elovolmr - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > elovolmr		Structured version Visualization version GIF version

Theorem elovolmr 25804

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 25802	. . 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 2766	. . . . . . . . . . 11 ⊢ (𝑓 = 𝐹 → 𝐹 = 𝑓)
5	4	coeq2d 5840	. . . . . . . . . 10 ⊢ (𝑓 = 𝐹 → ((abs ∘ − ) ∘ 𝐹) = ((abs ∘ − ) ∘ 𝑓))
6	5	seqeq3d 14157	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → seq1( + , ((abs ∘ − ) ∘ 𝐹)) = seq1( + , ((abs ∘ − ) ∘ 𝑓)))
7	2, 6	eqtrid 2807	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝑓)))
8	7	rneqd 5920	. . . . . . 7 ⊢ (𝑓 = 𝐹 → ran 𝑆 = ran seq1( + , ((abs ∘ − ) ∘ 𝑓)))
9	8	supeq1d 9443	. . . . . 6 ⊢ (𝑓 = 𝐹 → sup(ran 𝑆, ℝ^, < ) = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^, < ))
10	9	biantrud 541	. . . . 5 ⊢ (𝑓 = 𝐹 → (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ↔ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ sup(ran 𝑆, ℝ^, < ) = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^, < ))))
11		coeq2 5836	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → ((,) ∘ 𝑓) = ((,) ∘ 𝐹))
12	11	rneqd 5920	. . . . . . 7 ⊢ (𝑓 = 𝐹 → ran ((,) ∘ 𝑓) = ran ((,) ∘ 𝐹))
13	12	unieqd 4879	. . . . . 6 ⊢ (𝑓 = 𝐹 → ∪ ran ((,) ∘ 𝑓) = ∪ ran ((,) ∘ 𝐹))
14	13	sseq2d 3941	. . . . 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 3576	. . 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 594	. 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 25803	. 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

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1569 ∈ wcel 2144 ∃wrex 3086 {crab 3412 ⊆ wss 3896 ∩ cin 4088 ∪ cuni 4866 × cxp 5648 ran crn 5651 ∘ ccom 5654 ⟶wf 6537 (class class class)co 7422 ↑_m cmap 8852 supcsup 9437 ℝcr 11204 1c1 11206 + caddc 11208 ℝ^*cxr 11347 < clt 11348 ≤ cle 11349 − cmin 11546 ℕcn 12340 (,)cioo 13481 seqcseq 14149 abscabs 15408

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7753 ax-cnex 11261 ax-resscn 11262 ax-1cn 11263 ax-icn 11264 ax-addcl 11265 ax-addrcl 11266 ax-mulcl 11267 ax-mulrcl 11268 ax-mulcom 11269 ax-addass 11270 ax-mulass 11271 ax-distr 11272 ax-i2m1 11273 ax-1ne0 11274 ax-1rid 11275 ax-rnegex 11276 ax-rrecex 11277 ax-cnre 11278 ax-pre-lttri 11279 ax-pre-lttrn 11280 ax-pre-ltadd 11281 ax-pre-mulgt0 11282 ax-pre-sup 11283

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3848 df-ss 3897 df-pss 4013 df-dif 4046 df-in 4090 df-un 4150 df-nul 4278 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-op 4835 df-uni 4867 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7379 df-ov 7425 df-oprab 7426 df-mpo 7427 df-om 7880 df-1st 8003 df-2nd 8004 df-frecs 8304 df-wrecs 8335 df-recs 8384 df-rdg 8423 df-er 8722 df-map 8854 df-en 8979 df-dom 8980 df-sdom 8981 df-sup 9439 df-pnf 11350 df-mnf 11351 df-xr 11352 df-ltxr 11353 df-le 11354 df-sub 11548 df-neg 11549 df-div 11979 df-nn 12341 df-2 12410 df-3 12411 df-n0 12612 df-z 12699 df-uz 12971 df-rp 13126 df-ico 13487 df-fz 13645 df-seq 14150 df-exp 14210 df-cj 15273 df-re 15274 df-im 15275 df-sqrt 15409 df-abs 15410

This theorem is used by: ovollb 25807 ovolshftlem1 25837

W3C validator