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Theorem ssps 30418
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 2731 . . . . . . . . . . 11 (BaseSet‘𝑈) = (BaseSet‘𝑈)
2 ssps.s . . . . . . . . . . 11 𝑆 = ( ·𝑠OLD𝑈)
31, 2nvsf 30307 . . . . . . . . . 10 (𝑈 ∈ NrmCVec → 𝑆:(ℂ × (BaseSet‘𝑈))⟶(BaseSet‘𝑈))
43ffund 6721 . . . . . . . . 9 (𝑈 ∈ NrmCVec → Fun 𝑆)
54funresd 6591 . . . . . . . 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 30414 . . . . . . . . 9 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑊 ∈ NrmCVec)
9 ssps.y . . . . . . . . . 10 𝑌 = (BaseSet‘𝑊)
10 ssps.r . . . . . . . . . 10 𝑅 = ( ·𝑠OLD𝑊)
119, 10nvsf 30307 . . . . . . . . 9 (𝑊 ∈ NrmCVec → 𝑅:(ℂ × 𝑌)⟶𝑌)
128, 11syl 17 . . . . . . . 8 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑅:(ℂ × 𝑌)⟶𝑌)
1312ffnd 6718 . . . . . . 7 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑅 Fn (ℂ × 𝑌))
14 fnresdm 6669 . . . . . . . . 9 (𝑅 Fn (ℂ × 𝑌) → (𝑅 ↾ (ℂ × 𝑌)) = 𝑅)
1513, 14syl 17 . . . . . . . 8 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → (𝑅 ↾ (ℂ × 𝑌)) = 𝑅)
16 eqid 2731 . . . . . . . . . . . 12 ( +𝑣𝑈) = ( +𝑣𝑈)
17 eqid 2731 . . . . . . . . . . . 12 ( +𝑣𝑊) = ( +𝑣𝑊)
18 eqid 2731 . . . . . . . . . . . 12 (normCV𝑈) = (normCV𝑈)
19 eqid 2731 . . . . . . . . . . . 12 (normCV𝑊) = (normCV𝑊)
2016, 17, 2, 10, 18, 19, 7isssp 30412 . . . . . . . . . . 11 (𝑈 ∈ NrmCVec → (𝑊𝐻 ↔ (𝑊 ∈ NrmCVec ∧ (( +𝑣𝑊) ⊆ ( +𝑣𝑈) ∧ 𝑅𝑆 ∧ (normCV𝑊) ⊆ (normCV𝑈)))))
2120simplbda 499 . . . . . . . . . 10 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → (( +𝑣𝑊) ⊆ ( +𝑣𝑈) ∧ 𝑅𝑆 ∧ (normCV𝑊) ⊆ (normCV𝑈)))
2221simp2d 1142 . . . . . . . . 9 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑅𝑆)
23 ssres 6008 . . . . . . . . 9 (𝑅𝑆 → (𝑅 ↾ (ℂ × 𝑌)) ⊆ (𝑆 ↾ (ℂ × 𝑌)))
2422, 23syl 17 . . . . . . . 8 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → (𝑅 ↾ (ℂ × 𝑌)) ⊆ (𝑆 ↾ (ℂ × 𝑌)))
2515, 24eqsstrrd 4021 . . . . . . 7 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑅 ⊆ (𝑆 ↾ (ℂ × 𝑌)))
266, 13, 253jca 1127 . . . . . 6 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → (Fun (𝑆 ↾ (ℂ × 𝑌)) ∧ 𝑅 Fn (ℂ × 𝑌) ∧ 𝑅 ⊆ (𝑆 ↾ (ℂ × 𝑌))))
27 oprssov 7580 . . . . . 6 (((Fun (𝑆 ↾ (ℂ × 𝑌)) ∧ 𝑅 Fn (ℂ × 𝑌) ∧ 𝑅 ⊆ (𝑆 ↾ (ℂ × 𝑌))) ∧ (𝑥 ∈ ℂ ∧ 𝑦𝑌)) → (𝑥(𝑆 ↾ (ℂ × 𝑌))𝑦) = (𝑥𝑅𝑦))
2826, 27sylan 579 . . . . 5 (((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) ∧ (𝑥 ∈ ℂ ∧ 𝑦𝑌)) → (𝑥(𝑆 ↾ (ℂ × 𝑌))𝑦) = (𝑥𝑅𝑦))
2928eqcomd 2737 . . . 4 (((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) ∧ (𝑥 ∈ ℂ ∧ 𝑦𝑌)) → (𝑥𝑅𝑦) = (𝑥(𝑆 ↾ (ℂ × 𝑌))𝑦))
3029ralrimivva 3199 . . 3 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → ∀𝑥 ∈ ℂ ∀𝑦𝑌 (𝑥𝑅𝑦) = (𝑥(𝑆 ↾ (ℂ × 𝑌))𝑦))
31 eqid 2731 . . 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 6718 . . . . 5 (𝑈 ∈ NrmCVec → 𝑆 Fn (ℂ × (BaseSet‘𝑈)))
3433adantr 480 . . . 4 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑆 Fn (ℂ × (BaseSet‘𝑈)))
35 ssid 4004 . . . . 5 ℂ ⊆ ℂ
361, 9, 7sspba 30415 . . . . 5 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑌 ⊆ (BaseSet‘𝑈))
37 xpss12 5691 . . . . 5 ((ℂ ⊆ ℂ ∧ 𝑌 ⊆ (BaseSet‘𝑈)) → (ℂ × 𝑌) ⊆ (ℂ × (BaseSet‘𝑈)))
3835, 36, 37sylancr 586 . . . 4 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → (ℂ × 𝑌) ⊆ (ℂ × (BaseSet‘𝑈)))
39 fnssres 6673 . . . 4 ((𝑆 Fn (ℂ × (BaseSet‘𝑈)) ∧ (ℂ × 𝑌) ⊆ (ℂ × (BaseSet‘𝑈))) → (𝑆 ↾ (ℂ × 𝑌)) Fn (ℂ × 𝑌))
4034, 38, 39syl2anc 583 . . 3 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → (𝑆 ↾ (ℂ × 𝑌)) Fn (ℂ × 𝑌))
41 eqfnov 7541 . . 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 583 . 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 205  wa 395  w3a 1086   = wceq 1540  wcel 2105  wral 3060  wss 3948   × cxp 5674  cres 5678  Fun wfun 6537   Fn wfn 6538  wf 6539  cfv 6543  (class class class)co 7412  cc 11114  NrmCVeccnv 30272   +𝑣 cpv 30273  BaseSetcba 30274   ·𝑠OLD cns 30275  normCVcnmcv 30278  SubSpcss 30409
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7415  df-oprab 7416  df-1st 7979  df-2nd 7980  df-vc 30247  df-nv 30280  df-va 30283  df-ba 30284  df-sm 30285  df-0v 30286  df-nmcv 30288  df-ssp 30410
This theorem is referenced by:  sspsval  30419
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