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Theorem ovolval4 41389
 Description: The value of the Lebesgue outer measure for subsets of the reals. Similar to ovolval3 41385, 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 fveq2 6353 . . . . 5 (𝑘 = 𝑛 → (𝑓𝑘) = (𝑓𝑛))
43fveq2d 6357 . . . 4 (𝑘 = 𝑛 → (1st ‘(𝑓𝑘)) = (1st ‘(𝑓𝑛)))
53fveq2d 6357 . . . . . 6 (𝑘 = 𝑛 → (2nd ‘(𝑓𝑘)) = (2nd ‘(𝑓𝑛)))
64, 5breq12d 4817 . . . . 5 (𝑘 = 𝑛 → ((1st ‘(𝑓𝑘)) ≤ (2nd ‘(𝑓𝑘)) ↔ (1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛))))
76, 5, 4ifbieq12d 4257 . . . 4 (𝑘 = 𝑛 → if((1st ‘(𝑓𝑘)) ≤ (2nd ‘(𝑓𝑘)), (2nd ‘(𝑓𝑘)), (1st ‘(𝑓𝑘))) = if((1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛)), (2nd ‘(𝑓𝑛)), (1st ‘(𝑓𝑛))))
84, 7opeq12d 4561 . . 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)))⟩)
98cbvmptv 4902 . 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)))⟩)
101, 2, 9ovolval4lem2 41388 1 (𝜑 → (vol*‘𝐴) = inf(𝑀, ℝ*, < ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1632  ∃wrex 3051  {crab 3054   ⊆ wss 3715  ifcif 4230  ⟨cop 4327  ∪ cuni 4588   class class class wbr 4804   ↦ cmpt 4881   × cxp 5264  ran crn 5267   ∘ ccom 5270  ‘cfv 6049  (class class class)co 6814  1st c1st 7332  2nd c2nd 7333   ↑𝑚 cmap 8025  infcinf 8514  ℝcr 10147  ℝ*cxr 10285   < clt 10286   ≤ cle 10287  ℕcn 11232  (,)cioo 12388  vol*covol 23451  volcvol 23452  Σ^csumge0 41100 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-inf2 8713  ax-cnex 10204  ax-resscn 10205  ax-1cn 10206  ax-icn 10207  ax-addcl 10208  ax-addrcl 10209  ax-mulcl 10210  ax-mulrcl 10211  ax-mulcom 10212  ax-addass 10213  ax-mulass 10214  ax-distr 10215  ax-i2m1 10216  ax-1ne0 10217  ax-1rid 10218  ax-rnegex 10219  ax-rrecex 10220  ax-cnre 10221  ax-pre-lttri 10222  ax-pre-lttrn 10223  ax-pre-ltadd 10224  ax-pre-mulgt0 10225  ax-pre-sup 10226 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-se 5226  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-isom 6058  df-riota 6775  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-of 7063  df-om 7232  df-1st 7334  df-2nd 7335  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-1o 7730  df-2o 7731  df-oadd 7734  df-er 7913  df-map 8027  df-pm 8028  df-en 8124  df-dom 8125  df-sdom 8126  df-fin 8127  df-fi 8484  df-sup 8515  df-inf 8516  df-oi 8582  df-card 8975  df-cda 9202  df-pnf 10288  df-mnf 10289  df-xr 10290  df-ltxr 10291  df-le 10292  df-sub 10480  df-neg 10481  df-div 10897  df-nn 11233  df-2 11291  df-3 11292  df-n0 11505  df-z 11590  df-uz 11900  df-q 12002  df-rp 12046  df-xneg 12159  df-xadd 12160  df-xmul 12161  df-ioo 12392  df-ico 12394  df-icc 12395  df-fz 12540  df-fzo 12680  df-fl 12807  df-seq 13016  df-exp 13075  df-hash 13332  df-cj 14058  df-re 14059  df-im 14060  df-sqrt 14194  df-abs 14195  df-clim 14438  df-rlim 14439  df-sum 14636  df-rest 16305  df-topgen 16326  df-psmet 19960  df-xmet 19961  df-met 19962  df-bl 19963  df-mopn 19964  df-top 20921  df-topon 20938  df-bases 20972  df-cmp 21412  df-ovol 23453  df-vol 23454  df-sumge0 41101 This theorem is referenced by:  ovolval5  41393
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