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Mirrors > Home > MPE Home > Th. List > minveclem3a | Structured version Visualization version GIF version |
Description: Lemma for minvec 24041. 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 2823 | . . . 4 ⊢ (Base‘(𝑈 ↾s 𝑌)) = (Base‘(𝑈 ↾s 𝑌)) | |
3 | eqid 2823 | . . . 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 23951 | . . 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 5851 | . . 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 2823 | . . . . . . . 8 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
11 | 9, 10 | lssss 19710 | . . . . . . 7 ⊢ (𝑌 ∈ (LSubSp‘𝑈) → 𝑌 ⊆ 𝑋) |
12 | 8, 11 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑌 ⊆ 𝑋) |
13 | xpss12 5572 | . . . . . 6 ⊢ ((𝑌 ⊆ 𝑋 ∧ 𝑌 ⊆ 𝑋) → (𝑌 × 𝑌) ⊆ (𝑋 × 𝑋)) | |
14 | 12, 12, 13 | syl2anc 586 | . . . . 5 ⊢ (𝜑 → (𝑌 × 𝑌) ⊆ (𝑋 × 𝑋)) |
15 | 14 | resabs1d 5886 | . . . 4 ⊢ (𝜑 → (((dist‘𝑈) ↾ (𝑋 × 𝑋)) ↾ (𝑌 × 𝑌)) = ((dist‘𝑈) ↾ (𝑌 × 𝑌))) |
16 | eqid 2823 | . . . . . . 7 ⊢ (𝑈 ↾s 𝑌) = (𝑈 ↾s 𝑌) | |
17 | eqid 2823 | . . . . . . 7 ⊢ (dist‘𝑈) = (dist‘𝑈) | |
18 | 16, 17 | ressds 16688 | . . . . . 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 16557 | . . . . . . 7 ⊢ (𝑌 ⊆ 𝑋 → 𝑌 = (Base‘(𝑈 ↾s 𝑌))) |
21 | 12, 20 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑌 = (Base‘(𝑈 ↾s 𝑌))) |
22 | 21 | sqxpeqd 5589 | . . . . 5 ⊢ (𝜑 → (𝑌 × 𝑌) = ((Base‘(𝑈 ↾s 𝑌)) × (Base‘(𝑈 ↾s 𝑌)))) |
23 | 19, 22 | reseq12d 5856 | . . . 4 ⊢ (𝜑 → ((dist‘𝑈) ↾ (𝑌 × 𝑌)) = ((dist‘(𝑈 ↾s 𝑌)) ↾ ((Base‘(𝑈 ↾s 𝑌)) × (Base‘(𝑈 ↾s 𝑌))))) |
24 | 15, 23 | eqtrd 2858 | . . 3 ⊢ (𝜑 → (((dist‘𝑈) ↾ (𝑋 × 𝑋)) ↾ (𝑌 × 𝑌)) = ((dist‘(𝑈 ↾s 𝑌)) ↾ ((Base‘(𝑈 ↾s 𝑌)) × (Base‘(𝑈 ↾s 𝑌))))) |
25 | 7, 24 | syl5eq 2870 | . 2 ⊢ (𝜑 → (𝐷 ↾ (𝑌 × 𝑌)) = ((dist‘(𝑈 ↾s 𝑌)) ↾ ((Base‘(𝑈 ↾s 𝑌)) × (Base‘(𝑈 ↾s 𝑌))))) |
26 | 21 | fveq2d 6676 | . 2 ⊢ (𝜑 → (CMet‘𝑌) = (CMet‘(Base‘(𝑈 ↾s 𝑌)))) |
27 | 5, 25, 26 | 3eltr4d 2930 | 1 ⊢ (𝜑 → (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ⊆ wss 3938 ↦ cmpt 5148 × cxp 5555 ran crn 5558 ↾ cres 5559 ‘cfv 6357 (class class class)co 7158 infcinf 8907 ℝcr 10538 < clt 10677 Basecbs 16485 ↾s cress 16486 distcds 16576 TopOpenctopn 16697 -gcsg 18107 LSubSpclss 19705 normcnm 23188 ℂPreHilccph 23772 CMetccmet 23859 CMetSpccms 23937 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-ds 16589 df-lss 19706 df-cms 23940 |
This theorem is referenced by: minveclem3 24034 minveclem4a 24035 |
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