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Theorem sspph 27556
Description: A subspace of an inner product space is an inner product space. (Contributed by NM, 1-Feb-2008.) (New usage is discouraged.)
Hypothesis
Ref Expression
sspph.h 𝐻 = (SubSp‘𝑈)
Assertion
Ref Expression
sspph ((𝑈 ∈ CPreHilOLD𝑊𝐻) → 𝑊 ∈ CPreHilOLD)

Proof of Theorem sspph
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 phnv 27515 . . 3 (𝑈 ∈ CPreHilOLD𝑈 ∈ NrmCVec)
2 sspph.h . . . 4 𝐻 = (SubSp‘𝑈)
32sspnv 27427 . . 3 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑊 ∈ NrmCVec)
41, 3sylan 488 . 2 ((𝑈 ∈ CPreHilOLD𝑊𝐻) → 𝑊 ∈ NrmCVec)
5 eqid 2621 . . . . . . . . . 10 (BaseSet‘𝑈) = (BaseSet‘𝑈)
6 eqid 2621 . . . . . . . . . 10 (BaseSet‘𝑊) = (BaseSet‘𝑊)
75, 6, 2sspba 27428 . . . . . . . . 9 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → (BaseSet‘𝑊) ⊆ (BaseSet‘𝑈))
87sseld 3582 . . . . . . . 8 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → (𝑥 ∈ (BaseSet‘𝑊) → 𝑥 ∈ (BaseSet‘𝑈)))
97sseld 3582 . . . . . . . 8 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → (𝑦 ∈ (BaseSet‘𝑊) → 𝑦 ∈ (BaseSet‘𝑈)))
108, 9anim12d 585 . . . . . . 7 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → ((𝑥 ∈ (BaseSet‘𝑊) ∧ 𝑦 ∈ (BaseSet‘𝑊)) → (𝑥 ∈ (BaseSet‘𝑈) ∧ 𝑦 ∈ (BaseSet‘𝑈))))
111, 10sylan 488 . . . . . 6 ((𝑈 ∈ CPreHilOLD𝑊𝐻) → ((𝑥 ∈ (BaseSet‘𝑊) ∧ 𝑦 ∈ (BaseSet‘𝑊)) → (𝑥 ∈ (BaseSet‘𝑈) ∧ 𝑦 ∈ (BaseSet‘𝑈))))
1211imp 445 . . . . 5 (((𝑈 ∈ CPreHilOLD𝑊𝐻) ∧ (𝑥 ∈ (BaseSet‘𝑊) ∧ 𝑦 ∈ (BaseSet‘𝑊))) → (𝑥 ∈ (BaseSet‘𝑈) ∧ 𝑦 ∈ (BaseSet‘𝑈)))
13 eqid 2621 . . . . . . . 8 ( +𝑣𝑈) = ( +𝑣𝑈)
14 eqid 2621 . . . . . . . 8 ( −𝑣𝑈) = ( −𝑣𝑈)
15 eqid 2621 . . . . . . . 8 (normCV𝑈) = (normCV𝑈)
165, 13, 14, 15phpar2 27524 . . . . . . 7 ((𝑈 ∈ CPreHilOLD𝑥 ∈ (BaseSet‘𝑈) ∧ 𝑦 ∈ (BaseSet‘𝑈)) → ((((normCV𝑈)‘(𝑥( +𝑣𝑈)𝑦))↑2) + (((normCV𝑈)‘(𝑥( −𝑣𝑈)𝑦))↑2)) = (2 · ((((normCV𝑈)‘𝑥)↑2) + (((normCV𝑈)‘𝑦)↑2))))
17163expb 1263 . . . . . 6 ((𝑈 ∈ CPreHilOLD ∧ (𝑥 ∈ (BaseSet‘𝑈) ∧ 𝑦 ∈ (BaseSet‘𝑈))) → ((((normCV𝑈)‘(𝑥( +𝑣𝑈)𝑦))↑2) + (((normCV𝑈)‘(𝑥( −𝑣𝑈)𝑦))↑2)) = (2 · ((((normCV𝑈)‘𝑥)↑2) + (((normCV𝑈)‘𝑦)↑2))))
