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Theorem ovolval 25527

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 11275	. . 3 ⊢ ℝ ∈ V
2	1	elpw2 5352	. 2 ⊢ (𝐴 ∈ 𝒫 ℝ ↔ 𝐴 ⊆ ℝ)
3		cleq1lem 15031	. . . . . . 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 3185	. . . . . 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 3451	. . . . 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 2798	. . . 4 ⊢ (𝑥 = 𝐴 → {𝑦 ∈ ℝ^* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝑥 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^*, < ))} = 𝑀)
8	7	infeq1d 9546	. . 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 25518	. . 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 13203	. . . 4 ⊢ < Or ℝ^*
11	10	infex 9562	. . 3 ⊢ inf(𝑀, ℝ^*, < ) ∈ V
12	8, 9, 11	fvmpt 7029	. 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 1537 ∈ wcel 2108 ∃wrex 3076 {crab 3443 ∩ cin 3975 ⊆ wss 3976 𝒫 cpw 4622 ∪ cuni 4931 × cxp 5698 ran crn 5701 ∘ ccom 5704 ‘cfv 6573 (class class class)co 7448 ↑_m cmap 8884 supcsup 9509 infcinf 9510 ℝcr 11183 1c1 11185 + caddc 11187 ℝ^*cxr 11323 < clt 11324 ≤ cle 11325 − cmin 11520 ℕcn 12293 (,)cioo 13407 seqcseq 14052 abscabs 15283 vol*covol 25516

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 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-pre-lttri 11258 ax-pre-lttrn 11259

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-po 5607 df-so 5608 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-er 8763 df-en 9004 df-dom 9005 df-sdom 9006 df-sup 9511 df-inf 9512 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-ovol 25518

This theorem is referenced by: ovolcl 25532 ovollb 25533 ovolgelb 25534 ovolge0 25535 ovolsslem 25538 ovolshft 25565 ovolicc2 25576 ismblfin 37621 ovolval2 46565

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