ovolval2 - Mathbox for Glauco Siliprandi

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Theorem ovolval2 45360

Description: The value of the Lebesgue outer measure for subsets of the reals, expressed using Σ^{^}. See ovolval 24990 for an alternative expression. (Contributed by Glauco Siliprandi, 3-Mar-2021.)

**Hypotheses**
Ref	Expression
ovolval2.a	⊢ (𝜑 → 𝐴 ⊆ ℝ)
ovolval2.m	⊢ 𝑀 = {𝑦 ∈ ℝ^* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^{^}‘((abs ∘ − ) ∘ 𝑓)))}

**Assertion**
Ref	Expression
ovolval2	⊢ (𝜑 → (vol‘𝐴) = inf(𝑀, ℝ^, < ))

Distinct variable groups: 𝐴,𝑓,𝑦 𝜑,𝑓,𝑦 Allowed substitution hints: 𝑀(𝑦,𝑓)

**Proof of Theorem ovolval2**
Step	Hyp	Ref	Expression
1		ovolval2.a	. . 3 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
2		eqid 2733	. . . 4 ⊢ {𝑦 ∈ ℝ^* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^, < ))} = {𝑦 ∈ ℝ^ ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^*, < ))}
3	2	ovolval 24990	. . 3 ⊢ (𝐴 ⊆ ℝ → (vol‘𝐴) = inf({𝑦 ∈ ℝ^ ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^, < ))}, ℝ^, < ))
4	1, 3	syl 17	. 2 ⊢ (𝜑 → (vol‘𝐴) = inf({𝑦 ∈ ℝ^ ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^, < ))}, ℝ^, < ))
5	2	a1i 11	. . . 4 ⊢ (𝜑 → {𝑦 ∈ ℝ^* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^, < ))} = {𝑦 ∈ ℝ^ ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^*, < ))})
6		reex 11201	. . . . . . . . . . . . . . 15 ⊢ ℝ ∈ V
7	6, 6	xpex 7740	. . . . . . . . . . . . . 14 ⊢ (ℝ × ℝ) ∈ V
8		inss2 4230	. . . . . . . . . . . . . 14 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
9		mapss 8883	. . . . . . . . . . . . . 14 ⊢ (((ℝ × ℝ) ∈ V ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)) → (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ⊆ ((ℝ × ℝ) ↑_m ℕ))
10	7, 8, 9	mp2an 691	. . . . . . . . . . . . 13 ⊢ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ⊆ ((ℝ × ℝ) ↑_m ℕ)
11	10	sseli 3979	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) → 𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ))
12		1zzd 12593	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) → 1 ∈ ℤ)
13	11, 12	syl 17	. . . . . . . . . . 11 ⊢ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) → 1 ∈ ℤ)
14	13	adantl 483	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)) → 1 ∈ ℤ)
15		nnuz 12865	. . . . . . . . . 10 ⊢ ℕ = (ℤ_≥‘1)
16		absfico 43917	. . . . . . . . . . . . . 14 ⊢ abs:ℂ⟶(0[,)+∞)
17		subf 11462	. . . . . . . . . . . . . 14 ⊢ − :(ℂ × ℂ)⟶ℂ
18		fco 6742	. . . . . . . . . . . . . 14 ⊢ ((abs:ℂ⟶(0[,)+∞) ∧ − :(ℂ × ℂ)⟶ℂ) → (abs ∘ − ):(ℂ × ℂ)⟶(0[,)+∞))
19	16, 17, 18	mp2an 691	. . . . . . . . . . . . 13 ⊢ (abs ∘ − ):(ℂ × ℂ)⟶(0[,)+∞)
20	19	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) → (abs ∘ − ):(ℂ × ℂ)⟶(0[,)+∞))
21		rr2sscn2 44076	. . . . . . . . . . . . 13 ⊢ (ℝ × ℝ) ⊆ (ℂ × ℂ)
22	21	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) → (ℝ × ℝ) ⊆ (ℂ × ℂ))
23		elmapi 8843	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ) → 𝑓:ℕ⟶(ℝ × ℝ))
24	11, 23	syl 17	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) → 𝑓:ℕ⟶(ℝ × ℝ))
25	20, 22, 24	fcoss 43909	. . . . . . . . . . 11 ⊢ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) → ((abs ∘ − ) ∘ 𝑓):ℕ⟶(0[,)+∞))