1817adantlr 750 . . . . 5 (((𝑈 ∈ CPreHilOLD𝑊𝐻) ∧ (𝑥 ∈ (BaseSet‘𝑈) ∧ 𝑦 ∈ (BaseSet‘𝑈))) → ((((normCV𝑈)‘(𝑥( +𝑣𝑈)𝑦))↑2) + (((normCV𝑈)‘(𝑥( −𝑣𝑈)𝑦))↑2)) = (2 · ((((normCV𝑈)‘𝑥)↑2) + (((normCV𝑈)‘𝑦)↑2))))
1912, 18syldan 487 . . . 4 (((𝑈 ∈ CPreHilOLD𝑊𝐻) ∧ (𝑥 ∈ (BaseSet‘𝑊) ∧ 𝑦 ∈ (BaseSet‘𝑊))) → ((((normCV𝑈)‘(𝑥( +𝑣𝑈)𝑦))↑2) + (((normCV𝑈)‘(𝑥( −𝑣𝑈)𝑦))↑2)) = (2 · ((((normCV𝑈)‘𝑥)↑2) + (((normCV𝑈)‘𝑦)↑2))))
20 eqid 2621 . . . . . . . . . . . 12 ( +𝑣𝑊) = ( +𝑣𝑊)
216, 20nvgcl 27321 . . . . . . . . . . 11 ((𝑊 ∈ NrmCVec ∧ 𝑥 ∈ (BaseSet‘𝑊) ∧ 𝑦 ∈ (BaseSet‘𝑊)) → (𝑥( +𝑣𝑊)𝑦) ∈ (BaseSet‘𝑊))
22213expb 1263 . . . . . . . . . 10 ((𝑊 ∈ NrmCVec ∧ (𝑥 ∈ (BaseSet‘𝑊) ∧ 𝑦 ∈ (BaseSet‘𝑊))) → (𝑥( +𝑣𝑊)𝑦) ∈ (BaseSet‘𝑊))
233, 22sylan 488 . . . . . . . . 9 (((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) ∧ (𝑥 ∈ (BaseSet‘𝑊) ∧ 𝑦 ∈ (BaseSet‘𝑊))) → (𝑥( +𝑣𝑊)𝑦) ∈ (BaseSet‘𝑊))
24 eqid 2621 . . . . . . . . . . 11 (normCV𝑊) = (normCV𝑊)
256, 15, 24, 2sspnval 27438 . . . . . . . . . 10 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻 ∧ (𝑥( +𝑣𝑊)𝑦) ∈ (BaseSet‘𝑊)) → ((normCV𝑊)‘(𝑥( +𝑣𝑊)𝑦)) = ((normCV𝑈)‘(𝑥( +𝑣𝑊)𝑦)))
26253expa 1262 . . . . . . . . 9 (((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) ∧ (𝑥( +𝑣𝑊)𝑦) ∈ (BaseSet‘𝑊)) → ((normCV𝑊)‘(𝑥( +𝑣𝑊)𝑦)) = ((normCV𝑈)‘(𝑥( +𝑣𝑊)𝑦)))
2723, 26syldan 487 . . . . . . . 8 (((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) ∧ (𝑥 ∈ (BaseSet‘𝑊) ∧ 𝑦 ∈ (BaseSet‘𝑊))) → ((normCV𝑊)‘(𝑥( +𝑣𝑊)𝑦)) = ((normCV𝑈)‘(𝑥( +𝑣𝑊)𝑦)))
2827oveq1d 6619 . . . . . . 7 (((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) ∧ (𝑥 ∈ (BaseSet‘𝑊) ∧ 𝑦 ∈ (BaseSet‘𝑊))) → (((normCV𝑊)‘(𝑥( +𝑣𝑊)𝑦))↑2) = (((normCV𝑈)‘(𝑥( +𝑣𝑊)𝑦))↑2))
296, 13, 20, 2sspgval 27430 . . . . . . . . 9 (((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) ∧ (𝑥 ∈ (BaseSet‘𝑊) ∧ 𝑦 ∈ (BaseSet‘𝑊))) → (𝑥( +𝑣𝑊)𝑦) = (𝑥( +𝑣𝑈)𝑦))
3029fveq2d 6152 . . . . . . . 8 (((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) ∧ (𝑥 ∈ (BaseSet‘𝑊) ∧ 𝑦 ∈ (BaseSet‘𝑊))) → ((normCV𝑈)‘(𝑥( +𝑣𝑊)𝑦)) = ((normCV𝑈)‘(𝑥( +𝑣𝑈)𝑦)))
