Theorem ssps 30817

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 2737	. . . . . . . . . . 11 ⊢ (BaseSet‘𝑈) = (BaseSet‘𝑈)
2		ssps.s	. . . . . . . . . . 11 ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈)
3	1, 2	nvsf 30706	. . . . . . . . . 10 ⊢ (𝑈 ∈ NrmCVec → 𝑆:(ℂ × (BaseSet‘𝑈))⟶(BaseSet‘𝑈))
4	3	ffund 6674	. . . . . . . . 9 ⊢ (𝑈 ∈ NrmCVec → Fun 𝑆)
5	4	funresd 6543	. . . . . . . 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 30813	. . . . . . . . 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 30706	. . . . . . . . 9 ⊢ (𝑊 ∈ NrmCVec → 𝑅:(ℂ × 𝑌)⟶𝑌)
12	8, 11	syl 17	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑅:(ℂ × 𝑌)⟶𝑌)
13	12	ffnd 6671	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑅 Fn (ℂ × 𝑌))
14		fnresdm 6619	. . . . . . . . 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 2737	. . . . . . . . . . . 12 ⊢ ( +𝑣 ‘𝑈) = ( +𝑣 ‘𝑈)
17		eqid 2737	. . . . . . . . . . . 12 ⊢ ( +𝑣 ‘𝑊) = ( +𝑣 ‘𝑊)
18		eqid 2737	. . . . . . . . . . . 12 ⊢ (normCV‘𝑈) = (normCV‘𝑈)
19		eqid 2737	. . . . . . . . . . . 12 ⊢ (normCV‘𝑊) = (normCV‘𝑊)
20	16, 17, 2, 10, 18, 19, 7	isssp 30811	. . . . . . . . . . 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 1144	. . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑅 ⊆ 𝑆)
23		ssres 5970	. . . . . . . . 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 3971	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑅 ⊆ (𝑆 ↾ (ℂ × 𝑌)))
26	6, 13, 25	3jca 1129	. . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → (Fun (𝑆 ↾ (ℂ × 𝑌)) ∧ 𝑅 Fn (ℂ × 𝑌) ∧ 𝑅 ⊆ (𝑆 ↾ (ℂ × 𝑌))))
27		oprssov 7537	. . . . . 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 581	. . . . 5 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑌)) → (𝑥(𝑆 ↾ (ℂ × 𝑌))𝑦) = (𝑥𝑅𝑦))
29	28	eqcomd 2743	. . . 4 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑌)) → (𝑥𝑅𝑦) = (𝑥(𝑆 ↾ (ℂ × 𝑌))𝑦))
30	29	ralrimivva 3181	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → ∀𝑥 ∈ ℂ ∀𝑦 ∈ 𝑌 (𝑥𝑅𝑦) = (𝑥(𝑆 ↾ (ℂ × 𝑌))𝑦))
31		eqid 2737	. . 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 6671	. . . . 5 ⊢ (𝑈 ∈ NrmCVec → 𝑆 Fn (ℂ × (BaseSet‘𝑈)))
34	33	adantr 480	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑆 Fn (ℂ × (BaseSet‘𝑈)))
35		ssid 3958	. . . . 5 ⊢ ℂ ⊆ ℂ
36	1, 9, 7	sspba 30814	. . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑌 ⊆ (BaseSet‘𝑈))
37		xpss12 5647	. . . . 5 ⊢ ((ℂ ⊆ ℂ ∧ 𝑌 ⊆ (BaseSet‘𝑈)) → (ℂ × 𝑌) ⊆ (ℂ × (BaseSet‘𝑈)))
38	35, 36, 37	sylancr 588	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → (ℂ × 𝑌) ⊆ (ℂ × (BaseSet‘𝑈)))
39		fnssres 6623	. . . 4 ⊢ ((𝑆 Fn (ℂ × (BaseSet‘𝑈)) ∧ (ℂ × 𝑌) ⊆ (ℂ × (BaseSet‘𝑈))) → (𝑆 ↾ (ℂ × 𝑌)) Fn (ℂ × 𝑌))
40	34, 38, 39	syl2anc 585	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → (𝑆 ↾ (ℂ × 𝑌)) Fn (ℂ × 𝑌))
41		eqfnov 7497	. . 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 585	. 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 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052   ⊆ wss 3903   × cxp 5630   ↾ cres 5634  Fun wfun 6494   Fn wfn 6495  ⟶wf 6496  ‘cfv 6500  (class class class)co 7368  ℂcc 11036  NrmCVeccnv 30671   +_𝑣 cpv 30672  BaseSetcba 30673   ·_𝑠OLD cns 30674  norm_CVcnmcv 30677  SubSpcss 30808

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 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-oprab 7372  df-1st 7943  df-2nd 7944  df-vc 30646  df-nv 30679  df-va 30682  df-ba 30683  df-sm 30684  df-0v 30685  df-nmcv 30687  df-ssp 30809

This theorem is referenced by:  sspsval  30818