ovnn0val - Mathbox for Glauco Siliprandi

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Theorem ovnn0val 46595

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 46589	. 2 ⊢ (𝜑 → ((voln‘𝑋)‘𝐴) = if(𝑋 = ∅, 0, inf(𝑀, ℝ^, < )))
5		ovnn0val.2	. . . 4 ⊢ (𝜑 → 𝑋 ≠ ∅)
6	5	neneqd 2933	. . 3 ⊢ (𝜑 → ¬ 𝑋 = ∅)
7	6	iffalsed 4486	. 2 ⊢ (𝜑 → if(𝑋 = ∅, 0, inf(𝑀, ℝ^, < )) = inf(𝑀, ℝ^, < ))
8	4, 7	eqtrd 2766	1 ⊢ (𝜑 → ((voln‘𝑋)‘𝐴) = inf(𝑀, ℝ^, < ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2111 ≠ wne 2928 ∃wrex 3056 {crab 3395 ⊆ wss 3902 ∅c0 4283 ifcif 4475 ∪ ciun 4941 ↦ cmpt 5172 × cxp 5614 ∘ ccom 5620 ‘cfv 6481 (class class class)co 7346 ↑_m cmap 8750 Xcixp 8821 Fincfn 8869 infcinf 9325 ℝcr 11005 0cc0 11006 ℝ^*cxr 11145 < clt 11146 ℕcn 12125 [,)cico 13247 ∏cprod 15810 volcvol 25392 Σ^{^}csumge0 46406 voln*covoln 46580

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 5217 ax-sep 5234 ax-nul 5244 ax-pow 5303 ax-pr 5370 ax-un 7668 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-mulcl 11068 ax-i2m1 11074 ax-pre-lttri 11080 ax-pre-lttrn 11081

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 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-nul 4284 df-if 4476 df-pw 4552 df-sn 4577 df-pr 4579 df-op 4583 df-uni 4860 df-iun 4943 df-br 5092 df-opab 5154 df-mpt 5173 df-id 5511 df-po 5524 df-so 5525 df-xp 5622 df-rel 5623 df-cnv 5624 df-co 5625 df-dm 5626 df-rn 5627 df-res 5628 df-ima 5629 df-pred 6248 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-ov 7349 df-oprab 7350 df-mpo 7351 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-er 8622 df-ixp 8822 df-en 8870 df-dom 8871 df-sdom 8872 df-sup 9326 df-inf 9327 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-seq 13909 df-prod 15811 df-ovoln 46581

This theorem is referenced by: ovnlecvr 46602 ovnsslelem 46604 ovnlerp 46606 ovnhoilem2 46646 ovnlecvr2 46654

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