ovolval4 - Mathbox for Glauco Siliprandi

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Theorem ovolval4 47425

Description: The value of the Lebesgue outer measure for subsets of the reals. Similar to ovolval3 47421, 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 6890	. . . 4 ⊢ (𝑘 = 𝑛 → (1^st ‘(𝑓‘𝑘)) = (1^st ‘(𝑓‘𝑛)))
4		2fveq3 6890	. . . . . 6 ⊢ (𝑘 = 𝑛 → (2^nd ‘(𝑓‘𝑘)) = (2^nd ‘(𝑓‘𝑛)))
5	3, 4	breq12d 5124	. . . . 5 ⊢ (𝑘 = 𝑛 → ((1^st ‘(𝑓‘𝑘)) ≤ (2^nd ‘(𝑓‘𝑘)) ↔ (1^st ‘(𝑓‘𝑛)) ≤ (2^nd ‘(𝑓‘𝑛))))
6	5, 4, 3	ifbieq12d 4518	. . . 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 4848	. . 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 5217	. 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 47424	1 ⊢ (𝜑 → (vol‘𝐴) = inf(𝑀, ℝ^, < ))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∃wrex 3091 {crab 3418 ⊆ wss 3906 ifcif 4489 ⟨cop 4597 ∪ cuni 4874 class class class wbr 5111 ↦ cmpt 5194 × cxp 5661 ran crn 5664 ∘ ccom 5667 ‘cfv 6540 (class class class)co 7419 1^st c1st 7990 2^nd c2nd 7991 ↑_m cmap 8830 infcinf 9408 ℝcr 11116 ℝ^*cxr 11259 < clt 11260 ≤ cle 11261 ℕcn 12250 (,)cioo 13390 vol*covol 25674 volcvol 25675 Σ^{^}csumge0 47136

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742 ax-inf2 9617 ax-cnex 11173 ax-resscn 11174 ax-1cn 11175 ax-icn 11176 ax-addcl 11177 ax-addrcl 11178 ax-mulcl 11179 ax-mulrcl 11180 ax-mulcom 11181 ax-addass 11182 ax-mulass 11183 ax-distr 11184 ax-i2m1 11185 ax-1ne0 11186 ax-1rid 11187 ax-rnegex 11188 ax-rrecex 11189 ax-cnre 11190 ax-pre-lttri 11191 ax-pre-lttrn 11192 ax-pre-ltadd 11193 ax-pre-mulgt0 11194 ax-pre-sup 11195

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4875 df-int 4915 df-iun 4960 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-se 5617 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7376 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7684 df-om 7869 df-1st 7992 df-2nd 7993 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-2o 8460 df-er 8700 df-map 8832 df-pm 8833 df-en 8950 df-dom 8951 df-sdom 8952 df-fin 8953 df-fi 9378 df-sup 9409 df-inf 9410 df-oi 9479 df-dju 9903 df-card 9941 df-pnf 11262 df-mnf 11263 df-xr 11264 df-ltxr 11265 df-le 11266 df-sub 11460 df-neg 11461 df-div 11889 df-nn 12251 df-2 12320 df-3 12321 df-n0 12522 df-z 12609 df-uz 12881 df-q 12991 df-rp 13035 df-xneg 13155 df-xadd 13156 df-xmul 13157 df-ioo 13394 df-ico 13396 df-icc 13397 df-fz 13554 df-fzo 13702 df-fl 13845 df-seq 14058 df-exp 14118 df-hash 14387 df-cj 15176 df-re 15177 df-im 15178 df-sqrt 15312 df-abs 15313 df-clim 15565 df-rlim 15566 df-sum 15764 df-rest 17499 df-topgen 17520 df-psmet 21566 df-xmet 21567 df-met 21568 df-bl 21569 df-mopn 21570 df-top 23103 df-topon 23120 df-bases 23155 df-cmp 23596 df-ovol 25676 df-vol 25677 df-sumge0 47137

This theorem is used by: ovolval5 47429

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