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Theorem elovolm 23683

Description: Elementhood in the set 𝑀 of approximations to the outer measure. (Contributed by Mario Carneiro, 16-Mar-2014.)

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

**Assertion**
Ref	Expression
elovolm	⊢ (𝐵 ∈ 𝑀 ↔ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_𝑚 ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝐵 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^*, < )))

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

**Proof of Theorem elovolm**
Step	Hyp	Ref	Expression
1		eqeq1 2782	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^, < ) ↔ 𝐵 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^, < )))
2	1	anbi2d 622	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^, < )) ↔ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝐵 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^, < ))))
3	2	rexbidv 3237	. . 3 ⊢ (𝑦 = 𝐵 → (∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_𝑚 ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^, < )) ↔ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_𝑚 ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝐵 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^, < ))))
4		elovolm.1	. . 3 ⊢ 𝑀 = {𝑦 ∈ ℝ^* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_𝑚 ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^*, < ))}
5	3, 4	elrab2 3576	. 2 ⊢ (𝐵 ∈ 𝑀 ↔ (𝐵 ∈ ℝ^* ∧ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_𝑚 ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝐵 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^*, < ))))
6		elovolmlem 23682	. . . . . . . . . 10 ⊢ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_𝑚 ℕ) ↔ 𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
7		eqid 2778	. . . . . . . . . . 11 ⊢ ((abs ∘ − ) ∘ 𝑓) = ((abs ∘ − ) ∘ 𝑓)
8		eqid 2778	. . . . . . . . . . 11 ⊢ seq1( + , ((abs ∘ − ) ∘ 𝑓)) = seq1( + , ((abs ∘ − ) ∘ 𝑓))
9	7, 8	ovolsf 23680	. . . . . . . . . 10 ⊢ (𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → seq1( + , ((abs ∘ − ) ∘ 𝑓)):ℕ⟶(0[,)+∞))
10	6, 9	sylbi 209	. . . . . . . . 9 ⊢ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_𝑚 ℕ) → seq1( + , ((abs ∘ − ) ∘ 𝑓)):ℕ⟶(0[,)+∞))
11		icossxr 12574	. . . . . . . . 9 ⊢ (0[,)+∞) ⊆ ℝ^*
12		fss 6306	. . . . . . . . 9 ⊢ ((seq1( + , ((abs ∘ − ) ∘ 𝑓)):ℕ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ ℝ^) → seq1( + , ((abs ∘ − ) ∘ 𝑓)):ℕ⟶ℝ^)
13	10, 11, 12	sylancl 580	. . . . . . . 8 ⊢ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_𝑚 ℕ) → seq1( + , ((abs ∘ − ) ∘ 𝑓)):ℕ⟶ℝ^*)
14		frn 6299	. . . . . . . 8 ⊢ (seq1( + , ((abs ∘ − ) ∘ 𝑓)):ℕ⟶ℝ^* → ran seq1( + , ((abs ∘ − ) ∘ 𝑓)) ⊆ ℝ^*)
15		supxrcl 12461	. . . . . . . 8 ⊢ (ran seq1( + , ((abs ∘ − ) ∘ 𝑓)) ⊆ ℝ^* → sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^, < ) ∈ ℝ^)
16	13, 14, 15	3syl 18	. . . . . . 7 ⊢ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_𝑚 ℕ) → sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^, < ) ∈ ℝ^)
17		eleq1 2847	. . . . . . 7 ⊢ (𝐵 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^, < ) → (𝐵 ∈ ℝ^ ↔ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^, < ) ∈ ℝ^))
18	16, 17	syl5ibrcom 239	. . . . . 6 ⊢ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_𝑚 ℕ) → (𝐵 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^, < ) → 𝐵 ∈ ℝ^))
19	18	imp 397	. . . . 5 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_𝑚 ℕ) ∧ 𝐵 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^, < )) → 𝐵 ∈ ℝ^)
20	19	adantrl 706	. . . 4 ⊢ ((𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_𝑚 ℕ) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝐵 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^, < ))) → 𝐵 ∈ ℝ^)
21	20	rexlimiva 3210	. . 3 ⊢ (∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_𝑚 ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝐵 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^, < )) → 𝐵 ∈ ℝ^)
22	21	pm4.71ri 556	. 2 ⊢ (∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_𝑚 ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝐵 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^, < )) ↔ (𝐵 ∈ ℝ^ ∧ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_𝑚 ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝐵 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^*, < ))))
23	5, 22	bitr4i 270	1 ⊢ (𝐵 ∈ 𝑀 ↔ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_𝑚 ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝐵 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^*, < )))

Colors of variables: wff setvar class

Syntax hints: ↔ wb 198 ∧ wa 386 = wceq 1601 ∈ wcel 2107 ∃wrex 3091 {crab 3094 ∩ cin 3791 ⊆ wss 3792 ∪ cuni 4673 × cxp 5355 ran crn 5358 ∘ ccom 5361 ⟶wf 6133 (class class class)co 6924 ↑_𝑚 cmap 8142 supcsup 8636 ℝcr 10273 0cc0 10274 1c1 10275 + caddc 10277 +∞cpnf 10410 ℝ^*cxr 10412 < clt 10413 ≤ cle 10414 − cmin 10608 ℕcn 11378 (,)cioo 12491 [,)cico 12493 seqcseq 13123 abscabs 14385

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 ax-pre-sup 10352

This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-om 7346 df-1st 7447 df-2nd 7448 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-er 8028 df-map 8144 df-en 8244 df-dom 8245 df-sdom 8246 df-sup 8638 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-div 11035 df-nn 11379 df-2 11442 df-3 11443 df-n0 11647 df-z 11733 df-uz 11997 df-rp 12142 df-ico 12497 df-fz 12648 df-seq 13124 df-exp 13183 df-cj 14250 df-re 14251 df-im 14252 df-sqrt 14386 df-abs 14387

This theorem is referenced by: elovolmr 23684 ovolmge0 23685 ovolicc2 23730

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