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Theorem ovolfsval 25427

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 6835	. . 3 ⊢ (𝐺‘𝑁) = (((abs ∘ − ) ∘ 𝐹)‘𝑁)
3		fvco3 6933	. . 3 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐹)‘𝑁) = ((abs ∘ − )‘(𝐹‘𝑁)))
4	2, 3	eqtrid 2783	. 2 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (𝐺‘𝑁) = ((abs ∘ − )‘(𝐹‘𝑁)))
5		ffvelcdm 7026	. . . . . . 7 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (𝐹‘𝑁) ∈ ( ≤ ∩ (ℝ × ℝ)))
6	5	elin2d 4157	. . . . . 6 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (𝐹‘𝑁) ∈ (ℝ × ℝ))
7		1st2nd2 7972	. . . . . 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 6838	. . . 4 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → ((abs ∘ − )‘(𝐹‘𝑁)) = ((abs ∘ − )‘⟨(1^st ‘(𝐹‘𝑁)), (2^nd ‘(𝐹‘𝑁))⟩))
10		df-ov 7361	. . . 4 ⊢ ((1^st ‘(𝐹‘𝑁))(abs ∘ − )(2^nd ‘(𝐹‘𝑁))) = ((abs ∘ − )‘⟨(1^st ‘(𝐹‘𝑁)), (2^nd ‘(𝐹‘𝑁))⟩)
11	9, 10	eqtr4di 2789	. . 3 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → ((abs ∘ − )‘(𝐹‘𝑁)) = ((1^st ‘(𝐹‘𝑁))(abs ∘ − )(2^nd ‘(𝐹‘𝑁))))
12		ovolfcl 25423	. . . . . . 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 11160	. . . . 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 11160	. . . . 5 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (2^nd ‘(𝐹‘𝑁)) ∈ ℂ)
17		eqid 2736	. . . . . 6 ⊢ (abs ∘ − ) = (abs ∘ − )
18	17	cnmetdval 24714	. . . . 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 15252	. . . . 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 2771	. . 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 2771	. 2 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → ((abs ∘ − )‘(𝐹‘𝑁)) = ((2^nd ‘(𝐹‘𝑁)) − (1^st ‘(𝐹‘𝑁))))
24	4, 23	eqtrd 2771	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 3900 ⟨cop 4586 class class class wbr 5098 × cxp 5622 ∘ ccom 5628 ⟶wf 6488 ‘cfv 6492 (class class class)co 7358 1^st c1st 7931 2^nd c2nd 7932 ℂcc 11024 ℝcr 11025 ≤ cle 11167 − cmin 11364 ℕcn 12145 abscabs 15157

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 2184 ax-ext 2708 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103 ax-pre-sup 11104

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 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-iun 4948 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7809 df-1st 7933 df-2nd 7934 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-er 8635 df-en 8884 df-dom 8885 df-sdom 8886 df-sup 9345 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-div 11795 df-nn 12146 df-2 12208 df-3 12209 df-n0 12402 df-z 12489 df-uz 12752 df-rp 12906 df-seq 13925 df-exp 13985 df-cj 15022 df-re 15023 df-im 15024 df-sqrt 15158 df-abs 15159

This theorem is referenced by: ovolfsf 25428 ovollb2lem 25445 ovolunlem1a 25453 ovoliunlem1 25459 ovolshftlem1 25466 ovolscalem1 25470 ovolicc1 25473 ovolicc2lem4 25477 ioombl1lem3 25517 ovolfs2 25528 uniioovol 25536 uniioombllem3 25542

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