ovnf - Mathbox for Glauco Siliprandi

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Theorem ovnf 46671

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 13359	. . . . 5 ⊢ 0 ∈ (0[,]+∞)
2	1	a1i 11	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝒫 (ℝ ↑_m 𝑋)) → 0 ∈ (0[,]+∞))
3		0xr 11159	. . . . . 6 ⊢ 0 ∈ ℝ^*
4	3	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝒫 (ℝ ↑_m 𝑋)) → 0 ∈ ℝ^*)
5		pnfxr 11166	. . . . . 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 4554	. . . . . . 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 2731	. . . . . 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 46665	. . . . 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 46667	. . . . . 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 4289	. . . . 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 45644	. . . 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 4519	. . 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 2731	. . 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 7047	. 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 46649	. . 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 6633	. 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 1541 ∈ wcel 2111 ∃wrex 3056 {crab 3395 ⊆ wss 3897 ∅c0 4280 ifcif 4472 𝒫 cpw 4547 ∪ ciun 4939 ↦ cmpt 5170 × cxp 5612 ∘ ccom 5618 ⟶wf 6477 ‘cfv 6481 (class class class)co 7346 ↑_m cmap 8750 Xcixp 8821 Fincfn 8869 infcinf 9325 ℝcr 11005 0cc0 11006 +∞cpnf 11143 ℝ^*cxr 11145 < clt 11146 ℕcn 12125 [,)cico 13247 [,]cicc 13248 ∏cprod 15810 volcvol 25391 Σ^{^}csumge0 46470 voln*covoln 46644

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668 ax-inf2 9531 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083 ax-pre-sup 11084

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4857 df-int 4896 df-iun 4941 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-se 5568 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-isom 6490 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-of 7610 df-om 7797 df-1st 7921 df-2nd 7922 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-2o 8386 df-er 8622 df-map 8752 df-pm 8753 df-ixp 8822 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-fi 9295 df-sup 9326 df-inf 9327 df-oi 9396 df-dju 9794 df-card 9832 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-div 11775 df-nn 12126 df-2 12188 df-3 12189 df-n0 12382 df-z 12469 df-uz 12733 df-q 12847 df-rp 12891 df-xneg 13011 df-xadd 13012 df-xmul 13013 df-ioo 13249 df-ico 13251 df-icc 13252 df-fz 13408 df-fzo 13555 df-fl 13696 df-seq 13909 df-exp 13969 df-hash 14238 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 df-clim 15395 df-rlim 15396 df-sum 15594 df-prod 15811 df-rest 17326 df-topgen 17347 df-psmet 21283 df-xmet 21284 df-met 21285 df-bl 21286 df-mopn 21287 df-top 22809 df-topon 22826 df-bases 22861 df-cmp 23302 df-ovol 25392 df-vol 25393 df-sumge0 46471 df-ovoln 46645

This theorem is referenced by: ovn0 46674 ovncl 46675 ovn02 46676 ovnome 46681 dmovn 46712 hspmbl 46737

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