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Theorem phlssphl 21608
Description: A subspace of an inner product space (pre-Hilbert space) is an inner product space. (Contributed by AV, 25-Sep-2022.)
Hypotheses
Ref Expression
phlssphl.x 𝑋 = (𝑊s 𝑈)
phlssphl.s 𝑆 = (LSubSp‘𝑊)
Assertion
Ref Expression
phlssphl ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → 𝑋 ∈ PreHil)

Proof of Theorem phlssphl
Dummy variables 𝑞 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqidd 2726 . 2 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → (Base‘𝑋) = (Base‘𝑋))
2 eqidd 2726 . 2 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → (+g𝑋) = (+g𝑋))
3 eqidd 2726 . 2 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → ( ·𝑠𝑋) = ( ·𝑠𝑋))
4 eqidd 2726 . 2 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → (·𝑖𝑋) = (·𝑖𝑋))
5 phllmod 21579 . . . 4 (𝑊 ∈ PreHil → 𝑊 ∈ LMod)
6 phlssphl.x . . . . 5 𝑋 = (𝑊s 𝑈)
7 eqid 2725 . . . . 5 (0g𝑊) = (0g𝑊)
8 eqid 2725 . . . . 5 (0g𝑋) = (0g𝑋)
9 phlssphl.s . . . . 5 𝑆 = (LSubSp‘𝑊)
106, 7, 8, 9lss0v 20913 . . . 4 ((𝑊 ∈ LMod ∧ 𝑈𝑆) → (0g𝑋) = (0g𝑊))
115, 10sylan 578 . . 3 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → (0g𝑋) = (0g𝑊))
1211eqcomd 2731 . 2 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → (0g𝑊) = (0g𝑋))
13 eqidd 2726 . 2 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → (Scalar‘𝑋) = (Scalar‘𝑋))
14 eqidd 2726 . 2 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → (Base‘(Scalar‘𝑋)) = (Base‘(Scalar‘𝑋)))
15 eqidd 2726 . 2 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → (+g‘(Scalar‘𝑋)) = (+g‘(Scalar‘𝑋)))
16 eqidd 2726 . 2 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → (.r‘(Scalar‘𝑋)) = (.r‘(Scalar‘𝑋)))
17 eqidd 2726 . 2 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → (*𝑟‘(Scalar‘𝑋)) = (*𝑟‘(Scalar‘𝑋)))
18 eqidd 2726 . 2 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → (0g‘(Scalar‘𝑋)) = (0g‘(Scalar‘𝑋)))
19 phllvec 21578 . . 3 (𝑊 ∈ PreHil → 𝑊 ∈ LVec)
