ovnf - Mathbox for Glauco Siliprandi

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Theorem ovnf 43991

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 13120	. . . . 5 ⊢ 0 ∈ (0[,]+∞)
2	1	a1i 11	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝒫 (ℝ ↑_m 𝑋)) → 0 ∈ (0[,]+∞))
3		0xr 10953	. . . . . 6 ⊢ 0 ∈ ℝ^*
4	3	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝒫 (ℝ ↑_m 𝑋)) → 0 ∈ ℝ^*)
5		pnfxr 10960	. . . . . 6 ⊢ +∞ ∈ ℝ^*
6	5	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝒫 (ℝ ↑_m 𝑋)) → +∞ ∈ ℝ^*)
7		ovnf.1	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ Fin)
8	7	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝒫 (ℝ ↑_m 𝑋)) → 𝑋 ∈ Fin)
9		elpwi 4539	. . . . . . 7 ⊢ (𝑦 ∈ 𝒫 (ℝ ↑_m 𝑋) → 𝑦 ⊆ (ℝ ↑_m 𝑋))
10	9	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝒫 (ℝ ↑_m 𝑋)) → 𝑦 ⊆ (ℝ ↑_m 𝑋))
11		eqid 2738	. . . . . 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 43985	. . . . 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 43987	. . . . . 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 4266	. . . . 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 42962	. . . 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 4502	. . 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 2738	. . 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 6970	. 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 43969	. . 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 6569	. 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 256	1 ⊢ (𝜑 → (voln*‘𝑋):𝒫 (ℝ ↑_m 𝑋)⟶(0[,]+∞))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2108 ∃wrex 3064 {crab 3067 ⊆ wss 3883 ∅c0 4253 ifcif 4456 𝒫 cpw 4530 ∪ ciun 4921 ↦ cmpt 5153 × cxp 5578 ∘ ccom 5584 ⟶wf 6414 ‘cfv 6418 (class class class)co 7255 ↑_m cmap 8573 Xcixp 8643 Fincfn 8691 infcinf 9130 ℝcr 10801 0cc0 10802 +∞cpnf 10937 ℝ^*cxr 10939 < clt 10940 ℕcn 11903 [,)cico 13010 [,]cicc 13011 ∏cprod 15543 volcvol 24532 Σ^{^}csumge0 43790 voln*covoln 43964

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-inf2 9329 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-of 7511 df-om 7688 df-1st 7804 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-2o 8268 df-er 8456 df-map 8575 df-pm 8576 df-ixp 8644 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-fi 9100 df-sup 9131 df-inf 9132 df-oi 9199 df-dju 9590 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-n0 12164 df-z 12250 df-uz 12512 df-q 12618 df-rp 12660 df-xneg 12777 df-xadd 12778 df-xmul 12779 df-ioo 13012 df-ico 13014 df-icc 13015 df-fz 13169 df-fzo 13312 df-fl 13440 df-seq 13650 df-exp 13711 df-hash 13973 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-clim 15125 df-rlim 15126 df-sum 15326 df-prod 15544 df-rest 17050 df-topgen 17071 df-psmet 20502 df-xmet 20503 df-met 20504 df-bl 20505 df-mopn 20506 df-top 21951 df-topon 21968 df-bases 22004 df-cmp 22446 df-ovol 24533 df-vol 24534 df-sumge0 43791 df-ovoln 43965

This theorem is referenced by: ovn0 43994 ovncl 43995 ovn02 43996 ovnome 44001 dmovn 44032 hspmbl 44057

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