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Theorem ovolmge0 25469

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 25467	. 2 ⊢ (𝐵 ∈ 𝑀 ↔ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝐵 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^*, < )))
3		elovolmlem 25466	. . . . . 6 ⊢ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ↔ 𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
4		eqid 2740	. . . . . . . . . 10 ⊢ ((abs ∘ − ) ∘ 𝑓) = ((abs ∘ − ) ∘ 𝑓)
5		eqid 2740	. . . . . . . . . 10 ⊢ seq1( + , ((abs ∘ − ) ∘ 𝑓)) = seq1( + , ((abs ∘ − ) ∘ 𝑓))
6	4, 5	ovolsf 25464	. . . . . . . . 9 ⊢ (𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → seq1( + , ((abs ∘ − ) ∘ 𝑓)):ℕ⟶(0[,)+∞))
7		1nn 12183	. . . . . . . . 9 ⊢ 1 ∈ ℕ
8		ffvelcdm 7029	. . . . . . . . 9 ⊢ ((seq1( + , ((abs ∘ − ) ∘ 𝑓)):ℕ⟶(0[,)+∞) ∧ 1 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘1) ∈ (0[,)+∞))
9	6, 7, 8	sylancl 592	. . . . . . . 8 ⊢ (𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘1) ∈ (0[,)+∞))
10		elrege0 13405	. . . . . . . . 9 ⊢ ((seq1( + , ((abs ∘ − ) ∘ 𝑓))‘1) ∈ (0[,)+∞) ↔ ((seq1( + , ((abs ∘ − ) ∘ 𝑓))‘1) ∈ ℝ ∧ 0 ≤ (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘1)))
11	10	simprbi 498	. . . . . . . 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 6670	. . . . . . . . 9 ⊢ (𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ran seq1( + , ((abs ∘ − ) ∘ 𝑓)) ⊆ (0[,)+∞))
14		icossxr 13383	. . . . . . . . 9 ⊢ (0[,)+∞) ⊆ ℝ^*
15	13, 14	sstrdi 3934	. . . . . . . 8 ⊢ (𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ran seq1( + , ((abs ∘ − ) ∘ 𝑓)) ⊆ ℝ^*)
16	6	ffnd 6663	. . . . . . . . 9 ⊢ (𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → seq1( + , ((abs ∘ − ) ∘ 𝑓)) Fn ℕ)
17		fnfvelrn 7028	. . . . . . . . 9 ⊢ ((seq1( + , ((abs ∘ − ) ∘ 𝑓)) Fn ℕ ∧ 1 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘1) ∈ ran seq1( + , ((abs ∘ − ) ∘ 𝑓)))
18	16, 7, 17	sylancl 592	. . . . . . . 8 ⊢ (𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘1) ∈ ran seq1( + , ((abs ∘ − ) ∘ 𝑓)))
19		supxrub 13274	. . . . . . . 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 590	. . . . . . 7 ⊢ (𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘1) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^*, < ))
21		0xr 11190	. . . . . . . 8 ⊢ 0 ∈ ℝ^*
22	14, 9	sselid 3920	. . . . . . . 8 ⊢ (𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘1) ∈ ℝ^*)
23		supxrcl 13265	. . . . . . . . 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 13107	. . . . . . . 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 1474	. . . . . . 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 705	. . . . . 6 ⊢ (𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 0 ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^*, < ))
28	3, 27	sylbi 218	. . . . 5 ⊢ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) → 0 ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^*, < ))
29		breq2 5083	. . . . 5 ⊢ (𝐵 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^, < ) → (0 ≤ 𝐵 ↔ 0 ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^, < )))
30	28, 29	syl5ibrcom 248	. . . 4 ⊢ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) → (𝐵 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^*, < ) → 0 ≤ 𝐵))
31	30	adantld 491	. . 3 ⊢ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) → ((𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝐵 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^*, < )) → 0 ≤ 𝐵))
32	31	rexlimiv 3134	. 2 ⊢ (∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝐵 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^*, < )) → 0 ≤ 𝐵)
33	2, 32	sylbi 218	1 ⊢ (𝐵 ∈ 𝑀 → 0 ≤ 𝐵)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1547 ∈ wcel 2119 ∃wrex 3064 {crab 3392 ∩ cin 3889 ⊆ wss 3890 ∪ cuni 4845 class class class wbr 5079 × cxp 5623 ran crn 5626 ∘ ccom 5629 Fn wfn 6487 ⟶wf 6488 ‘cfv 6492 (class class class)co 7363 ↑_m cmap 8770 supcsup 9350 ℝcr 11035 0cc0 11036 1c1 11037 + caddc 11039 +∞cpnf 11174 ℝ^*cxr 11176 < clt 11177 ≤ cle 11178 − cmin 11375 ℕcn 12172 (,)cioo 13296 [,)cico 13298 seqcseq 13961 abscabs 15194

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2712 ax-sep 5225 ax-nul 5235 ax-pow 5301 ax-pr 5369 ax-un 7685 ax-cnex 11092 ax-resscn 11093 ax-1cn 11094 ax-icn 11095 ax-addcl 11096 ax-addrcl 11097 ax-mulcl 11098 ax-mulrcl 11099 ax-mulcom 11100 ax-addass 11101 ax-mulass 11102 ax-distr 11103 ax-i2m1 11104 ax-1ne0 11105 ax-1rid 11106 ax-rnegex 11107 ax-rrecex 11108 ax-cnre 11109 ax-pre-lttri 11110 ax-pre-lttrn 11111 ax-pre-ltadd 11112 ax-pre-mulgt0 11113 ax-pre-sup 11114

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2719 df-cleq 2732 df-clel 2815 df-nfc 2889 df-ne 2936 df-nel 3040 df-ral 3055 df-rex 3065 df-rmo 3345 df-reu 3346 df-rab 3393 df-v 3434 df-sbc 3731 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4269 df-if 4462 df-pw 4538 df-sn 4563 df-pr 4565 df-op 4569 df-uni 4846 df-iun 4930 df-br 5080 df-opab 5142 df-mpt 5161 df-tr 5187 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7320 df-ov 7366 df-oprab 7367 df-mpo 7368 df-om 7814 df-1st 7938 df-2nd 7939 df-frecs 8228 df-wrecs 8259 df-recs 8308 df-rdg 8346 df-er 8640 df-map 8772 df-en 8891 df-dom 8892 df-sdom 8893 df-sup 9352 df-pnf 11179 df-mnf 11180 df-xr 11181 df-ltxr 11182 df-le 11183 df-sub 11377 df-neg 11378 df-div 11806 df-nn 12173 df-2 12242 df-3 12243 df-n0 12436 df-z 12523 df-uz 12787 df-rp 12941 df-ico 13302 df-fz 13460 df-seq 13962 df-exp 14022 df-cj 15059 df-re 15060 df-im 15061 df-sqrt 15195 df-abs 15196

This theorem is referenced by: ovolge0 25473

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