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Theorem ssps 27894
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 = ( ·𝑠OLD𝑈)
ssps.r 𝑅 = ( ·𝑠OLD𝑊)
ssps.h 𝐻 = (SubSp‘𝑈)
Assertion
Ref Expression
ssps ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑅 = (𝑆 ↾ (ℂ × 𝑌)))

Proof of Theorem ssps
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2760 . . . . . . . . . . 11 (BaseSet‘𝑈) = (BaseSet‘𝑈)
2 ssps.s . . . . . . . . . . 11 𝑆 = ( ·𝑠OLD𝑈)
31, 2nvsf 27783 . . . . . . . . . 10 (𝑈 ∈ NrmCVec → 𝑆:(ℂ × (BaseSet‘𝑈))⟶(BaseSet‘𝑈))
4 ffun 6209 . . . . . . . . . 10 (𝑆:(ℂ × (BaseSet‘𝑈))⟶(BaseSet‘𝑈) → Fun 𝑆)
53, 4syl 17 . . . . . . . . 9 (𝑈 ∈ NrmCVec → Fun 𝑆)
6 funres 6090 . . . . . . . . 9 (Fun 𝑆 → Fun (𝑆 ↾ (ℂ × 𝑌)))
75, 6syl 17 . . . . . . . 8 (𝑈 ∈ NrmCVec → Fun (𝑆 ↾ (ℂ × 𝑌)))
87adantr 472 . . . . . . 7 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → Fun (𝑆 ↾ (ℂ × 𝑌)))
9 ssps.h . . . . . . . . . 10 𝐻 = (SubSp‘𝑈)
109sspnv 27890 . . . . . . . . 9 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑊 ∈ NrmCVec)
11 ssps.y . . . . . . . . . 10 𝑌 = (BaseSet‘𝑊)
12 ssps.r . . . . . . . . . 10 𝑅 = ( ·𝑠OLD𝑊)
1311, 12nvsf 27783 . . . . . . . . 9 (𝑊 ∈ NrmCVec → 𝑅:(ℂ × 𝑌)⟶𝑌)
1410, 13syl 17 . . . . . . . 8 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑅:(ℂ × 𝑌)⟶𝑌)
15 ffn 6206 . . . . . . . 8 (𝑅:(ℂ × 𝑌)⟶𝑌𝑅 Fn (ℂ × 𝑌))
1614, 15syl 17 . . . . . . 7 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑅 Fn (ℂ × 𝑌))
17 fnresdm 6161 . . . . . . . . 9 (𝑅 Fn (ℂ × 𝑌) → (𝑅 ↾ (ℂ × 𝑌)) = 𝑅)
1816, 17syl 17 . . . . . . . 8 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → (𝑅 ↾ (ℂ × 𝑌)) = 𝑅)
19 eqid 2760 . . . . . . . . . . . 12 ( +𝑣𝑈) = ( +𝑣𝑈)
20 eqid 2760 . . . . . . . . . . . 12 ( +𝑣𝑊) = ( +𝑣𝑊)
21 eqid 2760 . . . . . . . . . . . 12 (normCV𝑈) = (normCV𝑈)
22 eqid 2760 . . . . . . . . . . . 12 (normCV𝑊) = (normCV𝑊)
2319, 20, 2, 12, 21, 22, 9isssp 27888 . . . . . . . . . . 11 (𝑈 ∈ NrmCVec → (𝑊𝐻 ↔ (𝑊 ∈ NrmCVec ∧ (( +𝑣𝑊) ⊆ ( +𝑣𝑈) ∧ 𝑅𝑆 ∧ (normCV𝑊) ⊆ (normCV𝑈)))))
2423simplbda 655 . . . . . . . . . 10 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → (( +𝑣𝑊) ⊆ ( +𝑣𝑈) ∧ 𝑅𝑆 ∧ (normCV𝑊) ⊆ (normCV𝑈)))
2524simp2d 1138 . . . . . . . . 9 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑅𝑆)
26 ssres 5582 . . . . . . . . 9 (𝑅𝑆 → (𝑅 ↾ (ℂ × 𝑌)) ⊆ (𝑆 ↾ (ℂ × 𝑌)))
2725, 26syl 17 . . . . . . . 8 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → (𝑅 ↾ (ℂ × 𝑌)) ⊆ (𝑆 ↾ (ℂ × 𝑌)))
2818, 27eqsstr3d 3781 . . . . . . 7 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑅 ⊆ (𝑆 ↾ (ℂ × 𝑌)))
