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Mirrors > Home > MPE Home > Th. List > uniioombllem1 | Structured version Visualization version GIF version |
Description: Lemma for uniioombl 25105. (Contributed by Mario Carneiro, 25-Mar-2015.) |
Ref | Expression |
---|---|
uniioombl.1 | ⊢ (𝜑 → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) |
uniioombl.2 | ⊢ (𝜑 → Disj 𝑥 ∈ ℕ ((,)‘(𝐹‘𝑥))) |
uniioombl.3 | ⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹)) |
uniioombl.a | ⊢ 𝐴 = ∪ ran ((,) ∘ 𝐹) |
uniioombl.e | ⊢ (𝜑 → (vol*‘𝐸) ∈ ℝ) |
uniioombl.c | ⊢ (𝜑 → 𝐶 ∈ ℝ+) |
uniioombl.g | ⊢ (𝜑 → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ))) |
uniioombl.s | ⊢ (𝜑 → 𝐸 ⊆ ∪ ran ((,) ∘ 𝐺)) |
uniioombl.t | ⊢ 𝑇 = seq1( + , ((abs ∘ − ) ∘ 𝐺)) |
uniioombl.v | ⊢ (𝜑 → sup(ran 𝑇, ℝ*, < ) ≤ ((vol*‘𝐸) + 𝐶)) |
Ref | Expression |
---|---|
uniioombllem1 | ⊢ (𝜑 → sup(ran 𝑇, ℝ*, < ) ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uniioombl.g | . . . . 5 ⊢ (𝜑 → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ))) | |
2 | eqid 2732 | . . . . . 6 ⊢ ((abs ∘ − ) ∘ 𝐺) = ((abs ∘ − ) ∘ 𝐺) | |
3 | uniioombl.t | . . . . . 6 ⊢ 𝑇 = seq1( + , ((abs ∘ − ) ∘ 𝐺)) | |
4 | 2, 3 | ovolsf 24988 | . . . . 5 ⊢ (𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑇:ℕ⟶(0[,)+∞)) |
5 | 1, 4 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑇:ℕ⟶(0[,)+∞)) |
6 | 5 | frnd 6725 | . . 3 ⊢ (𝜑 → ran 𝑇 ⊆ (0[,)+∞)) |
7 | rge0ssre 13432 | . . 3 ⊢ (0[,)+∞) ⊆ ℝ | |
8 | 6, 7 | sstrdi 3994 | . 2 ⊢ (𝜑 → ran 𝑇 ⊆ ℝ) |
9 | 1nn 12222 | . . . . 5 ⊢ 1 ∈ ℕ | |
10 | 5 | fdmd 6728 | . . . . 5 ⊢ (𝜑 → dom 𝑇 = ℕ) |
11 | 9, 10 | eleqtrrid 2840 | . . . 4 ⊢ (𝜑 → 1 ∈ dom 𝑇) |
12 | 11 | ne0d 4335 | . . 3 ⊢ (𝜑 → dom 𝑇 ≠ ∅) |
13 | dm0rn0 5924 | . . . 4 ⊢ (dom 𝑇 = ∅ ↔ ran 𝑇 = ∅) | |
14 | 13 | necon3bii 2993 | . . 3 ⊢ (dom 𝑇 ≠ ∅ ↔ ran 𝑇 ≠ ∅) |
15 | 12, 14 | sylib 217 | . 2 ⊢ (𝜑 → ran 𝑇 ≠ ∅) |
16 | icossxr 13408 | . . . . 5 ⊢ (0[,)+∞) ⊆ ℝ* | |
17 | 6, 16 | sstrdi 3994 | . . . 4 ⊢ (𝜑 → ran 𝑇 ⊆ ℝ*) |
18 | supxrcl 13293 | . . . 4 ⊢ (ran 𝑇 ⊆ ℝ* → sup(ran 𝑇, ℝ*, < ) ∈ ℝ*) | |
19 | 17, 18 | syl 17 | . . 3 ⊢ (𝜑 → sup(ran 𝑇, ℝ*, < ) ∈ ℝ*) |
20 | uniioombl.e | . . . . 5 ⊢ (𝜑 → (vol*‘𝐸) ∈ ℝ) | |
21 | uniioombl.c | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ ℝ+) | |
22 | 21 | rpred 13015 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℝ) |
23 | 20, 22 | readdcld 11242 | . . . 4 ⊢ (𝜑 → ((vol*‘𝐸) + 𝐶) ∈ ℝ) |
24 | 23 | rexrd 11263 | . . 3 ⊢ (𝜑 → ((vol*‘𝐸) + 𝐶) ∈ ℝ*) |
25 | pnfxr 11267 | . . . 4 ⊢ +∞ ∈ ℝ* | |
26 | 25 | a1i 11 | . . 3 ⊢ (𝜑 → +∞ ∈ ℝ*) |
27 | uniioombl.v | . . 3 ⊢ (𝜑 → sup(ran 𝑇, ℝ*, < ) ≤ ((vol*‘𝐸) + 𝐶)) | |
28 | 23 | ltpnfd 13100 | . . 3 ⊢ (𝜑 → ((vol*‘𝐸) + 𝐶) < +∞) |
29 | 19, 24, 26, 27, 28 | xrlelttrd 13138 | . 2 ⊢ (𝜑 → sup(ran 𝑇, ℝ*, < ) < +∞) |
30 | supxrbnd 13306 | . 2 ⊢ ((ran 𝑇 ⊆ ℝ ∧ ran 𝑇 ≠ ∅ ∧ sup(ran 𝑇, ℝ*, < ) < +∞) → sup(ran 𝑇, ℝ*, < ) ∈ ℝ) | |
31 | 8, 15, 29, 30 | syl3anc 1371 | 1 ⊢ (𝜑 → sup(ran 𝑇, ℝ*, < ) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ∩ cin 3947 ⊆ wss 3948 ∅c0 4322 ∪ cuni 4908 Disj wdisj 5113 class class class wbr 5148 × cxp 5674 dom cdm 5676 ran crn 5677 ∘ ccom 5680 ⟶wf 6539 ‘cfv 6543 (class class class)co 7408 supcsup 9434 ℝcr 11108 0cc0 11109 1c1 11110 + caddc 11112 +∞cpnf 11244 ℝ*cxr 11246 < clt 11247 ≤ cle 11248 − cmin 11443 ℕcn 12211 ℝ+crp 12973 (,)cioo 13323 [,)cico 13325 seqcseq 13965 abscabs 15180 vol*covol 24978 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7855 df-1st 7974 df-2nd 7975 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-sup 9436 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-div 11871 df-nn 12212 df-2 12274 df-3 12275 df-n0 12472 df-z 12558 df-uz 12822 df-rp 12974 df-ico 13329 df-fz 13484 df-seq 13966 df-exp 14027 df-cj 15045 df-re 15046 df-im 15047 df-sqrt 15181 df-abs 15182 |
This theorem is referenced by: uniioombllem3 25101 uniioombllem4 25102 uniioombllem5 25103 uniioombllem6 25104 |
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