ovnf - Mathbox for Glauco Siliprandi

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Theorem ovnf 47394

Description: The Lebesgue outer measure is a function that maps sets to nonnegative extended reals. This is step (a)(i) of the proof of Proposition 115D (a) of [Fremlin1] p. 30. (Contributed by Glauco Siliprandi, 11-Oct-2020.)

**Hypothesis**
Ref	Expression
ovnf.1	⊢ (𝜑 → 𝑋 ∈ Fin)

**Assertion**
Ref	Expression
ovnf	⊢ (𝜑 → (voln*‘𝑋):𝒫 (ℝ ↑_m 𝑋)⟶(0[,]+∞))

Proof of Theorem ovnf Dummy variables 𝑖 𝑗 𝑘 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		0e0iccpnf 13515	. . . . 5 ⊢ 0 ∈ (0[,]+∞)
2	1	a1i 11	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝒫 (ℝ ↑_m 𝑋)) → 0 ∈ (0[,]+∞))
3		0xr 11283	. . . . . 6 ⊢ 0 ∈ ℝ^*
4	3	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝒫 (ℝ ↑_m 𝑋)) → 0 ∈ ℝ^*)
5		pnfxr 11290	. . . . . 6 ⊢ +∞ ∈ ℝ^*
6	5	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝒫 (ℝ ↑_m 𝑋)) → +∞ ∈ ℝ^*)
7		ovnf.1	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ Fin)
8	7	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝒫 (ℝ ↑_m 𝑋)) → 𝑋 ∈ Fin)
9		elpwi 4564	. . . . . . 7 ⊢ (𝑦 ∈ 𝒫 (ℝ ↑_m 𝑋) → 𝑦 ⊆ (ℝ ↑_m 𝑋))
10	9	adantl 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝒫 (ℝ ↑_m 𝑋)) → 𝑦 ⊆ (ℝ ↑_m 𝑋))
11		eqid 2760	. . . . . 6 ⊢ {𝑧 ∈ ℝ^* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ)(𝑦 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))} = {𝑧 ∈ ℝ^* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ)(𝑦 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))}
12	8, 10, 11	ovnsupge0 47388	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝒫 (ℝ ↑_m 𝑋)) → {𝑧 ∈ ℝ^* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ)(𝑦 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))} ⊆ (0[,]+∞))
13	8, 10, 11	ovnpnfelsup 47390	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝒫 (ℝ ↑_m 𝑋)) → +∞ ∈ {𝑧 ∈ ℝ^* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ)(𝑦 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))})
14	13	ne0d 4288	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝒫 (ℝ ↑_m 𝑋)) → {𝑧 ∈ ℝ^* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ)(𝑦 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))} ≠ ∅)
15	4, 6, 12, 14	inficc 46367	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝒫 (ℝ ↑_m 𝑋)) → inf({𝑧 ∈ ℝ^* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ)(𝑦 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))}, ℝ^*, < ) ∈ (0[,]+∞))
16	2, 15	ifcld 4529	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝒫 (ℝ ↑_m 𝑋)) → if(𝑋 = ∅, 0, inf({𝑧 ∈ ℝ^* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ)(𝑦 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))}, ℝ^*, < )) ∈ (0[,]+∞))
17		eqid 2760	. . 3 ⊢ (𝑦 ∈ 𝒫 (ℝ ↑_m 𝑋) ↦ if(𝑋 = ∅, 0, inf({𝑧 ∈ ℝ^* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ)(𝑦 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))}, ℝ^, < ))) = (𝑦 ∈ 𝒫 (ℝ ↑_m 𝑋) ↦ if(𝑋 = ∅, 0, inf({𝑧 ∈ ℝ^ ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ)(𝑦 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))}, ℝ^*, < )))
18	16, 17	fmptd 7108	. 2 ⊢ (𝜑 → (𝑦 ∈ 𝒫 (ℝ ↑_m 𝑋) ↦ if(𝑋 = ∅, 0, inf({𝑧 ∈ ℝ^* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ)(𝑦 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))}, ℝ^*, < ))):𝒫 (ℝ ↑_m 𝑋)⟶(0[,]+∞))
19	7	ovnval 47372	. . 3 ⊢ (𝜑 → (voln‘𝑋) = (𝑦 ∈ 𝒫 (ℝ ↑_m 𝑋) ↦ if(𝑋 = ∅, 0, inf({𝑧 ∈ ℝ^ ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ)(𝑦 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))}, ℝ^*, < ))))
20	19	feq1d 6685	. 2 ⊢ (𝜑 → ((voln‘𝑋):𝒫 (ℝ ↑_m 𝑋)⟶(0[,]+∞) ↔ (𝑦 ∈ 𝒫 (ℝ ↑_m 𝑋) ↦ if(𝑋 = ∅, 0, inf({𝑧 ∈ ℝ^ ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ)(𝑦 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))}, ℝ^*, < ))):𝒫 (ℝ ↑_m 𝑋)⟶(0[,]+∞)))
21	18, 20	mpbird 260	1 ⊢ (𝜑 → (voln*‘𝑋):𝒫 (ℝ ↑_m 𝑋)⟶(0[,]+∞))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ∃wrex 3086 {crab 3412 ⊆ wss 3899 ∅c0 4279 ifcif 4482 𝒫 cpw 4557 ∪ ciun 4951 ↦ cmpt 5186 × cxp 5653 ∘ ccom 5659 ⟶wf 6529 ‘cfv 6533 (class class class)co 7414 ↑_m cmap 8829 Xcixp 8907 Fincfn 8955 infcinf 9414 ℝcr 11126 0cc0 11127 +∞cpnf 11267 ℝ^*cxr 11269 < clt 11270 ℕcn 12260 [,)cico 13403 [,]cicc 13404 ∏cprod 15995 volcvol 25694 Σ^{^}csumge0 47193 voln*covoln 47367

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7737 ax-inf2 9623 ax-cnex 11183 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203 ax-pre-mulgt0 11204 ax-pre-sup 11205

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7371 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7679 df-om 7864 df-1st 7987 df-2nd 7988 df-frecs 8281 df-wrecs 8312 df-recs 8361 df-rdg 8400 df-1o 8458 df-2o 8459 df-er 8699 df-map 8831 df-pm 8832 df-ixp 8908 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-fi 9384 df-sup 9415 df-inf 9416 df-oi 9485 df-dju 9909 df-card 9947 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11470 df-neg 11471 df-div 11899 df-nn 12261 df-2 12330 df-3 12331 df-n0 12532 df-z 12619 df-uz 12891 df-q 13001 df-rp 13046 df-xneg 13166 df-xadd 13167 df-xmul 13168 df-ioo 13405 df-ico 13407 df-icc 13408 df-fz 13565 df-fzo 13713 df-fl 13856 df-seq 14069 df-exp 14129 df-hash 14398 df-cj 15189 df-re 15190 df-im 15191 df-sqrt 15325 df-abs 15326 df-clim 15578 df-rlim 15579 df-sum 15777 df-prod 15996 df-rest 17510 df-topgen 17531 df-psmet 21580 df-xmet 21581 df-met 21582 df-bl 21583 df-mopn 21584 df-top 23122 df-topon 23139 df-bases 23174 df-cmp 23615 df-ovol 25695 df-vol 25696 df-sumge0 47194 df-ovoln 47368

This theorem is used by: ovn0 47397 ovncl 47398 ovn02 47399 ovnome 47404 dmovn 47435 hspmbl 47460

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