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Mirrors > Home > MPE Home > Th. List > minveclem3a | Structured version Visualization version GIF version |
Description: Lemma for minvec 23605. 𝐷 is a complete metric when restricted to 𝑌. (Contributed by Mario Carneiro, 7-May-2014.) (Revised by Mario Carneiro, 15-Oct-2015.) |
Ref | Expression |
---|---|
minvec.x | ⊢ 𝑋 = (Base‘𝑈) |
minvec.m | ⊢ − = (-g‘𝑈) |
minvec.n | ⊢ 𝑁 = (norm‘𝑈) |
minvec.u | ⊢ (𝜑 → 𝑈 ∈ ℂPreHil) |
minvec.y | ⊢ (𝜑 → 𝑌 ∈ (LSubSp‘𝑈)) |
minvec.w | ⊢ (𝜑 → (𝑈 ↾s 𝑌) ∈ CMetSp) |
minvec.a | ⊢ (𝜑 → 𝐴 ∈ 𝑋) |
minvec.j | ⊢ 𝐽 = (TopOpen‘𝑈) |
minvec.r | ⊢ 𝑅 = ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦))) |
minvec.s | ⊢ 𝑆 = inf(𝑅, ℝ, < ) |
minvec.d | ⊢ 𝐷 = ((dist‘𝑈) ↾ (𝑋 × 𝑋)) |
Ref | Expression |
---|---|
minveclem3a | ⊢ (𝜑 → (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | minvec.w | . . 3 ⊢ (𝜑 → (𝑈 ↾s 𝑌) ∈ CMetSp) | |
2 | eqid 2826 | . . . 4 ⊢ (Base‘(𝑈 ↾s 𝑌)) = (Base‘(𝑈 ↾s 𝑌)) | |
3 | eqid 2826 | . . . 4 ⊢ ((dist‘(𝑈 ↾s 𝑌)) ↾ ((Base‘(𝑈 ↾s 𝑌)) × (Base‘(𝑈 ↾s 𝑌)))) = ((dist‘(𝑈 ↾s 𝑌)) ↾ ((Base‘(𝑈 ↾s 𝑌)) × (Base‘(𝑈 ↾s 𝑌)))) | |
4 | 2, 3 | cmscmet 23515 | . . 3 ⊢ ((𝑈 ↾s 𝑌) ∈ CMetSp → ((dist‘(𝑈 ↾s 𝑌)) ↾ ((Base‘(𝑈 ↾s 𝑌)) × (Base‘(𝑈 ↾s 𝑌)))) ∈ (CMet‘(Base‘(𝑈 ↾s 𝑌)))) |
5 | 1, 4 | syl 17 | . 2 ⊢ (𝜑 → ((dist‘(𝑈 ↾s 𝑌)) ↾ ((Base‘(𝑈 ↾s 𝑌)) × (Base‘(𝑈 ↾s 𝑌)))) ∈ (CMet‘(Base‘(𝑈 ↾s 𝑌)))) |
6 | minvec.d | . . . 4 ⊢ 𝐷 = ((dist‘𝑈) ↾ (𝑋 × 𝑋)) | |
7 | 6 | reseq1i 5626 | . . 3 ⊢ (𝐷 ↾ (𝑌 × 𝑌)) = (((dist‘𝑈) ↾ (𝑋 × 𝑋)) ↾ (𝑌 × 𝑌)) |
8 | minvec.y | . . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ (LSubSp‘𝑈)) | |
9 | minvec.x | . . . . . . . 8 ⊢ 𝑋 = (Base‘𝑈) | |
10 | eqid 2826 | . . . . . . . 8 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
11 | 9, 10 | lssss 19294 | . . . . . . 7 ⊢ (𝑌 ∈ (LSubSp‘𝑈) → 𝑌 ⊆ 𝑋) |
12 | 8, 11 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑌 ⊆ 𝑋) |
13 | xpss12 5358 | . . . . . 6 ⊢ ((𝑌 ⊆ 𝑋 ∧ 𝑌 ⊆ 𝑋) → (𝑌 × 𝑌) ⊆ (𝑋 × 𝑋)) | |
14 | 12, 12, 13 | syl2anc 581 | . . . . 5 ⊢ (𝜑 → (𝑌 × 𝑌) ⊆ (𝑋 × 𝑋)) |
15 | 14 | resabs1d 5665 | . . . 4 ⊢ (𝜑 → (((dist‘𝑈) ↾ (𝑋 × 𝑋)) ↾ (𝑌 × 𝑌)) = ((dist‘𝑈) ↾ (𝑌 × 𝑌))) |
16 | eqid 2826 | . . . . . . 7 ⊢ (𝑈 ↾s 𝑌) = (𝑈 ↾s 𝑌) | |
17 | eqid 2826 | . . . . . . 7 ⊢ (dist‘𝑈) = (dist‘𝑈) | |
18 | 16, 17 | ressds 16427 | . . . . . 6 ⊢ (𝑌 ∈ (LSubSp‘𝑈) → (dist‘𝑈) = (dist‘(𝑈 ↾s 𝑌))) |
19 | 8, 18 | syl 17 | . . . . 5 ⊢ (𝜑 → (dist‘𝑈) = (dist‘(𝑈 ↾s 𝑌))) |
20 | 16, 9 | ressbas2 16295 | . . . . . . 7 ⊢ (𝑌 ⊆ 𝑋 → 𝑌 = (Base‘(𝑈 ↾s 𝑌))) |
21 | 12, 20 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑌 = (Base‘(𝑈 ↾s 𝑌))) |
