ovnf - Mathbox for Glauco Siliprandi

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Theorem ovnf 46583

Description: The Lebesgue outer measure is a function that maps sets to nonnegative extended reals. This is step (a)(i) of the proof of Proposition 115D (a) of [Fremlin1] p. 30. (Contributed by Glauco Siliprandi, 11-Oct-2020.)

**Hypothesis**
Ref	Expression
ovnf.1	⊢ (𝜑 → 𝑋 ∈ Fin)

**Assertion**
Ref	Expression
ovnf	⊢ (𝜑 → (voln*‘𝑋):𝒫 (ℝ ↑_m 𝑋)⟶(0[,]+∞))

Proof of Theorem ovnf Dummy variables 𝑖 𝑗 𝑘 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		0e0iccpnf 13500	. . . . 5 ⊢ 0 ∈ (0[,]+∞)
2	1	a1i 11	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝒫 (ℝ ↑_m 𝑋)) → 0 ∈ (0[,]+∞))
3		0xr 11309	. . . . . 6 ⊢ 0 ∈ ℝ^*
4	3	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝒫 (ℝ ↑_m 𝑋)) → 0 ∈ ℝ^*)
5		pnfxr 11316	. . . . . 6 ⊢ +∞ ∈ ℝ^*
6	5	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝒫 (ℝ ↑_m 𝑋)) → +∞ ∈ ℝ^*)
7		ovnf.1	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ Fin)
8	7	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝒫 (ℝ ↑_m 𝑋)) → 𝑋 ∈ Fin)
9		elpwi 4606	. . . . . . 7 ⊢ (𝑦 ∈ 𝒫 (ℝ ↑_m 𝑋) → 𝑦 ⊆ (ℝ ↑_m 𝑋))
10	9	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝒫 (ℝ ↑_m 𝑋)) → 𝑦 ⊆ (ℝ ↑_m 𝑋))
11		eqid 2736	. . . . . 6 ⊢ {𝑧 ∈ ℝ^* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ)(𝑦 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))} = {𝑧 ∈ ℝ^* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ)(𝑦 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))}
12	8, 10, 11	ovnsupge0 46577	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝒫 (ℝ ↑_m 𝑋)) → {𝑧 ∈ ℝ^* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ)(𝑦 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))} ⊆ (0[,]+∞))
13	8, 10, 11	ovnpnfelsup 46579	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝒫 (ℝ ↑_m 𝑋)) → +∞ ∈ {𝑧 ∈ ℝ^* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ)(𝑦 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))})
14	13	ne0d 4341	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝒫 (ℝ ↑_m 𝑋)) → {𝑧 ∈ ℝ^* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ)(𝑦 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))} ≠ ∅)
15	4, 6, 12, 14	inficc 45552	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝒫 (ℝ ↑_m 𝑋)) → inf({𝑧 ∈ ℝ^* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ)(𝑦 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))}, ℝ^*, < ) ∈ (0[,]+∞))
16	2, 15	ifcld 4571	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝒫 (ℝ ↑_m 𝑋)) → if(𝑋 = ∅, 0, inf({𝑧 ∈ ℝ^* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ)(𝑦 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))}, ℝ^*, < )) ∈ (0[,]+∞))
17		eqid 2736	. . 3 ⊢ (𝑦 ∈ 𝒫 (ℝ ↑_m 𝑋) ↦ if(𝑋 = ∅, 0, inf({𝑧 ∈ ℝ^* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ)(𝑦 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))}, ℝ^, < ))) = (𝑦 ∈ 𝒫 (ℝ ↑_m 𝑋) ↦ if(𝑋 = ∅, 0, inf({𝑧 ∈ ℝ^ ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ)(𝑦 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))}, ℝ^*, < )))
18	16, 17	fmptd 7133	. 2 ⊢ (𝜑 → (𝑦 ∈ 𝒫 (ℝ ↑_m 𝑋) ↦ if(𝑋 = ∅, 0, inf({𝑧 ∈ ℝ^* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ)(𝑦 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))}, ℝ^*, < ))):𝒫 (ℝ ↑_m 𝑋)⟶(0[,]+∞))
19	7	ovnval 46561	. . 3 ⊢ (𝜑 → (voln‘𝑋) = (𝑦 ∈ 𝒫 (ℝ ↑_m 𝑋) ↦ if(𝑋 = ∅, 0, inf({𝑧 ∈ ℝ^ ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ)(𝑦 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))}, ℝ^*, < ))))
20	19	feq1d 6719	. 2 ⊢ (𝜑 → ((voln‘𝑋):𝒫 (ℝ ↑_m 𝑋)⟶(0[,]+∞) ↔ (𝑦 ∈ 𝒫 (ℝ ↑_m 𝑋) ↦ if(𝑋 = ∅, 0, inf({𝑧 ∈ ℝ^ ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ)(𝑦 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))}, ℝ^*, < ))):𝒫 (ℝ ↑_m 𝑋)⟶(0[,]+∞)))
21	18, 20	mpbird 257	1 ⊢ (𝜑 → (voln*‘𝑋):𝒫 (ℝ ↑_m 𝑋)⟶(0[,]+∞))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ∃wrex 3069 {crab 3435 ⊆ wss 3950 ∅c0 4332 ifcif 4524 𝒫 cpw 4599 ∪ ciun 4990 ↦ cmpt 5224 × cxp 5682 ∘ ccom 5688 ⟶wf 6556 ‘cfv 6560 (class class class)co 7432 ↑_m cmap 8867 Xcixp 8938 Fincfn 8986 infcinf 9482 ℝcr 11155 0cc0 11156 +∞cpnf 11293 ℝ^*cxr 11295 < clt 11296 ℕcn 12267 [,)cico 13390 [,]cicc 13391 ∏cprod 15940 volcvol 25499 Σ^{^}csumge0 46382 voln*covoln 46556

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-inf2 9682 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 ax-pre-sup 11234

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-se 5637 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-isom 6569 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-of 7698 df-om 7889 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-2o 8508 df-er 8746 df-map 8869 df-pm 8870 df-ixp 8939 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-fi 9452 df-sup 9483 df-inf 9484 df-oi 9551 df-dju 9942 df-card 9980 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-div 11922 df-nn 12268 df-2 12330 df-3 12331 df-n0 12529 df-z 12616 df-uz 12880 df-q 12992 df-rp 13036 df-xneg 13155 df-xadd 13156 df-xmul 13157 df-ioo 13392 df-ico 13394 df-icc 13395 df-fz 13549 df-fzo 13696 df-fl 13833 df-seq 14044 df-exp 14104 df-hash 14371 df-cj 15139 df-re 15140 df-im 15141 df-sqrt 15275 df-abs 15276 df-clim 15525 df-rlim 15526 df-sum 15724 df-prod 15941 df-rest 17468 df-topgen 17489 df-psmet 21357 df-xmet 21358 df-met 21359 df-bl 21360 df-mopn 21361 df-top 22901 df-topon 22918 df-bases 22954 df-cmp 23396 df-ovol 25500 df-vol 25501 df-sumge0 46383 df-ovoln 46557

This theorem is referenced by: ovn0 46586 ovncl 46587 ovn02 46588 ovnome 46593 dmovn 46624 hspmbl 46649

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