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Theorem ovolmge0 25393

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 25391	. 2 ⊢ (𝐵 ∈ 𝑀 ↔ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝐵 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^*, < )))
3		elovolmlem 25390	. . . . . 6 ⊢ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ↔ 𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
4		eqid 2727	. . . . . . . . . 10 ⊢ ((abs ∘ − ) ∘ 𝑓) = ((abs ∘ − ) ∘ 𝑓)
5		eqid 2727	. . . . . . . . . 10 ⊢ seq1( + , ((abs ∘ − ) ∘ 𝑓)) = seq1( + , ((abs ∘ − ) ∘ 𝑓))
6	4, 5	ovolsf 25388	. . . . . . . . 9 ⊢ (𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → seq1( + , ((abs ∘ − ) ∘ 𝑓)):ℕ⟶(0[,)+∞))
7		1nn 12245	. . . . . . . . 9 ⊢ 1 ∈ ℕ
8		ffvelcdm 7085	. . . . . . . . 9 ⊢ ((seq1( + , ((abs ∘ − ) ∘ 𝑓)):ℕ⟶(0[,)+∞) ∧ 1 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘1) ∈ (0[,)+∞))
9	6, 7, 8	sylancl 585	. . . . . . . 8 ⊢ (𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘1) ∈ (0[,)+∞))
10		elrege0 13455	. . . . . . . . 9 ⊢ ((seq1( + , ((abs ∘ − ) ∘ 𝑓))‘1) ∈ (0[,)+∞) ↔ ((seq1( + , ((abs ∘ − ) ∘ 𝑓))‘1) ∈ ℝ ∧ 0 ≤ (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘1)))
11	10	simprbi 496	. . . . . . . 8 ⊢ ((seq1( + , ((abs ∘ − ) ∘ 𝑓))‘1) ∈ (0[,)+∞) → 0 ≤ (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘1))
12	9, 11	syl 17	. . . . . . 7 ⊢ (𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 0 ≤ (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘1))
13	6	frnd 6724	. . . . . . . . 9 ⊢ (𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ran seq1( + , ((abs ∘ − ) ∘ 𝑓)) ⊆ (0[,)+∞))
14		icossxr 13433	. . . . . . . . 9 ⊢ (0[,)+∞) ⊆ ℝ^*
15	13, 14	sstrdi 3990	. . . . . . . 8 ⊢ (𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ran seq1( + , ((abs ∘ − ) ∘ 𝑓)) ⊆ ℝ^*)
16	6	ffnd 6717	. . . . . . . . 9 ⊢ (𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → seq1( + , ((abs ∘ − ) ∘ 𝑓)) Fn ℕ)
17		fnfvelrn 7084	. . . . . . . . 9 ⊢ ((seq1( + , ((abs ∘ − ) ∘ 𝑓)) Fn ℕ ∧ 1 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘1) ∈ ran seq1( + , ((abs ∘ − ) ∘ 𝑓)))
18	16, 7, 17	sylancl 585	. . . . . . . 8 ⊢ (𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘1) ∈ ran seq1( + , ((abs ∘ − ) ∘ 𝑓)))
19		supxrub 13327	. . . . . . . 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 583	. . . . . . 7 ⊢ (𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘1) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^*, < ))
21		0xr 11283	. . . . . . . 8 ⊢ 0 ∈ ℝ^*
22	14, 9	sselid 3976	. . . . . . . 8 ⊢ (𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘1) ∈ ℝ^*)
23		supxrcl 13318	. . . . . . . . 9 ⊢ (ran seq1( + , ((abs ∘ − ) ∘ 𝑓)) ⊆ ℝ^* → sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^, < ) ∈ ℝ^)
24	15, 23	syl 17	. . . . . . . 8 ⊢ (𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^, < ) ∈ ℝ^)
25		xrletr 13161	. . . . . . . 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 1463	. . . . . . 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 698	. . . . . 6 ⊢ (𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 0 ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^*, < ))
28	3, 27	sylbi 216	. . . . 5 ⊢ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) → 0 ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^*, < ))
29		breq2 5146	. . . . 5 ⊢ (𝐵 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^, < ) → (0 ≤ 𝐵 ↔ 0 ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^, < )))
30	28, 29	syl5ibrcom 246	. . . 4 ⊢ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) → (𝐵 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^*, < ) → 0 ≤ 𝐵))
31	30	adantld 490	. . 3 ⊢ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) → ((𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝐵 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^*, < )) → 0 ≤ 𝐵))
32	31	rexlimiv 3143	. 2 ⊢ (∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝐵 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^*, < )) → 0 ≤ 𝐵)
33	2, 32	sylbi 216	1 ⊢ (𝐵 ∈ 𝑀 → 0 ≤ 𝐵)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ∃wrex 3065 {crab 3427 ∩ cin 3943 ⊆ wss 3944 ∪ cuni 4903 class class class wbr 5142 × cxp 5670 ran crn 5673 ∘ ccom 5676 Fn wfn 6537 ⟶wf 6538 ‘cfv 6542 (class class class)co 7414 ↑_m cmap 8836 supcsup 9455 ℝcr 11129 0cc0 11130 1c1 11131 + caddc 11133 +∞cpnf 11267 ℝ^*cxr 11269 < clt 11270 ≤ cle 11271 − cmin 11466 ℕcn 12234 (,)cioo 13348 [,)cico 13350 seqcseq 13990 abscabs 15205

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207 ax-pre-sup 11208

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8718 df-map 8838 df-en 8956 df-dom 8957 df-sdom 8958 df-sup 9457 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-div 11894 df-nn 12235 df-2 12297 df-3 12298 df-n0 12495 df-z 12581 df-uz 12845 df-rp 12999 df-ico 13354 df-fz 13509 df-seq 13991 df-exp 14051 df-cj 15070 df-re 15071 df-im 15072 df-sqrt 15206 df-abs 15207

This theorem is referenced by: ovolge0 25397

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