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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 27593 | . 2 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → (𝑥𝐶𝑦) = (𝑥𝐷𝑦)) |
6 | 1, 4 | imsdf 27544 | . 2 ⊢ (𝑊 ∈ NrmCVec → 𝐶:(𝑌 × 𝑌)⟶ℝ) |
7 | eqid 2622 | . . 3 ⊢ (BaseSet‘𝑈) = (BaseSet‘𝑈) | |
8 | 7, 3 | imsdf 27544 | . 2 ⊢ (𝑈 ∈ NrmCVec → 𝐷:((BaseSet‘𝑈) × (BaseSet‘𝑈))⟶ℝ) |
9 | 1, 2, 5, 6, 8 | sspmlem 27587 | 1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝐶 = (𝐷 ↾ (𝑌 × 𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 × cxp 5112 ↾ cres 5116 ‘cfv 5888 ℝcr 9935 NrmCVeccnv 27439 BaseSetcba 27441 IndMetcims 27446 SubSpcss 27576 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-ltxr 10079 df-sub 10268 df-neg 10269 df-grpo 27347 df-gid 27348 df-ginv 27349 df-gdiv 27350 df-ablo 27399 df-vc 27414 df-nv 27447 df-va 27450 df-ba 27451 df-sm 27452 df-0v 27453 df-vs 27454 df-nmcv 27455 df-ims 27456 df-ssp 27577 |
This theorem is referenced by: bnsscmcl 27724 minvecolem4a 27733 |
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