Theorem sspn 30711

Description: The norm on a subspace is a restriction 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 = (normCV‘𝑈)
sspn.m	⊢ 𝑀 = (normCV‘𝑊)
sspn.h	⊢ 𝐻 = (SubSp‘𝑈)

Assertion

Ref	Expression
sspn	⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑀 = (𝑁 ↾ 𝑌))

Proof of Theorem sspn

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sspn.h	. . . . 5 ⊢ 𝐻 = (SubSp‘𝑈)
2	1	sspnv 30701	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑊 ∈ NrmCVec)
3		sspn.y	. . . . 5 ⊢ 𝑌 = (BaseSet‘𝑊)
4		sspn.m	. . . . 5 ⊢ 𝑀 = (normCV‘𝑊)
5	3, 4	nvf 30635	. . . 4 ⊢ (𝑊 ∈ NrmCVec → 𝑀:𝑌⟶ℝ)
6	2, 5	syl 17	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑀:𝑌⟶ℝ)
7	6	ffnd 6652	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑀 Fn 𝑌)
8		eqid 2731	. . . . . 6 ⊢ (BaseSet‘𝑈) = (BaseSet‘𝑈)
9		sspn.n	. . . . . 6 ⊢ 𝑁 = (normCV‘𝑈)
10	8, 9	nvf 30635	. . . . 5 ⊢ (𝑈 ∈ NrmCVec → 𝑁:(BaseSet‘𝑈)⟶ℝ)
11	10	ffnd 6652	. . . 4 ⊢ (𝑈 ∈ NrmCVec → 𝑁 Fn (BaseSet‘𝑈))
12	11	adantr 480	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑁 Fn (BaseSet‘𝑈))
13	8, 3, 1	sspba 30702	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑌 ⊆ (BaseSet‘𝑈))
14		fnssres 6604	. . 3 ⊢ ((𝑁 Fn (BaseSet‘𝑈) ∧ 𝑌 ⊆ (BaseSet‘𝑈)) → (𝑁 ↾ 𝑌) Fn 𝑌)
15	12, 13, 14	syl2anc 584	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → (𝑁 ↾ 𝑌) Fn 𝑌)
16	10	ffund 6655	. . . . . 6 ⊢ (𝑈 ∈ NrmCVec → Fun 𝑁)
17	16	funresd 6524	. . . . 5 ⊢ (𝑈 ∈ NrmCVec → Fun (𝑁 ↾ 𝑌))
18	17	ad2antrr 726	. . . 4 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ 𝑥 ∈ 𝑌) → Fun (𝑁 ↾ 𝑌))
19		fnresdm 6600	. . . . . . 7 ⊢ (𝑀 Fn 𝑌 → (𝑀 ↾ 𝑌) = 𝑀)
20	7, 19	syl 17	. . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → (𝑀 ↾ 𝑌) = 𝑀)
21		eqid 2731	. . . . . . . . . 10 ⊢ ( +𝑣 ‘𝑈) = ( +𝑣 ‘𝑈)
22		eqid 2731	. . . . . . . . . 10 ⊢ ( +𝑣 ‘𝑊) = ( +𝑣 ‘𝑊)
23		eqid 2731	. . . . . . . . . 10 ⊢ ( ·𝑠OLD ‘𝑈) = ( ·𝑠OLD ‘𝑈)
24		eqid 2731	. . . . . . . . . 10 ⊢ ( ·𝑠OLD ‘𝑊) = ( ·𝑠OLD ‘𝑊)
25	21, 22, 23, 24, 9, 4, 1	isssp 30699	. . . . . . . . 9 ⊢ (𝑈 ∈ NrmCVec → (𝑊 ∈ 𝐻 ↔ (𝑊 ∈ NrmCVec ∧ (( +𝑣 ‘𝑊) ⊆ ( +𝑣 ‘𝑈) ∧ ( ·𝑠OLD ‘𝑊) ⊆ ( ·𝑠OLD ‘𝑈) ∧ 𝑀 ⊆ 𝑁))))
26	25	simplbda 499	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → (( +𝑣 ‘𝑊) ⊆ ( +𝑣 ‘𝑈) ∧ ( ·𝑠OLD ‘𝑊) ⊆ ( ·𝑠OLD ‘𝑈) ∧ 𝑀 ⊆ 𝑁))
27	26	simp3d 1144	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑀 ⊆ 𝑁)
28		ssres 5952	. . . . . . 7 ⊢ (𝑀 ⊆ 𝑁 → (𝑀 ↾ 𝑌) ⊆ (𝑁 ↾ 𝑌))
29	27, 28	syl 17	. . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → (𝑀 ↾ 𝑌) ⊆ (𝑁 ↾ 𝑌))
30	20, 29	eqsstrrd 3970	. . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑀 ⊆ (𝑁 ↾ 𝑌))
31	30	adantr 480	. . . 4 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ 𝑥 ∈ 𝑌) → 𝑀 ⊆ (𝑁 ↾ 𝑌))
32	5	fdmd 6661	. . . . . . 7 ⊢ (𝑊 ∈ NrmCVec → dom 𝑀 = 𝑌)
33	32	eleq2d 2817	. . . . . 6 ⊢ (𝑊 ∈ NrmCVec → (𝑥 ∈ dom 𝑀 ↔ 𝑥 ∈ 𝑌))
34	33	biimpar 477	. . . . 5 ⊢ ((𝑊 ∈ NrmCVec ∧ 𝑥 ∈ 𝑌) → 𝑥 ∈ dom 𝑀)
35	2, 34	sylan 580	. . . 4 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ 𝑥 ∈ 𝑌) → 𝑥 ∈ dom 𝑀)
36		funssfv 6843	. . . 4 ⊢ ((Fun (𝑁 ↾ 𝑌) ∧ 𝑀 ⊆ (𝑁 ↾ 𝑌) ∧ 𝑥 ∈ dom 𝑀) → ((𝑁 ↾ 𝑌)‘𝑥) = (𝑀‘𝑥))
37	18, 31, 35, 36	syl3anc 1373	. . 3 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ 𝑥 ∈ 𝑌) → ((𝑁 ↾ 𝑌)‘𝑥) = (𝑀‘𝑥))
38	37	eqcomd 2737	. 2 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ 𝑥 ∈ 𝑌) → (𝑀‘𝑥) = ((𝑁 ↾ 𝑌)‘𝑥))
39	7, 15, 38	eqfnfvd 6967	1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑀 = (𝑁 ↾ 𝑌))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111   ⊆ wss 3902  dom cdm 5616   ↾ cres 5618  Fun wfun 6475   Fn wfn 6476  ⟶wf 6477  ‘cfv 6481  ℝcr 11002  NrmCVeccnv 30559   +_𝑣 cpv 30560  BaseSetcba 30561   ·_𝑠OLD cns 30562  norm_CVcnmcv 30565  SubSpcss 30696

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-ov 7349  df-oprab 7350  df-1st 7921  df-2nd 7922  df-vc 30534  df-nv 30567  df-va 30570  df-ba 30571  df-sm 30572  df-0v 30573  df-nmcv 30575  df-ssp 30697

This theorem is referenced by:  sspnval  30712