ovnn0val - Mathbox for Glauco Siliprandi

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Theorem ovnn0val 45026

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 45020	. 2 ⊢ (𝜑 → ((voln‘𝑋)‘𝐴) = if(𝑋 = ∅, 0, inf(𝑀, ℝ^, < )))
5		ovnn0val.2	. . . 4 ⊢ (𝜑 → 𝑋 ≠ ∅)
6	5	neneqd 2944	. . 3 ⊢ (𝜑 → ¬ 𝑋 = ∅)
7	6	iffalsed 4530	. 2 ⊢ (𝜑 → if(𝑋 = ∅, 0, inf(𝑀, ℝ^, < )) = inf(𝑀, ℝ^, < ))
8	4, 7	eqtrd 2771	1 ⊢ (𝜑 → ((voln‘𝑋)‘𝐴) = inf(𝑀, ℝ^, < ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2939 ∃wrex 3069 {crab 3429 ⊆ wss 3941 ∅c0 4315 ifcif 4519 ∪ ciun 4987 ↦ cmpt 5221 × cxp 5664 ∘ ccom 5670 ‘cfv 6529 (class class class)co 7390 ↑_m cmap 8800 Xcixp 8871 Fincfn 8919 infcinf 9415 ℝcr 11088 0cc0 11089 ℝ^*cxr 11226 < clt 11227 ℕcn 12191 [,)cico 13305 ∏cprod 15828 volcvol 24904 Σ^{^}csumge0 44837 voln*covoln 45011

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7705 ax-cnex 11145 ax-resscn 11146 ax-1cn 11147 ax-icn 11148 ax-addcl 11149 ax-mulcl 11151 ax-i2m1 11157 ax-pre-lttri 11163 ax-pre-lttrn 11164

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3430 df-v 3472 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4520 df-pw 4595 df-sn 4620 df-pr 4622 df-op 4626 df-uni 4899 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-po 5578 df-so 5579 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6286 df-iota 6481 df-fun 6531 df-fn 6532 df-f 6533 df-f1 6534 df-fo 6535 df-f1o 6536 df-fv 6537 df-ov 7393 df-oprab 7394 df-mpo 7395 df-frecs 8245 df-wrecs 8276 df-recs 8350 df-rdg 8389 df-er 8683 df-ixp 8872 df-en 8920 df-dom 8921 df-sdom 8922 df-sup 9416 df-inf 9417 df-pnf 11229 df-mnf 11230 df-xr 11231 df-ltxr 11232 df-seq 13946 df-prod 15829 df-ovoln 45012

This theorem is referenced by: ovnlecvr 45033 ovnsslelem 45035 ovnlerp 45037 ovnhoilem2 45077 ovnlecvr2 45085

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