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Mirrors > Home > MPE Home > Th. List > minveclem3a | Structured version Visualization version GIF version |
Description: Lemma for minvec 24033. 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 2821 | . . . 4 ⊢ (Base‘(𝑈 ↾s 𝑌)) = (Base‘(𝑈 ↾s 𝑌)) | |
3 | eqid 2821 | . . . 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 23943 | . . 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 5844 | . . 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 2821 | . . . . . . . 8 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
11 | 9, 10 | lssss 19702 | . . . . . . 7 ⊢ (𝑌 ∈ (LSubSp‘𝑈) → 𝑌 ⊆ 𝑋) |
12 | 8, 11 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑌 ⊆ 𝑋) |
13 | xpss12 5565 | . . . . . 6 ⊢ ((𝑌 ⊆ 𝑋 ∧ 𝑌 ⊆ 𝑋) → (𝑌 × 𝑌) ⊆ (𝑋 × 𝑋)) | |
14 | 12, 12, 13 | syl2anc 586 | . . . . 5 ⊢ (𝜑 → (𝑌 × 𝑌) ⊆ (𝑋 × 𝑋)) |
15 | 14 | resabs1d 5879 | . . . 4 ⊢ (𝜑 → (((dist‘𝑈) ↾ (𝑋 × 𝑋)) ↾ (𝑌 × 𝑌)) = ((dist‘𝑈) ↾ (𝑌 × 𝑌))) |
16 | eqid 2821 | . . . . . . 7 ⊢ (𝑈 ↾s 𝑌) = (𝑈 ↾s 𝑌) | |
17 | eqid 2821 | . . . . . . 7 ⊢ (dist‘𝑈) = (dist‘𝑈) | |
18 | 16, 17 | ressds 16680 | . . . . . 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 16549 | . . . . . . 7 ⊢ (𝑌 ⊆ 𝑋 → 𝑌 = (Base‘(𝑈 ↾s 𝑌))) |
21 | 12, 20 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑌 = (Base‘(𝑈 ↾s 𝑌))) |
22 | 21 | sqxpeqd 5582 | . . . . 5 ⊢ (𝜑 → (𝑌 × 𝑌) = ((Base‘(𝑈 ↾s 𝑌)) × (Base‘(𝑈 ↾s 𝑌)))) |
23 | 19, 22 | reseq12d 5849 | . . . 4 ⊢ (𝜑 → ((dist‘𝑈) ↾ (𝑌 × 𝑌)) = ((dist‘(𝑈 ↾s 𝑌)) ↾ ((Base‘(𝑈 ↾s 𝑌)) × (Base‘(𝑈 ↾s 𝑌))))) |
24 | 15, 23 | eqtrd 2856 | . . 3 ⊢ (𝜑 → (((dist‘𝑈) ↾ (𝑋 × 𝑋)) ↾ (𝑌 × 𝑌)) = ((dist‘(𝑈 ↾s 𝑌)) ↾ ((Base‘(𝑈 ↾s 𝑌)) × (Base‘(𝑈 ↾s 𝑌))))) |
25 | 7, 24 | syl5eq 2868 | . 2 ⊢ (𝜑 → (𝐷 ↾ (𝑌 × 𝑌)) = ((dist‘(𝑈 ↾s 𝑌)) ↾ ((Base‘(𝑈 ↾s 𝑌)) × (Base‘(𝑈 ↾s 𝑌))))) |
26 | 21 | fveq2d 6669 | . 2 ⊢ (𝜑 → (CMet‘𝑌) = (CMet‘(Base‘(𝑈 ↾s 𝑌)))) |
27 | 5, 25, 26 | 3eltr4d 2928 | 1 ⊢ (𝜑 → (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ⊆ wss 3936 ↦ cmpt 5139 × cxp 5548 ran crn 5551 ↾ cres 5552 ‘cfv 6350 (class class class)co 7150 infcinf 8899 ℝcr 10530 < clt 10669 Basecbs 16477 ↾s cress 16478 distcds 16568 TopOpenctopn 16689 -gcsg 18099 LSubSpclss 19697 normcnm 23180 ℂPreHilccph 23764 CMetccmet 23851 CMetSpccms 23929 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-ds 16581 df-lss 19698 df-cms 23932 |
This theorem is referenced by: minveclem3 24026 minveclem4a 24027 |
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