Theorem ssps 31119

Description: Scalar multiplication on a subspace is a restriction of scalar multiplication on the parent space. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.)

Hypotheses

Ref	Expression
ssps.y	⊢ 𝑌 = (BaseSet‘𝑊)
ssps.s	⊢ 𝑆 = ( ·𝑠OLD ‘𝑈)
ssps.r	⊢ 𝑅 = ( ·𝑠OLD ‘𝑊)
ssps.h	⊢ 𝐻 = (SubSp‘𝑈)

Assertion

Ref	Expression
ssps	⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑅 = (𝑆 ↾ (ℂ × 𝑌)))

Proof of Theorem ssps

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2766	. . . . . . . . . . 11 ⊢ (BaseSet‘𝑈) = (BaseSet‘𝑈)
2		ssps.s	. . . . . . . . . . 11 ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈)
3	1, 2	nvsf 31008	. . . . . . . . . 10 ⊢ (𝑈 ∈ NrmCVec → 𝑆:(ℂ × (BaseSet‘𝑈))⟶(BaseSet‘𝑈))
4	3	ffund 6717	. . . . . . . . 9 ⊢ (𝑈 ∈ NrmCVec → Fun 𝑆)
5	4	funresd 6586	. . . . . . . 8 ⊢ (𝑈 ∈ NrmCVec → Fun (𝑆 ↾ (ℂ × 𝑌)))
6	5	adantr 486	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → Fun (𝑆 ↾ (ℂ × 𝑌)))
7		ssps.h	. . . . . . . . . 10 ⊢ 𝐻 = (SubSp‘𝑈)
8	7	sspnv 31115	. . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑊 ∈ NrmCVec)
9		ssps.y	. . . . . . . . . 10 ⊢ 𝑌 = (BaseSet‘𝑊)
10		ssps.r	. . . . . . . . . 10 ⊢ 𝑅 = ( ·𝑠OLD ‘𝑊)
11	9, 10	nvsf 31008	. . . . . . . . 9 ⊢ (𝑊 ∈ NrmCVec → 𝑅:(ℂ × 𝑌)⟶𝑌)
12	8, 11	syl 18	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑅:(ℂ × 𝑌)⟶𝑌)
13	12	ffnd 6713	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑅 Fn (ℂ × 𝑌))
14		fnresdm 6661	. . . . . . . . 9 ⊢ (𝑅 Fn (ℂ × 𝑌) → (𝑅 ↾ (ℂ × 𝑌)) = 𝑅)
15	13, 14	syl 18	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → (𝑅 ↾ (ℂ × 𝑌)) = 𝑅)
16		eqid 2766	. . . . . . . . . . . 12 ⊢ ( +𝑣 ‘𝑈) = ( +𝑣 ‘𝑈)
17		eqid 2766	. . . . . . . . . . . 12 ⊢ ( +𝑣 ‘𝑊) = ( +𝑣 ‘𝑊)
18		eqid 2766	. . . . . . . . . . . 12 ⊢ (normCV‘𝑈) = (normCV‘𝑈)
19		eqid 2766	. . . . . . . . . . . 12 ⊢ (normCV‘𝑊) = (normCV‘𝑊)
20	16, 17, 2, 10, 18, 19, 7	isssp 31113	. . . . . . . . . . 11 ⊢ (𝑈 ∈ NrmCVec → (𝑊 ∈ 𝐻 ↔ (𝑊 ∈ NrmCVec ∧ (( +𝑣 ‘𝑊) ⊆ ( +𝑣 ‘𝑈) ∧ 𝑅 ⊆ 𝑆 ∧ (normCV‘𝑊) ⊆ (normCV‘𝑈)))))
21	20	simplbda 505	. . . . . . . . . 10 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → (( +𝑣 ‘𝑊) ⊆ ( +𝑣 ‘𝑈) ∧ 𝑅 ⊆ 𝑆 ∧ (normCV‘𝑊) ⊆ (normCV‘𝑈)))
22	21	simp2d 1161	. . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑅 ⊆ 𝑆)
23		ssres 6007	. . . . . . . . 9 ⊢ (𝑅 ⊆ 𝑆 → (𝑅 ↾ (ℂ × 𝑌)) ⊆ (𝑆 ↾ (ℂ × 𝑌)))
24	22, 23	syl 18	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → (𝑅 ↾ (ℂ × 𝑌)) ⊆ (𝑆 ↾ (ℂ × 𝑌)))
25	15, 24	eqsstrrd 3975	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑅 ⊆ (𝑆 ↾ (ℂ × 𝑌)))
