Theorem ssps 30712

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 2733	. . . . . . . . . . 11 ⊢ (BaseSet‘𝑈) = (BaseSet‘𝑈)
2		ssps.s	. . . . . . . . . . 11 ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈)
3	1, 2	nvsf 30601	. . . . . . . . . 10 ⊢ (𝑈 ∈ NrmCVec → 𝑆:(ℂ × (BaseSet‘𝑈))⟶(BaseSet‘𝑈))
4	3	ffund 6660	. . . . . . . . 9 ⊢ (𝑈 ∈ NrmCVec → Fun 𝑆)
5	4	funresd 6529	. . . . . . . 8 ⊢ (𝑈 ∈ NrmCVec → Fun (𝑆 ↾ (ℂ × 𝑌)))
6	5	adantr 480	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → Fun (𝑆 ↾ (ℂ × 𝑌)))
7		ssps.h	. . . . . . . . . 10 ⊢ 𝐻 = (SubSp‘𝑈)
8	7	sspnv 30708	. . . . . . . . 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 30601	. . . . . . . . 9 ⊢ (𝑊 ∈ NrmCVec → 𝑅:(ℂ × 𝑌)⟶𝑌)
12	8, 11	syl 17	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑅:(ℂ × 𝑌)⟶𝑌)
13	12	ffnd 6657	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑅 Fn (ℂ × 𝑌))
14		fnresdm 6605	. . . . . . . . 9 ⊢ (𝑅 Fn (ℂ × 𝑌) → (𝑅 ↾ (ℂ × 𝑌)) = 𝑅)
15	13, 14	syl 17	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → (𝑅 ↾ (ℂ × 𝑌)) = 𝑅)
16		eqid 2733	. . . . . . . . . . . 12 ⊢ ( +𝑣 ‘𝑈) = ( +𝑣 ‘𝑈)
17		eqid 2733	. . . . . . . . . . . 12 ⊢ ( +𝑣 ‘𝑊) = ( +𝑣 ‘𝑊)
18		eqid 2733	. . . . . . . . . . . 12 ⊢ (normCV‘𝑈) = (normCV‘𝑈)
19		eqid 2733	. . . . . . . . . . . 12 ⊢ (normCV‘𝑊) = (normCV‘𝑊)
20	16, 17, 2, 10, 18, 19, 7	isssp 30706	. . . . . . . . . . 11 ⊢ (𝑈 ∈ NrmCVec → (𝑊 ∈ 𝐻 ↔ (𝑊 ∈ NrmCVec ∧ (( +𝑣 ‘𝑊) ⊆ ( +𝑣 ‘𝑈) ∧ 𝑅 ⊆ 𝑆 ∧ (normCV‘𝑊) ⊆ (normCV‘𝑈)))))
21	20	simplbda 499	. . . . . . . . . 10 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → (( +𝑣 ‘𝑊) ⊆ ( +𝑣 ‘𝑈) ∧ 𝑅 ⊆ 𝑆 ∧ (normCV‘𝑊) ⊆ (normCV‘𝑈)))
22	21	simp2d 1143	. . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑅 ⊆ 𝑆)
23		ssres 5956	. . . . . . . . 9 ⊢ (𝑅 ⊆ 𝑆 → (𝑅 ↾ (ℂ × 𝑌)) ⊆ (𝑆 ↾ (ℂ × 𝑌)))
24	22, 23	syl 17	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → (𝑅 ↾ (ℂ × 𝑌)) ⊆ (𝑆 ↾ (ℂ × 𝑌)))
25	15, 24	eqsstrrd 3966	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑅 ⊆ (𝑆 ↾ (ℂ × 𝑌)))
26	6, 13, 25	3jca 1128	. . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → (Fun (𝑆 ↾ (ℂ × 𝑌)) ∧ 𝑅 Fn (ℂ × 𝑌) ∧ 𝑅 ⊆ (𝑆 ↾ (ℂ × 𝑌))))
27		oprssov 7521	. . . . . 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 580	. . . . 5 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑌)) → (𝑥(𝑆 ↾ (ℂ × 𝑌))𝑦) = (𝑥𝑅𝑦))
29	28	eqcomd 2739	. . . 4 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑌)) → (𝑥𝑅𝑦) = (𝑥(𝑆 ↾ (ℂ × 𝑌))𝑦))
30	29	ralrimivva 3176	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → ∀𝑥 ∈ ℂ ∀𝑦 ∈ 𝑌 (𝑥𝑅𝑦) = (𝑥(𝑆 ↾ (ℂ × 𝑌))𝑦))
31		eqid 2733	. . 3 ⊢ (ℂ × 𝑌) = (ℂ × 𝑌)
32	30, 31	jctil 519	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → ((ℂ × 𝑌) = (ℂ × 𝑌) ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ 𝑌 (𝑥𝑅𝑦) = (𝑥(𝑆 ↾ (ℂ × 𝑌))𝑦)))
33	3	ffnd 6657	. . . . 5 ⊢ (𝑈 ∈ NrmCVec → 𝑆 Fn (ℂ × (BaseSet‘𝑈)))
34	33	adantr 480	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑆 Fn (ℂ × (BaseSet‘𝑈)))
35		ssid 3953	. . . . 5 ⊢ ℂ ⊆ ℂ
36	1, 9, 7	sspba 30709	. . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑌 ⊆ (BaseSet‘𝑈))
37		xpss12 5634	. . . . 5 ⊢ ((ℂ ⊆ ℂ ∧ 𝑌 ⊆ (BaseSet‘𝑈)) → (ℂ × 𝑌) ⊆ (ℂ × (BaseSet‘𝑈)))
38	35, 36, 37	sylancr 587	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → (ℂ × 𝑌) ⊆ (ℂ × (BaseSet‘𝑈)))
39		fnssres 6609	. . . 4 ⊢ ((𝑆 Fn (ℂ × (BaseSet‘𝑈)) ∧ (ℂ × 𝑌) ⊆ (ℂ × (BaseSet‘𝑈))) → (𝑆 ↾ (ℂ × 𝑌)) Fn (ℂ × 𝑌))
40	34, 38, 39	syl2anc 584	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → (𝑆 ↾ (ℂ × 𝑌)) Fn (ℂ × 𝑌))
41		eqfnov 7481	. . 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 584	. 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 257	1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑅 = (𝑆 ↾ (ℂ × 𝑌)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  ∀wral 3048   ⊆ wss 3898   × cxp 5617   ↾ cres 5621  Fun wfun 6480   Fn wfn 6481  ⟶wf 6482  ‘cfv 6486  (class class class)co 7352  ℂcc 11011  NrmCVeccnv 30566   +_𝑣 cpv 30567  BaseSetcba 30568   ·_𝑠OLD cns 30569  norm_CVcnmcv 30572  SubSpcss 30703

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7355  df-oprab 7356  df-1st 7927  df-2nd 7928  df-vc 30541  df-nv 30574  df-va 30577  df-ba 30578  df-sm 30579  df-0v 30580  df-nmcv 30582  df-ssp 30704

This theorem is referenced by:  sspsval  30713