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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 28671 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑀 = (𝑁 ↾ 𝑌)) |
6 | 5 | fveq1d 6676 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → (𝑀‘𝐴) = ((𝑁 ↾ 𝑌)‘𝐴)) |
7 | fvres 6693 | . . 3 ⊢ (𝐴 ∈ 𝑌 → ((𝑁 ↾ 𝑌)‘𝐴) = (𝑁‘𝐴)) | |
8 | 6, 7 | sylan9eq 2793 | . 2 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ 𝐴 ∈ 𝑌) → (𝑀‘𝐴) = (𝑁‘𝐴)) |
9 | 8 | 3impa 1111 | 1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ∧ 𝐴 ∈ 𝑌) → (𝑀‘𝐴) = (𝑁‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1088 = wceq 1542 ∈ wcel 2114 ↾ cres 5527 ‘cfv 6339 NrmCVeccnv 28519 BaseSetcba 28521 normCVcnmcv 28525 SubSpcss 28656 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-op 4523 df-uni 4797 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5429 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-ov 7173 df-oprab 7174 df-1st 7714 df-2nd 7715 df-vc 28494 df-nv 28527 df-va 28530 df-ba 28531 df-sm 28532 df-0v 28533 df-nmcv 28535 df-ssp 28657 |
This theorem is referenced by: sspimsval 28673 |
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