206, 9lsslvec 21006 . . 3 ((𝑊 ∈ LVec ∧ 𝑈𝑆) → 𝑋 ∈ LVec)
2119, 20sylan 578 . 2 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → 𝑋 ∈ LVec)
22 eqid 2725 . . . . . 6 (Scalar‘𝑊) = (Scalar‘𝑊)
236, 22resssca 17327 . . . . 5 (𝑈𝑆 → (Scalar‘𝑊) = (Scalar‘𝑋))
2423eqcomd 2731 . . . 4 (𝑈𝑆 → (Scalar‘𝑋) = (Scalar‘𝑊))
2524adantl 480 . . 3 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → (Scalar‘𝑋) = (Scalar‘𝑊))
2622phlsrng 21580 . . . 4 (𝑊 ∈ PreHil → (Scalar‘𝑊) ∈ *-Ring)
2726adantr 479 . . 3 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → (Scalar‘𝑊) ∈ *-Ring)
2825, 27eqeltrd 2825 . 2 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → (Scalar‘𝑋) ∈ *-Ring)
29 simpl 481 . . . . 5 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → 𝑊 ∈ PreHil)
30 eqid 2725 . . . . . . 7 (Base‘𝑊) = (Base‘𝑊)
316, 30ressbasss 17222 . . . . . 6 (Base‘𝑋) ⊆ (Base‘𝑊)
3231sseli 3972 . . . . 5 (𝑥 ∈ (Base‘𝑋) → 𝑥 ∈ (Base‘𝑊))
3331sseli 3972 . . . . 5 (𝑦 ∈ (Base‘𝑋) → 𝑦 ∈ (Base‘𝑊))
34 eqid 2725 . . . . . 6 (·𝑖𝑊) = (·𝑖𝑊)
35 eqid 2725 . . . . . 6 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
3622, 34, 30, 35ipcl 21582 . . . . 5 ((𝑊 ∈ PreHil ∧ 𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊)) → (𝑥(·𝑖𝑊)𝑦) ∈ (Base‘(Scalar‘𝑊)))
3729, 32, 33, 36syl3an 1157 . . . 4 (((𝑊 ∈ PreHil ∧ 𝑈𝑆) ∧ 𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋)) → (𝑥(·𝑖𝑊)𝑦) ∈ (Base‘(Scalar‘𝑊)))
3824fveq2d 6900 . . . . . . 7 (𝑈𝑆 → (Base‘(Scalar‘𝑋)) = (Base‘(Scalar‘𝑊)))
3938eleq2d 2811 . . . . . 6 (𝑈𝑆 → ((𝑥(·𝑖𝑊)𝑦) ∈ (Base‘(Scalar‘𝑋)) ↔ (𝑥(·𝑖𝑊)𝑦) ∈ (Base‘(Scalar‘𝑊))))
4039adantl 480 . . . . 5 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → ((𝑥(·𝑖𝑊)𝑦) ∈ (Base‘(Scalar‘𝑋)) ↔ (𝑥(·𝑖𝑊)𝑦) ∈ (Base‘(Scalar‘𝑊))))
41403ad2ant1 1130 . . . 4 (((𝑊 ∈ PreHil ∧ 𝑈𝑆) ∧ 𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋)) → ((𝑥(·𝑖𝑊)𝑦) ∈ (Base‘(Scalar‘𝑋)) ↔ (𝑥(·𝑖𝑊)𝑦) ∈ (Base‘(Scalar‘𝑊))))
