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Theorem ovolfsf 25525

Description: Closure for the interval length function. (Contributed by Mario Carneiro, 16-Mar-2014.)

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

**Assertion**
Ref	Expression
ovolfsf	⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝐺:ℕ⟶(0[,)+∞))

Proof of Theorem ovolfsf Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		absf 15386	. . . . . 6 ⊢ abs:ℂ⟶ℝ
2		subf 11538	. . . . . 6 ⊢ − :(ℂ × ℂ)⟶ℂ
3		fco 6771	. . . . . 6 ⊢ ((abs:ℂ⟶ℝ ∧ − :(ℂ × ℂ)⟶ℂ) → (abs ∘ − ):(ℂ × ℂ)⟶ℝ)
4	1, 2, 3	mp2an 691	. . . . 5 ⊢ (abs ∘ − ):(ℂ × ℂ)⟶ℝ
5		inss2 4259	. . . . . . 7 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
6		ax-resscn 11241	. . . . . . . 8 ⊢ ℝ ⊆ ℂ
7		xpss12 5715	. . . . . . . 8 ⊢ ((ℝ ⊆ ℂ ∧ ℝ ⊆ ℂ) → (ℝ × ℝ) ⊆ (ℂ × ℂ))
8	6, 6, 7	mp2an 691	. . . . . . 7 ⊢ (ℝ × ℝ) ⊆ (ℂ × ℂ)
9	5, 8	sstri 4018	. . . . . 6 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℂ × ℂ)
10		fss 6763	. . . . . 6 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℂ × ℂ)) → 𝐹:ℕ⟶(ℂ × ℂ))
11	9, 10	mpan2 690	. . . . 5 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝐹:ℕ⟶(ℂ × ℂ))
12		fco 6771	. . . . 5 ⊢ (((abs ∘ − ):(ℂ × ℂ)⟶ℝ ∧ 𝐹:ℕ⟶(ℂ × ℂ)) → ((abs ∘ − ) ∘ 𝐹):ℕ⟶ℝ)
13	4, 11, 12	sylancr 586	. . . 4 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ((abs ∘ − ) ∘ 𝐹):ℕ⟶ℝ)
14		ovolfs.1	. . . . 5 ⊢ 𝐺 = ((abs ∘ − ) ∘ 𝐹)
15	14	feq1i 6738	. . . 4 ⊢ (𝐺:ℕ⟶ℝ ↔ ((abs ∘ − ) ∘ 𝐹):ℕ⟶ℝ)
16	13, 15	sylibr 234	. . 3 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝐺:ℕ⟶ℝ)
17	16	ffnd 6748	. 2 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝐺 Fn ℕ)
18	16	ffvelcdmda 7118	. . . 4 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (𝐺‘𝑥) ∈ ℝ)
19		ovolfcl 25520	. . . . . 6 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → ((1^st ‘(𝐹‘𝑥)) ∈ ℝ ∧ (2^nd ‘(𝐹‘𝑥)) ∈ ℝ ∧ (1^st ‘(𝐹‘𝑥)) ≤ (2^nd ‘(𝐹‘𝑥))))
20		subge0 11803	. . . . . . . 8 ⊢ (((2^nd ‘(𝐹‘𝑥)) ∈ ℝ ∧ (1^st ‘(𝐹‘𝑥)) ∈ ℝ) → (0 ≤ ((2^nd ‘(𝐹‘𝑥)) − (1^st ‘(𝐹‘𝑥))) ↔ (1^st ‘(𝐹‘𝑥)) ≤ (2^nd ‘(𝐹‘𝑥))))
21	20	ancoms 458	. . . . . . 7 ⊢ (((1^st ‘(𝐹‘𝑥)) ∈ ℝ ∧ (2^nd ‘(𝐹‘𝑥)) ∈ ℝ) → (0 ≤ ((2^nd ‘(𝐹‘𝑥)) − (1^st ‘(𝐹‘𝑥))) ↔ (1^st ‘(𝐹‘𝑥)) ≤ (2^nd ‘(𝐹‘𝑥))))
22	21	biimp3ar 1470	. . . . . 6 ⊢ (((1^st ‘(𝐹‘𝑥)) ∈ ℝ ∧ (2^nd ‘(𝐹‘𝑥)) ∈ ℝ ∧ (1^st ‘(𝐹‘𝑥)) ≤ (2^nd ‘(𝐹‘𝑥))) → 0 ≤ ((2^nd ‘(𝐹‘𝑥)) − (1^st ‘(𝐹‘𝑥))))
23	19, 22	syl 17	. . . . 5 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → 0 ≤ ((2^nd ‘(𝐹‘𝑥)) − (1^st ‘(𝐹‘𝑥))))
24	14	ovolfsval 25524	. . . . 5 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (𝐺‘𝑥) = ((2^nd ‘(𝐹‘𝑥)) − (1^st ‘(𝐹‘𝑥))))
25	23, 24	breqtrrd 5194	. . . 4 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → 0 ≤ (𝐺‘𝑥))
26		elrege0 13514	. . . 4 ⊢ ((𝐺‘𝑥) ∈ (0[,)+∞) ↔ ((𝐺‘𝑥) ∈ ℝ ∧ 0 ≤ (𝐺‘𝑥)))
27	18, 25, 26	sylanbrc 582	. . 3 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (𝐺‘𝑥) ∈ (0[,)+∞))
28	27	ralrimiva 3152	. 2 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ∀𝑥 ∈ ℕ (𝐺‘𝑥) ∈ (0[,)+∞))
29		ffnfv 7153	. 2 ⊢ (𝐺:ℕ⟶(0[,)+∞) ↔ (𝐺 Fn ℕ ∧ ∀𝑥 ∈ ℕ (𝐺‘𝑥) ∈ (0[,)+∞)))
30	17, 28, 29	sylanbrc 582	1 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝐺:ℕ⟶(0[,)+∞))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1537 ∈ wcel 2108 ∀wral 3067 ∩ cin 3975 ⊆ wss 3976 class class class wbr 5166 × cxp 5698 ∘ ccom 5704 Fn wfn 6568 ⟶wf 6569 ‘cfv 6573 (class class class)co 7448 1^st c1st 8028 2^nd c2nd 8029 ℂcc 11182 ℝcr 11183 0cc0 11184 +∞cpnf 11321 ≤ cle 11325 − cmin 11520 ℕcn 12293 [,)cico 13409 abscabs 15283

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-1st 8030 df-2nd 8031 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-er 8763 df-en 9004 df-dom 9005 df-sdom 9006 df-sup 9511 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-n0 12554 df-z 12640 df-uz 12904 df-rp 13058 df-ico 13413 df-seq 14053 df-exp 14113 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285

This theorem is referenced by: ovolsf 25526 ovollb2lem 25542 ovolunlem1a 25550 ovoliunlem1 25556 ovolshftlem1 25563 ovolicc2lem4 25574 ioombl1lem4 25615 ovolfs2 25625 uniioombllem2 25637 uniioombllem6 25642

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