Theorem ovolval 22962

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:   𝑦,𝑓,𝐴

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 9879	. . 3 ⊢ ℝ ∈ V
2	1	elpw2 4746	. 2 ⊢ (𝐴 ∈ 𝒫 ℝ ↔ 𝐴 ⊆ ℝ)
3		sseq1 3584	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑥 ⊆ ∪ ran ((,) ∘ 𝑓) ↔ 𝐴 ⊆ ∪ ran ((,) ∘ 𝑓)))
4	3	anbi1d 736	. . . . . . 7 ⊢ (𝑥 = 𝐴 → ((𝑥 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < )) ↔ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))))
5	4	rexbidv 3029	. . . . . 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*, < ))))
6	5	rabbidv 3159	. . . . 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*, < ))})
7		ovolval.1	. . . . 5 ⊢ 𝑀 = {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))}
8	6, 7	syl6eqr 2657	. . . 4 ⊢ (𝑥 = 𝐴 → {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ)(𝑥 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))} = 𝑀)
9	8	infeq1d 8239	. . 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*, < ))
10		df-ovol 22953	. . 3 ⊢ vol* = (𝑥 ∈ 𝒫 ℝ ↦ inf({𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ)(𝑥 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))}, ℝ*, < ))
11		xrltso 11805	. . . 4 ⊢ < Or ℝ*
12	11	infex 8255	. . 3 ⊢ inf(𝑀, ℝ*, < ) ∈ V
13	9, 10, 12	fvmpt 6172	. 2 ⊢ (𝐴 ∈ 𝒫 ℝ → (vol*‘𝐴) = inf(𝑀, ℝ*, < ))
14	2, 13	sylbir 223	1 ⊢ (𝐴 ⊆ ℝ → (vol*‘𝐴) = inf(𝑀, ℝ*, < ))

Syntax hints:   → wi 4   ∧ wa 382   = wceq 1474   ∈ wcel 1975  ∃wrex 2892  {crab 2895   ∩ cin 3534   ⊆ wss 3535  𝒫 cpw 4103  ∪ cuni 4362   × cxp 5022  ran crn 5025   ∘ ccom 5028  ‘cfv 5786  (class class class)co 6523   ↑_𝑚 cmap 7717  supcsup 8202  infcinf 8203  ℝcr 9787  1c1 9789   + caddc 9791  ℝ^*cxr 9925   < clt 9926   ≤ cle 9927   − cmin 10113  ℕcn 10863  (,)cioo 11998  seqcseq 12614  abscabs 13764  vol*covol 22951

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824  ax-un 6820  ax-cnex 9844  ax-resscn 9845  ax-pre-lttri 9862  ax-pre-lttrn 9863

This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-nel 2778  df-ral 2896  df-rex 2897  df-rmo 2899  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-op 4127  df-uni 4363  df-br 4574  df-opab 4634  df-mpt 4635  df-id 4939  df-po 4945  df-so 4946  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-iota 5750  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-fo 5792  df-f1o 5793  df-fv 5794  df-er 7602  df-en 7815  df-dom 7816  df-sdom 7817  df-sup 8204  df-inf 8205  df-pnf 9928  df-mnf 9929  df-xr 9930  df-ltxr 9931  df-ovol 22953

This theorem is referenced by:  ovolcl  22966  ovollb  22967  ovolgelb  22968  ovolge0  22969  ovolsslem  22972  ovolshft  22999  ovolicc2  23010  ismblfin  32419  ovolval2  39334