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Theorem ssps 28434
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 2818 . . . . . . . . . . 11 (BaseSet‘𝑈) = (BaseSet‘𝑈)
2 ssps.s . . . . . . . . . . 11 𝑆 = ( ·𝑠OLD𝑈)
31, 2nvsf 28323 . . . . . . . . . 10 (𝑈 ∈ NrmCVec → 𝑆:(ℂ × (BaseSet‘𝑈))⟶(BaseSet‘𝑈))
43ffund 6511 . . . . . . . . 9 (𝑈 ∈ NrmCVec → Fun 𝑆)
5 funres 6390 . . . . . . . . 9 (Fun 𝑆 → Fun (𝑆 ↾ (ℂ × 𝑌)))
64, 5syl 17 . . . . . . . 8 (𝑈 ∈ NrmCVec → Fun (𝑆 ↾ (ℂ × 𝑌)))
76adantr 481 . . . . . . 7 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → Fun (𝑆 ↾ (ℂ × 𝑌)))
8 ssps.h . . . . . . . . . 10 𝐻 = (SubSp‘𝑈)
98sspnv 28430 . . . . . . . . 9 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑊 ∈ NrmCVec)
10 ssps.y . . . . . . . . . 10 𝑌 = (BaseSet‘𝑊)
11 ssps.r . . . . . . . . . 10 𝑅 = ( ·𝑠OLD𝑊)
1210, 11nvsf 28323 . . . . . . . . 9 (𝑊 ∈ NrmCVec → 𝑅:(ℂ × 𝑌)⟶𝑌)
139, 12syl 17 . . . . . . . 8 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑅:(ℂ × 𝑌)⟶𝑌)
1413ffnd 6508 . . . . . . 7 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑅 Fn (ℂ × 𝑌))
15 fnresdm 6459 . . . . . . . . 9 (𝑅 Fn (ℂ × 𝑌) → (𝑅 ↾ (ℂ × 𝑌)) = 𝑅)
1614, 15syl 17 . . . . . . . 8 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → (𝑅 ↾ (ℂ × 𝑌)) = 𝑅)
17 eqid 2818 . . . . . . . . . . . 12 ( +𝑣𝑈) = ( +𝑣𝑈)
18 eqid 2818 . . . . . . . . . . . 12 ( +𝑣𝑊) = ( +𝑣𝑊)
19 eqid 2818 . . . . . . . . . . . 12 (normCV𝑈) = (normCV𝑈)
20 eqid 2818 . . . . . . . . . . . 12 (normCV𝑊) = (normCV𝑊)
2117, 18, 2, 11, 19, 20, 8isssp 28428 . . . . . . . . . . 11 (𝑈 ∈ NrmCVec → (𝑊𝐻 ↔ (𝑊 ∈ NrmCVec ∧ (( +𝑣𝑊) ⊆ ( +𝑣𝑈) ∧ 𝑅𝑆 ∧ (normCV𝑊) ⊆ (normCV𝑈)))))
2221simplbda 500 . . . . . . . . . 10 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → (( +𝑣𝑊) ⊆ ( +𝑣𝑈) ∧ 𝑅𝑆 ∧ (normCV𝑊) ⊆ (normCV𝑈)))
2322simp2d 1135 . . . . . . . . 9 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑅𝑆)
24 ssres 5873 . . . . . . . . 9 (𝑅𝑆 → (𝑅 ↾ (ℂ × 𝑌)) ⊆ (𝑆 ↾ (ℂ × 𝑌)))
2523, 24syl 17 . . . . . . . 8 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → (𝑅 ↾ (ℂ × 𝑌)) ⊆ (𝑆 ↾ (ℂ × 𝑌)))
2616, 25eqsstrrd 4003 . . . . . . 7 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑅 ⊆ (𝑆 ↾ (ℂ × 𝑌)))
277, 14, 263jca 1120 . . . . . 6 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → (Fun (𝑆 ↾ (ℂ × 𝑌)) ∧ 𝑅 Fn (ℂ × 𝑌) ∧ 𝑅 ⊆ (𝑆 ↾ (ℂ × 𝑌))))