3130oveq1d 6619 . . . . . . 7 (((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) ∧ (𝑥 ∈ (BaseSet‘𝑊) ∧ 𝑦 ∈ (BaseSet‘𝑊))) → (((normCV𝑈)‘(𝑥( +𝑣𝑊)𝑦))↑2) = (((normCV𝑈)‘(𝑥( +𝑣𝑈)𝑦))↑2))
3228, 31eqtrd 2655 . . . . . 6 (((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) ∧ (𝑥 ∈ (BaseSet‘𝑊) ∧ 𝑦 ∈ (BaseSet‘𝑊))) → (((normCV𝑊)‘(𝑥( +𝑣𝑊)𝑦))↑2) = (((normCV𝑈)‘(𝑥( +𝑣𝑈)𝑦))↑2))
33 eqid 2621 . . . . . . . . . . . 12 ( −𝑣𝑊) = ( −𝑣𝑊)
346, 33nvmcl 27347 . . . . . . . . . . 11 ((𝑊 ∈ NrmCVec ∧ 𝑥 ∈ (BaseSet‘𝑊) ∧ 𝑦 ∈ (BaseSet‘𝑊)) → (𝑥( −𝑣𝑊)𝑦) ∈ (BaseSet‘𝑊))
35343expb 1263 . . . . . . . . . 10 ((𝑊 ∈ NrmCVec ∧ (𝑥 ∈ (BaseSet‘𝑊) ∧ 𝑦 ∈ (BaseSet‘𝑊))) → (𝑥( −𝑣𝑊)𝑦) ∈ (BaseSet‘𝑊))
363, 35sylan 488 . . . . . . . . 9 (((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) ∧ (𝑥 ∈ (BaseSet‘𝑊) ∧ 𝑦 ∈ (BaseSet‘𝑊))) → (𝑥( −𝑣𝑊)𝑦) ∈ (BaseSet‘𝑊))
376, 15, 24, 2sspnval 27438 . . . . . . . . . 10 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻 ∧ (𝑥( −𝑣𝑊)𝑦) ∈ (BaseSet‘𝑊)) → ((normCV𝑊)‘(𝑥( −𝑣𝑊)𝑦)) = ((normCV𝑈)‘(𝑥( −𝑣𝑊)𝑦)))
38373expa 1262 . . . . . . . . 9 (((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) ∧ (𝑥( −𝑣𝑊)𝑦) ∈ (BaseSet‘𝑊)) → ((normCV𝑊)‘(𝑥( −𝑣𝑊)𝑦)) = ((normCV𝑈)‘(𝑥( −𝑣𝑊)𝑦)))
3936, 38syldan 487 . . . . . . . 8 (((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) ∧ (𝑥 ∈ (BaseSet‘𝑊) ∧ 𝑦 ∈ (BaseSet‘𝑊))) → ((normCV𝑊)‘(𝑥( −𝑣𝑊)𝑦)) = ((normCV𝑈)‘(𝑥( −𝑣𝑊)𝑦)))
406, 14, 33, 2sspmval 27434 . . . . . . . . 9 (((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) ∧ (𝑥 ∈ (BaseSet‘𝑊) ∧ 𝑦 ∈ (BaseSet‘𝑊))) → (𝑥( −𝑣𝑊)𝑦) = (𝑥( −𝑣𝑈)𝑦))
4140fveq2d 6152 . . . . . . . 8 (((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) ∧ (𝑥 ∈ (BaseSet‘𝑊) ∧ 𝑦 ∈ (BaseSet‘𝑊))) → ((normCV𝑈)‘(𝑥( −𝑣𝑊)𝑦)) = ((normCV𝑈)‘(𝑥( −𝑣𝑈)𝑦)))
4239, 41eqtrd 2655 . . . . . . 7 (((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) ∧ (𝑥 ∈ (BaseSet‘𝑊) ∧ 𝑦 ∈ (BaseSet‘𝑊))) → ((normCV𝑊)‘(𝑥( −𝑣𝑊)𝑦)) = ((normCV𝑈)‘(𝑥( −𝑣𝑈)𝑦)))
