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Theorem ovolmge0 25760

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 25758	. 2 ⊢ (𝐵 ∈ 𝑀 ↔ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝐵 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^*, < )))
3		elovolmlem 25757	. . . . . 6 ⊢ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ↔ 𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
4		eqid 2760	. . . . . . . . . 10 ⊢ ((abs ∘ − ) ∘ 𝑓) = ((abs ∘ − ) ∘ 𝑓)
5		eqid 2760	. . . . . . . . . 10 ⊢ seq1( + , ((abs ∘ − ) ∘ 𝑓)) = seq1( + , ((abs ∘ − ) ∘ 𝑓))
6	4, 5	ovolsf 25755	. . . . . . . . 9 ⊢ (𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → seq1( + , ((abs ∘ − ) ∘ 𝑓)):ℕ⟶(0[,)+∞))
7		1nn 12316	. . . . . . . . 9 ⊢ 1 ∈ ℕ
8		ffvelcdm 7069	. . . . . . . . 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 13555	. . . . . . . . 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 6706	. . . . . . . . 9 ⊢ (𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ran seq1( + , ((abs ∘ − ) ∘ 𝑓)) ⊆ (0[,)+∞))
14		icossxr 13533	. . . . . . . . 9 ⊢ (0[,)+∞) ⊆ ℝ^*
15	13, 14	sstrdi 3942	. . . . . . . 8 ⊢ (𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ran seq1( + , ((abs ∘ − ) ∘ 𝑓)) ⊆ ℝ^*)
16	6	ffnd 6698	. . . . . . . . 9 ⊢ (𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → seq1( + , ((abs ∘ − ) ∘ 𝑓)) Fn ℕ)
17		fnfvelrn 7068	. . . . . . . . 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 13424	. . . . . . . 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 11328	. . . . . . . 8 ⊢ 0 ∈ ℝ^*
22	14, 9	sselid 3928	. . . . . . . 8 ⊢ (𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘1) ∈ ℝ^*)
23		supxrcl 13415	. . . . . . . . 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 13257	. . . . . . . 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 1495	. . . . . . 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 5106	. . . . 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 3156	. 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 2145 ∃wrex 3086 {crab 3412 ∩ cin 3897 ⊆ wss 3898 ∪ cuni 4866 class class class wbr 5102 × cxp 5645 ran crn 5648 ∘ ccom 5651 Fn wfn 6522 ⟶wf 6523 ‘cfv 6527 (class class class)co 7408 ↑_m cmap 8825 supcsup 9410 ℝcr 11171 0cc0 11172 1c1 11173 + caddc 11175 +∞cpnf 11312 ℝ^*cxr 11314 < clt 11315 ≤ cle 11316 − cmin 11513 ℕcn 12305 (,)cioo 13446 [,)cico 13448 seqcseq 14113 abscabs 15369

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 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7734 ax-cnex 11228 ax-resscn 11229 ax-1cn 11230 ax-icn 11231 ax-addcl 11232 ax-addrcl 11233 ax-mulcl 11234 ax-mulrcl 11235 ax-mulcom 11236 ax-addass 11237 ax-mulass 11238 ax-distr 11239 ax-i2m1 11240 ax-1ne0 11241 ax-1rid 11242 ax-rnegex 11243 ax-rrecex 11244 ax-cnre 11245 ax-pre-lttri 11246 ax-pre-lttrn 11247 ax-pre-ltadd 11248 ax-pre-mulgt0 11249 ax-pre-sup 11250

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 2100 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 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-op 4590 df-uni 4867 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5542 df-eprel 5547 df-po 5555 df-so 5556 df-fr 5600 df-we 5602 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-ima 5660 df-pred 6293 df-ord 6354 df-on 6355 df-lim 6356 df-suc 6357 df-iota 6483 df-fun 6529 df-fn 6530 df-f 6531 df-f1 6532 df-fo 6533 df-f1o 6534 df-fv 6535 df-riota 7365 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7861 df-1st 7984 df-2nd 7985 df-frecs 8277 df-wrecs 8308 df-recs 8357 df-rdg 8396 df-er 8695 df-map 8827 df-en 8952 df-dom 8953 df-sdom 8954 df-sup 9412 df-pnf 11317 df-mnf 11318 df-xr 11319 df-ltxr 11320 df-le 11321 df-sub 11515 df-neg 11516 df-div 11944 df-nn 12306 df-2 12375 df-3 12376 df-n0 12577 df-z 12664 df-uz 12936 df-rp 13091 df-ico 13452 df-fz 13610 df-seq 14114 df-exp 14174 df-cj 15234 df-re 15235 df-im 15236 df-sqrt 15370 df-abs 15371

This theorem is used by: ovolge0 25764

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