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Theorem ovolmge0 25597

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 25595	. 2 ⊢ (𝐵 ∈ 𝑀 ↔ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝐵 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^*, < )))
3		elovolmlem 25594	. . . . . 6 ⊢ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ↔ 𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
4		eqid 2765	. . . . . . . . . 10 ⊢ ((abs ∘ − ) ∘ 𝑓) = ((abs ∘ − ) ∘ 𝑓)
5		eqid 2765	. . . . . . . . . 10 ⊢ seq1( + , ((abs ∘ − ) ∘ 𝑓)) = seq1( + , ((abs ∘ − ) ∘ 𝑓))
6	4, 5	ovolsf 25592	. . . . . . . . 9 ⊢ (𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → seq1( + , ((abs ∘ − ) ∘ 𝑓)):ℕ⟶(0[,)+∞))
7		1nn 12235	. . . . . . . . 9 ⊢ 1 ∈ ℕ
8		ffvelcdm 7066	. . . . . . . . 9 ⊢ ((seq1( + , ((abs ∘ − ) ∘ 𝑓)):ℕ⟶(0[,)+∞) ∧ 1 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘1) ∈ (0[,)+∞))
9	6, 7, 8	sylancl 597	. . . . . . . 8 ⊢ (𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘1) ∈ (0[,)+∞))
10		elrege0 13472	. . . . . . . . 9 ⊢ ((seq1( + , ((abs ∘ − ) ∘ 𝑓))‘1) ∈ (0[,)+∞) ↔ ((seq1( + , ((abs ∘ − ) ∘ 𝑓))‘1) ∈ ℝ ∧ 0 ≤ (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘1)))
11	10	simprbi 502	. . . . . . . 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 6704	. . . . . . . . 9 ⊢ (𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ran seq1( + , ((abs ∘ − ) ∘ 𝑓)) ⊆ (0[,)+∞))
14		icossxr 13450	. . . . . . . . 9 ⊢ (0[,)+∞) ⊆ ℝ^*
15	13, 14	sstrdi 3951	. . . . . . . 8 ⊢ (𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ran seq1( + , ((abs ∘ − ) ∘ 𝑓)) ⊆ ℝ^*)
16	6	ffnd 6696	. . . . . . . . 9 ⊢ (𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → seq1( + , ((abs ∘ − ) ∘ 𝑓)) Fn ℕ)
17		fnfvelrn 7065	. . . . . . . . 9 ⊢ ((seq1( + , ((abs ∘ − ) ∘ 𝑓)) Fn ℕ ∧ 1 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘1) ∈ ran seq1( + , ((abs ∘ − ) ∘ 𝑓)))
18	16, 7, 17	sylancl 597	. . . . . . . 8 ⊢ (𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘1) ∈ ran seq1( + , ((abs ∘ − ) ∘ 𝑓)))
19		supxrub 13341	. . . . . . . 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 595	. . . . . . 7 ⊢ (𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘1) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^*, < ))
21		0xr 11244	. . . . . . . 8 ⊢ 0 ∈ ℝ^*
22	14, 9	sselid 3937	. . . . . . . 8 ⊢ (𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘1) ∈ ℝ^*)
23		supxrcl 13332	. . . . . . . . 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 13174	. . . . . . . 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 1490	. . . . . . 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 711	. . . . . 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 5109	. . . . 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 495	. . 3 ⊢ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) → ((𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝐵 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^*, < )) → 0 ≤ 𝐵))
32	31	rexlimiv 3159	. 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

Syntax hints: → wi 4 ∧ wa 400 = wceq 1563 ∈ wcel 2145 ∃wrex 3089 {crab 3417 ∩ cin 3906 ⊆ wss 3907 ∪ cuni 4868 class class class wbr 5105 × cxp 5650 ran crn 5653 ∘ ccom 5656 Fn wfn 6520 ⟶wf 6521 ‘cfv 6525 (class class class)co 7400 ↑_m cmap 8812 supcsup 9388 ℝcr 11087 0cc0 11088 1c1 11089 + caddc 11091 +∞cpnf 11228 ℝ^*cxr 11230 < clt 11231 ≤ cle 11232 − cmin 11429 ℕcn 12224 (,)cioo 13363 [,)cico 13365 seqcseq 14028 abscabs 15275

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 5251 ax-nul 5261 ax-pow 5327 ax-pr 5395 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 4869 df-iun 4954 df-br 5106 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5547 df-eprel 5552 df-po 5560 df-so 5561 df-fr 5605 df-we 5607 df-xp 5658 df-rel 5659 df-cnv 5660 df-co 5661 df-dm 5662 df-rn 5663 df-res 5664 df-ima 5665 df-pred 6292 df-ord 6353 df-on 6354 df-lim 6355 df-suc 6356 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 12225 df-2 12294 df-3 12295 df-n0 12496 df-z 12583 df-uz 12854 df-rp 13008 df-ico 13369 df-fz 13527 df-seq 14029 df-exp 14089 df-cj 15140 df-re 15141 df-im 15142 df-sqrt 15276 df-abs 15277

This theorem is referenced by: ovolge0 25601

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