4237, 41mpbird 256 . . 3 (((𝑊 ∈ PreHil ∧ 𝑈𝑆) ∧ 𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋)) → (𝑥(·𝑖𝑊)𝑦) ∈ (Base‘(Scalar‘𝑋)))
43 eqid 2725 . . . . . . . 8 (·𝑖𝑋) = (·𝑖𝑋)
446, 34, 43ssipeq 21605 . . . . . . 7 (𝑈𝑆 → (·𝑖𝑋) = (·𝑖𝑊))
4544oveqd 7436 . . . . . 6 (𝑈𝑆 → (𝑥(·𝑖𝑋)𝑦) = (𝑥(·𝑖𝑊)𝑦))
4645eleq1d 2810 . . . . 5 (𝑈𝑆 → ((𝑥(·𝑖𝑋)𝑦) ∈ (Base‘(Scalar‘𝑋)) ↔ (𝑥(·𝑖𝑊)𝑦) ∈ (Base‘(Scalar‘𝑋))))
4746adantl 480 . . . 4 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → ((𝑥(·𝑖𝑋)𝑦) ∈ (Base‘(Scalar‘𝑋)) ↔ (𝑥(·𝑖𝑊)𝑦) ∈ (Base‘(Scalar‘𝑋))))
48473ad2ant1 1130 . . 3 (((𝑊 ∈ PreHil ∧ 𝑈𝑆) ∧ 𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋)) → ((𝑥(·𝑖𝑋)𝑦) ∈ (Base‘(Scalar‘𝑋)) ↔ (𝑥(·𝑖𝑊)𝑦) ∈ (Base‘(Scalar‘𝑋))))
4942, 48mpbird 256 . 2 (((𝑊 ∈ PreHil ∧ 𝑈𝑆) ∧ 𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋)) → (𝑥(·𝑖𝑋)𝑦) ∈ (Base‘(Scalar‘𝑋)))
50293ad2ant1 1130 . . . . 5 (((𝑊 ∈ PreHil ∧ 𝑈𝑆) ∧ 𝑞 ∈ (Base‘(Scalar‘𝑋)) ∧ (𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋) ∧ 𝑧 ∈ (Base‘𝑋))) → 𝑊 ∈ PreHil)
515adantr 479 . . . . . . 7 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → 𝑊 ∈ LMod)
52513ad2ant1 1130 . . . . . 6 (((𝑊 ∈ PreHil ∧ 𝑈𝑆) ∧ 𝑞 ∈ (Base‘(Scalar‘𝑋)) ∧ (𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋) ∧ 𝑧 ∈ (Base‘𝑋))) → 𝑊 ∈ LMod)
5325fveq2d 6900 . . . . . . . . 9 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → (Base‘(Scalar‘𝑋)) = (Base‘(Scalar‘𝑊)))
5453eleq2d 2811 . . . . . . . 8 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → (𝑞 ∈ (Base‘(Scalar‘𝑋)) ↔ 𝑞 ∈ (Base‘(Scalar‘𝑊))))
5554biimpa 475 . . . . . . 7 (((𝑊 ∈ PreHil ∧ 𝑈𝑆) ∧ 𝑞 ∈ (Base‘(Scalar‘𝑋))) → 𝑞 ∈ (Base‘(Scalar‘𝑊)))
56553adant3 1129 . . . . . 6 (((𝑊 ∈ PreHil ∧ 𝑈𝑆) ∧ 𝑞 ∈ (Base‘(Scalar‘𝑋)) ∧ (𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋) ∧ 𝑧 ∈ (Base‘𝑋))) → 𝑞 ∈ (Base‘(Scalar‘𝑊)))
57323ad2ant1 1130 . . . . . . 7 ((𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋) ∧ 𝑧 ∈ (Base‘𝑋)) → 𝑥 ∈ (Base‘𝑊))
