Theorem sspn 30807

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 30797	. . . 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 30731	. . . 4 ⊢ (𝑊 ∈ NrmCVec → 𝑀:𝑌⟶ℝ)
6	2, 5	syl 17	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑀:𝑌⟶ℝ)
7	6	ffnd 6669	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑀 Fn 𝑌)
8		eqid 2736	. . . . . 6 ⊢ (BaseSet‘𝑈) = (BaseSet‘𝑈)
9		sspn.n	. . . . . 6 ⊢ 𝑁 = (normCV‘𝑈)
10	8, 9	nvf 30731	. . . . 5 ⊢ (𝑈 ∈ NrmCVec → 𝑁:(BaseSet‘𝑈)⟶ℝ)
11	10	ffnd 6669	. . . 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 30798	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑌 ⊆ (BaseSet‘𝑈))
14		fnssres 6621	. . 3 ⊢ ((𝑁 Fn (BaseSet‘𝑈) ∧ 𝑌 ⊆ (BaseSet‘𝑈)) → (𝑁 ↾ 𝑌) Fn 𝑌)
15	12, 13, 14	syl2anc 585	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → (𝑁 ↾ 𝑌) Fn 𝑌)
16	10	ffund 6672	. . . . . 6 ⊢ (𝑈 ∈ NrmCVec → Fun 𝑁)
17	16	funresd 6541	. . . . 5 ⊢ (𝑈 ∈ NrmCVec → Fun (𝑁 ↾ 𝑌))
18	17	ad2antrr 727	. . . 4 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ 𝑥 ∈ 𝑌) → Fun (𝑁 ↾ 𝑌))
19		fnresdm 6617	. . . . . . 7 ⊢ (𝑀 Fn 𝑌 → (𝑀 ↾ 𝑌) = 𝑀)
20	7, 19	syl 17	. . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → (𝑀 ↾ 𝑌) = 𝑀)
21		eqid 2736	. . . . . . . . . 10 ⊢ ( +𝑣 ‘𝑈) = ( +𝑣 ‘𝑈)
22		eqid 2736	. . . . . . . . . 10 ⊢ ( +𝑣 ‘𝑊) = ( +𝑣 ‘𝑊)
23		eqid 2736	. . . . . . . . . 10 ⊢ ( ·𝑠OLD ‘𝑈) = ( ·𝑠OLD ‘𝑈)
24		eqid 2736	. . . . . . . . . 10 ⊢ ( ·𝑠OLD ‘𝑊) = ( ·𝑠OLD ‘𝑊)
25	21, 22, 23, 24, 9, 4, 1	isssp 30795	. . . . . . . . 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 1145	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑀 ⊆ 𝑁)
28		ssres 5968	. . . . . . 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 3957	. . . . 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 6678	. . . . . . 7 ⊢ (𝑊 ∈ NrmCVec → dom 𝑀 = 𝑌)
33	32	eleq2d 2822	. . . . . 6 ⊢ (𝑊 ∈ NrmCVec → (𝑥 ∈ dom 𝑀 ↔ 𝑥 ∈ 𝑌))
34	33	biimpar 477	. . . . 5 ⊢ ((𝑊 ∈ NrmCVec ∧ 𝑥 ∈ 𝑌) → 𝑥 ∈ dom 𝑀)
35	2, 34	sylan 581	. . . 4 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ 𝑥 ∈ 𝑌) → 𝑥 ∈ dom 𝑀)
36		funssfv 6861	. . . 4 ⊢ ((Fun (𝑁 ↾ 𝑌) ∧ 𝑀 ⊆ (𝑁 ↾ 𝑌) ∧ 𝑥 ∈ dom 𝑀) → ((𝑁 ↾ 𝑌)‘𝑥) = (𝑀‘𝑥))
37	18, 31, 35, 36	syl3anc 1374	. . 3 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ 𝑥 ∈ 𝑌) → ((𝑁 ↾ 𝑌)‘𝑥) = (𝑀‘𝑥))
38	37	eqcomd 2742	. 2 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ 𝑥 ∈ 𝑌) → (𝑀‘𝑥) = ((𝑁 ↾ 𝑌)‘𝑥))
39	7, 15, 38	eqfnfvd 6986	1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑀 = (𝑁 ↾ 𝑌))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ⊆ wss 3889  dom cdm 5631   ↾ cres 5633  Fun wfun 6492   Fn wfn 6493  ⟶wf 6494  ‘cfv 6498  ℝcr 11037  NrmCVeccnv 30655   +_𝑣 cpv 30656  BaseSetcba 30657   ·_𝑠OLD cns 30658  norm_CVcnmcv 30661  SubSpcss 30792

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 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-oprab 7371  df-1st 7942  df-2nd 7943  df-vc 30630  df-nv 30663  df-va 30666  df-ba 30667  df-sm 30668  df-0v 30669  df-nmcv 30671  df-ssp 30793

This theorem is referenced by:  sspnval  30808