ovnn0val - Mathbox for Glauco Siliprandi

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Theorem ovnn0val 46676

Description: The value of a (multidimensional) Lebesgue outer measure, defined on a nonzero-dimensional space of reals. (Contributed by Glauco Siliprandi, 11-Oct-2020.)

**Hypotheses**
Ref	Expression
ovnn0val.1	⊢ (𝜑 → 𝑋 ∈ Fin)
ovnn0val.2	⊢ (𝜑 → 𝑋 ≠ ∅)
ovnn0val.3	⊢ (𝜑 → 𝐴 ⊆ (ℝ ↑_m 𝑋))
ovnn0val.4	⊢ 𝑀 = {𝑧 ∈ ℝ^* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ)(𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))}

**Assertion**
Ref	Expression
ovnn0val	⊢ (𝜑 → ((voln‘𝑋)‘𝐴) = inf(𝑀, ℝ^, < ))

Distinct variable groups: 𝐴,𝑖,𝑧 𝑖,𝑋,𝑗,𝑘,𝑧 Allowed substitution hints: 𝜑(𝑧,𝑖,𝑗,𝑘) 𝐴(𝑗,𝑘) 𝑀(𝑧,𝑖,𝑗,𝑘)

**Proof of Theorem ovnn0val**
Step	Hyp	Ref	Expression
1		ovnn0val.1	. . 3 ⊢ (𝜑 → 𝑋 ∈ Fin)
2		ovnn0val.3	. . 3 ⊢ (𝜑 → 𝐴 ⊆ (ℝ ↑_m 𝑋))
3		ovnn0val.4	. . 3 ⊢ 𝑀 = {𝑧 ∈ ℝ^* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ)(𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))}
4	1, 2, 3	ovnval2 46670	. 2 ⊢ (𝜑 → ((voln‘𝑋)‘𝐴) = if(𝑋 = ∅, 0, inf(𝑀, ℝ^, < )))
5		ovnn0val.2	. . . 4 ⊢ (𝜑 → 𝑋 ≠ ∅)
6	5	neneqd 2934	. . 3 ⊢ (𝜑 → ¬ 𝑋 = ∅)
7	6	iffalsed 4487	. 2 ⊢ (𝜑 → if(𝑋 = ∅, 0, inf(𝑀, ℝ^, < )) = inf(𝑀, ℝ^, < ))
8	4, 7	eqtrd 2768	1 ⊢ (𝜑 → ((voln‘𝑋)‘𝐴) = inf(𝑀, ℝ^, < ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2113 ≠ wne 2929 ∃wrex 3057 {crab 3396 ⊆ wss 3898 ∅c0 4282 ifcif 4476 ∪ ciun 4943 ↦ cmpt 5176 × cxp 5619 ∘ ccom 5625 ‘cfv 6488 (class class class)co 7354 ↑_m cmap 8758 Xcixp 8829 Fincfn 8877 infcinf 9334 ℝcr 11014 0cc0 11015 ℝ^*cxr 11154 < clt 11155 ℕcn 12134 [,)cico 13251 ∏cprod 15814 volcvol 25394 Σ^{^}csumge0 46487 voln*covoln 46661

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 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7676 ax-cnex 11071 ax-resscn 11072 ax-1cn 11073 ax-icn 11074 ax-addcl 11075 ax-mulcl 11077 ax-i2m1 11083 ax-pre-lttri 11089 ax-pre-lttrn 11090

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 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-nel 3034 df-ral 3049 df-rex 3058 df-rmo 3347 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-nul 4283 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4861 df-iun 4945 df-br 5096 df-opab 5158 df-mpt 5177 df-id 5516 df-po 5529 df-so 5530 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6255 df-iota 6444 df-fun 6490 df-fn 6491 df-f 6492 df-f1 6493 df-fo 6494 df-f1o 6495 df-fv 6496 df-ov 7357 df-oprab 7358 df-mpo 7359 df-frecs 8219 df-wrecs 8250 df-recs 8299 df-rdg 8337 df-er 8630 df-ixp 8830 df-en 8878 df-dom 8879 df-sdom 8880 df-sup 9335 df-inf 9336 df-pnf 11157 df-mnf 11158 df-xr 11159 df-ltxr 11160 df-seq 13913 df-prod 15815 df-ovoln 46662

This theorem is referenced by: ovnlecvr 46683 ovnsslelem 46685 ovnlerp 46687 ovnhoilem2 46727 ovnlecvr2 46735

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