sspnval - Metamath Proof Explorer

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Theorem sspnval 31348

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 = (norm_CV‘𝑈)
sspn.m	⊢ 𝑀 = (norm_CV‘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 = (BaseSet‘𝑊)
2		sspn.n	. . . . 5 ⊢ 𝑁 = (norm_CV‘𝑈)
3		sspn.m	. . . . 5 ⊢ 𝑀 = (norm_CV‘𝑊)
4		sspn.h	. . . . 5 ⊢ 𝐻 = (SubSp‘𝑈)
5	1, 2, 3, 4	sspn 31347	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑀 = (𝑁 ↾ 𝑌))
6	5	fveq1d 6892	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → (𝑀‘𝐴) = ((𝑁 ↾ 𝑌)‘𝐴))
7		fvres 6909	. . 3 ⊢ (𝐴 ∈ 𝑌 → ((𝑁 ↾ 𝑌)‘𝐴) = (𝑁‘𝐴))
8	6, 7	sylan9eq 2815	. 2 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ 𝐴 ∈ 𝑌) → (𝑀‘𝐴) = (𝑁‘𝐴))
9	8	3impa 1127	1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ∧ 𝐴 ∈ 𝑌) → (𝑀‘𝐴) = (𝑁‘𝐴))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1569 ∈ wcel 2144 ↾ cres 5653 ‘cfv 6543 NrmCVeccnv 31195 BaseSetcba 31197 norm_CVcnmcv 31201 SubSpcss 31332

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5327 ax-pr 5391 ax-un 7755

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-ral 3077 df-rex 3087 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3848 df-ss 3897 df-dif 4046 df-in 4090 df-un 4150 df-nul 4278 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-op 4835 df-uni 4867 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-id 5546 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-iota 6497 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7427 df-oprab 7428 df-1st 8005 df-2nd 8006 df-vc 31170 df-nv 31203 df-va 31206 df-ba 31207 df-sm 31208 df-0v 31209 df-nmcv 31211 df-ssp 31333

This theorem is used by: sspimsval 31349

W3C validator