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Theorem ovolval 25374

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 11159	. . 3 ⊢ ℝ ∈ V
2	1	elpw2 5289	. 2 ⊢ (𝐴 ∈ 𝒫 ℝ ↔ 𝐴 ⊆ ℝ)
3		cleq1lem 14948	. . . . . . 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 3157	. . . . . 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 3413	. . . . 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 2782	. . . 4 ⊢ (𝑥 = 𝐴 → {𝑦 ∈ ℝ^* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝑥 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^*, < ))} = 𝑀)
8	7	infeq1d 9429	. . 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 25365	. . 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 13101	. . . 4 ⊢ < Or ℝ^*
11	10	infex 9446	. . 3 ⊢ inf(𝑀, ℝ^*, < ) ∈ V
12	8, 9, 11	fvmpt 6968	. 2 ⊢ (𝐴 ∈ 𝒫 ℝ → (vol‘𝐴) = inf(𝑀, ℝ^, < ))
13	2, 12	sylbir 235	1 ⊢ (𝐴 ⊆ ℝ → (vol‘𝐴) = inf(𝑀, ℝ^, < ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ∃wrex 3053 {crab 3405 ∩ cin 3913 ⊆ wss 3914 𝒫 cpw 4563 ∪ cuni 4871 × cxp 5636 ran crn 5639 ∘ ccom 5642 ‘cfv 6511 (class class class)co 7387 ↑_m cmap 8799 supcsup 9391 infcinf 9392 ℝcr 11067 1c1 11069 + caddc 11071 ℝ^*cxr 11207 < clt 11208 ≤ cle 11209 − cmin 11405 ℕcn 12186 (,)cioo 13306 seqcseq 13966 abscabs 15200 vol*covol 25363

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-pre-lttri 11142 ax-pre-lttrn 11143

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-br 5108 df-opab 5170 df-mpt 5189 df-id 5533 df-po 5546 df-so 5547 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-er 8671 df-en 8919 df-dom 8920 df-sdom 8921 df-sup 9393 df-inf 9394 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-ovol 25365

This theorem is referenced by: ovolcl 25379 ovollb 25380 ovolgelb 25381 ovolge0 25382 ovolsslem 25385 ovolshft 25412 ovolicc2 25423 ismblfin 37655 ovolval2 46642

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