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Mirrors > Home > MPE Home > Th. List > metnrmlem2 | Structured version Visualization version GIF version |
Description: Lemma for metnrm 22473. (Contributed by Mario Carneiro, 14-Jan-2014.) (Revised by Mario Carneiro, 5-Sep-2015.) |
Ref | Expression |
---|---|
metdscn.f | ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ inf(ran (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < )) |
metdscn.j | ⊢ 𝐽 = (MetOpen‘𝐷) |
metnrmlem.1 | ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋)) |
metnrmlem.2 | ⊢ (𝜑 → 𝑆 ∈ (Clsd‘𝐽)) |
metnrmlem.3 | ⊢ (𝜑 → 𝑇 ∈ (Clsd‘𝐽)) |
metnrmlem.4 | ⊢ (𝜑 → (𝑆 ∩ 𝑇) = ∅) |
metnrmlem.u | ⊢ 𝑈 = ∪ 𝑡 ∈ 𝑇 (𝑡(ball‘𝐷)(if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) / 2)) |
Ref | Expression |
---|---|
metnrmlem2 | ⊢ (𝜑 → (𝑈 ∈ 𝐽 ∧ 𝑇 ⊆ 𝑈)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | metnrmlem.u | . . 3 ⊢ 𝑈 = ∪ 𝑡 ∈ 𝑇 (𝑡(ball‘𝐷)(if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) / 2)) | |
2 | metnrmlem.1 | . . . . 5 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋)) | |
3 | metdscn.j | . . . . . 6 ⊢ 𝐽 = (MetOpen‘𝐷) | |
4 | 3 | mopntop 22055 | . . . . 5 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top) |
5 | 2, 4 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐽 ∈ Top) |
6 | 2 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝐷 ∈ (∞Met‘𝑋)) |
7 | metnrmlem.3 | . . . . . . . . 9 ⊢ (𝜑 → 𝑇 ∈ (Clsd‘𝐽)) | |
8 | eqid 2610 | . . . . . . . . . 10 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
9 | 8 | cldss 20643 | . . . . . . . . 9 ⊢ (𝑇 ∈ (Clsd‘𝐽) → 𝑇 ⊆ ∪ 𝐽) |
10 | 7, 9 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑇 ⊆ ∪ 𝐽) |
11 | 3 | mopnuni 22056 | . . . . . . . . 9 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = ∪ 𝐽) |
12 | 2, 11 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑋 = ∪ 𝐽) |
13 | 10, 12 | sseqtr4d 3605 | . . . . . . 7 ⊢ (𝜑 → 𝑇 ⊆ 𝑋) |
14 | 13 | sselda 3568 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝑡 ∈ 𝑋) |
15 | metdscn.f | . . . . . . . . . 10 ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ inf(ran (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < )) | |
16 | metnrmlem.2 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑆 ∈ (Clsd‘𝐽)) | |
17 | metnrmlem.4 | . . . . . . . . . 10 ⊢ (𝜑 → (𝑆 ∩ 𝑇) = ∅) | |
18 | 15, 3, 2, 16, 7, 17 | metnrmlem1a 22469 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (0 < (𝐹‘𝑡) ∧ if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) ∈ ℝ+)) |
19 | 18 | simprd 478 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) ∈ ℝ+) |
20 | 19 | rphalfcld 11760 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) / 2) ∈ ℝ+) |
21 | 20 | rpxrd 11749 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) / 2) ∈ ℝ*) |
22 | 3 | blopn 22115 | . . . . . 6 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑡 ∈ 𝑋 ∧ (if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) / 2) ∈ ℝ*) → (𝑡(ball‘𝐷)(if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) / 2)) ∈ 𝐽) |
