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Theorem ssps 28510
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 2824 . . . . . . . . . . 11 (BaseSet‘𝑈) = (BaseSet‘𝑈)
2 ssps.s . . . . . . . . . . 11 𝑆 = ( ·𝑠OLD𝑈)
31, 2nvsf 28399 . . . . . . . . . 10 (𝑈 ∈ NrmCVec → 𝑆:(ℂ × (BaseSet‘𝑈))⟶(BaseSet‘𝑈))
43ffund 6521 . . . . . . . . 9 (𝑈 ∈ NrmCVec → Fun 𝑆)
5 funres 6400 . . . . . . . . 9 (Fun 𝑆 → Fun (𝑆 ↾ (ℂ × 𝑌)))
64, 5syl 17 . . . . . . . 8 (𝑈 ∈ NrmCVec → Fun (𝑆 ↾ (ℂ × 𝑌)))
76adantr 483 . . . . . . 7 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → Fun (𝑆 ↾ (ℂ × 𝑌)))
8 ssps.h . . . . . . . . . 10 𝐻 = (SubSp‘𝑈)
98sspnv 28506 . . . . . . . . 9 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑊 ∈ NrmCVec)
10 ssps.y . . . . . . . . . 10 𝑌 = (BaseSet‘𝑊)
11 ssps.r . . . . . . . . . 10 𝑅 = ( ·𝑠OLD𝑊)
1210, 11nvsf 28399 . . . . . . . . 9 (𝑊 ∈ NrmCVec → 𝑅:(ℂ × 𝑌)⟶𝑌)
139, 12syl 17 . . . . . . . 8 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑅:(ℂ × 𝑌)⟶𝑌)
1413ffnd 6518 . . . . . . 7 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑅 Fn (ℂ × 𝑌))
15 fnresdm 6469 . . . . . . . . 9 (𝑅 Fn (ℂ × 𝑌) → (𝑅 ↾ (ℂ × 𝑌)) = 𝑅)
1614, 15syl 17 . . . . . . . 8 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → (𝑅 ↾ (ℂ × 𝑌)) = 𝑅)
17 eqid 2824 . . . . . . . . . . . 12 ( +𝑣𝑈) = ( +𝑣𝑈)
18 eqid 2824 . . . . . . . . . . . 12 ( +𝑣𝑊) = ( +𝑣𝑊)
19 eqid 2824 . . . . . . . . . . . 12 (normCV𝑈) = (normCV𝑈)
20 eqid 2824 . . . . . . . . . . . 12 (normCV𝑊) = (normCV𝑊)
2117, 18, 2, 11, 19, 20, 8isssp 28504 . . . . . . . . . . 11 (𝑈 ∈ NrmCVec → (𝑊𝐻 ↔ (𝑊 ∈ NrmCVec ∧ (( +𝑣𝑊) ⊆ ( +𝑣𝑈) ∧ 𝑅𝑆 ∧ (normCV𝑊) ⊆ (normCV𝑈)))))
2221simplbda 502 . . . . . . . . . 10 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → (( +𝑣𝑊) ⊆ ( +𝑣𝑈) ∧ 𝑅𝑆 ∧ (normCV𝑊) ⊆ (normCV𝑈)))
2322simp2d 1139 . . . . . . . . 9 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑅𝑆)
24 ssres 5883 . . . . . . . . 9 (𝑅𝑆 → (𝑅 ↾ (ℂ × 𝑌)) ⊆ (𝑆 ↾ (ℂ × 𝑌)))
2523, 24syl 17 . . . . . . . 8 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → (𝑅 ↾ (ℂ × 𝑌)) ⊆ (𝑆 ↾ (ℂ × 𝑌)))
2616, 25eqsstrrd 4009 . . . . . . 7 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑅 ⊆ (𝑆 ↾ (ℂ × 𝑌)))
277, 14, 263jca 1124 . . . . . 6 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → (Fun (𝑆 ↾ (ℂ × 𝑌)) ∧ 𝑅 Fn (ℂ × 𝑌) ∧ 𝑅 ⊆ (𝑆 ↾ (ℂ × 𝑌))))
28 oprssov 7320 . . . . . 6 (((Fun (𝑆 ↾ (ℂ × 𝑌)) ∧ 𝑅 Fn (ℂ × 𝑌) ∧ 𝑅 ⊆ (𝑆 ↾ (ℂ × 𝑌))) ∧ (𝑥 ∈ ℂ ∧ 𝑦𝑌)) → (𝑥(𝑆 ↾ (ℂ × 𝑌))𝑦) = (𝑥𝑅𝑦))
