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Theorem ovolfsval 25399

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 6829	. . 3 ⊢ (𝐺‘𝑁) = (((abs ∘ − ) ∘ 𝐹)‘𝑁)
3		fvco3 6927	. . 3 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐹)‘𝑁) = ((abs ∘ − )‘(𝐹‘𝑁)))
4	2, 3	eqtrid 2780	. 2 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (𝐺‘𝑁) = ((abs ∘ − )‘(𝐹‘𝑁)))
5		ffvelcdm 7020	. . . . . . 7 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (𝐹‘𝑁) ∈ ( ≤ ∩ (ℝ × ℝ)))
6	5	elin2d 4154	. . . . . 6 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (𝐹‘𝑁) ∈ (ℝ × ℝ))
7		1st2nd2 7966	. . . . . 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 6832	. . . 4 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → ((abs ∘ − )‘(𝐹‘𝑁)) = ((abs ∘ − )‘⟨(1^st ‘(𝐹‘𝑁)), (2^nd ‘(𝐹‘𝑁))⟩))
10		df-ov 7355	. . . 4 ⊢ ((1^st ‘(𝐹‘𝑁))(abs ∘ − )(2^nd ‘(𝐹‘𝑁))) = ((abs ∘ − )‘⟨(1^st ‘(𝐹‘𝑁)), (2^nd ‘(𝐹‘𝑁))⟩)
11	9, 10	eqtr4di 2786	. . 3 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → ((abs ∘ − )‘(𝐹‘𝑁)) = ((1^st ‘(𝐹‘𝑁))(abs ∘ − )(2^nd ‘(𝐹‘𝑁))))
12		ovolfcl 25395	. . . . . . 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 1142	. . . . . 6 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (1^st ‘(𝐹‘𝑁)) ∈ ℝ)
14	13	recnd 11147	. . . . 5 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (1^st ‘(𝐹‘𝑁)) ∈ ℂ)
15	12	simp2d 1143	. . . . . 6 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (2^nd ‘(𝐹‘𝑁)) ∈ ℝ)
16	15	recnd 11147	. . . . 5 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (2^nd ‘(𝐹‘𝑁)) ∈ ℂ)
17		eqid 2733	. . . . . 6 ⊢ (abs ∘ − ) = (abs ∘ − )
18	17	cnmetdval 24686	. . . . 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 15238	. . . . 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 2768	. . 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 2768	. 2 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → ((abs ∘ − )‘(𝐹‘𝑁)) = ((2^nd ‘(𝐹‘𝑁)) − (1^st ‘(𝐹‘𝑁))))
24	4, 23	eqtrd 2768	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 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2113 ∩ cin 3897 ⟨cop 4581 class class class wbr 5093 × cxp 5617 ∘ ccom 5623 ⟶wf 6482 ‘cfv 6486 (class class class)co 7352 1^st c1st 7925 2^nd c2nd 7926 ℂcc 11011 ℝcr 11012 ≤ cle 11154 − cmin 11351 ℕcn 12132 abscabs 15143

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 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-sep 5236 ax-nul 5246 ax-pow 5305 ax-pr 5372 ax-un 7674 ax-cnex 11069 ax-resscn 11070 ax-1cn 11071 ax-icn 11072 ax-addcl 11073 ax-addrcl 11074 ax-mulcl 11075 ax-mulrcl 11076 ax-mulcom 11077 ax-addass 11078 ax-mulass 11079 ax-distr 11080 ax-i2m1 11081 ax-1ne0 11082 ax-1rid 11083 ax-rnegex 11084 ax-rrecex 11085 ax-cnre 11086 ax-pre-lttri 11087 ax-pre-lttrn 11088 ax-pre-ltadd 11089 ax-pre-mulgt0 11090 ax-pre-sup 11091

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-nel 3034 df-ral 3049 df-rex 3058 df-rmo 3347 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4283 df-if 4475 df-pw 4551 df-sn 4576 df-pr 4578 df-op 4582 df-uni 4859 df-iun 4943 df-br 5094 df-opab 5156 df-mpt 5175 df-tr 5201 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-riota 7309 df-ov 7355 df-oprab 7356 df-mpo 7357 df-om 7803 df-1st 7927 df-2nd 7928 df-frecs 8217 df-wrecs 8248 df-recs 8297 df-rdg 8335 df-er 8628 df-en 8876 df-dom 8877 df-sdom 8878 df-sup 9333 df-pnf 11155 df-mnf 11156 df-xr 11157 df-ltxr 11158 df-le 11159 df-sub 11353 df-neg 11354 df-div 11782 df-nn 12133 df-2 12195 df-3 12196 df-n0 12389 df-z 12476 df-uz 12739 df-rp 12893 df-seq 13911 df-exp 13971 df-cj 15008 df-re 15009 df-im 15010 df-sqrt 15144 df-abs 15145

This theorem is referenced by: ovolfsf 25400 ovollb2lem 25417 ovolunlem1a 25425 ovoliunlem1 25431 ovolshftlem1 25438 ovolscalem1 25442 ovolicc1 25445 ovolicc2lem4 25449 ioombl1lem3 25489 ovolfs2 25500 uniioovol 25508 uniioombllem3 25514

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