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Mirrors > Home > MPE Home > Th. List > ressust | Structured version Visualization version GIF version |
Description: The uniform structure of a restricted space. (Contributed by Thierry Arnoux, 22-Jan-2018.) |
Ref | Expression |
---|---|
ressust.x | ⊢ 𝑋 = (Base‘𝑊) |
ressust.t | ⊢ 𝑇 = (UnifSt‘(𝑊 ↾s 𝐴)) |
Ref | Expression |
---|---|
ressust | ⊢ ((𝑊 ∈ UnifSp ∧ 𝐴 ⊆ 𝑋) → 𝑇 ∈ (UnifOn‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ressust.t | . . 3 ⊢ 𝑇 = (UnifSt‘(𝑊 ↾s 𝐴)) | |
2 | ressust.x | . . . . . . 7 ⊢ 𝑋 = (Base‘𝑊) | |
3 | 2 | fvexi 6460 | . . . . . 6 ⊢ 𝑋 ∈ V |
4 | 3 | ssex 5039 | . . . . 5 ⊢ (𝐴 ⊆ 𝑋 → 𝐴 ∈ V) |
5 | 4 | adantl 475 | . . . 4 ⊢ ((𝑊 ∈ UnifSp ∧ 𝐴 ⊆ 𝑋) → 𝐴 ∈ V) |
6 | ressuss 22475 | . . . 4 ⊢ (𝐴 ∈ V → (UnifSt‘(𝑊 ↾s 𝐴)) = ((UnifSt‘𝑊) ↾t (𝐴 × 𝐴))) | |
7 | 5, 6 | syl 17 | . . 3 ⊢ ((𝑊 ∈ UnifSp ∧ 𝐴 ⊆ 𝑋) → (UnifSt‘(𝑊 ↾s 𝐴)) = ((UnifSt‘𝑊) ↾t (𝐴 × 𝐴))) |
8 | 1, 7 | syl5eq 2825 | . 2 ⊢ ((𝑊 ∈ UnifSp ∧ 𝐴 ⊆ 𝑋) → 𝑇 = ((UnifSt‘𝑊) ↾t (𝐴 × 𝐴))) |
9 | eqid 2777 | . . . . 5 ⊢ (UnifSt‘𝑊) = (UnifSt‘𝑊) | |
10 | eqid 2777 | . . . . 5 ⊢ (TopOpen‘𝑊) = (TopOpen‘𝑊) | |
11 | 2, 9, 10 | isusp 22473 | . . . 4 ⊢ (𝑊 ∈ UnifSp ↔ ((UnifSt‘𝑊) ∈ (UnifOn‘𝑋) ∧ (TopOpen‘𝑊) = (unifTop‘(UnifSt‘𝑊)))) |
12 | 11 | simplbi 493 | . . 3 ⊢ (𝑊 ∈ UnifSp → (UnifSt‘𝑊) ∈ (UnifOn‘𝑋)) |
13 | trust 22441 | . . 3 ⊢ (((UnifSt‘𝑊) ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → ((UnifSt‘𝑊) ↾t (𝐴 × 𝐴)) ∈ (UnifOn‘𝐴)) | |
14 | 12, 13 | sylan 575 | . 2 ⊢ ((𝑊 ∈ UnifSp ∧ 𝐴 ⊆ 𝑋) → ((UnifSt‘𝑊) ↾t (𝐴 × 𝐴)) ∈ (UnifOn‘𝐴)) |
15 | 8, 14 | eqeltrd 2858 | 1 ⊢ ((𝑊 ∈ UnifSp ∧ 𝐴 ⊆ 𝑋) → 𝑇 ∈ (UnifOn‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2106 Vcvv 3397 ⊆ wss 3791 × cxp 5353 ‘cfv 6135 (class class class)co 6922 Basecbs 16255 ↾s cress 16256 ↾t crest 16467 TopOpenctopn 16468 UnifOncust 22411 unifTopcutop 22442 UnifStcuss 22465 UnifSpcusp 22466 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-7 11443 df-8 11444 df-9 11445 df-n0 11643 df-z 11729 df-dec 11846 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-ress 16263 df-unif 16361 df-rest 16469 df-ust 22412 df-uss 22468 df-usp 22469 |
This theorem is referenced by: ucnextcn 22516 |
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