Theorem sspn 31069

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 31059	. . . 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 30993	. . . 4 ⊢ (𝑊 ∈ NrmCVec → 𝑀:𝑌⟶ℝ)
6	2, 5	syl 18	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑀:𝑌⟶ℝ)
7	6	ffnd 6708	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑀 Fn 𝑌)
8		eqid 2763	. . . . . 6 ⊢ (BaseSet‘𝑈) = (BaseSet‘𝑈)
9		sspn.n	. . . . . 6 ⊢ 𝑁 = (normCV‘𝑈)
10	8, 9	nvf 30993	. . . . 5 ⊢ (𝑈 ∈ NrmCVec → 𝑁:(BaseSet‘𝑈)⟶ℝ)
11	10	ffnd 6708	. . . 4 ⊢ (𝑈 ∈ NrmCVec → 𝑁 Fn (BaseSet‘𝑈))
12	11	adantr 485	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑁 Fn (BaseSet‘𝑈))
13	8, 3, 1	sspba 31060	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑌 ⊆ (BaseSet‘𝑈))
14		fnssres 6660	. . 3 ⊢ ((𝑁 Fn (BaseSet‘𝑈) ∧ 𝑌 ⊆ (BaseSet‘𝑈)) → (𝑁 ↾ 𝑌) Fn 𝑌)
15	12, 13, 14	syl2anc 595	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → (𝑁 ↾ 𝑌) Fn 𝑌)
16	10	ffund 6712	. . . . . 6 ⊢ (𝑈 ∈ NrmCVec → Fun 𝑁)
17	16	funresd 6581	. . . . 5 ⊢ (𝑈 ∈ NrmCVec → Fun (𝑁 ↾ 𝑌))
18	17	ad2antrr 738	. . . 4 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ 𝑥 ∈ 𝑌) → Fun (𝑁 ↾ 𝑌))
19		fnresdm 6656	. . . . . . 7 ⊢ (𝑀 Fn 𝑌 → (𝑀 ↾ 𝑌) = 𝑀)
20	7, 19	syl 18	. . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → (𝑀 ↾ 𝑌) = 𝑀)
21		eqid 2763	. . . . . . . . . 10 ⊢ ( +𝑣 ‘𝑈) = ( +𝑣 ‘𝑈)
22		eqid 2763	. . . . . . . . . 10 ⊢ ( +𝑣 ‘𝑊) = ( +𝑣 ‘𝑊)
23		eqid 2763	. . . . . . . . . 10 ⊢ ( ·𝑠OLD ‘𝑈) = ( ·𝑠OLD ‘𝑈)
24		eqid 2763	. . . . . . . . . 10 ⊢ ( ·𝑠OLD ‘𝑊) = ( ·𝑠OLD ‘𝑊)
25	21, 22, 23, 24, 9, 4, 1	isssp 31057	. . . . . . . . 9 ⊢ (𝑈 ∈ NrmCVec → (𝑊 ∈ 𝐻 ↔ (𝑊 ∈ NrmCVec ∧ (( +𝑣 ‘𝑊) ⊆ ( +𝑣 ‘𝑈) ∧ ( ·𝑠OLD ‘𝑊) ⊆ ( ·𝑠OLD ‘𝑈) ∧ 𝑀 ⊆ 𝑁))))
26	25	simplbda 504	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → (( +𝑣 ‘𝑊) ⊆ ( +𝑣 ‘𝑈) ∧ ( ·𝑠OLD ‘𝑊) ⊆ ( ·𝑠OLD ‘𝑈) ∧ 𝑀 ⊆ 𝑁))
27	26	simp3d 1162	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑀 ⊆ 𝑁)
28		ssres 6004	. . . . . . 7 ⊢ (𝑀 ⊆ 𝑁 → (𝑀 ↾ 𝑌) ⊆ (𝑁 ↾ 𝑌))
29	27, 28	syl 18	. . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → (𝑀 ↾ 𝑌) ⊆ (𝑁 ↾ 𝑌))
30	20, 29	eqsstrrd 3973	. . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑀 ⊆ (𝑁 ↾ 𝑌))
31	30	adantr 485	. . . 4 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ 𝑥 ∈ 𝑌) → 𝑀 ⊆ (𝑁 ↾ 𝑌))
32	5	fdmd 6718	. . . . . . 7 ⊢ (𝑊 ∈ NrmCVec → dom 𝑀 = 𝑌)
33	32	eleq2d 2849	. . . . . 6 ⊢ (𝑊 ∈ NrmCVec → (𝑥 ∈ dom 𝑀 ↔ 𝑥 ∈ 𝑌))
34	33	biimpar 482	. . . . 5 ⊢ ((𝑊 ∈ NrmCVec ∧ 𝑥 ∈ 𝑌) → 𝑥 ∈ dom 𝑀)
35	2, 34	sylan 591	. . . 4 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ 𝑥 ∈ 𝑌) → 𝑥 ∈ dom 𝑀)
36		funssfv 6904	. . . 4 ⊢ ((Fun (𝑁 ↾ 𝑌) ∧ 𝑀 ⊆ (𝑁 ↾ 𝑌) ∧ 𝑥 ∈ dom 𝑀) → ((𝑁 ↾ 𝑌)‘𝑥) = (𝑀‘𝑥))
37	18, 31, 35, 36	syl3anc 1398	. . 3 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ 𝑥 ∈ 𝑌) → ((𝑁 ↾ 𝑌)‘𝑥) = (𝑀‘𝑥))
38	37	eqcomd 2769	. 2 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ 𝑥 ∈ 𝑌) → (𝑀‘𝑥) = ((𝑁 ↾ 𝑌)‘𝑥))
39	7, 15, 38	eqfnfvd 7030	1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑀 = (𝑁 ↾ 𝑌))

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143   ⊆ wss 3906  dom cdm 5663   ↾ cres 5665  Fun wfun 6532   Fn wfn 6533  ⟶wf 6534  ‘cfv 6538  ℝcr 11100  NrmCVeccnv 30917   +_𝑣 cpv 30918  BaseSetcba 30919   ·_𝑠OLD cns 30920  norm_CVcnmcv 30923  SubSpcss 31054

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7415  df-oprab 7416  df-1st 7987  df-2nd 7988  df-vc 30892  df-nv 30925  df-va 30928  df-ba 30929  df-sm 30930  df-0v 30931  df-nmcv 30933  df-ssp 31055

This theorem is referenced by:  sspnval  31070