28 oprssov 7306 . . . . . 6 (((Fun (𝑆 ↾ (ℂ × 𝑌)) ∧ 𝑅 Fn (ℂ × 𝑌) ∧ 𝑅 ⊆ (𝑆 ↾ (ℂ × 𝑌))) ∧ (𝑥 ∈ ℂ ∧ 𝑦𝑌)) → (𝑥(𝑆 ↾ (ℂ × 𝑌))𝑦) = (𝑥𝑅𝑦))
2927, 28sylan 580 . . . . 5 (((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) ∧ (𝑥 ∈ ℂ ∧ 𝑦𝑌)) → (𝑥(𝑆 ↾ (ℂ × 𝑌))𝑦) = (𝑥𝑅𝑦))
3029eqcomd 2824 . . . 4 (((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) ∧ (𝑥 ∈ ℂ ∧ 𝑦𝑌)) → (𝑥𝑅𝑦) = (𝑥(𝑆 ↾ (ℂ × 𝑌))𝑦))
3130ralrimivva 3188 . . 3 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → ∀𝑥 ∈ ℂ ∀𝑦𝑌 (𝑥𝑅𝑦) = (𝑥(𝑆 ↾ (ℂ × 𝑌))𝑦))
32 eqid 2818 . . 3 (ℂ × 𝑌) = (ℂ × 𝑌)
3331, 32jctil 520 . 2 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → ((ℂ × 𝑌) = (ℂ × 𝑌) ∧ ∀𝑥 ∈ ℂ ∀𝑦𝑌 (𝑥𝑅𝑦) = (𝑥(𝑆 ↾ (ℂ × 𝑌))𝑦)))
343ffnd 6508 . . . . 5 (𝑈 ∈ NrmCVec → 𝑆 Fn (ℂ × (BaseSet‘𝑈)))
3534adantr 481 . . . 4 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑆 Fn (ℂ × (BaseSet‘𝑈)))
36 ssid 3986 . . . . 5 ℂ ⊆ ℂ
371, 10, 8sspba 28431 . . . . 5 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑌 ⊆ (BaseSet‘𝑈))
38 xpss12 5563 . . . . 5 ((ℂ ⊆ ℂ ∧ 𝑌 ⊆ (BaseSet‘𝑈)) → (ℂ × 𝑌) ⊆ (ℂ × (BaseSet‘𝑈)))
3936, 37, 38sylancr 587 . . . 4 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → (ℂ × 𝑌) ⊆ (ℂ × (BaseSet‘𝑈)))
40 fnssres 6463 . . . 4 ((𝑆 Fn (ℂ × (BaseSet‘𝑈)) ∧ (ℂ × 𝑌) ⊆ (ℂ × (BaseSet‘𝑈))) → (𝑆 ↾ (ℂ × 𝑌)) Fn (ℂ × 𝑌))
4135, 39, 40syl2anc 584 . . 3 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → (𝑆 ↾ (ℂ × 𝑌)) Fn (ℂ × 𝑌))
42 eqfnov 7269 . . 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))))
4314, 41, 42syl2anc 584 . 2 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → (𝑅 = (𝑆 ↾ (ℂ × 𝑌)) ↔ ((ℂ × 𝑌) = (ℂ × 𝑌) ∧ ∀𝑥 ∈ ℂ ∀𝑦𝑌 (𝑥𝑅𝑦) = (𝑥(𝑆 ↾ (ℂ × 𝑌))𝑦))))
4433, 43mpbird 258 1 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑅 = (𝑆 ↾ (ℂ × 𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  wral 3135  wss 3933   × cxp 5546  cres 5550  Fun wfun 6342   Fn wfn 6343  wf 6344  cfv 6348  (class class class)co 7145  cc 10523  NrmCVeccnv 28288   +𝑣 cpv 28289  BaseSetcba 28290   ·𝑠OLD cns 28291  normCVcnmcv 28294  SubSpcss 28425
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-1st 7678  df-2nd 7679  df-vc 28263  df-nv 28296  df-va 28299  df-ba 28300  df-sm 28301  df-0v 28302  df-nmcv 28304  df-ssp 28426
This theorem is referenced by:  sspsval  28435
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