58573ad2ant3 1132 . . . . . 6 (((𝑊 ∈ PreHil ∧ 𝑈𝑆) ∧ 𝑞 ∈ (Base‘(Scalar‘𝑋)) ∧ (𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋) ∧ 𝑧 ∈ (Base‘𝑋))) → 𝑥 ∈ (Base‘𝑊))
59 eqid 2725 . . . . . . 7 ( ·𝑠𝑊) = ( ·𝑠𝑊)
6030, 22, 59, 35lmodvscl 20773 . . . . . 6 ((𝑊 ∈ LMod ∧ 𝑞 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑥 ∈ (Base‘𝑊)) → (𝑞( ·𝑠𝑊)𝑥) ∈ (Base‘𝑊))
6152, 56, 58, 60syl3anc 1368 . . . . 5 (((𝑊 ∈ PreHil ∧ 𝑈𝑆) ∧ 𝑞 ∈ (Base‘(Scalar‘𝑋)) ∧ (𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋) ∧ 𝑧 ∈ (Base‘𝑋))) → (𝑞( ·𝑠𝑊)𝑥) ∈ (Base‘𝑊))
62333ad2ant2 1131 . . . . . 6 ((𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋) ∧ 𝑧 ∈ (Base‘𝑋)) → 𝑦 ∈ (Base‘𝑊))
63623ad2ant3 1132 . . . . 5 (((𝑊 ∈ PreHil ∧ 𝑈𝑆) ∧ 𝑞 ∈ (Base‘(Scalar‘𝑋)) ∧ (𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋) ∧ 𝑧 ∈ (Base‘𝑋))) → 𝑦 ∈ (Base‘𝑊))
6431sseli 3972 . . . . . . 7 (𝑧 ∈ (Base‘𝑋) → 𝑧 ∈ (Base‘𝑊))
65643ad2ant3 1132 . . . . . 6 ((𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋) ∧ 𝑧 ∈ (Base‘𝑋)) → 𝑧 ∈ (Base‘𝑊))
66653ad2ant3 1132 . . . . 5 (((𝑊 ∈ PreHil ∧ 𝑈𝑆) ∧ 𝑞 ∈ (Base‘(Scalar‘𝑋)) ∧ (𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋) ∧ 𝑧 ∈ (Base‘𝑋))) → 𝑧 ∈ (Base‘𝑊))
67 eqid 2725 . . . . . 6 (+g𝑊) = (+g𝑊)
68 eqid 2725 . . . . . 6 (+g‘(Scalar‘𝑊)) = (+g‘(Scalar‘𝑊))
6922, 34, 30, 67, 68ipdir 21588 . . . . 5 ((𝑊 ∈ PreHil ∧ ((𝑞( ·𝑠𝑊)𝑥) ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (((𝑞( ·𝑠𝑊)𝑥)(+g𝑊)𝑦)(·𝑖𝑊)𝑧) = (((𝑞( ·𝑠𝑊)𝑥)(·𝑖𝑊)𝑧)(+g‘(Scalar‘𝑊))(𝑦(·𝑖𝑊)𝑧)))
7050, 61, 63, 66, 69syl13anc 1369 . . . 4 (((𝑊 ∈ PreHil ∧ 𝑈𝑆) ∧ 𝑞 ∈ (Base‘(Scalar‘𝑋)) ∧ (𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋) ∧ 𝑧 ∈ (Base‘𝑋))) → (((𝑞( ·𝑠𝑊)𝑥)(+g𝑊)𝑦)(·𝑖𝑊)𝑧) = (((𝑞( ·𝑠𝑊)𝑥)(·𝑖𝑊)𝑧)(+g‘(Scalar‘𝑊))(𝑦(·𝑖𝑊)𝑧)))
71 eqid 2725 . . . . . . 7 (.r‘(Scalar‘𝑊)) = (.r‘(Scalar‘𝑊))
7222, 34, 30, 35, 59, 71ipass 21594 . . . . . 6 ((𝑊 ∈ PreHil ∧ (𝑞 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑥 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → ((𝑞( ·𝑠𝑊)𝑥)(·𝑖𝑊)𝑧) = (𝑞(.r‘(Scalar‘𝑊))(𝑥(·𝑖𝑊)𝑧)))
