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Theorem elovolmr 24545

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 24543	. . 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 2744	. . . . . . . . . . 11 ⊢ (𝑓 = 𝐹 → 𝐹 = 𝑓)
5	4	coeq2d 5760	. . . . . . . . . 10 ⊢ (𝑓 = 𝐹 → ((abs ∘ − ) ∘ 𝐹) = ((abs ∘ − ) ∘ 𝑓))
6	5	seqeq3d 13657	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → seq1( + , ((abs ∘ − ) ∘ 𝐹)) = seq1( + , ((abs ∘ − ) ∘ 𝑓)))
7	2, 6	syl5eq 2791	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝑓)))
8	7	rneqd 5836	. . . . . . 7 ⊢ (𝑓 = 𝐹 → ran 𝑆 = ran seq1( + , ((abs ∘ − ) ∘ 𝑓)))
9	8	supeq1d 9135	. . . . . 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 5756	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → ((,) ∘ 𝑓) = ((,) ∘ 𝐹))
12	11	rneqd 5836	. . . . . . 7 ⊢ (𝑓 = 𝐹 → ran ((,) ∘ 𝑓) = ran ((,) ∘ 𝐹))
13	12	unieqd 4850	. . . . . 6 ⊢ (𝑓 = 𝐹 → ∪ ran ((,) ∘ 𝑓) = ∪ ran ((,) ∘ 𝐹))
14	13	sseq2d 3949	. . . . 5 ⊢ (𝑓 = 𝐹 → (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ↔ 𝐴 ⊆ ∪ ran ((,) ∘ 𝐹)))
15	10, 14	bitr3d 280	. . . 4 ⊢ (𝑓 = 𝐹 → ((𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ sup(ran 𝑆, ℝ^, < ) = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^, < )) ↔ 𝐴 ⊆ ∪ ran ((,) ∘ 𝐹)))
16	15	rspcev 3552	. . 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 24544	. 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 1539 ∈ wcel 2108 ∃wrex 3064 {crab 3067 ∩ cin 3882 ⊆ wss 3883 ∪ cuni 4836 × cxp 5578 ran crn 5581 ∘ ccom 5584 ⟶wf 6414 (class class class)co 7255 ↑_m cmap 8573 supcsup 9129 ℝcr 10801 1c1 10803 + caddc 10805 ℝ^*cxr 10939 < clt 10940 ≤ cle 10941 − cmin 11135 ℕcn 11903 (,)cioo 13008 seqcseq 13649 abscabs 14873

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-er 8456 df-map 8575 df-en 8692 df-dom 8693 df-sdom 8694 df-sup 9131 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-n0 12164 df-z 12250 df-uz 12512 df-rp 12660 df-ico 13014 df-fz 13169 df-seq 13650 df-exp 13711 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875

This theorem is referenced by: ovollb 24548 ovolshftlem1 24578

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