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Theorem ovolmge0 25828

Description: The set 𝑀 is composed of nonnegative extended real numbers. (Contributed by Mario Carneiro, 16-Mar-2014.)

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

**Assertion**
Ref	Expression
ovolmge0	⊢ (𝐵 ∈ 𝑀 → 0 ≤ 𝐵)

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

**Proof of Theorem ovolmge0**
Step	Hyp	Ref	Expression
1		elovolm.1	. . 3 ⊢ 𝑀 = {𝑦 ∈ ℝ^* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^*, < ))}
2	1	elovolm 25826	. 2 ⊢ (𝐵 ∈ 𝑀 ↔ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝐵 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^*, < )))
3		elovolmlem 25825	. . . . . 6 ⊢ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ↔ 𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
4		eqid 2763	. . . . . . . . . 10 ⊢ ((abs ∘ − ) ∘ 𝑓) = ((abs ∘ − ) ∘ 𝑓)
5		eqid 2763	. . . . . . . . . 10 ⊢ seq1( + , ((abs ∘ − ) ∘ 𝑓)) = seq1( + , ((abs ∘ − ) ∘ 𝑓))
6	4, 5	ovolsf 25823	. . . . . . . . 9 ⊢ (𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → seq1( + , ((abs ∘ − ) ∘ 𝑓)):ℕ⟶(0[,)+∞))
7		1nn 12373	. . . . . . . . 9 ⊢ 1 ∈ ℕ
8		ffvelcdm 7105	. . . . . . . . 9 ⊢ ((seq1( + , ((abs ∘ − ) ∘ 𝑓)):ℕ⟶(0[,)+∞) ∧ 1 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘1) ∈ (0[,)+∞))
9	6, 7, 8	sylancl 598	. . . . . . . 8 ⊢ (𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘1) ∈ (0[,)+∞))
10		elrege0 13612	. . . . . . . . 9 ⊢ ((seq1( + , ((abs ∘ − ) ∘ 𝑓))‘1) ∈ (0[,)+∞) ↔ ((seq1( + , ((abs ∘ − ) ∘ 𝑓))‘1) ∈ ℝ ∧ 0 ≤ (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘1)))
11	10	simprbi 503	. . . . . . . 8 ⊢ ((seq1( + , ((abs ∘ − ) ∘ 𝑓))‘1) ∈ (0[,)+∞) → 0 ≤ (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘1))
12	9, 11	syl 18	. . . . . . 7 ⊢ (𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 0 ≤ (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘1))
13	6	frnd 6740	. . . . . . . . 9 ⊢ (𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ran seq1( + , ((abs ∘ − ) ∘ 𝑓)) ⊆ (0[,)+∞))
14		icossxr 13590	. . . . . . . . 9 ⊢ (0[,)+∞) ⊆ ℝ^*
15	13, 14	sstrdi 3925	. . . . . . . 8 ⊢ (𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ran seq1( + , ((abs ∘ − ) ∘ 𝑓)) ⊆ ℝ^*)
16	6	ffnd 6732	. . . . . . . . 9 ⊢ (𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → seq1( + , ((abs ∘ − ) ∘ 𝑓)) Fn ℕ)
17		fnfvelrn 7104	. . . . . . . . 9 ⊢ ((seq1( + , ((abs ∘ − ) ∘ 𝑓)) Fn ℕ ∧ 1 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘1) ∈ ran seq1( + , ((abs ∘ − ) ∘ 𝑓)))
18	16, 7, 17	sylancl 598	. . . . . . . 8 ⊢ (𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘1) ∈ ran seq1( + , ((abs ∘ − ) ∘ 𝑓)))
19		supxrub 13481	. . . . . . . 8 ⊢ ((ran seq1( + , ((abs ∘ − ) ∘ 𝑓)) ⊆ ℝ^* ∧ (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘1) ∈ ran seq1( + , ((abs ∘ − ) ∘ 𝑓))) → (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘1) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^*, < ))
20	15, 18, 19	syl2anc 596	. . . . . . 7 ⊢ (𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘1) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^*, < ))
21		0xr 11383	. . . . . . . 8 ⊢ 0 ∈ ℝ^*
22	14, 9	sselid 3911	. . . . . . . 8 ⊢ (𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘1) ∈ ℝ^*)
23		supxrcl 13472	. . . . . . . . 9 ⊢ (ran seq1( + , ((abs ∘ − ) ∘ 𝑓)) ⊆ ℝ^* → sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^, < ) ∈ ℝ^)
24	15, 23	syl 18	. . . . . . . 8 ⊢ (𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^, < ) ∈ ℝ^)
25		xrletr 13314	. . . . . . . 8 ⊢ ((0 ∈ ℝ^* ∧ (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘1) ∈ ℝ^* ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^, < ) ∈ ℝ^) → ((0 ≤ (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘1) ∧ (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘1) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^, < )) → 0 ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^, < )))
26	21, 22, 24, 25	mp3an2i 1496	. . . . . . 7 ⊢ (𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ((0 ≤ (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘1) ∧ (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘1) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^, < )) → 0 ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^, < )))
27	12, 20, 26	mp2and 712	. . . . . 6 ⊢ (𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 0 ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^*, < ))
28	3, 27	sylbi 220	. . . . 5 ⊢ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) → 0 ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^*, < ))
29		breq2 5110	. . . . 5 ⊢ (𝐵 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^, < ) → (0 ≤ 𝐵 ↔ 0 ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^, < )))
30	28, 29	syl5ibrcom 250	. . . 4 ⊢ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) → (𝐵 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^*, < ) → 0 ≤ 𝐵))
31	30	adantld 496	. . 3 ⊢ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) → ((𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝐵 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^*, < )) → 0 ≤ 𝐵))
32	31	rexlimiv 3160	. 2 ⊢ (∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝐵 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^*, < )) → 0 ≤ 𝐵)
33	2, 32	sylbi 220	1 ⊢ (𝐵 ∈ 𝑀 → 0 ≤ 𝐵)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2147 ∃wrex 3090 {crab 3416 ⊆ wss 3900 ∩ cin 4092 ∪ cuni 4870 class class class wbr 5106 × cxp 5653 ran crn 5656 ∘ ccom 5659 Fn wfn 6553 ⟶wf 6554 ‘cfv 6558 (class class class)co 7443 ↑_m cmap 8875 supcsup 9459 ℝcr 11226 0cc0 11227 1c1 11228 + caddc 11230 +∞cpnf 11367 ℝ^*cxr 11369 < clt 11370 ≤ cle 11371 − cmin 11568 ℕcn 12362 (,)cioo 13503 [,)cico 13505 seqcseq 14171 abscabs 15430