7350, 56, 58, 66, 72syl13anc 1369 . . . . 5 (((𝑊 ∈ PreHil ∧ 𝑈𝑆) ∧ 𝑞 ∈ (Base‘(Scalar‘𝑋)) ∧ (𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋) ∧ 𝑧 ∈ (Base‘𝑋))) → ((𝑞( ·𝑠𝑊)𝑥)(·𝑖𝑊)𝑧) = (𝑞(.r‘(Scalar‘𝑊))(𝑥(·𝑖𝑊)𝑧)))
7473oveq1d 7434 . . . 4 (((𝑊 ∈ PreHil ∧ 𝑈𝑆) ∧ 𝑞 ∈ (Base‘(Scalar‘𝑋)) ∧ (𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋) ∧ 𝑧 ∈ (Base‘𝑋))) → (((𝑞( ·𝑠𝑊)𝑥)(·𝑖𝑊)𝑧)(+g‘(Scalar‘𝑊))(𝑦(·𝑖𝑊)𝑧)) = ((𝑞(.r‘(Scalar‘𝑊))(𝑥(·𝑖𝑊)𝑧))(+g‘(Scalar‘𝑊))(𝑦(·𝑖𝑊)𝑧)))
7570, 74eqtrd 2765 . . 3 (((𝑊 ∈ PreHil ∧ 𝑈𝑆) ∧ 𝑞 ∈ (Base‘(Scalar‘𝑋)) ∧ (𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋) ∧ 𝑧 ∈ (Base‘𝑋))) → (((𝑞( ·𝑠𝑊)𝑥)(+g𝑊)𝑦)(·𝑖𝑊)𝑧) = ((𝑞(.r‘(Scalar‘𝑊))(𝑥(·𝑖𝑊)𝑧))(+g‘(Scalar‘𝑊))(𝑦(·𝑖𝑊)𝑧)))
766, 67ressplusg 17274 . . . . . . . . 9 (𝑈𝑆 → (+g𝑊) = (+g𝑋))
7776eqcomd 2731 . . . . . . . 8 (𝑈𝑆 → (+g𝑋) = (+g𝑊))
786, 59ressvsca 17328 . . . . . . . . . 10 (𝑈𝑆 → ( ·𝑠𝑊) = ( ·𝑠𝑋))
7978eqcomd 2731 . . . . . . . . 9 (𝑈𝑆 → ( ·𝑠𝑋) = ( ·𝑠𝑊))
8079oveqd 7436 . . . . . . . 8 (𝑈𝑆 → (𝑞( ·𝑠𝑋)𝑥) = (𝑞( ·𝑠𝑊)𝑥))
81 eqidd 2726 . . . . . . . 8 (𝑈𝑆𝑦 = 𝑦)
8277, 80, 81oveq123d 7440 . . . . . . 7 (𝑈𝑆 → ((𝑞( ·𝑠𝑋)𝑥)(+g𝑋)𝑦) = ((𝑞( ·𝑠𝑊)𝑥)(+g𝑊)𝑦))
83 eqidd 2726 . . . . . . 7 (𝑈𝑆𝑧 = 𝑧)
8444, 82, 83oveq123d 7440 . . . . . 6 (𝑈𝑆 → (((𝑞( ·𝑠𝑋)𝑥)(+g𝑋)𝑦)(·𝑖𝑋)𝑧) = (((𝑞( ·𝑠𝑊)𝑥)(+g𝑊)𝑦)(·𝑖𝑊)𝑧))
8524fveq2d 6900 . . . . . . 7 (𝑈𝑆 → (+g‘(Scalar‘𝑋)) = (+g‘(Scalar‘𝑊)))
8624fveq2d 6900 . . . . . . . 8 (𝑈𝑆 → (.r‘(Scalar‘𝑋)) = (.r‘(Scalar‘𝑊)))
87 eqidd 2726 . . . . . . . 8 (𝑈𝑆𝑞 = 𝑞)
8844oveqd 7436 . . . . . . . 8 (𝑈𝑆 → (𝑥(·𝑖𝑋)𝑧) = (𝑥(·𝑖𝑊)𝑧))
8986, 87, 88oveq123d 7440 . . . . . . 7 (𝑈𝑆 → (𝑞(.r‘(Scalar‘𝑋))(𝑥(·𝑖𝑋)𝑧)) = (𝑞(.r‘(Scalar‘𝑊))(𝑥(·𝑖𝑊)𝑧)))
9044oveqd 7436 . . . . . . 7 (𝑈𝑆 → (𝑦(·𝑖𝑋)𝑧) = (𝑦(·𝑖𝑊)𝑧))
9185, 89, 90oveq123d 7440 . . . . . 6 (𝑈𝑆 → ((𝑞(.r‘(Scalar‘𝑋))(𝑥(·𝑖𝑋)𝑧))(+g‘(Scalar‘𝑋))(𝑦(·𝑖𝑋)𝑧)) = ((𝑞(.r‘(Scalar‘𝑊))(𝑥(·𝑖𝑊)𝑧))(+g‘(Scalar‘𝑊))(𝑦(·𝑖𝑊)𝑧)))
9284, 91eqeq12d 2741 . . . . 5 (𝑈𝑆 → ((((𝑞( ·𝑠𝑋)𝑥)(+g𝑋)𝑦)(·𝑖𝑋)𝑧) = ((𝑞(.r‘(Scalar‘𝑋))(𝑥(·𝑖𝑋)𝑧))(+g‘(Scalar‘𝑋))(𝑦(·𝑖𝑋)𝑧)) ↔ (((𝑞( ·𝑠𝑊)𝑥)(+g𝑊)𝑦)(·𝑖𝑊)𝑧) = ((𝑞(.r‘(Scalar‘𝑊))(𝑥(·𝑖𝑊)𝑧))(+g‘(Scalar‘𝑊))(𝑦(·𝑖𝑊)𝑧))))
