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Theorem elovolmr 25592

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 25590	. . 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 2771	. . . . . . . . . . 11 ⊢ (𝑓 = 𝐹 → 𝐹 = 𝑓)
5	4	coeq2d 5838	. . . . . . . . . 10 ⊢ (𝑓 = 𝐹 → ((abs ∘ − ) ∘ 𝐹) = ((abs ∘ − ) ∘ 𝑓))
6	5	seqeq3d 14033	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → seq1( + , ((abs ∘ − ) ∘ 𝐹)) = seq1( + , ((abs ∘ − ) ∘ 𝑓)))
7	2, 6	eqtrid 2812	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝑓)))
8	7	rneqd 5918	. . . . . . 7 ⊢ (𝑓 = 𝐹 → ran 𝑆 = ran seq1( + , ((abs ∘ − ) ∘ 𝑓)))
9	8	supeq1d 9394	. . . . . 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 5834	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → ((,) ∘ 𝑓) = ((,) ∘ 𝐹))
12	11	rneqd 5918	. . . . . . 7 ⊢ (𝑓 = 𝐹 → ran ((,) ∘ 𝑓) = ran ((,) ∘ 𝐹))
13	12	unieqd 4880	. . . . . 6 ⊢ (𝑓 = 𝐹 → ∪ ran ((,) ∘ 𝑓) = ∪ ran ((,) ∘ 𝐹))
14	13	sseq2d 3971	. . . . 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 3584	. . 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 25591	. 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 1563 ∈ wcel 2145 ∃wrex 3089 {crab 3417 ∩ cin 3906 ⊆ wss 3907 ∪ cuni 4867 × cxp 5649 ran crn 5652 ∘ ccom 5655 ⟶wf 6521 (class class class)co 7400 ↑_m cmap 8812 supcsup 9388 ℝcr 11087 1c1 11089 + caddc 11091 ℝ^*cxr 11230 < clt 11231 ≤ cle 11232 − cmin 11429 ℕcn 12221 (,)cioo 13360 seqcseq 14025 abscabs 15273

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1818 ax-4 1832 ax-5 1933 ax-6 1990 ax-7 2031 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2737 ax-sep 5250 ax-nul 5260 ax-pow 5326 ax-pr 5394 ax-un 7722 ax-cnex 11144 ax-resscn 11145 ax-1cn 11146 ax-icn 11147 ax-addcl 11148 ax-addrcl 11149 ax-mulcl 11150 ax-mulrcl 11151 ax-mulcom 11152 ax-addass 11153 ax-mulass 11154 ax-distr 11155 ax-i2m1 11156 ax-1ne0 11157 ax-1rid 11158 ax-rnegex 11159 ax-rrecex 11160 ax-cnre 11161 ax-pre-lttri 11162 ax-pre-lttrn 11163 ax-pre-ltadd 11164 ax-pre-mulgt0 11165 ax-pre-sup 11166

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1566 df-fal 1576 df-ex 1803 df-nf 1807 df-sb 2094 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3370 df-reu 3371 df-rab 3418 df-v 3459 df-sbc 3748 df-csb 3856 df-dif 3910 df-un 3912 df-in 3914 df-ss 3924 df-pss 3927 df-nul 4289 df-if 4484 df-pw 4560 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4868 df-iun 4953 df-br 5105 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6291 df-ord 6352 df-on 6353 df-lim 6354 df-suc 6355 df-iota 6481 df-fun 6527 df-fn 6528 df-f 6529 df-f1 6530 df-fo 6531 df-f1o 6532 df-fv 6533 df-riota 7357 df-ov 7403 df-oprab 7404 df-mpo 7405 df-om 7851 df-1st 7974 df-2nd 7975 df-frecs 8266 df-wrecs 8297 df-recs 8346 df-rdg 8385 df-er 8682 df-map 8814 df-en 8932 df-dom 8933 df-sdom 8934 df-sup 9390 df-pnf 11233 df-mnf 11234 df-xr 11235 df-ltxr 11236 df-le 11237 df-sub 11431 df-neg 11432 df-div 11860 df-nn 12222 df-2 12291 df-3 12292 df-n0 12493 df-z 12580 df-uz 12851 df-rp 13005 df-ico 13366 df-fz 13524 df-seq 14026 df-exp 14086 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275

This theorem is referenced by: ovollb 25595 ovolshftlem1 25625

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