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Theorem sspnval 30567
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 π‘Œ = (BaseSetβ€˜π‘Š)
sspn.n 𝑁 = (normCVβ€˜π‘ˆ)
sspn.m 𝑀 = (normCVβ€˜π‘Š)
sspn.h 𝐻 = (SubSpβ€˜π‘ˆ)
Assertion
Ref Expression
sspnval ((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ 𝐻 ∧ 𝐴 ∈ π‘Œ) β†’ (π‘€β€˜π΄) = (π‘β€˜π΄))

Proof of Theorem sspnval
StepHypRef Expression
1 sspn.y . . . . 5 π‘Œ = (BaseSetβ€˜π‘Š)
2 sspn.n . . . . 5 𝑁 = (normCVβ€˜π‘ˆ)
3 sspn.m . . . . 5 𝑀 = (normCVβ€˜π‘Š)
4 sspn.h . . . . 5 𝐻 = (SubSpβ€˜π‘ˆ)
51, 2, 3, 4sspn 30566 . . . 4 ((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ 𝐻) β†’ 𝑀 = (𝑁 β†Ύ π‘Œ))
65fveq1d 6904 . . 3 ((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ 𝐻) β†’ (π‘€β€˜π΄) = ((𝑁 β†Ύ π‘Œ)β€˜π΄))
7 fvres 6921 . . 3 (𝐴 ∈ π‘Œ β†’ ((𝑁 β†Ύ π‘Œ)β€˜π΄) = (π‘β€˜π΄))
86, 7sylan9eq 2788 . 2 (((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ 𝐻) ∧ 𝐴 ∈ π‘Œ) β†’ (π‘€β€˜π΄) = (π‘β€˜π΄))
983impa 1107 1 ((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ 𝐻 ∧ 𝐴 ∈ π‘Œ) β†’ (π‘€β€˜π΄) = (π‘β€˜π΄))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   β†Ύ cres 5684  β€˜cfv 6553  NrmCVeccnv 30414  BaseSetcba 30416  normCVcnmcv 30420  SubSpcss 30551
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7429  df-oprab 7430  df-1st 7999  df-2nd 8000  df-vc 30389  df-nv 30422  df-va 30425  df-ba 30426  df-sm 30427  df-0v 30428  df-nmcv 30430  df-ssp 30552
This theorem is referenced by:  sspimsval  30568
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