8	6, 7	syl 17	. . . . 5 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (𝐹‘𝑁) = ⟨(1^st ‘(𝐹‘𝑁)), (2^nd ‘(𝐹‘𝑁))⟩)
9	8	fveq2d 6829	. . . 4 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → ((abs ∘ − )‘(𝐹‘𝑁)) = ((abs ∘ − )‘⟨(1^st ‘(𝐹‘𝑁)), (2^nd ‘(𝐹‘𝑁))⟩))
10		df-ov 7340	. . . 4 ⊢ ((1^st ‘(𝐹‘𝑁))(abs ∘ − )(2^nd ‘(𝐹‘𝑁))) = ((abs ∘ − )‘⟨(1^st ‘(𝐹‘𝑁)), (2^nd ‘(𝐹‘𝑁))⟩)
11	9, 10	eqtr4di 2794	. . 3 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → ((abs ∘ − )‘(𝐹‘𝑁)) = ((1^st ‘(𝐹‘𝑁))(abs ∘ − )(2^nd ‘(𝐹‘𝑁))))
12		ovolfcl 24736	. . . . . . 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 1141	. . . . . 6 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (1^st ‘(𝐹‘𝑁)) ∈ ℝ)
14	13	recnd 11104	. . . . 5 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (1^st ‘(𝐹‘𝑁)) ∈ ℂ)
15	12	simp2d 1142	. . . . . 6 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (2^nd ‘(𝐹‘𝑁)) ∈ ℝ)
16	15	recnd 11104	. . . . 5 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (2^nd ‘(𝐹‘𝑁)) ∈ ℂ)
17		eqid 2736	. . . . . 6 ⊢ (abs ∘ − ) = (abs ∘ − )
18	17	cnmetdval 24040	. . . . 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 584	. . . 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 15139	. . . . 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 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))))
22	19, 21	eqtrd 2776	. . 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 2776	. 2 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → ((abs ∘ − )‘(𝐹‘𝑁)) = ((2^nd ‘(𝐹‘𝑁)) − (1^st ‘(𝐹‘𝑁))))
24	4, 23	eqtrd 2776	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 396 ∧ w3a 1086 = wceq 1540 ∈ wcel 2105 ∩ cin 3897 ⟨cop 4579 class class class wbr 5092 × cxp 5618 ∘ ccom 5624 ⟶wf 6475 ‘cfv 6479 (class class class)co 7337 1^st c1st 7897 2^nd c2nd 7898 ℂcc 10970 ℝcr 10971 ≤ cle 11111 − cmin 11306 ℕcn 12074 abscabs 15044

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-sep 5243 ax-nul 5250 ax-pow 5308 ax-pr 5372 ax-un 7650 ax-cnex 11028 ax-resscn 11029 ax-1cn 11030 ax-icn 11031 ax-addcl 11032 ax-addrcl 11033 ax-mulcl 11034 ax-mulrcl 11035 ax-mulcom 11036 ax-addass 11037 ax-mulass 11038 ax-distr 11039 ax-i2m1 11040 ax-1ne0 11041 ax-1rid 11042 ax-rnegex 11043 ax-rrecex 11044 ax-cnre 11045 ax-pre-lttri 11046 ax-pre-lttrn 11047 ax-pre-ltadd 11048 ax-pre-mulgt0 11049 ax-pre-sup 11050

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3349 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3728 df-csb 3844 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3917 df-nul 4270 df-if 4474 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4853 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5176 df-tr 5210 df-id 5518 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5575 df-we 5577 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-pred 6238 df-ord 6305 df-on 6306 df-lim 6307 df-suc 6308 df-iota 6431 df-fun 6481 df-fn 6482 df-f 6483 df-f1 6484 df-fo 6485 df-f1o 6486 df-fv 6487 df-riota 7293 df-ov 7340 df-oprab 7341 df-mpo 7342 df-om 7781 df-1st 7899 df-2nd 7900 df-frecs 8167 df-wrecs 8198 df-recs 8272 df-rdg 8311 df-er 8569 df-en 8805 df-dom 8806 df-sdom 8807 df-sup 9299 df-pnf 11112 df-mnf 11113 df-xr 11114 df-ltxr 11115 df-le 11116 df-sub 11308 df-neg 11309 df-div 11734 df-nn 12075 df-2 12137 df-3 12138 df-n0 12335 df-z 12421 df-uz 12684 df-rp 12832 df-seq 13823 df-exp 13884 df-cj 14909 df-re 14910 df-im 14911 df-sqrt 15045 df-abs 15046

This theorem is referenced by: ovolfsf 24741 ovollb2lem 24758 ovolunlem1a 24766 ovoliunlem1 24772 ovolshftlem1 24779 ovolscalem1 24783 ovolicc1 24786 ovolicc2lem4 24790 ioombl1lem3 24830 ovolfs2 24841 uniioovol 24849 uniioombllem3 24855

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