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Theorem sspnval 30673

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 30672	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑀 = (𝑁 ↾ 𝑌))
6	5	fveq1d 6863	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → (𝑀‘𝐴) = ((𝑁 ↾ 𝑌)‘𝐴))
7		fvres 6880	. . 3 ⊢ (𝐴 ∈ 𝑌 → ((𝑁 ↾ 𝑌)‘𝐴) = (𝑁‘𝐴))
8	6, 7	sylan9eq 2785	. 2 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ 𝐴 ∈ 𝑌) → (𝑀‘𝐴) = (𝑁‘𝐴))
9	8	3impa 1109	1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ∧ 𝐴 ∈ 𝑌) → (𝑀‘𝐴) = (𝑁‘𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ↾ cres 5643 ‘cfv 6514 NrmCVeccnv 30520 BaseSetcba 30522 norm_CVcnmcv 30526 SubSpcss 30657

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-ral 3046 df-rex 3055 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-iun 4960 df-br 5111 df-opab 5173 df-mpt 5192 df-id 5536 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-ov 7393 df-oprab 7394 df-1st 7971 df-2nd 7972 df-vc 30495 df-nv 30528 df-va 30531 df-ba 30532 df-sm 30533 df-0v 30534 df-nmcv 30536 df-ssp 30658

This theorem is referenced by: sspimsval 30674

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