4342oveq1d 6619 . . . . . 6 (((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) ∧ (𝑥 ∈ (BaseSet‘𝑊) ∧ 𝑦 ∈ (BaseSet‘𝑊))) → (((normCV𝑊)‘(𝑥( −𝑣𝑊)𝑦))↑2) = (((normCV𝑈)‘(𝑥( −𝑣𝑈)𝑦))↑2))
4432, 43oveq12d 6622 . . . . 5 (((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) ∧ (𝑥 ∈ (BaseSet‘𝑊) ∧ 𝑦 ∈ (BaseSet‘𝑊))) → ((((normCV𝑊)‘(𝑥( +𝑣𝑊)𝑦))↑2) + (((normCV𝑊)‘(𝑥( −𝑣𝑊)𝑦))↑2)) = ((((normCV𝑈)‘(𝑥( +𝑣𝑈)𝑦))↑2) + (((normCV𝑈)‘(𝑥( −𝑣𝑈)𝑦))↑2)))
451, 44sylanl1 681 . . . 4 (((𝑈 ∈ CPreHilOLD𝑊𝐻) ∧ (𝑥 ∈ (BaseSet‘𝑊) ∧ 𝑦 ∈ (BaseSet‘𝑊))) → ((((normCV𝑊)‘(𝑥( +𝑣𝑊)𝑦))↑2) + (((normCV𝑊)‘(𝑥( −𝑣𝑊)𝑦))↑2)) = ((((normCV𝑈)‘(𝑥( +𝑣𝑈)𝑦))↑2) + (((normCV𝑈)‘(𝑥( −𝑣𝑈)𝑦))↑2)))
466, 15, 24, 2sspnval 27438 . . . . . . . . . 10 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻𝑥 ∈ (BaseSet‘𝑊)) → ((normCV𝑊)‘𝑥) = ((normCV𝑈)‘𝑥))
47463expa 1262 . . . . . . . . 9 (((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) ∧ 𝑥 ∈ (BaseSet‘𝑊)) → ((normCV𝑊)‘𝑥) = ((normCV𝑈)‘𝑥))
4847adantrr 752 . . . . . . . 8 (((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) ∧ (𝑥 ∈ (BaseSet‘𝑊) ∧ 𝑦 ∈ (BaseSet‘𝑊))) → ((normCV𝑊)‘𝑥) = ((normCV𝑈)‘𝑥))
4948oveq1d 6619 . . . . . . 7 (((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) ∧ (𝑥 ∈ (BaseSet‘𝑊) ∧ 𝑦 ∈ (BaseSet‘𝑊))) → (((normCV𝑊)‘𝑥)↑2) = (((normCV𝑈)‘𝑥)↑2))
506, 15, 24, 2sspnval 27438 . . . . . . . . . 10 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻𝑦 ∈ (BaseSet‘𝑊)) → ((normCV𝑊)‘𝑦) = ((normCV𝑈)‘𝑦))
51503expa 1262 . . . . . . . . 9 (((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) ∧ 𝑦 ∈ (BaseSet‘𝑊)) → ((normCV𝑊)‘𝑦) = ((normCV𝑈)‘𝑦))
5251adantrl 751 . . . . . . . 8 (((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) ∧ (𝑥 ∈ (BaseSet‘𝑊) ∧ 𝑦 ∈ (BaseSet‘𝑊))) → ((normCV𝑊)‘𝑦) = ((normCV𝑈)‘𝑦))
5352oveq1d 6619 . . . . . . 7 (((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) ∧ (𝑥 ∈ (BaseSet‘𝑊) ∧ 𝑦 ∈ (BaseSet‘𝑊))) → (((normCV𝑊)‘𝑦)↑2) = (((normCV𝑈)‘𝑦)↑2))
5449, 53oveq12d 6622 . . . . . 6 (((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) ∧ (𝑥 ∈ (BaseSet‘𝑊) ∧ 𝑦 ∈ (BaseSet‘𝑊))) → ((((normCV𝑊)‘𝑥)↑2) + (((normCV𝑊)‘𝑦)↑2)) = ((((normCV𝑈)‘𝑥)↑2) + (((normCV𝑈)‘𝑦)↑2)))
