Theorem ssps 31060

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 2763	. . . . . . . . . . 11 ⊢ (BaseSet‘𝑈) = (BaseSet‘𝑈)
2		ssps.s	. . . . . . . . . . 11 ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈)
3	1, 2	nvsf 30949	. . . . . . . . . 10 ⊢ (𝑈 ∈ NrmCVec → 𝑆:(ℂ × (BaseSet‘𝑈))⟶(BaseSet‘𝑈))
4	3	ffund 6712	. . . . . . . . 9 ⊢ (𝑈 ∈ NrmCVec → Fun 𝑆)
5	4	funresd 6581	. . . . . . . 8 ⊢ (𝑈 ∈ NrmCVec → Fun (𝑆 ↾ (ℂ × 𝑌)))
6	5	adantr 485	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → Fun (𝑆 ↾ (ℂ × 𝑌)))
7		ssps.h	. . . . . . . . . 10 ⊢ 𝐻 = (SubSp‘𝑈)
8	7	sspnv 31056	. . . . . . . . 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 30949	. . . . . . . . 9 ⊢ (𝑊 ∈ NrmCVec → 𝑅:(ℂ × 𝑌)⟶𝑌)
12	8, 11	syl 18	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑅:(ℂ × 𝑌)⟶𝑌)
13	12	ffnd 6708	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑅 Fn (ℂ × 𝑌))
14		fnresdm 6656	. . . . . . . . 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 2763	. . . . . . . . . . . 12 ⊢ ( +𝑣 ‘𝑈) = ( +𝑣 ‘𝑈)
17		eqid 2763	. . . . . . . . . . . 12 ⊢ ( +𝑣 ‘𝑊) = ( +𝑣 ‘𝑊)
18		eqid 2763	. . . . . . . . . . . 12 ⊢ (normCV‘𝑈) = (normCV‘𝑈)
19		eqid 2763	. . . . . . . . . . . 12 ⊢ (normCV‘𝑊) = (normCV‘𝑊)
20	16, 17, 2, 10, 18, 19, 7	isssp 31054	. . . . . . . . . . 11 ⊢ (𝑈 ∈ NrmCVec → (𝑊 ∈ 𝐻 ↔ (𝑊 ∈ NrmCVec ∧ (( +𝑣 ‘𝑊) ⊆ ( +𝑣 ‘𝑈) ∧ 𝑅 ⊆ 𝑆 ∧ (normCV‘𝑊) ⊆ (normCV‘𝑈)))))
21	20	simplbda 504	. . . . . . . . . 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 6004	. . . . . . . . 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 3973	. . . . . . 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 7581	. . . . . 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 591	. . . . 5 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑌)) → (𝑥(𝑆 ↾ (ℂ × 𝑌))𝑦) = (𝑥𝑅𝑦))
29	28	eqcomd 2769	. . . 4 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑌)) → (𝑥𝑅𝑦) = (𝑥(𝑆 ↾ (ℂ × 𝑌))𝑦))
30	29	ralrimivva 3208	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → ∀𝑥 ∈ ℂ ∀𝑦 ∈ 𝑌 (𝑥𝑅𝑦) = (𝑥(𝑆 ↾ (ℂ × 𝑌))𝑦))
31		eqid 2763	. . 3 ⊢ (ℂ × 𝑌) = (ℂ × 𝑌)
32	30, 31	jctil 528	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → ((ℂ × 𝑌) = (ℂ × 𝑌) ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ 𝑌 (𝑥𝑅𝑦) = (𝑥(𝑆 ↾ (ℂ × 𝑌))𝑦)))
33	3	ffnd 6708	. . . . 5 ⊢ (𝑈 ∈ NrmCVec → 𝑆 Fn (ℂ × (BaseSet‘𝑈)))
34	33	adantr 485	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑆 Fn (ℂ × (BaseSet‘𝑈)))
35		ssid 3960	. . . . 5 ⊢ ℂ ⊆ ℂ
36	1, 9, 7	sspba 31057	. . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑌 ⊆ (BaseSet‘𝑈))
37		xpss12 5678	. . . . 5 ⊢ ((ℂ ⊆ ℂ ∧ 𝑌 ⊆ (BaseSet‘𝑈)) → (ℂ × 𝑌) ⊆ (ℂ × (BaseSet‘𝑈)))
38	35, 36, 37	sylancr 598	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → (ℂ × 𝑌) ⊆ (ℂ × (BaseSet‘𝑈)))
39		fnssres 6660	. . . 4 ⊢ ((𝑆 Fn (ℂ × (BaseSet‘𝑈)) ∧ (ℂ × 𝑌) ⊆ (ℂ × (BaseSet‘𝑈))) → (𝑆 ↾ (ℂ × 𝑌)) Fn (ℂ × 𝑌))
40	34, 38, 39	syl2anc 595	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → (𝑆 ↾ (ℂ × 𝑌)) Fn (ℂ × 𝑌))
41		eqfnov 7541	. . 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 595	. 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)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  ∀wral 3079   ⊆ wss 3906   × cxp 5661   ↾ cres 5665  Fun wfun 6532   Fn wfn 6533  ⟶wf 6534  ‘cfv 6538  (class class class)co 7412  ℂcc 11099  NrmCVeccnv 30914   +_𝑣 cpv 30915  BaseSetcba 30916   ·_𝑠OLD cns 30917  norm_CVcnmcv 30920  SubSpcss 31051

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 30889  df-nv 30922  df-va 30925  df-ba 30926  df-sm 30927  df-0v 30928  df-nmcv 30930  df-ssp 31052

This theorem is referenced by:  sspsval  31061