ovnn0val - Mathbox for Glauco Siliprandi

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Theorem ovnn0val 46906

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 46900	. 2 ⊢ (𝜑 → ((voln‘𝑋)‘𝐴) = if(𝑋 = ∅, 0, inf(𝑀, ℝ^, < )))
5		ovnn0val.2	. . . 4 ⊢ (𝜑 → 𝑋 ≠ ∅)
6	5	neneqd 2938	. . 3 ⊢ (𝜑 → ¬ 𝑋 = ∅)
7	6	iffalsed 4492	. 2 ⊢ (𝜑 → if(𝑋 = ∅, 0, inf(𝑀, ℝ^, < )) = inf(𝑀, ℝ^, < ))
8	4, 7	eqtrd 2772	1 ⊢ (𝜑 → ((voln‘𝑋)‘𝐴) = inf(𝑀, ℝ^, < ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 ∃wrex 3062 {crab 3401 ⊆ wss 3903 ∅c0 4287 ifcif 4481 ∪ ciun 4948 ↦ cmpt 5181 × cxp 5630 ∘ ccom 5636 ‘cfv 6500 (class class class)co 7368 ↑_m cmap 8775 Xcixp 8847 Fincfn 8895 infcinf 9356 ℝcr 11037 0cc0 11038 ℝ^*cxr 11177 < clt 11178 ℕcn 12157 [,)cico 13275 ∏cprod 15838 volcvol 25432 Σ^{^}csumge0 46717 voln*covoln 46891

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-mulcl 11100 ax-i2m1 11106 ax-pre-lttri 11112 ax-pre-lttrn 11113

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-id 5527 df-po 5540 df-so 5541 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-ov 7371 df-oprab 7372 df-mpo 7373 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-er 8645 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-sup 9357 df-inf 9358 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-seq 13937 df-prod 15839 df-ovoln 46892

This theorem is referenced by: ovnlecvr 46913 ovnsslelem 46915 ovnlerp 46917 ovnhoilem2 46957 ovnlecvr2 46965

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