Theorem sspnval 8392

Description: The norm on a subspace in terms of the norm on the parent space.

Hypotheses

Ref	Expression
sspn.y	⊢ Y = (Base ‘W)
sspn.n	⊢ N = (norm ‘U)
sspn.m	⊢ M = (norm ‘W)
sspn.h	⊢ H = (SubSp ‘U)

Assertion

Ref	Expression
sspnval	⊢ ((U ∈ NrmCVec ⋀ W ∈ H ⋀ A ∈ Y) → (M ‘A) = (N ‘A))

Proof of Theorem sspnval

Step	Hyp	Ref	Expression
1		sspn.y	. . . . 5 ⊢ Y = (Base ‘W)
2		sspn.n	. . . . 5 ⊢ N = (norm ‘U)
3		sspn.m	. . . . 5 ⊢ M = (norm ‘W)
4		sspn.h	. . . . 5 ⊢ H = (SubSp ‘U)
5	1, 2, 3, 4	sspn 8391	. . . 4 ⊢ ((U ∈ NrmCVec ⋀ W ∈ H) → M = (N ↾ Y))
6	5	fveq1d 3732	. . 3 ⊢ ((U ∈ NrmCVec ⋀ W ∈ H) → (M ‘A) = ((N ↾ Y) ‘A))
7		fvres 3740	. . 3 ⊢ (A ∈ Y → ((N ↾ Y) ‘A) = (N ‘A))
8	6, 7	sylan9eq 1530	. 2 ⊢ (((U ∈ NrmCVec ⋀ W ∈ H) ⋀ A ∈ Y) → (M ‘A) = (N ‘A))
9	8	3impa 830	1 ⊢ ((U ∈ NrmCVec ⋀ W ∈ H ⋀ A ∈ Y) → (M ‘A) = (N ‘A))

Syntax hints:   → wi 3   ⋀ wa 223   ⋀ w3a 777   = wceq 958   ∈ wcel 960   ↾ cres 3178   ‘cfv 3188  NrmCVeccnv 8199  Basecba 8201  normcnm 8205  SubSpcss 8376

This theorem is referenced by:  sspival 8393  sspimsval 8395  sspph 8511  minveclem28 8568

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-rab 1655  df-v 1815  df-sbc 1945  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-fo 3202  df-fv 3204  df-opr 3971  df-oprab 3972  df-1st 4085  df-2nd 4086  df-grp 8034  df-gid 8035  df-nv 8207  df-va 8210  df-ba 8211  df-sm 8212  df-0v 8213  df-nm 8215  df-ssp 8377

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