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Theorem sspnval 30824

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 30823	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑀 = (𝑁 ↾ 𝑌))
6	5	fveq1d 6844	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → (𝑀‘𝐴) = ((𝑁 ↾ 𝑌)‘𝐴))
7		fvres 6861	. . 3 ⊢ (𝐴 ∈ 𝑌 → ((𝑁 ↾ 𝑌)‘𝐴) = (𝑁‘𝐴))
8	6, 7	sylan9eq 2792	. 2 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ 𝐴 ∈ 𝑌) → (𝑀‘𝐴) = (𝑁‘𝐴))
9	8	3impa 1110	1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ∧ 𝐴 ∈ 𝑌) → (𝑀‘𝐴) = (𝑁‘𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ↾ cres 5634 ‘cfv 6500 NrmCVeccnv 30671 BaseSetcba 30673 norm_CVcnmcv 30677 SubSpcss 30808

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3063 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-id 5527 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-ov 7371 df-oprab 7372 df-1st 7943 df-2nd 7944 df-vc 30646 df-nv 30679 df-va 30682 df-ba 30683 df-sm 30684 df-0v 30685 df-nmcv 30687 df-ssp 30809

This theorem is referenced by: sspimsval 30825

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