9392adantl 480 . . . 4 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → ((((𝑞( ·𝑠𝑋)𝑥)(+g𝑋)𝑦)(·𝑖𝑋)𝑧) = ((𝑞(.r‘(Scalar‘𝑋))(𝑥(·𝑖𝑋)𝑧))(+g‘(Scalar‘𝑋))(𝑦(·𝑖𝑋)𝑧)) ↔ (((𝑞( ·𝑠𝑊)𝑥)(+g𝑊)𝑦)(·𝑖𝑊)𝑧) = ((𝑞(.r‘(Scalar‘𝑊))(𝑥(·𝑖𝑊)𝑧))(+g‘(Scalar‘𝑊))(𝑦(·𝑖𝑊)𝑧))))
94933ad2ant1 1130 . . 3 (((𝑊 ∈ PreHil ∧ 𝑈𝑆) ∧ 𝑞 ∈ (Base‘(Scalar‘𝑋)) ∧ (𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋) ∧ 𝑧 ∈ (Base‘𝑋))) → ((((𝑞( ·𝑠𝑋)𝑥)(+g𝑋)𝑦)(·𝑖𝑋)𝑧) = ((𝑞(.r‘(Scalar‘𝑋))(𝑥(·𝑖𝑋)𝑧))(+g‘(Scalar‘𝑋))(𝑦(·𝑖𝑋)𝑧)) ↔ (((𝑞( ·𝑠𝑊)𝑥)(+g𝑊)𝑦)(·𝑖𝑊)𝑧) = ((𝑞(.r‘(Scalar‘𝑊))(𝑥(·𝑖𝑊)𝑧))(+g‘(Scalar‘𝑊))(𝑦(·𝑖𝑊)𝑧))))
9575, 94mpbird 256 . 2 (((𝑊 ∈ PreHil ∧ 𝑈𝑆) ∧ 𝑞 ∈ (Base‘(Scalar‘𝑋)) ∧ (𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋) ∧ 𝑧 ∈ (Base‘𝑋))) → (((𝑞( ·𝑠𝑋)𝑥)(+g𝑋)𝑦)(·𝑖𝑋)𝑧) = ((𝑞(.r‘(Scalar‘𝑋))(𝑥(·𝑖𝑋)𝑧))(+g‘(Scalar‘𝑋))(𝑦(·𝑖𝑋)𝑧)))
9644adantl 480 . . . . . 6 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → (·𝑖𝑋) = (·𝑖𝑊))
9796oveqdr 7447 . . . . 5 (((𝑊 ∈ PreHil ∧ 𝑈𝑆) ∧ 𝑥 ∈ (Base‘𝑋)) → (𝑥(·𝑖𝑋)𝑥) = (𝑥(·𝑖𝑊)𝑥))
9824fveq2d 6900 . . . . . . 7 (𝑈𝑆 → (0g‘(Scalar‘𝑋)) = (0g‘(Scalar‘𝑊)))
9998adantl 480 . . . . . 6 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → (0g‘(Scalar‘𝑋)) = (0g‘(Scalar‘𝑊)))
10099adantr 479 . . . . 5 (((𝑊 ∈ PreHil ∧ 𝑈𝑆) ∧ 𝑥 ∈ (Base‘𝑋)) → (0g‘(Scalar‘𝑋)) = (0g‘(Scalar‘𝑊)))
10197, 100eqeq12d 2741 . . . 4 (((𝑊 ∈ PreHil ∧ 𝑈𝑆) ∧ 𝑥 ∈ (Base‘𝑋)) → ((𝑥(·𝑖𝑋)𝑥) = (0g‘(Scalar‘𝑋)) ↔ (𝑥(·𝑖𝑊)𝑥) = (0g‘(Scalar‘𝑊))))
102 eqid 2725 . . . . . . 7 (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
10322, 34, 30, 102, 7ipeq0 21587 . . . . . 6 ((𝑊 ∈ PreHil ∧ 𝑥 ∈ (Base‘𝑊)) → ((𝑥(·𝑖𝑊)𝑥) = (0g‘(Scalar‘𝑊)) ↔ 𝑥 = (0g𝑊)))
10429, 32, 103syl2an 594 . . . . 5 (((𝑊 ∈ PreHil ∧ 𝑈𝑆) ∧ 𝑥 ∈ (Base‘𝑋)) → ((𝑥(·𝑖𝑊)𝑥) = (0g‘(Scalar‘𝑊)) ↔ 𝑥 = (0g𝑊)))
105104biimpd 228 . . . 4 (((𝑊 ∈ PreHil ∧ 𝑈𝑆) ∧ 𝑥 ∈ (Base‘𝑋)) → ((𝑥(·𝑖𝑊)𝑥) = (0g‘(Scalar‘𝑊)) → 𝑥 = (0g𝑊)))
106101, 105sylbid 239 . . 3 (((𝑊 ∈ PreHil ∧ 𝑈𝑆) ∧ 𝑥 ∈ (Base‘𝑋)) → ((𝑥(·𝑖𝑋)𝑥) = (0g‘(Scalar‘𝑋)) → 𝑥 = (0g𝑊)))