551, 54sylanl1 681 . . . . 5 (((𝑈 ∈ CPreHilOLD𝑊𝐻) ∧ (𝑥 ∈ (BaseSet‘𝑊) ∧ 𝑦 ∈ (BaseSet‘𝑊))) → ((((normCV𝑊)‘𝑥)↑2) + (((normCV𝑊)‘𝑦)↑2)) = ((((normCV𝑈)‘𝑥)↑2) + (((normCV𝑈)‘𝑦)↑2)))
5655oveq2d 6620 . . . 4 (((𝑈 ∈ CPreHilOLD𝑊𝐻) ∧ (𝑥 ∈ (BaseSet‘𝑊) ∧ 𝑦 ∈ (BaseSet‘𝑊))) → (2 · ((((normCV𝑊)‘𝑥)↑2) + (((normCV𝑊)‘𝑦)↑2))) = (2 · ((((normCV𝑈)‘𝑥)↑2) + (((normCV𝑈)‘𝑦)↑2))))
5719, 45, 563eqtr4d 2665 . . 3 (((𝑈 ∈ CPreHilOLD𝑊𝐻) ∧ (𝑥 ∈ (BaseSet‘𝑊) ∧ 𝑦 ∈ (BaseSet‘𝑊))) → ((((normCV𝑊)‘(𝑥( +𝑣𝑊)𝑦))↑2) + (((normCV𝑊)‘(𝑥( −𝑣𝑊)𝑦))↑2)) = (2 · ((((normCV𝑊)‘𝑥)↑2) + (((normCV𝑊)‘𝑦)↑2))))
5857ralrimivva 2965 . 2 ((𝑈 ∈ CPreHilOLD𝑊𝐻) → ∀𝑥 ∈ (BaseSet‘𝑊)∀𝑦 ∈ (BaseSet‘𝑊)((((normCV𝑊)‘(𝑥( +𝑣𝑊)𝑦))↑2) + (((normCV𝑊)‘(𝑥( −𝑣𝑊)𝑦))↑2)) = (2 · ((((normCV𝑊)‘𝑥)↑2) + (((normCV𝑊)‘𝑦)↑2))))
596, 20, 33, 24isph 27523 . 2 (𝑊 ∈ CPreHilOLD ↔ (𝑊 ∈ NrmCVec ∧ ∀𝑥 ∈ (BaseSet‘𝑊)∀𝑦 ∈ (BaseSet‘𝑊)((((normCV𝑊)‘(𝑥( +𝑣𝑊)𝑦))↑2) + (((normCV𝑊)‘(𝑥( −𝑣𝑊)𝑦))↑2)) = (2 · ((((normCV𝑊)‘𝑥)↑2) + (((normCV𝑊)‘𝑦)↑2)))))
604, 58, 59sylanbrc 697 1 ((𝑈 ∈ CPreHilOLD𝑊𝐻) → 𝑊 ∈ CPreHilOLD)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  wral 2907  cfv 5847  (class class class)co 6604   + caddc 9883   · cmul 9885  2c2 11014  cexp 12800  NrmCVeccnv 27285   +𝑣 cpv 27286  BaseSetcba 27287  𝑣 cnsb 27290  normCVcnmcv 27291  SubSpcss 27422  CPreHilOLDccphlo 27513
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-po 4995  df-so 4996  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-1st 7113  df-2nd 7114  df-er 7687  df-en 7900  df-dom 7901  df-sdom 7902  df-pnf 10020  df-mnf 10021  df-ltxr 10023  df-sub 10212  df-neg 10213  df-grpo 27193  df-gid 27194  df-ginv 27195  df-gdiv 27196  df-ablo 27245  df-vc 27260  df-nv 27293  df-va 27296  df-ba 27297  df-sm 27298  df-0v 27299  df-vs 27300  df-nmcv 27301  df-ssp 27423  df-ph 27514
This theorem is referenced by:  ssphl  27619  hhssph  27977
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