![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > sspnval | Structured version Visualization version GIF version |
Description: The norm on a subspace in terms of the norm on the parent space. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.) |
Ref | Expression |
---|---|
sspn.y | ⊢ 𝑌 = (BaseSet‘𝑊) |
sspn.n | ⊢ 𝑁 = (normCV‘𝑈) |
sspn.m | ⊢ 𝑀 = (normCV‘𝑊) |
sspn.h | ⊢ 𝐻 = (SubSp‘𝑈) |
Ref | Expression |
---|---|
sspnval | ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ∧ 𝐴 ∈ 𝑌) → (𝑀‘𝐴) = (𝑁‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sspn.y | . . . . 5 ⊢ 𝑌 = (BaseSet‘𝑊) | |
2 | sspn.n | . . . . 5 ⊢ 𝑁 = (normCV‘𝑈) | |
3 | sspn.m | . . . . 5 ⊢ 𝑀 = (normCV‘𝑊) | |
4 | sspn.h | . . . . 5 ⊢ 𝐻 = (SubSp‘𝑈) | |
5 | 1, 2, 3, 4 | sspn 29852 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑀 = (𝑁 ↾ 𝑌)) |
6 | 5 | fveq1d 6880 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → (𝑀‘𝐴) = ((𝑁 ↾ 𝑌)‘𝐴)) |
7 | fvres 6897 | . . 3 ⊢ (𝐴 ∈ 𝑌 → ((𝑁 ↾ 𝑌)‘𝐴) = (𝑁‘𝐴)) | |
8 | 6, 7 | sylan9eq 2791 | . 2 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ 𝐴 ∈ 𝑌) → (𝑀‘𝐴) = (𝑁‘𝐴)) |
9 | 8 | 3impa 1110 | 1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ∧ 𝐴 ∈ 𝑌) → (𝑀‘𝐴) = (𝑁‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ↾ cres 5671 ‘cfv 6532 NrmCVeccnv 29700 BaseSetcba 29702 normCVcnmcv 29706 SubSpcss 29837 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-ov 7396 df-oprab 7397 df-1st 7957 df-2nd 7958 df-vc 29675 df-nv 29708 df-va 29711 df-ba 29712 df-sm 29713 df-0v 29714 df-nmcv 29716 df-ssp 29838 |
This theorem is referenced by: sspimsval 29854 |
Copyright terms: Public domain | W3C validator |