Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > uniioombllem1 | Structured version Visualization version GIF version |
Description: Lemma for uniioombl 23163. (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 2610 | . . . . . 6 ⊢ ((abs ∘ − ) ∘ 𝐺) = ((abs ∘ − ) ∘ 𝐺) | |
3 | uniioombl.t | . . . . . 6 ⊢ 𝑇 = seq1( + , ((abs ∘ − ) ∘ 𝐺)) | |
4 | 2, 3 | ovolsf 23048 | . . . . 5 ⊢ (𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑇:ℕ⟶(0[,)+∞)) |
5 | 1, 4 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑇:ℕ⟶(0[,)+∞)) |
6 | frn 5966 | . . . 4 ⊢ (𝑇:ℕ⟶(0[,)+∞) → ran 𝑇 ⊆ (0[,)+∞)) | |
7 | 5, 6 | syl 17 | . . 3 ⊢ (𝜑 → ran 𝑇 ⊆ (0[,)+∞)) |
8 | rge0ssre 12151 | . . 3 ⊢ (0[,)+∞) ⊆ ℝ | |
9 | 7, 8 | syl6ss 3580 | . 2 ⊢ (𝜑 → ran 𝑇 ⊆ ℝ) |
10 | 1nn 10908 | . . . . 5 ⊢ 1 ∈ ℕ | |
11 | fdm 5964 | . . . . . 6 ⊢ (𝑇:ℕ⟶(0[,)+∞) → dom 𝑇 = ℕ) | |
12 | 5, 11 | syl 17 | . . . . 5 ⊢ (𝜑 → dom 𝑇 = ℕ) |
13 | 10, 12 | syl5eleqr 2695 | . . . 4 ⊢ (𝜑 → 1 ∈ dom 𝑇) |
14 | ne0i 3880 | . . . 4 ⊢ (1 ∈ dom 𝑇 → dom 𝑇 ≠ ∅) | |
15 | 13, 14 | syl 17 | . . 3 ⊢ (𝜑 → dom 𝑇 ≠ ∅) |
16 | dm0rn0 5263 | . . . 4 ⊢ (dom 𝑇 = ∅ ↔ ran 𝑇 = ∅) | |
17 | 16 | necon3bii 2834 | . . 3 ⊢ (dom 𝑇 ≠ ∅ ↔ ran 𝑇 ≠ ∅) |
18 | 15, 17 | sylib 207 | . 2 ⊢ (𝜑 → ran 𝑇 ≠ ∅) |
19 | icossxr 12129 | . . . . 5 ⊢ (0[,)+∞) ⊆ ℝ* | |
20 | 7, 19 | syl6ss 3580 | . . . 4 ⊢ (𝜑 → ran 𝑇 ⊆ ℝ*) |
21 | supxrcl 12017 | . . . 4 ⊢ (ran 𝑇 ⊆ ℝ* → sup(ran 𝑇, ℝ*, < ) ∈ ℝ*) | |
22 | 20, 21 | syl 17 | . . 3 ⊢ (𝜑 → sup(ran 𝑇, ℝ*, < ) ∈ ℝ*) |
23 | uniioombl.e | . . . . 5 ⊢ (𝜑 → (vol*‘𝐸) ∈ ℝ) | |
24 | uniioombl.c | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ ℝ+) | |
25 | 24 | rpred 11748 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℝ) |
26 | 23, 25 | readdcld 9948 | . . . 4 ⊢ (𝜑 → ((vol*‘𝐸) + 𝐶) ∈ ℝ) |
27 | 26 | rexrd 9968 | . . 3 ⊢ (𝜑 → ((vol*‘𝐸) + 𝐶) ∈ ℝ*) |
28 | pnfxr 9971 | . . . 4 ⊢ +∞ ∈ ℝ* | |
29 | 28 | a1i 11 | . . 3 ⊢ (𝜑 → +∞ ∈ ℝ*) |
30 | uniioombl.v | . . 3 ⊢ (𝜑 → sup(ran 𝑇, ℝ*, < ) ≤ ((vol*‘𝐸) + 𝐶)) | |
31 | ltpnf 11830 | . . . 4 ⊢ (((vol*‘𝐸) + 𝐶) ∈ ℝ → ((vol*‘𝐸) + 𝐶) < +∞) | |
32 | 26, 31 | syl 17 | . . 3 ⊢ (𝜑 → ((vol*‘𝐸) + 𝐶) < +∞) |
33 | 22, 27, 29, 30, 32 | xrlelttrd 11867 | . 2 ⊢ (𝜑 → sup(ran 𝑇, ℝ*, < ) < +∞) |
34 | supxrbnd 12030 | . 2 ⊢ ((ran 𝑇 ⊆ ℝ ∧ ran 𝑇 ≠ ∅ ∧ sup(ran 𝑇, ℝ*, < ) < +∞) → sup(ran 𝑇, ℝ*, < ) ∈ ℝ) | |
35 | 9, 18, 33, 34 | syl3anc 1318 | 1 ⊢ (𝜑 → sup(ran 𝑇, ℝ*, < ) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∩ cin 3539 ⊆ wss 3540 ∅c0 3874 ∪ cuni 4372 Disj wdisj 4553 class class class wbr 4583 × cxp 5036 dom cdm 5038 ran crn 5039 ∘ ccom 5042 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 supcsup 8229 ℝcr 9814 0cc0 9815 1c1 9816 + caddc 9818 +∞cpnf 9950 ℝ*cxr 9952 < clt 9953 ≤ cle 9954 − cmin 10145 ℕcn 10897 ℝ+crp 11708 (,)cioo 12046 [,)cico 12048 seqcseq 12663 abscabs 13822 vol*covol 23038 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-sup 8231 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-ico 12052 df-fz 12198 df-seq 12664 df-exp 12723 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 |
This theorem is referenced by: uniioombllem3 23159 uniioombllem4 23160 uniioombllem5 23161 uniioombllem6 23162 |
Copyright terms: Public domain | W3C validator |