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Theorem ssps 30749
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 2737 . . . . . . . . . . 11 (BaseSet‘𝑈) = (BaseSet‘𝑈)
2 ssps.s . . . . . . . . . . 11 𝑆 = ( ·𝑠OLD𝑈)
31, 2nvsf 30638 . . . . . . . . . 10 (𝑈 ∈ NrmCVec → 𝑆:(ℂ × (BaseSet‘𝑈))⟶(BaseSet‘𝑈))
43ffund 6740 . . . . . . . . 9 (𝑈 ∈ NrmCVec → Fun 𝑆)
54funresd 6609 . . . . . . . 8 (𝑈 ∈ NrmCVec → Fun (𝑆 ↾ (ℂ × 𝑌)))
65adantr 480 . . . . . . 7 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → Fun (𝑆 ↾ (ℂ × 𝑌)))
7 ssps.h . . . . . . . . . 10 𝐻 = (SubSp‘𝑈)
87sspnv 30745 . . . . . . . . 9 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑊 ∈ NrmCVec)
9 ssps.y . . . . . . . . . 10 𝑌 = (BaseSet‘𝑊)
10 ssps.r . . . . . . . . . 10 𝑅 = ( ·𝑠OLD𝑊)
119, 10nvsf 30638 . . . . . . . . 9 (𝑊 ∈ NrmCVec → 𝑅:(ℂ × 𝑌)⟶𝑌)
128, 11syl 17 . . . . . . . 8 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑅:(ℂ × 𝑌)⟶𝑌)
1312ffnd 6737 . . . . . . 7 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑅 Fn (ℂ × 𝑌))
14 fnresdm 6687 . . . . . . . . 9 (𝑅 Fn (ℂ × 𝑌) → (𝑅 ↾ (ℂ × 𝑌)) = 𝑅)
1513, 14syl 17 . . . . . . . 8 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → (𝑅 ↾ (ℂ × 𝑌)) = 𝑅)
16 eqid 2737 . . . . . . . . . . . 12 ( +𝑣𝑈) = ( +𝑣𝑈)
17 eqid 2737 . . . . . . . . . . . 12 ( +𝑣𝑊) = ( +𝑣𝑊)
18 eqid 2737 . . . . . . . . . . . 12 (normCV𝑈) = (normCV𝑈)
19 eqid 2737 . . . . . . . . . . . 12 (normCV𝑊) = (normCV𝑊)
2016, 17, 2, 10, 18, 19, 7isssp 30743 . . . . . . . . . . 11 (𝑈 ∈ NrmCVec → (𝑊𝐻 ↔ (𝑊 ∈ NrmCVec ∧ (( +𝑣𝑊) ⊆ ( +𝑣𝑈) ∧ 𝑅𝑆 ∧ (normCV𝑊) ⊆ (normCV𝑈)))))
2120simplbda 499 . . . . . . . . . 10 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → (( +𝑣𝑊) ⊆ ( +𝑣𝑈) ∧ 𝑅𝑆 ∧ (normCV𝑊) ⊆ (normCV𝑈)))
2221simp2d 1144 . . . . . . . . 9 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑅𝑆)
23 ssres 6021 . . . . . . . . 9 (𝑅𝑆 → (𝑅 ↾ (ℂ × 𝑌)) ⊆ (𝑆 ↾ (ℂ × 𝑌)))
2422, 23syl 17 . . . . . . . 8 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → (𝑅 ↾ (ℂ × 𝑌)) ⊆ (𝑆 ↾ (ℂ × 𝑌)))
2515, 24eqsstrrd 4019 . . . . . . 7 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑅 ⊆ (𝑆 ↾ (ℂ × 𝑌)))
266, 13, 253jca 1129 . . . . . 6 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → (Fun (𝑆 ↾ (ℂ × 𝑌)) ∧ 𝑅 Fn (ℂ × 𝑌) ∧ 𝑅 ⊆ (𝑆 ↾ (ℂ × 𝑌))))