1071063impia 1114 . 2 (((𝑊 ∈ PreHil ∧ 𝑈𝑆) ∧ 𝑥 ∈ (Base‘𝑋) ∧ (𝑥(·𝑖𝑋)𝑥) = (0g‘(Scalar‘𝑋))) → 𝑥 = (0g𝑊))
10824fveq2d 6900 . . . . . . 7 (𝑈𝑆 → (*𝑟‘(Scalar‘𝑋)) = (*𝑟‘(Scalar‘𝑊)))
109108fveq1d 6898 . . . . . 6 (𝑈𝑆 → ((*𝑟‘(Scalar‘𝑋))‘(𝑥(·𝑖𝑊)𝑦)) = ((*𝑟‘(Scalar‘𝑊))‘(𝑥(·𝑖𝑊)𝑦)))
110109adantl 480 . . . . 5 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → ((*𝑟‘(Scalar‘𝑋))‘(𝑥(·𝑖𝑊)𝑦)) = ((*𝑟‘(Scalar‘𝑊))‘(𝑥(·𝑖𝑊)𝑦)))
1111103ad2ant1 1130 . . . 4 (((𝑊 ∈ PreHil ∧ 𝑈𝑆) ∧ 𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋)) → ((*𝑟‘(Scalar‘𝑋))‘(𝑥(·𝑖𝑊)𝑦)) = ((*𝑟‘(Scalar‘𝑊))‘(𝑥(·𝑖𝑊)𝑦)))
112 eqid 2725 . . . . . 6 (*𝑟‘(Scalar‘𝑊)) = (*𝑟‘(Scalar‘𝑊))
11322, 34, 30, 112ipcj 21583 . . . . 5 ((𝑊 ∈ PreHil ∧ 𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊)) → ((*𝑟‘(Scalar‘𝑊))‘(𝑥(·𝑖𝑊)𝑦)) = (𝑦(·𝑖𝑊)𝑥))
11429, 32, 33, 113syl3an 1157 . . . 4 (((𝑊 ∈ PreHil ∧ 𝑈𝑆) ∧ 𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋)) → ((*𝑟‘(Scalar‘𝑊))‘(𝑥(·𝑖𝑊)𝑦)) = (𝑦(·𝑖𝑊)𝑥))
115111, 114eqtrd 2765 . . 3 (((𝑊 ∈ PreHil ∧ 𝑈𝑆) ∧ 𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋)) → ((*𝑟‘(Scalar‘𝑋))‘(𝑥(·𝑖𝑊)𝑦)) = (𝑦(·𝑖𝑊)𝑥))
11645fveq2d 6900 . . . . . 6 (𝑈𝑆 → ((*𝑟‘(Scalar‘𝑋))‘(𝑥(·𝑖𝑋)𝑦)) = ((*𝑟‘(Scalar‘𝑋))‘(𝑥(·𝑖𝑊)𝑦)))
11744oveqd 7436 . . . . . 6 (𝑈𝑆 → (𝑦(·𝑖𝑋)𝑥) = (𝑦(·𝑖𝑊)𝑥))
118116, 117eqeq12d 2741 . . . . 5 (𝑈𝑆 → (((*𝑟‘(Scalar‘𝑋))‘(𝑥(·𝑖𝑋)𝑦)) = (𝑦(·𝑖𝑋)𝑥) ↔ ((*𝑟‘(Scalar‘𝑋))‘(𝑥(·𝑖𝑊)𝑦)) = (𝑦(·𝑖𝑊)𝑥)))
119118adantl 480 . . . 4 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → (((*𝑟‘(Scalar‘𝑋))‘(𝑥(·𝑖𝑋)𝑦)) = (𝑦(·𝑖𝑋)𝑥) ↔ ((*𝑟‘(Scalar‘𝑋))‘(𝑥(·𝑖𝑊)𝑦)) = (𝑦(·𝑖𝑊)𝑥)))
1201193ad2ant1 1130 . . 3 (((𝑊 ∈ PreHil ∧ 𝑈𝑆) ∧ 𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋)) → (((*𝑟‘(Scalar‘𝑋))‘(𝑥(·𝑖𝑋)𝑦)) = (𝑦(·𝑖𝑋)𝑥) ↔ ((*𝑟‘(Scalar‘𝑋))‘(𝑥(·𝑖𝑊)𝑦)) = (𝑦(·𝑖𝑊)𝑥)))
121115, 120mpbird 256 . 2 (((𝑊 ∈ PreHil ∧ 𝑈𝑆) ∧ 𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋)) → ((*𝑟‘(Scalar‘𝑋))‘(𝑥(·𝑖𝑋)𝑦)) = (𝑦(·𝑖𝑋)𝑥))
