ovnn0val - Mathbox for Glauco Siliprandi

	Mathbox for Glauco Siliprandi		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > ovnn0val		Structured version Visualization version GIF version

Theorem ovnn0val 46077

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 46071	. 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 4541	. 2 ⊢ (𝜑 → if(𝑋 = ∅, 0, inf(𝑀, ℝ^, < )) = inf(𝑀, ℝ^, < ))
8	4, 7	eqtrd 2765	1 ⊢ (𝜑 → ((voln‘𝑋)‘𝐴) = inf(𝑀, ℝ^, < ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ≠ wne 2929 ∃wrex 3059 {crab 3418 ⊆ wss 3944 ∅c0 4322 ifcif 4530 ∪ ciun 4997 ↦ cmpt 5232 × cxp 5676 ∘ ccom 5682 ‘cfv 6549 (class class class)co 7419 ↑_m cmap 8845 Xcixp 8916 Fincfn 8964 infcinf 9466 ℝcr 11139 0cc0 11140 ℝ^*cxr 11279 < clt 11280 ℕcn 12245 [,)cico 13361 ∏cprod 15885 volcvol 25436 Σ^{^}csumge0 45888 voln*covoln 46062

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-mulcl 11202 ax-i2m1 11208 ax-pre-lttri 11214 ax-pre-lttrn 11215

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-po 5590 df-so 5591 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-ov 7422 df-oprab 7423 df-mpo 7424 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-sup 9467 df-inf 9468 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-seq 14003 df-prod 15886 df-ovoln 46063

This theorem is referenced by: ovnlecvr 46084 ovnsslelem 46086 ovnlerp 46088 ovnhoilem2 46128 ovnlecvr2 46136

W3C validator

	
		OSZAR »