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Theorem ovolval4 43229
 Description: The value of the Lebesgue outer measure for subsets of the reals. Similar to ovolval3 43225, but here 𝑓 may represent unordered interval bounds. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
Hypotheses
Ref Expression
ovolval4.a (𝜑𝐴 ⊆ 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.
StepHypRef Expression
1 ovolval4.a . 2 (𝜑𝐴 ⊆ ℝ)
2 ovolval4.m . 2 𝑀 = {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ ((ℝ × ℝ) ↑m ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))}
3 2fveq3 6657 . . . 4 (𝑘 = 𝑛 → (1st ‘(𝑓𝑘)) = (1st ‘(𝑓𝑛)))
4 2fveq3 6657 . . . . . 6 (𝑘 = 𝑛 → (2nd ‘(𝑓𝑘)) = (2nd ‘(𝑓𝑛)))
53, 4breq12d 5055 . . . . 5 (𝑘 = 𝑛 → ((1st ‘(𝑓𝑘)) ≤ (2nd ‘(𝑓𝑘)) ↔ (1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛))))
65, 4, 3ifbieq12d 4466 . . . 4 (𝑘 = 𝑛 → if((1st ‘(𝑓𝑘)) ≤ (2nd ‘(𝑓𝑘)), (2nd ‘(𝑓𝑘)), (1st ‘(𝑓𝑘))) = if((1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛)), (2nd ‘(𝑓𝑛)), (1st ‘(𝑓𝑛))))
73, 6opeq12d 4786 . . 3 (𝑘 = 𝑛 → ⟨(1st ‘(𝑓𝑘)), if((1st ‘(𝑓𝑘)) ≤ (2nd ‘(𝑓𝑘)), (2nd ‘(𝑓𝑘)), (1st ‘(𝑓𝑘)))⟩ = ⟨(1st ‘(𝑓𝑛)), if((1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛)), (2nd ‘(𝑓𝑛)), (1st ‘(𝑓𝑛)))⟩)
87cbvmptv 5145 . 2 (𝑘 ∈ ℕ ↦ ⟨(1st ‘(𝑓𝑘)), if((1st ‘(𝑓𝑘)) ≤ (2nd ‘(𝑓𝑘)), (2nd ‘(𝑓𝑘)), (1st ‘(𝑓𝑘)))⟩) = (𝑛 ∈ ℕ ↦ ⟨(1st ‘(𝑓𝑛)), if((1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛)), (2nd ‘(𝑓𝑛)), (1st ‘(𝑓𝑛)))⟩)
91, 2, 8ovolval4lem2 43228 1 (𝜑 → (vol*‘𝐴) = inf(𝑀, ℝ*, < ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538  ∃wrex 3131  {crab 3134   ⊆ wss 3908  ifcif 4439  ⟨cop 4545  ∪ cuni 4813   class class class wbr 5042   ↦ cmpt 5122   × cxp 5530  ran crn 5533   ∘ ccom 5536  ‘cfv 6334  (class class class)co 7140  1st c1st 7673  2nd c2nd 7674   ↑m cmap 8393  infcinf 8893  ℝcr 10525  ℝ*cxr 10663   < clt 10664   ≤ cle 10665  ℕcn 11625  (,)cioo 12726  vol*covol 24064  volcvol 24065  Σ^csumge0 42940 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 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446  ax-inf2 9092  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 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-nel 3116  df-ral 3135  df-rex 3136  df-reu 3137  df-rmo 3138  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-int 4852  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5437  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-se 5492  df-we 5493  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-isom 6343  df-riota 7098  df-ov 7143  df-oprab 7144  df-mpo 7145  df-of 7394  df-om 7566  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-2o 8090  df-oadd 8093  df-er 8276  df-map 8395  df-pm 8396  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-fi 8863  df-sup 8894  df-inf 8895  df-oi 8962  df-dju 9318  df-card 9356  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 14449  df-re 14450  df-im 14451  df-sqrt 14585  df-abs 14586  df-clim 14836  df-rlim 14837  df-sum 15034  df-rest 16687  df-topgen 16708  df-psmet 20081  df-xmet 20082  df-met 20083  df-bl 20084  df-mopn 20085  df-top 21497  df-topon 21514  df-bases 21549  df-cmp 21990  df-ovol 24066  df-vol 24067  df-sumge0 42941 This theorem is referenced by:  ovolval5  43233
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