26	6, 13, 25	3jca 1146	. . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → (Fun (𝑆 ↾ (ℂ × 𝑌)) ∧ 𝑅 Fn (ℂ × 𝑌) ∧ 𝑅 ⊆ (𝑆 ↾ (ℂ × 𝑌))))
27		oprssov 7592	. . . . . 6 ⊢ (((Fun (𝑆 ↾ (ℂ × 𝑌)) ∧ 𝑅 Fn (ℂ × 𝑌) ∧ 𝑅 ⊆ (𝑆 ↾ (ℂ × 𝑌))) ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑌)) → (𝑥(𝑆 ↾ (ℂ × 𝑌))𝑦) = (𝑥𝑅𝑦))
28	26, 27	sylan 592	. . . . 5 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑌)) → (𝑥(𝑆 ↾ (ℂ × 𝑌))𝑦) = (𝑥𝑅𝑦))
29	28	eqcomd 2772	. . . 4 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑌)) → (𝑥𝑅𝑦) = (𝑥(𝑆 ↾ (ℂ × 𝑌))𝑦))
30	29	ralrimivva 3211	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → ∀𝑥 ∈ ℂ ∀𝑦 ∈ 𝑌 (𝑥𝑅𝑦) = (𝑥(𝑆 ↾ (ℂ × 𝑌))𝑦))
31		eqid 2766	. . 3 ⊢ (ℂ × 𝑌) = (ℂ × 𝑌)
32	30, 31	jctil 529	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → ((ℂ × 𝑌) = (ℂ × 𝑌) ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ 𝑌 (𝑥𝑅𝑦) = (𝑥(𝑆 ↾ (ℂ × 𝑌))𝑦)))
33	3	ffnd 6713	. . . . 5 ⊢ (𝑈 ∈ NrmCVec → 𝑆 Fn (ℂ × (BaseSet‘𝑈)))
34	33	adantr 486	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑆 Fn (ℂ × (BaseSet‘𝑈)))
35		ssid 3962	. . . . 5 ⊢ ℂ ⊆ ℂ
36	1, 9, 7	sspba 31116	. . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑌 ⊆ (BaseSet‘𝑈))
37		xpss12 5681	. . . . 5 ⊢ ((ℂ ⊆ ℂ ∧ 𝑌 ⊆ (BaseSet‘𝑈)) → (ℂ × 𝑌) ⊆ (ℂ × (BaseSet‘𝑈)))
38	35, 36, 37	sylancr 599	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → (ℂ × 𝑌) ⊆ (ℂ × (BaseSet‘𝑈)))
39		fnssres 6665	. . . 4 ⊢ ((𝑆 Fn (ℂ × (BaseSet‘𝑈)) ∧ (ℂ × 𝑌) ⊆ (ℂ × (BaseSet‘𝑈))) → (𝑆 ↾ (ℂ × 𝑌)) Fn (ℂ × 𝑌))
40	34, 38, 39	syl2anc 596	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → (𝑆 ↾ (ℂ × 𝑌)) Fn (ℂ × 𝑌))
41		eqfnov 7552	. . 3 ⊢ ((𝑅 Fn (ℂ × 𝑌) ∧ (𝑆 ↾ (ℂ × 𝑌)) Fn (ℂ × 𝑌)) → (𝑅 = (𝑆 ↾ (ℂ × 𝑌)) ↔ ((ℂ × 𝑌) = (ℂ × 𝑌) ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ 𝑌 (𝑥𝑅𝑦) = (𝑥(𝑆 ↾ (ℂ × 𝑌))𝑦))))
42	13, 40, 41	syl2anc 596	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → (𝑅 = (𝑆 ↾ (ℂ × 𝑌)) ↔ ((ℂ × 𝑌) = (ℂ × 𝑌) ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ 𝑌 (𝑥𝑅𝑦) = (𝑥(𝑆 ↾ (ℂ × 𝑌))𝑦))))
43	32, 42	mpbird 260	1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑅 = (𝑆 ↾ (ℂ × 𝑌)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  ∀wral 3082   ⊆ wss 3908   × cxp 5664   ↾ cres 5668  Fun wfun 6537   Fn wfn 6538  ⟶wf 6539  ‘cfv 6543  (class class class)co 7423  ℂcc 11116  NrmCVeccnv 30973   +_𝑣 cpv 30974  BaseSetcba 30975   ·_𝑠OLD cns 30976  norm_CVcnmcv 30979  SubSpcss 31110

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7426  df-oprab 7427  df-1st 7995  df-2nd 7996  df-vc 30948  df-nv 30981  df-va 30984  df-ba 30985  df-sm 30986  df-0v 30987  df-nmcv 30989  df-ssp 31111

This theorem is used by:  sspsval  31120