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Theorem elovolmr 25600

Description: Sufficient condition for elementhood in the set 𝑀. (Contributed by Mario Carneiro, 16-Mar-2014.)

**Hypotheses**
Ref	Expression
elovolm.1	⊢ 𝑀 = {𝑦 ∈ ℝ^* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^*, < ))}
elovolmr.2	⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))

**Assertion**
Ref	Expression
elovolmr	⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ⊆ ∪ ran ((,) ∘ 𝐹)) → sup(ran 𝑆, ℝ^*, < ) ∈ 𝑀)

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

**Proof of Theorem elovolmr**
Step	Hyp	Ref	Expression
1		elovolmlem 25598	. . 3 ⊢ (𝐹 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ↔ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
2		elovolmr.2	. . . . . . . . 9 ⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
3		id 23	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝐹 → 𝑓 = 𝐹)
4	3	eqcomd 2775	. . . . . . . . . . 11 ⊢ (𝑓 = 𝐹 → 𝐹 = 𝑓)
5	4	coeq2d 5846	. . . . . . . . . 10 ⊢ (𝑓 = 𝐹 → ((abs ∘ − ) ∘ 𝐹) = ((abs ∘ − ) ∘ 𝑓))
6	5	seqeq3d 14041	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → seq1( + , ((abs ∘ − ) ∘ 𝐹)) = seq1( + , ((abs ∘ − ) ∘ 𝑓)))
7	2, 6	eqtrid 2816	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝑓)))
8	7	rneqd 5926	. . . . . . 7 ⊢ (𝑓 = 𝐹 → ran 𝑆 = ran seq1( + , ((abs ∘ − ) ∘ 𝑓)))
9	8	supeq1d 9402	. . . . . 6 ⊢ (𝑓 = 𝐹 → sup(ran 𝑆, ℝ^, < ) = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^, < ))
10	9	biantrud 540	. . . . 5 ⊢ (𝑓 = 𝐹 → (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ↔ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ sup(ran 𝑆, ℝ^, < ) = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^, < ))))
11		coeq2 5842	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → ((,) ∘ 𝑓) = ((,) ∘ 𝐹))
12	11	rneqd 5926	. . . . . . 7 ⊢ (𝑓 = 𝐹 → ran ((,) ∘ 𝑓) = ran ((,) ∘ 𝐹))
13	12	unieqd 4886	. . . . . 6 ⊢ (𝑓 = 𝐹 → ∪ ran ((,) ∘ 𝑓) = ∪ ran ((,) ∘ 𝐹))
14	13	sseq2d 3977	. . . . 5 ⊢ (𝑓 = 𝐹 → (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ↔ 𝐴 ⊆ ∪ ran ((,) ∘ 𝐹)))
15	10, 14	bitr3d 284	. . . 4 ⊢ (𝑓 = 𝐹 → ((𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ sup(ran 𝑆, ℝ^, < ) = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^, < )) ↔ 𝐴 ⊆ ∪ ran ((,) ∘ 𝐹)))
16	15	rspcev 3590	. . 3 ⊢ ((𝐹 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ 𝐴 ⊆ ∪ ran ((,) ∘ 𝐹)) → ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ sup(ran 𝑆, ℝ^, < ) = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^, < )))
17	1, 16	sylanbr 593	. 2 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ⊆ ∪ ran ((,) ∘ 𝐹)) → ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ sup(ran 𝑆, ℝ^, < ) = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^, < )))
18		elovolm.1	. . 3 ⊢ 𝑀 = {𝑦 ∈ ℝ^* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^*, < ))}
19	18	elovolm 25599	. 2 ⊢ (sup(ran 𝑆, ℝ^, < ) ∈ 𝑀 ↔ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ sup(ran 𝑆, ℝ^, < ) = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^*, < )))
20	17, 19	sylibr 237	1 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ⊆ ∪ ran ((,) ∘ 𝐹)) → sup(ran 𝑆, ℝ^*, < ) ∈ 𝑀)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 400 = wceq 1567 ∈ wcel 2149 ∃wrex 3095 {crab 3423 ∩ cin 3912 ⊆ wss 3913 ∪ cuni 4873 × cxp 5657 ran crn 5660 ∘ ccom 5663 ⟶wf 6530 (class class class)co 7408 ↑_m cmap 8820 supcsup 9396 ℝcr 11095 1c1 11097 + caddc 11099 ℝ^*cxr 11238 < clt 11239 ≤ cle 11240 − cmin 11437 ℕcn 12229 (,)cioo 13368 seqcseq 14033 abscabs 15281

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-sep 5258 ax-nul 5268 ax-pow 5334 ax-pr 5402 ax-un 7730 ax-cnex 11152 ax-resscn 11153 ax-1cn 11154 ax-icn 11155 ax-addcl 11156 ax-addrcl 11157 ax-mulcl 11158 ax-mulrcl 11159 ax-mulcom 11160 ax-addass 11161 ax-mulass 11162 ax-distr 11163 ax-i2m1 11164 ax-1ne0 11165 ax-1rid 11166 ax-rnegex 11167 ax-rrecex 11168 ax-cnre 11169 ax-pre-lttri 11170 ax-pre-lttrn 11171 ax-pre-ltadd 11172 ax-pre-mulgt0 11173 ax-pre-sup 11174

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4874 df-iun 4959 df-br 5111 df-opab 5175 df-mpt 5194 df-tr 5220 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6300 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6490 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7365 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7859 df-1st 7982 df-2nd 7983 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-er 8690 df-map 8822 df-en 8940 df-dom 8941 df-sdom 8942 df-sup 9398 df-pnf 11241 df-mnf 11242 df-xr 11243 df-ltxr 11244 df-le 11245 df-sub 11439 df-neg 11440 df-div 11868 df-nn 12230 df-2 12299 df-3 12300 df-n0 12501 df-z 12588 df-uz 12859 df-rp 13013 df-ico 13374 df-fz 13532 df-seq 14034 df-exp 14094 df-cj 15146 df-re 15147 df-im 15148 df-sqrt 15282 df-abs 15283

This theorem is referenced by: ovollb 25603 ovolshftlem1 25633

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