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Theorem sspnval 26976
 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 26975 . . . 4 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑀 = (𝑁𝑌))
65fveq1d 6105 . . 3 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → (𝑀𝐴) = ((𝑁𝑌)‘𝐴))
7 fvres 6117 . . 3 (𝐴𝑌 → ((𝑁𝑌)‘𝐴) = (𝑁𝐴))
86, 7sylan9eq 2664 . 2 (((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) ∧ 𝐴𝑌) → (𝑀𝐴) = (𝑁𝐴))
983impa 1251 1 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻𝐴𝑌) → (𝑀𝐴) = (𝑁𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ↾ cres 5040  ‘cfv 5804  NrmCVeccnv 26823  BaseSetcba 26825  normCVcnmcv 26829  SubSpcss 26960 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-1st 7059  df-2nd 7060  df-vc 26798  df-nv 26831  df-va 26834  df-ba 26835  df-sm 26836  df-0v 26837  df-nmcv 26839  df-ssp 26961 This theorem is referenced by:  sspimsval  26977  sspph  27094
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