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Theorem ovolfsf 24635

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 15049	. . . . . 6 ⊢ abs:ℂ⟶ℝ
2		subf 11223	. . . . . 6 ⊢ − :(ℂ × ℂ)⟶ℂ
3		fco 6624	. . . . . 6 ⊢ ((abs:ℂ⟶ℝ ∧ − :(ℂ × ℂ)⟶ℂ) → (abs ∘ − ):(ℂ × ℂ)⟶ℝ)
4	1, 2, 3	mp2an 689	. . . . 5 ⊢ (abs ∘ − ):(ℂ × ℂ)⟶ℝ
5		inss2 4163	. . . . . . 7 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
6		ax-resscn 10928	. . . . . . . 8 ⊢ ℝ ⊆ ℂ
7		xpss12 5604	. . . . . . . 8 ⊢ ((ℝ ⊆ ℂ ∧ ℝ ⊆ ℂ) → (ℝ × ℝ) ⊆ (ℂ × ℂ))
8	6, 6, 7	mp2an 689	. . . . . . 7 ⊢ (ℝ × ℝ) ⊆ (ℂ × ℂ)
9	5, 8	sstri 3930	. . . . . 6 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℂ × ℂ)
10		fss 6617	. . . . . 6 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℂ × ℂ)) → 𝐹:ℕ⟶(ℂ × ℂ))
11	9, 10	mpan2 688	. . . . 5 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝐹:ℕ⟶(ℂ × ℂ))
12		fco 6624	. . . . 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 6591	. . . 4 ⊢ (𝐺:ℕ⟶ℝ ↔ ((abs ∘ − ) ∘ 𝐹):ℕ⟶ℝ)
16	13, 15	sylibr 233	. . 3 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝐺:ℕ⟶ℝ)
17	16	ffnd 6601	. 2 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝐺 Fn ℕ)
18	16	ffvelrnda 6961	. . . 4 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (𝐺‘𝑥) ∈ ℝ)
19		ovolfcl 24630	. . . . . 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 11488	. . . . . . . 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 459	. . . . . . 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 1469	. . . . . 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 24634	. . . . 5 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (𝐺‘𝑥) = ((2^nd ‘(𝐹‘𝑥)) − (1^st ‘(𝐹‘𝑥))))
25	23, 24	breqtrrd 5102	. . . 4 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → 0 ≤ (𝐺‘𝑥))
26		elrege0 13186	. . . 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 3103	. 2 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ∀𝑥 ∈ ℕ (𝐺‘𝑥) ∈ (0[,)+∞))
29		ffnfv 6992	. 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 205 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ∀wral 3064 ∩ cin 3886 ⊆ wss 3887 class class class wbr 5074 × cxp 5587 ∘ ccom 5593 Fn wfn 6428 ⟶wf 6429 ‘cfv 6433 (class class class)co 7275 1^st c1st 7829 2^nd c2nd 7830 ℂcc 10869 ℝcr 10870 0cc0 10871 +∞cpnf 11006 ≤ cle 11010 − cmin 11205 ℕcn 11973 [,)cico 13081 abscabs 14945

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-sup 9201 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-n0 12234 df-z 12320 df-uz 12583 df-rp 12731 df-ico 13085 df-seq 13722 df-exp 13783 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947

This theorem is referenced by: ovolsf 24636 ovollb2lem 24652 ovolunlem1a 24660 ovoliunlem1 24666 ovolshftlem1 24673 ovolicc2lem4 24684 ioombl1lem4 24725 ovolfs2 24735 uniioombllem2 24747 uniioombllem6 24752

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