ovolval4 - Mathbox for Glauco Siliprandi

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Theorem ovolval4 44440

Description: The value of the Lebesgue outer measure for subsets of the reals. Similar to ovolval3 44436, 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 6817	. . . 4 ⊢ (𝑘 = 𝑛 → (1^st ‘(𝑓‘𝑘)) = (1^st ‘(𝑓‘𝑛)))
4		2fveq3 6817	. . . . . 6 ⊢ (𝑘 = 𝑛 → (2^nd ‘(𝑓‘𝑘)) = (2^nd ‘(𝑓‘𝑛)))
5	3, 4	breq12d 5100	. . . . 5 ⊢ (𝑘 = 𝑛 → ((1^st ‘(𝑓‘𝑘)) ≤ (2^nd ‘(𝑓‘𝑘)) ↔ (1^st ‘(𝑓‘𝑛)) ≤ (2^nd ‘(𝑓‘𝑛))))
6	5, 4, 3	ifbieq12d 4499	. . . 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 4823	. . 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 5200	. 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 44439	1 ⊢ (𝜑 → (vol‘𝐴) = inf(𝑀, ℝ^, < ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1540 ∃wrex 3071 {crab 3404 ⊆ wss 3897 ifcif 4471 ⟨cop 4577 ∪ cuni 4850 class class class wbr 5087 ↦ cmpt 5170 × cxp 5606 ran crn 5609 ∘ ccom 5612 ‘cfv 6466 (class class class)co 7317 1^st c1st 7876 2^nd c2nd 7877 ↑_m cmap 8665 infcinf 9277 ℝcr 10950 ℝ^*cxr 11088 < clt 11089 ≤ cle 11090 ℕcn 12053 (,)cioo 13159 vol*covol 24709 volcvol 24710 Σ^{^}csumge0 44151

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2708 ax-rep 5224 ax-sep 5238 ax-nul 5245 ax-pow 5303 ax-pr 5367 ax-un 7630 ax-inf2 9477 ax-cnex 11007 ax-resscn 11008 ax-1cn 11009 ax-icn 11010 ax-addcl 11011 ax-addrcl 11012 ax-mulcl 11013 ax-mulrcl 11014 ax-mulcom 11015 ax-addass 11016 ax-mulass 11017 ax-distr 11018 ax-i2m1 11019 ax-1ne0 11020 ax-1rid 11021 ax-rnegex 11022 ax-rrecex 11023 ax-cnre 11024 ax-pre-lttri 11025 ax-pre-lttrn 11026 ax-pre-ltadd 11027 ax-pre-mulgt0 11028 ax-pre-sup 11029

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3350 df-reu 3351 df-rab 3405 df-v 3443 df-sbc 3727 df-csb 3843 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3916 df-nul 4268 df-if 4472 df-pw 4547 df-sn 4572 df-pr 4574 df-op 4578 df-uni 4851 df-int 4893 df-iun 4939 df-br 5088 df-opab 5150 df-mpt 5171 df-tr 5205 df-id 5507 df-eprel 5513 df-po 5521 df-so 5522 df-fr 5563 df-se 5564 df-we 5565 df-xp 5614 df-rel 5615 df-cnv 5616 df-co 5617 df-dm 5618 df-rn 5619 df-res 5620 df-ima 5621 df-pred 6225 df-ord 6292 df-on 6293 df-lim 6294 df-suc 6295 df-iota 6418 df-fun 6468 df-fn 6469 df-f 6470 df-f1 6471 df-fo 6472 df-f1o 6473 df-fv 6474 df-isom 6475 df-riota 7274 df-ov 7320 df-oprab 7321 df-mpo 7322 df-of 7575 df-om 7760 df-1st 7878 df-2nd 7879 df-frecs 8146 df-wrecs 8177 df-recs 8251 df-rdg 8290 df-1o 8346 df-2o 8347 df-er 8548 df-map 8667 df-pm 8668 df-en 8784 df-dom 8785 df-sdom 8786 df-fin 8787 df-fi 9247 df-sup 9278 df-inf 9279 df-oi 9346 df-dju 9737 df-card 9775 df-pnf 11091 df-mnf 11092 df-xr 11093 df-ltxr 11094 df-le 11095 df-sub 11287 df-neg 11288 df-div 11713 df-nn 12054 df-2 12116 df-3 12117 df-n0 12314 df-z 12400 df-uz 12663 df-q 12769 df-rp 12811 df-xneg 12928 df-xadd 12929 df-xmul 12930 df-ioo 13163 df-ico 13165 df-icc 13166 df-fz 13320 df-fzo 13463 df-fl 13592 df-seq 13802 df-exp 13863 df-hash 14125 df-cj 14889 df-re 14890 df-im 14891 df-sqrt 15025 df-abs 15026 df-clim 15276 df-rlim 15277 df-sum 15477 df-rest 17210 df-topgen 17231 df-psmet 20672 df-xmet 20673 df-met 20674 df-bl 20675 df-mopn 20676 df-top 22126 df-topon 22143 df-bases 22179 df-cmp 22621 df-ovol 24711 df-vol 24712 df-sumge0 44152

This theorem is referenced by: ovolval5 44444

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