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Theorem ovolfsf 25350

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 15287	. . . . . 6 ⊢ abs:ℂ⟶ℝ
2		subf 11463	. . . . . 6 ⊢ − :(ℂ × ℂ)⟶ℂ
3		fco 6734	. . . . . 6 ⊢ ((abs:ℂ⟶ℝ ∧ − :(ℂ × ℂ)⟶ℂ) → (abs ∘ − ):(ℂ × ℂ)⟶ℝ)
4	1, 2, 3	mp2an 689	. . . . 5 ⊢ (abs ∘ − ):(ℂ × ℂ)⟶ℝ
5		inss2 4224	. . . . . . 7 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
6		ax-resscn 11166	. . . . . . . 8 ⊢ ℝ ⊆ ℂ
7		xpss12 5684	. . . . . . . 8 ⊢ ((ℝ ⊆ ℂ ∧ ℝ ⊆ ℂ) → (ℝ × ℝ) ⊆ (ℂ × ℂ))
8	6, 6, 7	mp2an 689	. . . . . . 7 ⊢ (ℝ × ℝ) ⊆ (ℂ × ℂ)
9	5, 8	sstri 3986	. . . . . 6 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℂ × ℂ)
10		fss 6727	. . . . . 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 6734	. . . . 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 6701	. . . 4 ⊢ (𝐺:ℕ⟶ℝ ↔ ((abs ∘ − ) ∘ 𝐹):ℕ⟶ℝ)
16	13, 15	sylibr 233	. . 3 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝐺:ℕ⟶ℝ)
17	16	ffnd 6711	. 2 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝐺 Fn ℕ)
18	16	ffvelcdmda 7079	. . . 4 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (𝐺‘𝑥) ∈ ℝ)
19		ovolfcl 25345	. . . . . 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 11728	. . . . . . . 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 1466	. . . . . 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 25349	. . . . 5 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (𝐺‘𝑥) = ((2^nd ‘(𝐹‘𝑥)) − (1^st ‘(𝐹‘𝑥))))
25	23, 24	breqtrrd 5169	. . . 4 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → 0 ≤ (𝐺‘𝑥))
26		elrege0 13434	. . . 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 3140	. 2 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ∀𝑥 ∈ ℕ (𝐺‘𝑥) ∈ (0[,)+∞))
29		ffnfv 7113	. 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 205 ∧ wa 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∀wral 3055 ∩ cin 3942 ⊆ wss 3943 class class class wbr 5141 × cxp 5667 ∘ ccom 5673 Fn wfn 6531 ⟶wf 6532 ‘cfv 6536 (class class class)co 7404 1^st c1st 7969 2^nd c2nd 7970 ℂcc 11107 ℝcr 11108 0cc0 11109 +∞cpnf 11246 ≤ cle 11250 − cmin 11445 ℕcn 12213 [,)cico 13329 abscabs 15184

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-sup 9436 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-div 11873 df-nn 12214 df-2 12276 df-3 12277 df-n0 12474 df-z 12560 df-uz 12824 df-rp 12978 df-ico 13333 df-seq 13970 df-exp 14030 df-cj 15049 df-re 15050 df-im 15051 df-sqrt 15185 df-abs 15186

This theorem is referenced by: ovolsf 25351 ovollb2lem 25367 ovolunlem1a 25375 ovoliunlem1 25381 ovolshftlem1 25388 ovolicc2lem4 25399 ioombl1lem4 25440 ovolfs2 25450 uniioombllem2 25462 uniioombllem6 25467

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