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Theorem sspnval 29721
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 29720 . . . 4 ((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ 𝐻) β†’ 𝑀 = (𝑁 β†Ύ π‘Œ))
65fveq1d 6845 . . 3 ((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ 𝐻) β†’ (π‘€β€˜π΄) = ((𝑁 β†Ύ π‘Œ)β€˜π΄))
7 fvres 6862 . . 3 (𝐴 ∈ π‘Œ β†’ ((𝑁 β†Ύ π‘Œ)β€˜π΄) = (π‘β€˜π΄))
86, 7sylan9eq 2793 . 2 (((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ 𝐻) ∧ 𝐴 ∈ π‘Œ) β†’ (π‘€β€˜π΄) = (π‘β€˜π΄))
983impa 1111 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β†Ύ cres 5636  β€˜cfv 6497  NrmCVeccnv 29568  BaseSetcba 29570  normCVcnmcv 29574  SubSpcss 29705
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-ov 7361  df-oprab 7362  df-1st 7922  df-2nd 7923  df-vc 29543  df-nv 29576  df-va 29579  df-ba 29580  df-sm 29581  df-0v 29582  df-nmcv 29584  df-ssp 29706
This theorem is referenced by:  sspimsval  29722
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