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Mirrors > Home > MPE Home > Th. List > uniioombllem1 | Structured version Visualization version GIF version |
Description: Lemma for uniioombl 24193. (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 2798 | . . . . . 6 ⊢ ((abs ∘ − ) ∘ 𝐺) = ((abs ∘ − ) ∘ 𝐺) | |
3 | uniioombl.t | . . . . . 6 ⊢ 𝑇 = seq1( + , ((abs ∘ − ) ∘ 𝐺)) | |
4 | 2, 3 | ovolsf 24076 | . . . . 5 ⊢ (𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑇:ℕ⟶(0[,)+∞)) |
5 | 1, 4 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑇:ℕ⟶(0[,)+∞)) |
6 | 5 | frnd 6494 | . . 3 ⊢ (𝜑 → ran 𝑇 ⊆ (0[,)+∞)) |
7 | rge0ssre 12834 | . . 3 ⊢ (0[,)+∞) ⊆ ℝ | |
8 | 6, 7 | sstrdi 3927 | . 2 ⊢ (𝜑 → ran 𝑇 ⊆ ℝ) |
9 | 1nn 11636 | . . . . 5 ⊢ 1 ∈ ℕ | |
10 | 5 | fdmd 6497 | . . . . 5 ⊢ (𝜑 → dom 𝑇 = ℕ) |
11 | 9, 10 | eleqtrrid 2897 | . . . 4 ⊢ (𝜑 → 1 ∈ dom 𝑇) |
12 | 11 | ne0d 4251 | . . 3 ⊢ (𝜑 → dom 𝑇 ≠ ∅) |
13 | dm0rn0 5759 | . . . 4 ⊢ (dom 𝑇 = ∅ ↔ ran 𝑇 = ∅) | |
14 | 13 | necon3bii 3039 | . . 3 ⊢ (dom 𝑇 ≠ ∅ ↔ ran 𝑇 ≠ ∅) |
15 | 12, 14 | sylib 221 | . 2 ⊢ (𝜑 → ran 𝑇 ≠ ∅) |
16 | icossxr 12810 | . . . . 5 ⊢ (0[,)+∞) ⊆ ℝ* | |
17 | 6, 16 | sstrdi 3927 | . . . 4 ⊢ (𝜑 → ran 𝑇 ⊆ ℝ*) |
18 | supxrcl 12696 | . . . 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 12419 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℝ) |
23 | 20, 22 | readdcld 10659 | . . . 4 ⊢ (𝜑 → ((vol*‘𝐸) + 𝐶) ∈ ℝ) |
24 | 23 | rexrd 10680 | . . 3 ⊢ (𝜑 → ((vol*‘𝐸) + 𝐶) ∈ ℝ*) |
25 | pnfxr 10684 | . . . 4 ⊢ +∞ ∈ ℝ* | |
26 | 25 | a1i 11 | . . 3 ⊢ (𝜑 → +∞ ∈ ℝ*) |
27 | uniioombl.v | . . 3 ⊢ (𝜑 → sup(ran 𝑇, ℝ*, < ) ≤ ((vol*‘𝐸) + 𝐶)) | |
28 | 23 | ltpnfd 12504 | . . 3 ⊢ (𝜑 → ((vol*‘𝐸) + 𝐶) < +∞) |
29 | 19, 24, 26, 27, 28 | xrlelttrd 12541 | . 2 ⊢ (𝜑 → sup(ran 𝑇, ℝ*, < ) < +∞) |
30 | supxrbnd 12709 | . 2 ⊢ ((ran 𝑇 ⊆ ℝ ∧ ran 𝑇 ≠ ∅ ∧ sup(ran 𝑇, ℝ*, < ) < +∞) → sup(ran 𝑇, ℝ*, < ) ∈ ℝ) | |
31 | 8, 15, 29, 30 | syl3anc 1368 | 1 ⊢ (𝜑 → sup(ran 𝑇, ℝ*, < ) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∩ cin 3880 ⊆ wss 3881 ∅c0 4243 ∪ cuni 4800 Disj wdisj 4995 class class class wbr 5030 × cxp 5517 dom cdm 5519 ran crn 5520 ∘ ccom 5523 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 supcsup 8888 ℝcr 10525 0cc0 10526 1c1 10527 + caddc 10529 +∞cpnf 10661 ℝ*cxr 10663 < clt 10664 ≤ cle 10665 − cmin 10859 ℕcn 11625 ℝ+crp 12377 (,)cioo 12726 [,)cico 12728 seqcseq 13364 abscabs 14585 vol*covol 24066 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-sup 8890 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-ico 12732 df-fz 12886 df-seq 13365 df-exp 13426 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 |
This theorem is referenced by: uniioombllem3 24189 uniioombllem4 24190 uniioombllem5 24191 uniioombllem6 24192 |
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