1221, 2, 3, 4, 12, 13, 14, 15, 16, 17, 18, 21, 28, 49, 95, 107, 121isphld 21603 1 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → 𝑋 ∈ PreHil)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  w3a 1084   = wceq 1533  wcel 2098  cfv 6549  (class class class)co 7419  Basecbs 17183  s cress 17212  +gcplusg 17236  .rcmulr 17237  *𝑟cstv 17238  Scalarcsca 17239   ·𝑠 cvsca 17240  ·𝑖cip 17241  0gc0g 17424  *-Ringcsr 20736  LModclmod 20755  LSubSpclss 20827  LVecclvec 20999  PreHilcphl 21573
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-cnex 11196  ax-resscn 11197  ax-1cn 11198  ax-icn 11199  ax-addcl 11200  ax-addrcl 11201  ax-mulcl 11202  ax-mulrcl 11203  ax-mulcom 11204  ax-addass 11205  ax-mulass 11206  ax-distr 11207  ax-i2m1 11208  ax-1ne0 11209  ax-1rid 11210  ax-rnegex 11211  ax-rrecex 11212  ax-cnre 11213  ax-pre-lttri 11214  ax-pre-lttrn 11215  ax-pre-ltadd 11216  ax-pre-mulgt0 11217
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-int 4951  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7872  df-1st 7994  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-er 8725  df-map 8847  df-en 8965  df-dom 8966  df-sdom 8967  df-pnf 11282  df-mnf 11283  df-xr 11284  df-ltxr 11285  df-le 11286  df-sub 11478  df-neg 11479  df-nn 12246  df-2 12308  df-3 12309  df-4 12310  df-5 12311  df-6 12312  df-7 12313  df-8 12314  df-sets 17136  df-slot 17154  df-ndx 17166  df-base 17184  df-ress 17213  df-plusg 17249  df-mulr 17250  df-sca 17252  df-vsca 17253  df-ip 17254  df-0g 17426  df-mgm 18603  df-sgrp 18682  df-mnd 18698  df-grp 18901  df-minusg 18902  df-sbg 18903  df-subg 19086  df-ghm 19176  df-cmn 19749  df-abl 19750  df-mgp 20087  df-rng 20105  df-ur 20134  df-ring 20187  df-subrg 20520  df-lmod 20757  df-lss 20828  df-lsp 20868  df-lmhm 20919  df-lvec 21000  df-sra 21070  df-rgmod 21071  df-phl 21575
This theorem is referenced by:  cphsscph  25223
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