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Theorem sspnval 28672
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.)
Hypotheses
Ref Expression
sspn.y 𝑌 = (BaseSet‘𝑊)
sspn.n 𝑁 = (normCV𝑈)
sspn.m 𝑀 = (normCV𝑊)
sspn.h 𝐻 = (SubSp‘𝑈)
Assertion
Ref Expression
sspnval ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻𝐴𝑌) → (𝑀𝐴) = (𝑁𝐴))

Proof of Theorem sspnval
StepHypRef Expression
1 sspn.y . . . . 5 𝑌 = (BaseSet‘𝑊)
2 sspn.n . . . . 5 𝑁 = (normCV𝑈)
3 sspn.m . . . . 5 𝑀 = (normCV𝑊)
4 sspn.h . . . . 5 𝐻 = (SubSp‘𝑈)
51, 2, 3, 4sspn 28671 . . . 4 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑀 = (𝑁𝑌))
65fveq1d 6676 . . 3 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → (𝑀𝐴) = ((𝑁𝑌)‘𝐴))
7 fvres 6693 . . 3 (𝐴𝑌 → ((𝑁𝑌)‘𝐴) = (𝑁𝐴))
86, 7sylan9eq 2793 . 2 (((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) ∧ 𝐴𝑌) → (𝑀𝐴) = (𝑁𝐴))
983impa 1111 1 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻𝐴𝑌) → (𝑀𝐴) = (𝑁𝐴))
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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