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Mirrors > Home > MPE Home > Th. List > sspims | Structured version Visualization version GIF version |
Description: The induced metric on a subspace is a restriction of the induced metric on the parent space. (Contributed by NM, 1-Feb-2008.) (New usage is discouraged.) |
Ref | Expression |
---|---|
sspims.y | ⊢ 𝑌 = (BaseSet‘𝑊) |
sspims.d | ⊢ 𝐷 = (IndMet‘𝑈) |
sspims.c | ⊢ 𝐶 = (IndMet‘𝑊) |
sspims.h | ⊢ 𝐻 = (SubSp‘𝑈) |
Ref | Expression |
---|---|
sspims | ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝐶 = (𝐷 ↾ (𝑌 × 𝑌))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sspims.y | . 2 ⊢ 𝑌 = (BaseSet‘𝑊) | |
2 | sspims.h | . 2 ⊢ 𝐻 = (SubSp‘𝑈) | |
3 | sspims.d | . . 3 ⊢ 𝐷 = (IndMet‘𝑈) | |
4 | sspims.c | . . 3 ⊢ 𝐶 = (IndMet‘𝑊) | |
5 | 1, 3, 4, 2 | sspimsval 28282 | . 2 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → (𝑥𝐶𝑦) = (𝑥𝐷𝑦)) |
6 | 1, 4 | imsdf 28233 | . 2 ⊢ (𝑊 ∈ NrmCVec → 𝐶:(𝑌 × 𝑌)⟶ℝ) |
7 | eqid 2772 | . . 3 ⊢ (BaseSet‘𝑈) = (BaseSet‘𝑈) | |
8 | 7, 3 | imsdf 28233 | . 2 ⊢ (𝑈 ∈ NrmCVec → 𝐷:((BaseSet‘𝑈) × (BaseSet‘𝑈))⟶ℝ) |
9 | 1, 2, 5, 6, 8 | sspmlem 28276 | 1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝐶 = (𝐷 ↾ (𝑌 × 𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 = wceq 1507 ∈ wcel 2048 × cxp 5398 ↾ cres 5402 ‘cfv 6182 ℝcr 10326 NrmCVeccnv 28128 BaseSetcba 28130 IndMetcims 28135 SubSpcss 28265 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-resscn 10384 ax-1cn 10385 ax-icn 10386 ax-addcl 10387 ax-addrcl 10388 ax-mulcl 10389 ax-mulrcl 10390 ax-mulcom 10391 ax-addass 10392 ax-mulass 10393 ax-distr 10394 ax-i2m1 10395 ax-1ne0 10396 ax-1rid 10397 ax-rnegex 10398 ax-rrecex 10399 ax-cnre 10400 ax-pre-lttri 10401 ax-pre-lttrn 10402 ax-pre-ltadd 10403 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rab 3091 df-v 3411 df-sbc 3678 df-csb 3783 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4707 df-iun 4788 df-br 4924 df-opab 4986 df-mpt 5003 df-id 5305 df-po 5319 df-so 5320 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-1st 7494 df-2nd 7495 df-er 8081 df-en 8299 df-dom 8300 df-sdom 8301 df-pnf 10468 df-mnf 10469 df-ltxr 10471 df-sub 10664 df-neg 10665 df-grpo 28037 df-gid 28038 df-ginv 28039 df-gdiv 28040 df-ablo 28089 df-vc 28103 df-nv 28136 df-va 28139 df-ba 28140 df-sm 28141 df-0v 28142 df-vs 28143 df-nmcv 28144 df-ims 28145 df-ssp 28266 |
This theorem is referenced by: bnsscmcl 28413 minvecolem4a 28422 |
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