Theorem sspn 27591

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 27581	. . . 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 27515	. . . 4 ⊢ (𝑊 ∈ NrmCVec → 𝑀:𝑌⟶ℝ)
6	2, 5	syl 17	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑀:𝑌⟶ℝ)
7		ffn 6045	. . 3 ⊢ (𝑀:𝑌⟶ℝ → 𝑀 Fn 𝑌)
8	6, 7	syl 17	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑀 Fn 𝑌)
9		eqid 2622	. . . . . 6 ⊢ (BaseSet‘𝑈) = (BaseSet‘𝑈)
10		sspn.n	. . . . . 6 ⊢ 𝑁 = (normCV‘𝑈)
11	9, 10	nvf 27515	. . . . 5 ⊢ (𝑈 ∈ NrmCVec → 𝑁:(BaseSet‘𝑈)⟶ℝ)
12		ffn 6045	. . . . 5 ⊢ (𝑁:(BaseSet‘𝑈)⟶ℝ → 𝑁 Fn (BaseSet‘𝑈))
13	11, 12	syl 17	. . . 4 ⊢ (𝑈 ∈ NrmCVec → 𝑁 Fn (BaseSet‘𝑈))
14	13	adantr 481	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑁 Fn (BaseSet‘𝑈))
15	9, 3, 1	sspba 27582	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑌 ⊆ (BaseSet‘𝑈))
16		fnssres 6004	. . 3 ⊢ ((𝑁 Fn (BaseSet‘𝑈) ∧ 𝑌 ⊆ (BaseSet‘𝑈)) → (𝑁 ↾ 𝑌) Fn 𝑌)
17	14, 15, 16	syl2anc 693	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → (𝑁 ↾ 𝑌) Fn 𝑌)
18		ffun 6048	. . . . . . 7 ⊢ (𝑁:(BaseSet‘𝑈)⟶ℝ → Fun 𝑁)
19	11, 18	syl 17	. . . . . 6 ⊢ (𝑈 ∈ NrmCVec → Fun 𝑁)
20		funres 5929	. . . . . 6 ⊢ (Fun 𝑁 → Fun (𝑁 ↾ 𝑌))
21	19, 20	syl 17	. . . . 5 ⊢ (𝑈 ∈ NrmCVec → Fun (𝑁 ↾ 𝑌))
22	21	ad2antrr 762	. . . 4 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ 𝑥 ∈ 𝑌) → Fun (𝑁 ↾ 𝑌))
23		fnresdm 6000	. . . . . . 7 ⊢ (𝑀 Fn 𝑌 → (𝑀 ↾ 𝑌) = 𝑀)
24	8, 23	syl 17	. . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → (𝑀 ↾ 𝑌) = 𝑀)
25		eqid 2622	. . . . . . . . . 10 ⊢ ( +𝑣 ‘𝑈) = ( +𝑣 ‘𝑈)
26		eqid 2622	. . . . . . . . . 10 ⊢ ( +𝑣 ‘𝑊) = ( +𝑣 ‘𝑊)
27		eqid 2622	. . . . . . . . . 10 ⊢ ( ·𝑠OLD ‘𝑈) = ( ·𝑠OLD ‘𝑈)
28		eqid 2622	. . . . . . . . . 10 ⊢ ( ·𝑠OLD ‘𝑊) = ( ·𝑠OLD ‘𝑊)
29	25, 26, 27, 28, 10, 4, 1	isssp 27579	. . . . . . . . 9 ⊢ (𝑈 ∈ NrmCVec → (𝑊 ∈ 𝐻 ↔ (𝑊 ∈ NrmCVec ∧ (( +𝑣 ‘𝑊) ⊆ ( +𝑣 ‘𝑈) ∧ ( ·𝑠OLD ‘𝑊) ⊆ ( ·𝑠OLD ‘𝑈) ∧ 𝑀 ⊆ 𝑁))))
30	29	simplbda 654	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → (( +𝑣 ‘𝑊) ⊆ ( +𝑣 ‘𝑈) ∧ ( ·𝑠OLD ‘𝑊) ⊆ ( ·𝑠OLD ‘𝑈) ∧ 𝑀 ⊆ 𝑁))
31	30	simp3d 1075	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑀 ⊆ 𝑁)
32		ssres 5424	. . . . . . 7 ⊢ (𝑀 ⊆ 𝑁 → (𝑀 ↾ 𝑌) ⊆ (𝑁 ↾ 𝑌))
33	31, 32	syl 17	. . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → (𝑀 ↾ 𝑌) ⊆ (𝑁 ↾ 𝑌))
34	24, 33	eqsstr3d 3640	. . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑀 ⊆ (𝑁 ↾ 𝑌))
35	34	adantr 481	. . . 4 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ 𝑥 ∈ 𝑌) → 𝑀 ⊆ (𝑁 ↾ 𝑌))
36		fdm 6051	. . . . . . . 8 ⊢ (𝑀:𝑌⟶ℝ → dom 𝑀 = 𝑌)
37	5, 36	syl 17	. . . . . . 7 ⊢ (𝑊 ∈ NrmCVec → dom 𝑀 = 𝑌)
38	37	eleq2d 2687	. . . . . 6 ⊢ (𝑊 ∈ NrmCVec → (𝑥 ∈ dom 𝑀 ↔ 𝑥 ∈ 𝑌))
39	38	biimpar 502	. . . . 5 ⊢ ((𝑊 ∈ NrmCVec ∧ 𝑥 ∈ 𝑌) → 𝑥 ∈ dom 𝑀)
40	2, 39	sylan 488	. . . 4 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ 𝑥 ∈ 𝑌) → 𝑥 ∈ dom 𝑀)
41		funssfv 6209	. . . 4 ⊢ ((Fun (𝑁 ↾ 𝑌) ∧ 𝑀 ⊆ (𝑁 ↾ 𝑌) ∧ 𝑥 ∈ dom 𝑀) → ((𝑁 ↾ 𝑌)‘𝑥) = (𝑀‘𝑥))
42	22, 35, 40, 41	syl3anc 1326	. . 3 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ 𝑥 ∈ 𝑌) → ((𝑁 ↾ 𝑌)‘𝑥) = (𝑀‘𝑥))
43	42	eqcomd 2628	. 2 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ 𝑥 ∈ 𝑌) → (𝑀‘𝑥) = ((𝑁 ↾ 𝑌)‘𝑥))
44	8, 17, 43	eqfnfvd 6314	1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑀 = (𝑁 ↾ 𝑌))

Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   ⊆ wss 3574  dom cdm 5114   ↾ cres 5116  Fun wfun 5882   Fn wfn 5883  ⟶wf 5884  ‘cfv 5888  ℝcr 9935  NrmCVeccnv 27439   +_𝑣 cpv 27440  BaseSetcba 27441   ·_𝑠OLD cns 27442  norm_CVcnmcv 27445  SubSpcss 27576

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-1st 7168  df-2nd 7169  df-vc 27414  df-nv 27447  df-va 27450  df-ba 27451  df-sm 27452  df-0v 27453  df-nmcv 27455  df-ssp 27577

This theorem is referenced by:  sspnval  27592