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Mirrors > Home > MPE Home > Th. List > minveclem3a | Structured version Visualization version GIF version |
Description: Lemma for minvec 23253. 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 2651 | . . . 4 ⊢ (Base‘(𝑈 ↾s 𝑌)) = (Base‘(𝑈 ↾s 𝑌)) | |
3 | eqid 2651 | . . . 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 23189 | . . 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 5424 | . . 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 2651 | . . . . . . . 8 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
11 | 9, 10 | lssss 18985 | . . . . . . 7 ⊢ (𝑌 ∈ (LSubSp‘𝑈) → 𝑌 ⊆ 𝑋) |
12 | 8, 11 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑌 ⊆ 𝑋) |
13 | xpss12 5158 | . . . . . 6 ⊢ ((𝑌 ⊆ 𝑋 ∧ 𝑌 ⊆ 𝑋) → (𝑌 × 𝑌) ⊆ (𝑋 × 𝑋)) | |
14 | 12, 12, 13 | syl2anc 694 | . . . . 5 ⊢ (𝜑 → (𝑌 × 𝑌) ⊆ (𝑋 × 𝑋)) |
15 | 14 | resabs1d 5463 | . . . 4 ⊢ (𝜑 → (((dist‘𝑈) ↾ (𝑋 × 𝑋)) ↾ (𝑌 × 𝑌)) = ((dist‘𝑈) ↾ (𝑌 × 𝑌))) |
16 | eqid 2651 | . . . . . . 7 ⊢ (𝑈 ↾s 𝑌) = (𝑈 ↾s 𝑌) | |
17 | eqid 2651 | . . . . . . 7 ⊢ (dist‘𝑈) = (dist‘𝑈) | |
18 | 16, 17 | ressds 16120 | . . . . . 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 15978 | . . . . . . 7 ⊢ (𝑌 ⊆ 𝑋 → 𝑌 = (Base‘(𝑈 ↾s 𝑌))) |
21 | 12, 20 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑌 = (Base‘(𝑈 ↾s 𝑌))) |
22 | 21 | sqxpeqd 5175 | . . . . 5 ⊢ (𝜑 → (𝑌 × 𝑌) = ((Base‘(𝑈 ↾s 𝑌)) × (Base‘(𝑈 ↾s 𝑌)))) |
23 | 19, 22 | reseq12d 5429 | . . . 4 ⊢ (𝜑 → ((dist‘𝑈) ↾ (𝑌 × 𝑌)) = ((dist‘(𝑈 ↾s 𝑌)) ↾ ((Base‘(𝑈 ↾s 𝑌)) × (Base‘(𝑈 ↾s 𝑌))))) |
24 | 15, 23 | eqtrd 2685 | . . 3 ⊢ (𝜑 → (((dist‘𝑈) ↾ (𝑋 × 𝑋)) ↾ (𝑌 × 𝑌)) = ((dist‘(𝑈 ↾s 𝑌)) ↾ ((Base‘(𝑈 ↾s 𝑌)) × (Base‘(𝑈 ↾s 𝑌))))) |
25 | 7, 24 | syl5eq 2697 | . 2 ⊢ (𝜑 → (𝐷 ↾ (𝑌 × 𝑌)) = ((dist‘(𝑈 ↾s 𝑌)) ↾ ((Base‘(𝑈 ↾s 𝑌)) × (Base‘(𝑈 ↾s 𝑌))))) |
26 | 21 | fveq2d 6233 | . 2 ⊢ (𝜑 → (CMet‘𝑌) = (CMet‘(Base‘(𝑈 ↾s 𝑌)))) |
27 | 5, 25, 26 | 3eltr4d 2745 | 1 ⊢ (𝜑 → (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1523 ∈ wcel 2030 ⊆ wss 3607 ↦ cmpt 4762 × cxp 5141 ran crn 5144 ↾ cres 5145 ‘cfv 5926 (class class class)co 6690 infcinf 8388 ℝcr 9973 < clt 10112 Basecbs 15904 ↾s cress 15905 distcds 15997 TopOpenctopn 16129 -gcsg 17471 LSubSpclss 18980 normcnm 22428 ℂPreHilccph 23012 CMetcms 23098 CMetSpccms 23175 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-nn 11059 df-2 11117 df-3 11118 df-4 11119 df-5 11120 df-6 11121 df-7 11122 df-8 11123 df-9 11124 df-n0 11331 df-z 11416 df-dec 11532 df-ndx 15907 df-slot 15908 df-base 15910 df-sets 15911 df-ress 15912 df-ds 16011 df-lss 18981 df-cms 23178 |
This theorem is referenced by: minveclem3 23246 minveclem4a 23247 |
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