298, 16, 283jca 1123 . . . . . 6 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → (Fun (𝑆 ↾ (ℂ × 𝑌)) ∧ 𝑅 Fn (ℂ × 𝑌) ∧ 𝑅 ⊆ (𝑆 ↾ (ℂ × 𝑌))))
30 oprssov 6968 . . . . . 6 (((Fun (𝑆 ↾ (ℂ × 𝑌)) ∧ 𝑅 Fn (ℂ × 𝑌) ∧ 𝑅 ⊆ (𝑆 ↾ (ℂ × 𝑌))) ∧ (𝑥 ∈ ℂ ∧ 𝑦𝑌)) → (𝑥(𝑆 ↾ (ℂ × 𝑌))𝑦) = (𝑥𝑅𝑦))
3129, 30sylan 489 . . . . 5 (((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) ∧ (𝑥 ∈ ℂ ∧ 𝑦𝑌)) → (𝑥(𝑆 ↾ (ℂ × 𝑌))𝑦) = (𝑥𝑅𝑦))
3231eqcomd 2766 . . . 4 (((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) ∧ (𝑥 ∈ ℂ ∧ 𝑦𝑌)) → (𝑥𝑅𝑦) = (𝑥(𝑆 ↾ (ℂ × 𝑌))𝑦))
3332ralrimivva 3109 . . 3 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → ∀𝑥 ∈ ℂ ∀𝑦𝑌 (𝑥𝑅𝑦) = (𝑥(𝑆 ↾ (ℂ × 𝑌))𝑦))
34 eqid 2760 . . 3 (ℂ × 𝑌) = (ℂ × 𝑌)
3533, 34jctil 561 . 2 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → ((ℂ × 𝑌) = (ℂ × 𝑌) ∧ ∀𝑥 ∈ ℂ ∀𝑦𝑌 (𝑥𝑅𝑦) = (𝑥(𝑆 ↾ (ℂ × 𝑌))𝑦)))
36 ffn 6206 . . . . . 6 (𝑆:(ℂ × (BaseSet‘𝑈))⟶(BaseSet‘𝑈) → 𝑆 Fn (ℂ × (BaseSet‘𝑈)))
373, 36syl 17 . . . . 5 (𝑈 ∈ NrmCVec → 𝑆 Fn (ℂ × (BaseSet‘𝑈)))
3837adantr 472 . . . 4 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑆 Fn (ℂ × (BaseSet‘𝑈)))
39 ssid 3765 . . . . 5 ℂ ⊆ ℂ
401, 11, 9sspba 27891 . . . . 5 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑌 ⊆ (BaseSet‘𝑈))
41 xpss12 5281 . . . . 5 ((ℂ ⊆ ℂ ∧ 𝑌 ⊆ (BaseSet‘𝑈)) → (ℂ × 𝑌) ⊆ (ℂ × (BaseSet‘𝑈)))
4239, 40, 41sylancr 698 . . . 4 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → (ℂ × 𝑌) ⊆ (ℂ × (BaseSet‘𝑈)))
43 fnssres 6165 . . . 4 ((𝑆 Fn (ℂ × (BaseSet‘𝑈)) ∧ (ℂ × 𝑌) ⊆ (ℂ × (BaseSet‘𝑈))) → (𝑆 ↾ (ℂ × 𝑌)) Fn (ℂ × 𝑌))
4438, 42, 43syl2anc 696 . . 3 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → (𝑆 ↾ (ℂ × 𝑌)) Fn (ℂ × 𝑌))
45 eqfnov 6931 . . 3 ((𝑅 Fn (ℂ × 𝑌) ∧ (𝑆 ↾ (ℂ × 𝑌)) Fn (ℂ × 𝑌)) → (𝑅 = (𝑆 ↾ (ℂ × 𝑌)) ↔ ((ℂ × 𝑌) = (ℂ × 𝑌) ∧ ∀𝑥 ∈ ℂ ∀𝑦𝑌 (𝑥𝑅𝑦) = (𝑥(𝑆 ↾ (ℂ × 𝑌))𝑦))))
4616, 44, 45syl2anc 696 . 2 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → (𝑅 = (𝑆 ↾ (ℂ × 𝑌)) ↔ ((ℂ × 𝑌) = (ℂ × 𝑌) ∧ ∀𝑥 ∈ ℂ ∀𝑦𝑌 (𝑥𝑅𝑦) = (𝑥(𝑆 ↾ (ℂ × 𝑌))𝑦))))
4735, 46mpbird 247 1 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑅 = (𝑆 ↾ (ℂ × 𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1632  wcel 2139  wral 3050  wss 3715   × cxp 5264  cres 5268  Fun wfun 6043   Fn wfn 6044  wf 6045  cfv 6049  (class class class)co 6813  cc 10126  NrmCVeccnv 27748   +𝑣 cpv 27749  BaseSetcba 27750   ·𝑠OLD cns 27751  normCVcnmcv 27754  SubSpcss 27885
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-oprab 6817  df-1st 7333  df-2nd 7334  df-vc 27723  df-nv 27756  df-va 27759  df-ba 27760  df-sm 27761  df-0v 27762  df-nmcv 27764  df-ssp 27886
This theorem is referenced by:  sspsval  27895
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