![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > minveclem3a | Structured version Visualization version GIF version |
Description: Lemma for minvec 25284. 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 2731 | . . . 4 ⊢ (Base‘(𝑈 ↾s 𝑌)) = (Base‘(𝑈 ↾s 𝑌)) | |
3 | eqid 2731 | . . . 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 25194 | . . 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 5977 | . . 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 2731 | . . . . . . . 8 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
11 | 9, 10 | lssss 20779 | . . . . . . 7 ⊢ (𝑌 ∈ (LSubSp‘𝑈) → 𝑌 ⊆ 𝑋) |
12 | 8, 11 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑌 ⊆ 𝑋) |
13 | xpss12 5691 | . . . . . 6 ⊢ ((𝑌 ⊆ 𝑋 ∧ 𝑌 ⊆ 𝑋) → (𝑌 × 𝑌) ⊆ (𝑋 × 𝑋)) | |
14 | 12, 12, 13 | syl2anc 583 | . . . . 5 ⊢ (𝜑 → (𝑌 × 𝑌) ⊆ (𝑋 × 𝑋)) |
15 | 14 | resabs1d 6012 | . . . 4 ⊢ (𝜑 → (((dist‘𝑈) ↾ (𝑋 × 𝑋)) ↾ (𝑌 × 𝑌)) = ((dist‘𝑈) ↾ (𝑌 × 𝑌))) |
16 | eqid 2731 | . . . . . . 7 ⊢ (𝑈 ↾s 𝑌) = (𝑈 ↾s 𝑌) | |
17 | eqid 2731 | . . . . . . 7 ⊢ (dist‘𝑈) = (dist‘𝑈) | |
18 | 16, 17 | ressds 17362 | . . . . . 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 17189 | . . . . . . 7 ⊢ (𝑌 ⊆ 𝑋 → 𝑌 = (Base‘(𝑈 ↾s 𝑌))) |
21 | 12, 20 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑌 = (Base‘(𝑈 ↾s 𝑌))) |
22 | 21 | sqxpeqd 5708 | . . . . 5 ⊢ (𝜑 → (𝑌 × 𝑌) = ((Base‘(𝑈 ↾s 𝑌)) × (Base‘(𝑈 ↾s 𝑌)))) |
23 | 19, 22 | reseq12d 5982 | . . . 4 ⊢ (𝜑 → ((dist‘𝑈) ↾ (𝑌 × 𝑌)) = ((dist‘(𝑈 ↾s 𝑌)) ↾ ((Base‘(𝑈 ↾s 𝑌)) × (Base‘(𝑈 ↾s 𝑌))))) |
24 | 15, 23 | eqtrd 2771 | . . 3 ⊢ (𝜑 → (((dist‘𝑈) ↾ (𝑋 × 𝑋)) ↾ (𝑌 × 𝑌)) = ((dist‘(𝑈 ↾s 𝑌)) ↾ ((Base‘(𝑈 ↾s 𝑌)) × (Base‘(𝑈 ↾s 𝑌))))) |
25 | 7, 24 | eqtrid 2783 | . 2 ⊢ (𝜑 → (𝐷 ↾ (𝑌 × 𝑌)) = ((dist‘(𝑈 ↾s 𝑌)) ↾ ((Base‘(𝑈 ↾s 𝑌)) × (Base‘(𝑈 ↾s 𝑌))))) |
26 | 21 | fveq2d 6895 | . 2 ⊢ (𝜑 → (CMet‘𝑌) = (CMet‘(Base‘(𝑈 ↾s 𝑌)))) |
27 | 5, 25, 26 | 3eltr4d 2847 | 1 ⊢ (𝜑 → (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2105 ⊆ wss 3948 ↦ cmpt 5231 × cxp 5674 ran crn 5677 ↾ cres 5678 ‘cfv 6543 (class class class)co 7412 infcinf 9442 ℝcr 11115 < clt 11255 Basecbs 17151 ↾s cress 17180 distcds 17213 TopOpenctopn 17374 -gcsg 18863 LSubSpclss 20774 normcnm 24405 ℂPreHilccph 25014 CMetccmet 25102 CMetSpccms 25180 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-er 8709 df-en 8946 df-dom 8947 df-sdom 8948 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-nn 12220 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-7 12287 df-8 12288 df-9 12289 df-n0 12480 df-z 12566 df-dec 12685 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-ds 17226 df-lss 20775 df-cms 25183 |
This theorem is referenced by: minveclem3 25277 minveclem4a 25278 |
Copyright terms: Public domain | W3C validator |