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Theorem elovolmr 25315

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 25313	. . 3 ⊢ (𝐹 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ↔ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
2		elovolmr.2	. . . . . . . . 9 ⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
3		id 22	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝐹 → 𝑓 = 𝐹)
4	3	eqcomd 2730	. . . . . . . . . . 11 ⊢ (𝑓 = 𝐹 → 𝐹 = 𝑓)
5	4	coeq2d 5852	. . . . . . . . . 10 ⊢ (𝑓 = 𝐹 → ((abs ∘ − ) ∘ 𝐹) = ((abs ∘ − ) ∘ 𝑓))
6	5	seqeq3d 13970	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → seq1( + , ((abs ∘ − ) ∘ 𝐹)) = seq1( + , ((abs ∘ − ) ∘ 𝑓)))
7	2, 6	eqtrid 2776	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝑓)))
8	7	rneqd 5927	. . . . . . 7 ⊢ (𝑓 = 𝐹 → ran 𝑆 = ran seq1( + , ((abs ∘ − ) ∘ 𝑓)))
9	8	supeq1d 9436	. . . . . 6 ⊢ (𝑓 = 𝐹 → sup(ran 𝑆, ℝ^, < ) = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^, < ))
10	9	biantrud 531	. . . . 5 ⊢ (𝑓 = 𝐹 → (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ↔ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ sup(ran 𝑆, ℝ^, < ) = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^, < ))))
11		coeq2 5848	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → ((,) ∘ 𝑓) = ((,) ∘ 𝐹))
12	11	rneqd 5927	. . . . . . 7 ⊢ (𝑓 = 𝐹 → ran ((,) ∘ 𝑓) = ran ((,) ∘ 𝐹))
13	12	unieqd 4912	. . . . . 6 ⊢ (𝑓 = 𝐹 → ∪ ran ((,) ∘ 𝑓) = ∪ ran ((,) ∘ 𝐹))
14	13	sseq2d 4006	. . . . 5 ⊢ (𝑓 = 𝐹 → (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ↔ 𝐴 ⊆ ∪ ran ((,) ∘ 𝐹)))
15	10, 14	bitr3d 281	. . . 4 ⊢ (𝑓 = 𝐹 → ((𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ sup(ran 𝑆, ℝ^, < ) = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^, < )) ↔ 𝐴 ⊆ ∪ ran ((,) ∘ 𝐹)))
16	15	rspcev 3604	. . 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 581	. 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 25314	. 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 233	1 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ⊆ ∪ ran ((,) ∘ 𝐹)) → sup(ran 𝑆, ℝ^*, < ) ∈ 𝑀)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ∃wrex 3062 {crab 3424 ∩ cin 3939 ⊆ wss 3940 ∪ cuni 4899 × cxp 5664 ran crn 5667 ∘ ccom 5670 ⟶wf 6529 (class class class)co 7401 ↑_m cmap 8815 supcsup 9430 ℝcr 11104 1c1 11106 + caddc 11108 ℝ^*cxr 11243 < clt 11244 ≤ cle 11245 − cmin 11440 ℕcn 12208 (,)cioo 13320 seqcseq 13962 abscabs 15177

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 ax-pre-sup 11183

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-1st 7968 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-er 8698 df-map 8817 df-en 8935 df-dom 8936 df-sdom 8937 df-sup 9432 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-n0 12469 df-z 12555 df-uz 12819 df-rp 12971 df-ico 13326 df-fz 13481 df-seq 13963 df-exp 14024 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179

This theorem is referenced by: ovollb 25318 ovolshftlem1 25348

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