2927, 28sylan 582 . . . . 5 (((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) ∧ (𝑥 ∈ ℂ ∧ 𝑦𝑌)) → (𝑥(𝑆 ↾ (ℂ × 𝑌))𝑦) = (𝑥𝑅𝑦))
3029eqcomd 2830 . . . 4 (((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) ∧ (𝑥 ∈ ℂ ∧ 𝑦𝑌)) → (𝑥𝑅𝑦) = (𝑥(𝑆 ↾ (ℂ × 𝑌))𝑦))
3130ralrimivva 3194 . . 3 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → ∀𝑥 ∈ ℂ ∀𝑦𝑌 (𝑥𝑅𝑦) = (𝑥(𝑆 ↾ (ℂ × 𝑌))𝑦))
32 eqid 2824 . . 3 (ℂ × 𝑌) = (ℂ × 𝑌)
3331, 32jctil 522 . 2 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → ((ℂ × 𝑌) = (ℂ × 𝑌) ∧ ∀𝑥 ∈ ℂ ∀𝑦𝑌 (𝑥𝑅𝑦) = (𝑥(𝑆 ↾ (ℂ × 𝑌))𝑦)))
343ffnd 6518 . . . . 5 (𝑈 ∈ NrmCVec → 𝑆 Fn (ℂ × (BaseSet‘𝑈)))
3534adantr 483 . . . 4 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑆 Fn (ℂ × (BaseSet‘𝑈)))
36 ssid 3992 . . . . 5 ℂ ⊆ ℂ
371, 10, 8sspba 28507 . . . . 5 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑌 ⊆ (BaseSet‘𝑈))
38 xpss12 5573 . . . . 5 ((ℂ ⊆ ℂ ∧ 𝑌 ⊆ (BaseSet‘𝑈)) → (ℂ × 𝑌) ⊆ (ℂ × (BaseSet‘𝑈)))
3936, 37, 38sylancr 589 . . . 4 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → (ℂ × 𝑌) ⊆ (ℂ × (BaseSet‘𝑈)))
40 fnssres 6473 . . . 4 ((𝑆 Fn (ℂ × (BaseSet‘𝑈)) ∧ (ℂ × 𝑌) ⊆ (ℂ × (BaseSet‘𝑈))) → (𝑆 ↾ (ℂ × 𝑌)) Fn (ℂ × 𝑌))
4135, 39, 40syl2anc 586 . . 3 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → (𝑆 ↾ (ℂ × 𝑌)) Fn (ℂ × 𝑌))
42 eqfnov 7283 . . 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 586 . 2 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → (𝑅 = (𝑆 ↾ (ℂ × 𝑌)) ↔ ((ℂ × 𝑌) = (ℂ × 𝑌) ∧ ∀𝑥 ∈ ℂ ∀𝑦𝑌 (𝑥𝑅𝑦) = (𝑥(𝑆 ↾ (ℂ × 𝑌))𝑦))))
4433, 43mpbird 259 1 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑅 = (𝑆 ↾ (ℂ × 𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  w3a 1083   = wceq 1536  wcel 2113  wral 3141  wss 3939   × cxp 5556  cres 5560  Fun wfun 6352   Fn wfn 6353  wf 6354  cfv 6358  (class class class)co 7159  cc 10538  NrmCVeccnv 28364   +𝑣 cpv 28365  BaseSetcba 28366   ·𝑠OLD cns 28367  normCVcnmcv 28370  SubSpcss 28501
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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-rep 5193  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-ral 3146  df-rex 3147  df-reu 3148  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-iun 4924  df-br 5070  df-opab 5132  df-mpt 5150  df-id 5463  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-ov 7162  df-oprab 7163  df-1st 7692  df-2nd 7693  df-vc 28339  df-nv 28372  df-va 28375  df-ba 28376  df-sm 28377  df-0v 28378  df-nmcv 28380  df-ssp 28502
This theorem is referenced by:  sspsval  28511
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