22 | 21 | sqxpeqd 5375 | . . . . 5 ⊢ (𝜑 → (𝑌 × 𝑌) = ((Base‘(𝑈 ↾s 𝑌)) × (Base‘(𝑈 ↾s 𝑌)))) |
23 | 19, 22 | reseq12d 5631 | . . . 4 ⊢ (𝜑 → ((dist‘𝑈) ↾ (𝑌 × 𝑌)) = ((dist‘(𝑈 ↾s 𝑌)) ↾ ((Base‘(𝑈 ↾s 𝑌)) × (Base‘(𝑈 ↾s 𝑌))))) |
24 | 15, 23 | eqtrd 2862 | . . 3 ⊢ (𝜑 → (((dist‘𝑈) ↾ (𝑋 × 𝑋)) ↾ (𝑌 × 𝑌)) = ((dist‘(𝑈 ↾s 𝑌)) ↾ ((Base‘(𝑈 ↾s 𝑌)) × (Base‘(𝑈 ↾s 𝑌))))) |
25 | 7, 24 | syl5eq 2874 | . 2 ⊢ (𝜑 → (𝐷 ↾ (𝑌 × 𝑌)) = ((dist‘(𝑈 ↾s 𝑌)) ↾ ((Base‘(𝑈 ↾s 𝑌)) × (Base‘(𝑈 ↾s 𝑌))))) |
26 | 21 | fveq2d 6438 | . 2 ⊢ (𝜑 → (CMet‘𝑌) = (CMet‘(Base‘(𝑈 ↾s 𝑌)))) |
27 | 5, 25, 26 | 3eltr4d 2922 | 1 ⊢ (𝜑 → (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1658 ∈ wcel 2166 ⊆ wss 3799 ↦ cmpt 4953 × cxp 5341 ran crn 5344 ↾ cres 5345 ‘cfv 6124 (class class class)co 6906 infcinf 8617 ℝcr 10252 < clt 10392 Basecbs 16223 ↾s cress 16224 distcds 16315 TopOpenctopn 16436 -gcsg 17779 LSubSpclss 19289 normcnm 22752 ℂPreHilccph 23336 CMetccmet 23423 CMetSpccms 23501 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 ax-un 7210 ax-cnex 10309 ax-resscn 10310 ax-1cn 10311 ax-icn 10312 ax-addcl 10313 ax-addrcl 10314 ax-mulcl 10315 ax-mulrcl 10316 ax-mulcom 10317 ax-addass 10318 ax-mulass 10319 ax-distr 10320 ax-i2m1 10321 ax-1ne0 10322 ax-1rid 10323 ax-rnegex 10324 ax-rrecex 10325 ax-cnre 10326 ax-pre-lttri 10327 ax-pre-lttrn 10328 ax-pre-ltadd 10329 ax-pre-mulgt0 10330 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-nel 3104 df-ral 3123 df-rex 3124 df-reu 3125 df-rab 3127 df-v 3417 df-sbc 3664 df-csb 3759 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-pss 3815 df-nul 4146 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4660 df-iun 4743 df-br 4875 df-opab 4937 df-mpt 4954 df-tr 4977 df-id 5251 df-eprel 5256 df-po 5264 df-so 5265 df-fr 5302 df-we 5304 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 df-pred 5921 df-ord 5967 df-on 5968 df-lim 5969 df-suc 5970 df-iota 6087 df-fun 6126 df-fn 6127 df-f 6128 df-f1 6129 df-fo 6130 df-f1o 6131 df-fv 6132 df-riota 6867 df-ov 6909 df-oprab 6910 df-mpt2 6911 df-om 7328 df-wrecs 7673 df-recs 7735 df-rdg 7773 df-er 8010 df-en 8224 df-dom 8225 df-sdom 8226 df-pnf 10394 df-mnf 10395 df-xr 10396 df-ltxr 10397 df-le 10398 df-sub 10588 df-neg 10589 df-nn 11352 df-2 11415 df-3 11416 df-4 11417 df-5 11418 df-6 11419 df-7 11420 df-8 11421 df-9 11422 df-n0 11620 df-z 11706 df-dec 11823 df-ndx 16226 df-slot 16227 df-base 16229 df-sets 16230 df-ress 16231 df-ds 16328 df-lss 19290 df-cms 23504 |
This theorem is referenced by: minveclem3 23598 minveclem4a 23599 |
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