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Mirrors > Home > MPE Home > Th. List > cmpcmet | Structured version Visualization version GIF version |
Description: A compact metric space is complete. One half of heibor 32790. (Contributed by Mario Carneiro, 15-Oct-2015.) |
Ref | Expression |
---|---|
relcmpcmet.1 | ⊢ 𝐽 = (MetOpen‘𝐷) |
relcmpcmet.2 | ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋)) |
cmpcmet.3 | ⊢ (𝜑 → 𝐽 ∈ Comp) |
Ref | Expression |
---|---|
cmpcmet | ⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relcmpcmet.1 | . 2 ⊢ 𝐽 = (MetOpen‘𝐷) | |
2 | relcmpcmet.2 | . 2 ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋)) | |
3 | 1rp 11712 | . . 3 ⊢ 1 ∈ ℝ+ | |
4 | 3 | a1i 11 | . 2 ⊢ (𝜑 → 1 ∈ ℝ+) |
5 | cmpcmet.3 | . . . 4 ⊢ (𝜑 → 𝐽 ∈ Comp) | |
6 | 5 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐽 ∈ Comp) |
7 | metxmet 21949 | . . . . . . 7 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋)) | |
8 | 2, 7 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋)) |
9 | 8 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐷 ∈ (∞Met‘𝑋)) |
10 | 1 | mopntop 22055 | . . . . 5 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top) |
11 | 9, 10 | syl 17 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐽 ∈ Top) |
12 | simpr 476 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋) | |
13 | rpxr 11716 | . . . . . . 7 ⊢ (1 ∈ ℝ+ → 1 ∈ ℝ*) | |
14 | 3, 13 | mp1i 13 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 1 ∈ ℝ*) |
15 | blssm 22033 | . . . . . 6 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ 1 ∈ ℝ*) → (𝑥(ball‘𝐷)1) ⊆ 𝑋) | |
16 | 9, 12, 14, 15 | syl3anc 1318 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑥(ball‘𝐷)1) ⊆ 𝑋) |
17 | 1 | mopnuni 22056 | . . . . . 6 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = ∪ 𝐽) |
18 | 9, 17 | syl 17 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑋 = ∪ 𝐽) |
19 | 16, 18 | sseqtrd 3604 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑥(ball‘𝐷)1) ⊆ ∪ 𝐽) |
20 | eqid 2610 | . . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
21 | 20 | clscld 20661 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ (𝑥(ball‘𝐷)1) ⊆ ∪ 𝐽) → ((cls‘𝐽)‘(𝑥(ball‘𝐷)1)) ∈ (Clsd‘𝐽)) |
22 | 11, 19, 21 | syl2anc 691 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((cls‘𝐽)‘(𝑥(ball‘𝐷)1)) ∈ (Clsd‘𝐽)) |
23 | cmpcld 21015 | . . 3 ⊢ ((𝐽 ∈ Comp ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)1)) ∈ (Clsd‘𝐽)) → (𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)1))) ∈ Comp) | |
24 | 6, 22, 23 | syl2anc 691 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)1))) ∈ Comp) |
25 | 1, 2, 4, 24 | relcmpcmet 22923 | 1 ⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ⊆ wss 3540 ∪ cuni 4372 ‘cfv 5804 (class class class)co 6549 1c1 9816 ℝ*cxr 9952 ℝ+crp 11708 ↾t crest 15904 ∞Metcxmt 19552 Metcme 19553 ballcbl 19554 MetOpencmopn 19557 Topctop 20517 Clsdccld 20630 clsccl 20632 Compccmp 20999 CMetcms 22860 |
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-rep 4699 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-int 4411 df-iun 4457 df-iin 4458 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-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fi 8200 df-sup 8231 df-inf 8232 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-n0 11170 df-z 11255 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-ico 12052 df-rest 15906 df-topgen 15927 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 df-mopn 19563 df-fbas 19564 df-fg 19565 df-top 20521 df-bases 20522 df-topon 20523 df-cld 20633 df-ntr 20634 df-cls 20635 df-nei 20712 df-cmp 21000 df-fil 21460 df-flim 21553 df-fcls 21555 df-cfil 22861 df-cmet 22863 |
This theorem is referenced by: (None) |
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