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Theorem ovolval 24665

Description: The value of the outer measure. (Contributed by Mario Carneiro, 16-Mar-2014.) (Revised by AV, 17-Sep-2020.)

**Hypothesis**
Ref	Expression
ovolval.1	⊢ 𝑀 = {𝑦 ∈ ℝ^* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^*, < ))}

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

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

Proof of Theorem ovolval Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		reex 10990	. . 3 ⊢ ℝ ∈ V
2	1	elpw2 5272	. 2 ⊢ (𝐴 ∈ 𝒫 ℝ ↔ 𝐴 ⊆ ℝ)
3		cleq1lem 14721	. . . . . . 7 ⊢ (𝑥 = 𝐴 → ((𝑥 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^, < )) ↔ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^, < ))))
4	3	rexbidv 3169	. . . . . 6 ⊢ (𝑥 = 𝐴 → (∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝑥 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^, < )) ↔ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^, < ))))
5	4	rabbidv 3416	. . . . 5 ⊢ (𝑥 = 𝐴 → {𝑦 ∈ ℝ^* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝑥 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^, < ))} = {𝑦 ∈ ℝ^ ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^*, < ))})
6		ovolval.1	. . . . 5 ⊢ 𝑀 = {𝑦 ∈ ℝ^* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^*, < ))}
7	5, 6	eqtr4di 2791	. . . 4 ⊢ (𝑥 = 𝐴 → {𝑦 ∈ ℝ^* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝑥 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^*, < ))} = 𝑀)
8	7	infeq1d 9264	. . 3 ⊢ (𝑥 = 𝐴 → inf({𝑦 ∈ ℝ^* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝑥 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^, < ))}, ℝ^, < ) = inf(𝑀, ℝ^*, < ))
9		df-ovol 24656	. . 3 ⊢ vol* = (𝑥 ∈ 𝒫 ℝ ↦ inf({𝑦 ∈ ℝ^* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝑥 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^, < ))}, ℝ^, < ))
10		xrltso 12903	. . . 4 ⊢ < Or ℝ^*
11	10	infex 9280	. . 3 ⊢ inf(𝑀, ℝ^*, < ) ∈ V
12	8, 9, 11	fvmpt 6895	. 2 ⊢ (𝐴 ∈ 𝒫 ℝ → (vol‘𝐴) = inf(𝑀, ℝ^, < ))
13	2, 12	sylbir 234	1 ⊢ (𝐴 ⊆ ℝ → (vol‘𝐴) = inf(𝑀, ℝ^, < ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2101 ∃wrex 3068 {crab 3221 ∩ cin 3888 ⊆ wss 3889 𝒫 cpw 4536 ∪ cuni 4841 × cxp 5589 ran crn 5592 ∘ ccom 5595 ‘cfv 6447 (class class class)co 7295 ↑_m cmap 8635 supcsup 9227 infcinf 9228 ℝcr 10898 1c1 10900 + caddc 10902 ℝ^*cxr 11036 < clt 11037 ≤ cle 11038 − cmin 11233 ℕcn 12001 (,)cioo 13107 seqcseq 13749 abscabs 14973 vol*covol 24654

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2103 ax-9 2111 ax-10 2132 ax-11 2149 ax-12 2166 ax-ext 2704 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7608 ax-cnex 10955 ax-resscn 10956 ax-pre-lttri 10973 ax-pre-lttrn 10974

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2063 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2884 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3222 df-rab 3224 df-v 3436 df-sbc 3719 df-csb 3835 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-nul 4260 df-if 4463 df-pw 4538 df-sn 4565 df-pr 4567 df-op 4571 df-uni 4842 df-br 5078 df-opab 5140 df-mpt 5161 df-id 5491 df-po 5505 df-so 5506 df-xp 5597 df-rel 5598 df-cnv 5599 df-co 5600 df-dm 5601 df-rn 5602 df-res 5603 df-ima 5604 df-iota 6399 df-fun 6449 df-fn 6450 df-f 6451 df-f1 6452 df-fo 6453 df-f1o 6454 df-fv 6455 df-er 8518 df-en 8754 df-dom 8755 df-sdom 8756 df-sup 9229 df-inf 9230 df-pnf 11039 df-mnf 11040 df-xr 11041 df-ltxr 11042 df-ovol 24656

This theorem is referenced by: ovolcl 24670 ovollb 24671 ovolgelb 24672 ovolge0 24673 ovolsslem 24676 ovolshft 24703 ovolicc2 24714 ismblfin 35846 ovolval2 44218

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