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7775 ax-cnex 11283 ax-resscn 11284 ax-1cn 11285 ax-icn 11286 ax-addcl 11287 ax-addrcl 11288 ax-mulcl 11289 ax-mulrcl 11290 ax-mulcom 11291 ax-addass 11292 ax-mulass 11293 ax-distr 11294 ax-i2m1 11295 ax-1ne0 11296 ax-1rid 11297 ax-rnegex 11298 ax-rrecex 11299 ax-cnre 11300 ax-pre-lttri 11301 ax-pre-lttrn 11302 ax-pre-ltadd 11303 ax-pre-mulgt0 11304 ax-pre-sup 11305

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4839 df-uni 4871 df-iun 4956 df-br 5107 df-opab 5171 df-mpt 5190 df-tr 5216 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6322 df-ord 6383 df-on 6384 df-lim 6385 df-suc 6386 df-iota 6512 df-fun 6560 df-fn 6561 df-f 6562 df-f1 6563 df-fo 6564 df-f1o 6565 df-fv 6566 df-riota 7400 df-ov 7446 df-oprab 7447 df-mpo 7448 df-om 7902 df-1st 8025 df-2nd 8026 df-frecs 8327 df-wrecs 8358 df-recs 8407 df-rdg 8446 df-er 8745 df-map 8877 df-en 9002 df-dom 9003 df-sdom 9004 df-sup 9461 df-pnf 11372 df-mnf 11373 df-xr 11374 df-ltxr 11375 df-le 11376 df-sub 11570 df-neg 11571 df-div 12001 df-nn 12363 df-2 12432 df-3 12433 df-n0 12634 df-z 12721 df-uz 12993 df-rp 13148 df-ico 13509 df-fz 13667 df-seq 14172 df-exp 14232 df-cj 15295 df-re 15296 df-im 15297 df-sqrt 15431 df-abs 15432

This theorem is used by: ovolge0 25832

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