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Theorem ovolfsf 25506

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 15376	. . . . . 6 ⊢ abs:ℂ⟶ℝ
2		subf 11510	. . . . . 6 ⊢ − :(ℂ × ℂ)⟶ℂ
3		fco 6760	. . . . . 6 ⊢ ((abs:ℂ⟶ℝ ∧ − :(ℂ × ℂ)⟶ℂ) → (abs ∘ − ):(ℂ × ℂ)⟶ℝ)
4	1, 2, 3	mp2an 692	. . . . 5 ⊢ (abs ∘ − ):(ℂ × ℂ)⟶ℝ
5		inss2 4238	. . . . . . 7 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
6		ax-resscn 11212	. . . . . . . 8 ⊢ ℝ ⊆ ℂ
7		xpss12 5700	. . . . . . . 8 ⊢ ((ℝ ⊆ ℂ ∧ ℝ ⊆ ℂ) → (ℝ × ℝ) ⊆ (ℂ × ℂ))
8	6, 6, 7	mp2an 692	. . . . . . 7 ⊢ (ℝ × ℝ) ⊆ (ℂ × ℂ)
9	5, 8	sstri 3993	. . . . . 6 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℂ × ℂ)
10		fss 6752	. . . . . 6 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℂ × ℂ)) → 𝐹:ℕ⟶(ℂ × ℂ))
11	9, 10	mpan2 691	. . . . 5 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝐹:ℕ⟶(ℂ × ℂ))
12		fco 6760	. . . . 5 ⊢ (((abs ∘ − ):(ℂ × ℂ)⟶ℝ ∧ 𝐹:ℕ⟶(ℂ × ℂ)) → ((abs ∘ − ) ∘ 𝐹):ℕ⟶ℝ)
13	4, 11, 12	sylancr 587	. . . 4 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ((abs ∘ − ) ∘ 𝐹):ℕ⟶ℝ)
14		ovolfs.1	. . . . 5 ⊢ 𝐺 = ((abs ∘ − ) ∘ 𝐹)
15	14	feq1i 6727	. . . 4 ⊢ (𝐺:ℕ⟶ℝ ↔ ((abs ∘ − ) ∘ 𝐹):ℕ⟶ℝ)
16	13, 15	sylibr 234	. . 3 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝐺:ℕ⟶ℝ)
17	16	ffnd 6737	. 2 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝐺 Fn ℕ)
18	16	ffvelcdmda 7104	. . . 4 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (𝐺‘𝑥) ∈ ℝ)
19		ovolfcl 25501	. . . . . 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 11776	. . . . . . . 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 1472	. . . . . 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 25505	. . . . 5 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (𝐺‘𝑥) = ((2^nd ‘(𝐹‘𝑥)) − (1^st ‘(𝐹‘𝑥))))
25	23, 24	breqtrrd 5171	. . . 4 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → 0 ≤ (𝐺‘𝑥))
26		elrege0 13494	. . . 4 ⊢ ((𝐺‘𝑥) ∈ (0[,)+∞) ↔ ((𝐺‘𝑥) ∈ ℝ ∧ 0 ≤ (𝐺‘𝑥)))
27	18, 25, 26	sylanbrc 583	. . 3 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (𝐺‘𝑥) ∈ (0[,)+∞))
28	27	ralrimiva 3146	. 2 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ∀𝑥 ∈ ℕ (𝐺‘𝑥) ∈ (0[,)+∞))
29		ffnfv 7139	. 2 ⊢ (𝐺:ℕ⟶(0[,)+∞) ↔ (𝐺 Fn ℕ ∧ ∀𝑥 ∈ ℕ (𝐺‘𝑥) ∈ (0[,)+∞)))
30	17, 28, 29	sylanbrc 583	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 1540 ∈ wcel 2108 ∀wral 3061 ∩ cin 3950 ⊆ wss 3951 class class class wbr 5143 × cxp 5683 ∘ ccom 5689 Fn wfn 6556 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 1^st c1st 8012 2^nd c2nd 8013 ℂcc 11153 ℝcr 11154 0cc0 11155 +∞cpnf 11292 ≤ cle 11296 − cmin 11492 ℕcn 12266 [,)cico 13389 abscabs 15273

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-sup 9482 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-z 12614 df-uz 12879 df-rp 13035 df-ico 13393 df-seq 14043 df-exp 14103 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275

This theorem is referenced by: ovolsf 25507 ovollb2lem 25523 ovolunlem1a 25531 ovoliunlem1 25537 ovolshftlem1 25544 ovolicc2lem4 25555 ioombl1lem4 25596 ovolfs2 25606 uniioombllem2 25618 uniioombllem6 25623

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