ovolval4 - Mathbox for Glauco Siliprandi

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Theorem ovolval4 43290

Description: The value of the Lebesgue outer measure for subsets of the reals. Similar to ovolval3 43286, but here 𝑓 may represent unordered interval bounds. (Contributed by Glauco Siliprandi, 3-Mar-2021.)

**Hypotheses**
Ref	Expression
ovolval4.a	⊢ (𝜑 → 𝐴 ⊆ ℝ)
ovolval4.m	⊢ 𝑀 = {𝑦 ∈ ℝ^* ∣ ∃𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑓)))}

**Assertion**
Ref	Expression
ovolval4	⊢ (𝜑 → (vol‘𝐴) = inf(𝑀, ℝ^, < ))

Distinct variable groups: 𝐴,𝑓,𝑦 𝜑,𝑦 Allowed substitution hints: 𝜑(𝑓) 𝑀(𝑦,𝑓)

Proof of Theorem ovolval4 Dummy variables 𝑘 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ovolval4.a	. 2 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
2		ovolval4.m	. 2 ⊢ 𝑀 = {𝑦 ∈ ℝ^* ∣ ∃𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^{^}‘((vol ∘ (,)) ∘ 𝑓)))}
3		2fveq3 6650	. . . 4 ⊢ (𝑘 = 𝑛 → (1^st ‘(𝑓‘𝑘)) = (1^st ‘(𝑓‘𝑛)))
4		2fveq3 6650	. . . . . 6 ⊢ (𝑘 = 𝑛 → (2^nd ‘(𝑓‘𝑘)) = (2^nd ‘(𝑓‘𝑛)))
5	3, 4	breq12d 5043	. . . . 5 ⊢ (𝑘 = 𝑛 → ((1^st ‘(𝑓‘𝑘)) ≤ (2^nd ‘(𝑓‘𝑘)) ↔ (1^st ‘(𝑓‘𝑛)) ≤ (2^nd ‘(𝑓‘𝑛))))
6	5, 4, 3	ifbieq12d 4452	. . . 4 ⊢ (𝑘 = 𝑛 → if((1^st ‘(𝑓‘𝑘)) ≤ (2^nd ‘(𝑓‘𝑘)), (2^nd ‘(𝑓‘𝑘)), (1^st ‘(𝑓‘𝑘))) = if((1^st ‘(𝑓‘𝑛)) ≤ (2^nd ‘(𝑓‘𝑛)), (2^nd ‘(𝑓‘𝑛)), (1^st ‘(𝑓‘𝑛))))
7	3, 6	opeq12d 4773	. . 3 ⊢ (𝑘 = 𝑛 → ⟨(1^st ‘(𝑓‘𝑘)), if((1^st ‘(𝑓‘𝑘)) ≤ (2^nd ‘(𝑓‘𝑘)), (2^nd ‘(𝑓‘𝑘)), (1^st ‘(𝑓‘𝑘)))⟩ = ⟨(1^st ‘(𝑓‘𝑛)), if((1^st ‘(𝑓‘𝑛)) ≤ (2^nd ‘(𝑓‘𝑛)), (2^nd ‘(𝑓‘𝑛)), (1^st ‘(𝑓‘𝑛)))⟩)
8	7	cbvmptv 5133	. 2 ⊢ (𝑘 ∈ ℕ ↦ ⟨(1^st ‘(𝑓‘𝑘)), if((1^st ‘(𝑓‘𝑘)) ≤ (2^nd ‘(𝑓‘𝑘)), (2^nd ‘(𝑓‘𝑘)), (1^st ‘(𝑓‘𝑘)))⟩) = (𝑛 ∈ ℕ ↦ ⟨(1^st ‘(𝑓‘𝑛)), if((1^st ‘(𝑓‘𝑛)) ≤ (2^nd ‘(𝑓‘𝑛)), (2^nd ‘(𝑓‘𝑛)), (1^st ‘(𝑓‘𝑛)))⟩)
9	1, 2, 8	ovolval4lem2 43289	1 ⊢ (𝜑 → (vol‘𝐴) = inf(𝑀, ℝ^, < ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∃wrex 3107 {crab 3110 ⊆ wss 3881 ifcif 4425 ⟨cop 4531 ∪ cuni 4800 class class class wbr 5030 ↦ cmpt 5110 × cxp 5517 ran crn 5520 ∘ ccom 5523 ‘cfv 6324 (class class class)co 7135 1^st c1st 7669 2^nd c2nd 7670 ↑_m cmap 8389 infcinf 8889 ℝcr 10525 ℝ^*cxr 10663 < clt 10664 ≤ cle 10665 ℕcn 11625 (,)cioo 12726 vol*covol 24066 volcvol 24067 Σ^{^}csumge0 43001

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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-map 8391 df-pm 8392 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fi 8859 df-sup 8890 df-inf 8891 df-oi 8958 df-dju 9314 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-ioo 12730 df-ico 12732 df-icc 12733 df-fz 12886 df-fzo 13029 df-fl 13157 df-seq 13365 df-exp 13426 df-hash 13687 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-clim 14837 df-rlim 14838 df-sum 15035 df-rest 16688 df-topgen 16709 df-psmet 20083 df-xmet 20084 df-met 20085 df-bl 20086 df-mopn 20087 df-top 21499 df-topon 21516 df-bases 21551 df-cmp 21992 df-ovol 24068 df-vol 24069 df-sumge0 43002

This theorem is referenced by: ovolval5 43294

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