23 | 6, 14, 21, 22 | syl3anc 1318 | . . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝑡(ball‘𝐷)(if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) / 2)) ∈ 𝐽) |
24 | 23 | ralrimiva 2949 | . . . 4 ⊢ (𝜑 → ∀𝑡 ∈ 𝑇 (𝑡(ball‘𝐷)(if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) / 2)) ∈ 𝐽) |
25 | iunopn 20528 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ ∀𝑡 ∈ 𝑇 (𝑡(ball‘𝐷)(if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) / 2)) ∈ 𝐽) → ∪ 𝑡 ∈ 𝑇 (𝑡(ball‘𝐷)(if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) / 2)) ∈ 𝐽) | |
26 | 5, 24, 25 | syl2anc 691 | . . 3 ⊢ (𝜑 → ∪ 𝑡 ∈ 𝑇 (𝑡(ball‘𝐷)(if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) / 2)) ∈ 𝐽) |
27 | 1, 26 | syl5eqel 2692 | . 2 ⊢ (𝜑 → 𝑈 ∈ 𝐽) |
28 | blcntr 22028 | . . . . . . 7 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑡 ∈ 𝑋 ∧ (if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) / 2) ∈ ℝ+) → 𝑡 ∈ (𝑡(ball‘𝐷)(if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) / 2))) | |
29 | 6, 14, 20, 28 | syl3anc 1318 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝑡 ∈ (𝑡(ball‘𝐷)(if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) / 2))) |
30 | 29 | snssd 4281 | . . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → {𝑡} ⊆ (𝑡(ball‘𝐷)(if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) / 2))) |
31 | 30 | ralrimiva 2949 | . . . 4 ⊢ (𝜑 → ∀𝑡 ∈ 𝑇 {𝑡} ⊆ (𝑡(ball‘𝐷)(if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) / 2))) |
32 | ss2iun 4472 | . . . 4 ⊢ (∀𝑡 ∈ 𝑇 {𝑡} ⊆ (𝑡(ball‘𝐷)(if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) / 2)) → ∪ 𝑡 ∈ 𝑇 {𝑡} ⊆ ∪ 𝑡 ∈ 𝑇 (𝑡(ball‘𝐷)(if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) / 2))) | |
33 | 31, 32 | syl 17 | . . 3 ⊢ (𝜑 → ∪ 𝑡 ∈ 𝑇 {𝑡} ⊆ ∪ 𝑡 ∈ 𝑇 (𝑡(ball‘𝐷)(if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) / 2))) |
34 | iunid 4511 | . . . 4 ⊢ ∪ 𝑡 ∈ 𝑇 {𝑡} = 𝑇 | |
35 | 34 | eqcomi 2619 | . . 3 ⊢ 𝑇 = ∪ 𝑡 ∈ 𝑇 {𝑡} |
36 | 33, 35, 1 | 3sstr4g 3609 | . 2 ⊢ (𝜑 → 𝑇 ⊆ 𝑈) |
37 | 27, 36 | jca 553 | 1 ⊢ (𝜑 → (𝑈 ∈ 𝐽 ∧ 𝑇 ⊆ 𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ∩ cin 3539 ⊆ wss 3540 ∅c0 3874 ifcif 4036 {csn 4125 ∪ cuni 4372 ∪ ciun 4455 class class class wbr 4583 ↦ cmpt 4643 ran crn 5039 ‘cfv 5804 (class class class)co 6549 infcinf 8230 0cc0 9815 1c1 9816 ℝ*cxr 9952 < clt 9953 ≤ cle 9954 / cdiv 10563 2c2 10947 ℝ+crp 11708 ∞Metcxmt 19552 ballcbl 19554 MetOpencmopn 19557 Topctop 20517 Clsdccld 20630 |
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-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 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-icc 12053 df-topgen 15927 df-psmet 19559 df-xmet 19560 df-bl 19562 df-mopn 19563 df-top 20521 df-bases 20522 df-topon 20523 df-cld 20633 df-ntr 20634 df-cls 20635 |
This theorem is referenced by: metnrmlem3 22472 |
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