27 oprssov 7602 . . . . . 6 (((Fun (𝑆 ↾ (ℂ × 𝑌)) ∧ 𝑅 Fn (ℂ × 𝑌) ∧ 𝑅 ⊆ (𝑆 ↾ (ℂ × 𝑌))) ∧ (𝑥 ∈ ℂ ∧ 𝑦𝑌)) → (𝑥(𝑆 ↾ (ℂ × 𝑌))𝑦) = (𝑥𝑅𝑦))
2826, 27sylan 580 . . . . 5 (((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) ∧ (𝑥 ∈ ℂ ∧ 𝑦𝑌)) → (𝑥(𝑆 ↾ (ℂ × 𝑌))𝑦) = (𝑥𝑅𝑦))
2928eqcomd 2743 . . . 4 (((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) ∧ (𝑥 ∈ ℂ ∧ 𝑦𝑌)) → (𝑥𝑅𝑦) = (𝑥(𝑆 ↾ (ℂ × 𝑌))𝑦))
3029ralrimivva 3202 . . 3 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → ∀𝑥 ∈ ℂ ∀𝑦𝑌 (𝑥𝑅𝑦) = (𝑥(𝑆 ↾ (ℂ × 𝑌))𝑦))
31 eqid 2737 . . 3 (ℂ × 𝑌) = (ℂ × 𝑌)
3230, 31jctil 519 . 2 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → ((ℂ × 𝑌) = (ℂ × 𝑌) ∧ ∀𝑥 ∈ ℂ ∀𝑦𝑌 (𝑥𝑅𝑦) = (𝑥(𝑆 ↾ (ℂ × 𝑌))𝑦)))
333ffnd 6737 . . . . 5 (𝑈 ∈ NrmCVec → 𝑆 Fn (ℂ × (BaseSet‘𝑈)))
3433adantr 480 . . . 4 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑆 Fn (ℂ × (BaseSet‘𝑈)))
35 ssid 4006 . . . . 5 ℂ ⊆ ℂ
361, 9, 7sspba 30746 . . . . 5 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑌 ⊆ (BaseSet‘𝑈))
37 xpss12 5700 . . . . 5 ((ℂ ⊆ ℂ ∧ 𝑌 ⊆ (BaseSet‘𝑈)) → (ℂ × 𝑌) ⊆ (ℂ × (BaseSet‘𝑈)))
3835, 36, 37sylancr 587 . . . 4 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → (ℂ × 𝑌) ⊆ (ℂ × (BaseSet‘𝑈)))
39 fnssres 6691 . . . 4 ((𝑆 Fn (ℂ × (BaseSet‘𝑈)) ∧ (ℂ × 𝑌) ⊆ (ℂ × (BaseSet‘𝑈))) → (𝑆 ↾ (ℂ × 𝑌)) Fn (ℂ × 𝑌))
4034, 38, 39syl2anc 584 . . 3 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → (𝑆 ↾ (ℂ × 𝑌)) Fn (ℂ × 𝑌))
41 eqfnov 7562 . . 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))))
4213, 40, 41syl2anc 584 . 2 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → (𝑅 = (𝑆 ↾ (ℂ × 𝑌)) ↔ ((ℂ × 𝑌) = (ℂ × 𝑌) ∧ ∀𝑥 ∈ ℂ ∀𝑦𝑌 (𝑥𝑅𝑦) = (𝑥(𝑆 ↾ (ℂ × 𝑌))𝑦))))
4332, 42mpbird 257 1 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑅 = (𝑆 ↾ (ℂ × 𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wral 3061  wss 3951   × cxp 5683  cres 5687  Fun wfun 6555   Fn wfn 6556  wf 6557  cfv 6561  (class class class)co 7431  cc 11153  NrmCVeccnv 30603   +𝑣 cpv 30604  BaseSetcba 30605   ·𝑠OLD cns 30606  normCVcnmcv 30609  SubSpcss 30740
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-1st 8014  df-2nd 8015  df-vc 30578  df-nv 30611  df-va 30614  df-ba 30615  df-sm 30616  df-0v 30617  df-nmcv 30619  df-ssp 30741
This theorem is referenced by:  sspsval  30750
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