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Mirrors > Home > MPE Home > Th. List > uniioombllem1 | Structured version Visualization version GIF version |
Description: Lemma for uniioombl 24976. (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 2733 | . . . . . 6 ⊢ ((abs ∘ − ) ∘ 𝐺) = ((abs ∘ − ) ∘ 𝐺) | |
3 | uniioombl.t | . . . . . 6 ⊢ 𝑇 = seq1( + , ((abs ∘ − ) ∘ 𝐺)) | |
4 | 2, 3 | ovolsf 24859 | . . . . 5 ⊢ (𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑇:ℕ⟶(0[,)+∞)) |
5 | 1, 4 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑇:ℕ⟶(0[,)+∞)) |
6 | 5 | frnd 6680 | . . 3 ⊢ (𝜑 → ran 𝑇 ⊆ (0[,)+∞)) |
7 | rge0ssre 13382 | . . 3 ⊢ (0[,)+∞) ⊆ ℝ | |
8 | 6, 7 | sstrdi 3960 | . 2 ⊢ (𝜑 → ran 𝑇 ⊆ ℝ) |
9 | 1nn 12172 | . . . . 5 ⊢ 1 ∈ ℕ | |
10 | 5 | fdmd 6683 | . . . . 5 ⊢ (𝜑 → dom 𝑇 = ℕ) |
11 | 9, 10 | eleqtrrid 2841 | . . . 4 ⊢ (𝜑 → 1 ∈ dom 𝑇) |
12 | 11 | ne0d 4299 | . . 3 ⊢ (𝜑 → dom 𝑇 ≠ ∅) |
13 | dm0rn0 5884 | . . . 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 13358 | . . . . 5 ⊢ (0[,)+∞) ⊆ ℝ* | |
17 | 6, 16 | sstrdi 3960 | . . . 4 ⊢ (𝜑 → ran 𝑇 ⊆ ℝ*) |
18 | supxrcl 13243 | . . . 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 12965 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℝ) |
23 | 20, 22 | readdcld 11192 | . . . 4 ⊢ (𝜑 → ((vol*‘𝐸) + 𝐶) ∈ ℝ) |
24 | 23 | rexrd 11213 | . . 3 ⊢ (𝜑 → ((vol*‘𝐸) + 𝐶) ∈ ℝ*) |
25 | pnfxr 11217 | . . . 4 ⊢ +∞ ∈ ℝ* | |
26 | 25 | a1i 11 | . . 3 ⊢ (𝜑 → +∞ ∈ ℝ*) |
27 | uniioombl.v | . . 3 ⊢ (𝜑 → sup(ran 𝑇, ℝ*, < ) ≤ ((vol*‘𝐸) + 𝐶)) | |
28 | 23 | ltpnfd 13050 | . . 3 ⊢ (𝜑 → ((vol*‘𝐸) + 𝐶) < +∞) |
29 | 19, 24, 26, 27, 28 | xrlelttrd 13088 | . 2 ⊢ (𝜑 → sup(ran 𝑇, ℝ*, < ) < +∞) |
30 | supxrbnd 13256 | . 2 ⊢ ((ran 𝑇 ⊆ ℝ ∧ ran 𝑇 ≠ ∅ ∧ sup(ran 𝑇, ℝ*, < ) < +∞) → sup(ran 𝑇, ℝ*, < ) ∈ ℝ) | |
31 | 8, 15, 29, 30 | syl3anc 1372 | 1 ⊢ (𝜑 → sup(ran 𝑇, ℝ*, < ) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 ≠ wne 2940 ∩ cin 3913 ⊆ wss 3914 ∅c0 4286 ∪ cuni 4869 Disj wdisj 5074 class class class wbr 5109 × cxp 5635 dom cdm 5637 ran crn 5638 ∘ ccom 5641 ⟶wf 6496 ‘cfv 6500 (class class class)co 7361 supcsup 9384 ℝcr 11058 0cc0 11059 1c1 11060 + caddc 11062 +∞cpnf 11194 ℝ*cxr 11196 < clt 11197 ≤ cle 11198 − cmin 11393 ℕcn 12161 ℝ+crp 12923 (,)cioo 13273 [,)cico 13275 seqcseq 13915 abscabs 15128 vol*covol 24849 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 ax-pre-sup 11137 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-1st 7925 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-er 8654 df-en 8890 df-dom 8891 df-sdom 8892 df-sup 9386 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-div 11821 df-nn 12162 df-2 12224 df-3 12225 df-n0 12422 df-z 12508 df-uz 12772 df-rp 12924 df-ico 13279 df-fz 13434 df-seq 13916 df-exp 13977 df-cj 14993 df-re 14994 df-im 14995 df-sqrt 15129 df-abs 15130 |
This theorem is referenced by: uniioombllem3 24972 uniioombllem4 24973 uniioombllem5 24974 uniioombllem6 24975 |
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