26	25	adantl 483	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)) → ((abs ∘ − ) ∘ 𝑓):ℕ⟶(0[,)+∞))
27		eqid 2733	. . . . . . . . . 10 ⊢ seq1( + , ((abs ∘ − ) ∘ 𝑓)) = seq1( + , ((abs ∘ − ) ∘ 𝑓))
28	14, 15, 26, 27	sge0seq 45162	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)) → (Σ^{^}‘((abs ∘ − ) ∘ 𝑓)) = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^*, < ))
29	28	eqcomd 2739	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)) → sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^*, < ) = (Σ^{^}‘((abs ∘ − ) ∘ 𝑓)))
30	29	eqeq2d 2744	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)) → (𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^*, < ) ↔ 𝑦 = (Σ^{^}‘((abs ∘ − ) ∘ 𝑓))))
31	30	anbi2d 630	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)) → ((𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^*, < )) ↔ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^{^}‘((abs ∘ − ) ∘ 𝑓)))))
32	31	rexbidva 3177	. . . . 5 ⊢ (𝜑 → (∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^*, < )) ↔ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^{^}‘((abs ∘ − ) ∘ 𝑓)))))
33	32	rabbidv 3441	. . . 4 ⊢ (𝜑 → {𝑦 ∈ ℝ^* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^, < ))} = {𝑦 ∈ ℝ^ ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^{^}‘((abs ∘ − ) ∘ 𝑓)))})
34		ovolval2.m	. . . . . 6 ⊢ 𝑀 = {𝑦 ∈ ℝ^* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^{^}‘((abs ∘ − ) ∘ 𝑓)))}
35	34	eqcomi 2742	. . . . 5 ⊢ {𝑦 ∈ ℝ^* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^{^}‘((abs ∘ − ) ∘ 𝑓)))} = 𝑀
36	35	a1i 11	. . . 4 ⊢ (𝜑 → {𝑦 ∈ ℝ^* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^{^}‘((abs ∘ − ) ∘ 𝑓)))} = 𝑀)
37	5, 33, 36	3eqtrd 2777	. . 3 ⊢ (𝜑 → {𝑦 ∈ ℝ^* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^*, < ))} = 𝑀)
38	37	infeq1d 9472	. 2 ⊢ (𝜑 → inf({𝑦 ∈ ℝ^* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^, < ))}, ℝ^, < ) = inf(𝑀, ℝ^*, < ))
39	4, 38	eqtrd 2773	1 ⊢ (𝜑 → (vol‘𝐴) = inf(𝑀, ℝ^, < ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∃wrex 3071 {crab 3433 Vcvv 3475 ∩ cin 3948 ⊆ wss 3949 ∪ cuni 4909 × cxp 5675 ran crn 5678 ∘ ccom 5681 ⟶wf 6540 ‘cfv 6544 (class class class)co 7409 ↑_m cmap 8820 supcsup 9435 infcinf 9436 ℂcc 11108 ℝcr 11109 0cc0 11110 1c1 11111 + caddc 11113 +∞cpnf 11245 ℝ^*cxr 11247 < clt 11248 ≤ cle 11249 − cmin 11444 ℕcn 12212 ℤcz 12558 (,)cioo 13324 [,)cico 13326 seqcseq 13966 abscabs 15181 vol*covol 24979 Σ^{^}csumge0 45078

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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-inf2 9636 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-er 8703 df-map 8822 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-sup 9437 df-inf 9438 df-oi 9505 df-card 9934 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-n0 12473 df-z 12559 df-uz 12823 df-rp 12975 df-ico 13330 df-icc 13331 df-fz 13485 df-fzo 13628 df-seq 13967 df-exp 14028 df-hash 14291 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-clim 15432 df-sum 15633 df-ovol 24981 df-sumge0 45079

This theorem is referenced by: ovolval3 45363

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