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Theorem phlssphl 20719
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 2827 . 2 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → (Base‘𝑋) = (Base‘𝑋))
2 eqidd 2827 . 2 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → (+g𝑋) = (+g𝑋))
3 eqidd 2827 . 2 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → ( ·𝑠𝑋) = ( ·𝑠𝑋))
4 eqidd 2827 . 2 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → (·𝑖𝑋) = (·𝑖𝑋))
5 phllmod 20690 . . . 4 (𝑊 ∈ PreHil → 𝑊 ∈ LMod)
6 phlssphl.x . . . . 5 𝑋 = (𝑊s 𝑈)
7 eqid 2826 . . . . 5 (0g𝑊) = (0g𝑊)
8 eqid 2826 . . . . 5 (0g𝑋) = (0g𝑋)
9 phlssphl.s . . . . 5 𝑆 = (LSubSp‘𝑊)
106, 7, 8, 9lss0v 19708 . . . 4 ((𝑊 ∈ LMod ∧ 𝑈𝑆) → (0g𝑋) = (0g𝑊))
115, 10sylan 580 . . 3 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → (0g𝑋) = (0g𝑊))
1211eqcomd 2832 . 2 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → (0g𝑊) = (0g𝑋))
13 eqidd 2827 . 2 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → (Scalar‘𝑋) = (Scalar‘𝑋))
14 eqidd 2827 . 2 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → (Base‘(Scalar‘𝑋)) = (Base‘(Scalar‘𝑋)))
15 eqidd 2827 . 2 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → (+g‘(Scalar‘𝑋)) = (+g‘(Scalar‘𝑋)))
16 eqidd 2827 . 2 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → (.r‘(Scalar‘𝑋)) = (.r‘(Scalar‘𝑋)))
17 eqidd 2827 . 2 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → (*𝑟‘(Scalar‘𝑋)) = (*𝑟‘(Scalar‘𝑋)))
18 eqidd 2827 . 2 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → (0g‘(Scalar‘𝑋)) = (0g‘(Scalar‘𝑋)))
19 phllvec 20689 . . 3 (𝑊 ∈ PreHil → 𝑊 ∈ LVec)
206, 9lsslvec 19799 . . 3 ((𝑊 ∈ LVec ∧ 𝑈𝑆) → 𝑋 ∈ LVec)
2119, 20sylan 580 . 2 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → 𝑋 ∈ LVec)
22 eqid 2826 . . . . . 6 (Scalar‘𝑊) = (Scalar‘𝑊)
236, 22resssca 16640 . . . . 5 (𝑈𝑆 → (Scalar‘𝑊) = (Scalar‘𝑋))
2423eqcomd 2832 . . . 4 (𝑈𝑆 → (Scalar‘𝑋) = (Scalar‘𝑊))
2524adantl 482 . . 3 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → (Scalar‘𝑋) = (Scalar‘𝑊))
2622phlsrng 20691 . . . 4 (𝑊 ∈ PreHil → (Scalar‘𝑊) ∈ *-Ring)
2726adantr 481 . . 3 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → (Scalar‘𝑊) ∈ *-Ring)
2825, 27eqeltrd 2918 . 2 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → (Scalar‘𝑋) ∈ *-Ring)
29 simpl 483 . . . . 5 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → 𝑊 ∈ PreHil)
30 eqid 2826 . . . . . . 7 (Base‘𝑊) = (Base‘𝑊)
316, 30ressbasss 16546 . . . . . 6 (Base‘𝑋) ⊆ (Base‘𝑊)
3231sseli 3967 . . . . 5 (𝑥 ∈ (Base‘𝑋) → 𝑥 ∈ (Base‘𝑊))
3331sseli 3967 . . . . 5 (𝑦 ∈ (Base‘𝑋) → 𝑦 ∈ (Base‘𝑊))
34 eqid 2826 . . . . . 6 (·𝑖𝑊) = (·𝑖𝑊)
35 eqid 2826 . . . . . 6 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
3622, 34, 30, 35ipcl 20693 . . . . 5 ((𝑊 ∈ PreHil ∧ 𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊)) → (𝑥(·𝑖𝑊)𝑦) ∈ (Base‘(Scalar‘𝑊)))
3729, 32, 33, 36syl3an 1154 . . . 4 (((𝑊 ∈ PreHil ∧ 𝑈𝑆) ∧ 𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋)) → (𝑥(·𝑖𝑊)𝑦) ∈ (Base‘(Scalar‘𝑊)))
3824fveq2d 6671 . . . . . . 7 (𝑈𝑆 → (Base‘(Scalar‘𝑋)) = (Base‘(Scalar‘𝑊)))
3938eleq2d 2903 . . . . . 6 (𝑈𝑆 → ((𝑥(·𝑖𝑊)𝑦) ∈ (Base‘(Scalar‘𝑋)) ↔ (𝑥(·𝑖𝑊)𝑦) ∈ (Base‘(Scalar‘𝑊))))
4039adantl 482 . . . . 5 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → ((𝑥(·𝑖𝑊)𝑦) ∈ (Base‘(Scalar‘𝑋)) ↔ (𝑥(·𝑖𝑊)𝑦) ∈ (Base‘(Scalar‘𝑊))))
41403ad2ant1 1127 . . . 4 (((𝑊 ∈ PreHil ∧ 𝑈𝑆) ∧ 𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋)) → ((𝑥(·𝑖𝑊)𝑦) ∈ (Base‘(Scalar‘𝑋)) ↔ (𝑥(·𝑖𝑊)𝑦) ∈ (Base‘(Scalar‘𝑊))))
4237, 41mpbird 258 . . 3 (((𝑊 ∈ PreHil ∧ 𝑈𝑆) ∧ 𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋)) → (𝑥(·𝑖𝑊)𝑦) ∈ (Base‘(Scalar‘𝑋)))
43 eqid 2826 . . . . . . . 8 (·𝑖𝑋) = (·𝑖𝑋)
446, 34, 43ssipeq 20716 . . . . . . 7 (𝑈𝑆 → (·𝑖𝑋) = (·𝑖𝑊))
4544oveqd 7165 . . . . . 6 (𝑈𝑆 → (𝑥(·𝑖𝑋)𝑦) = (𝑥(·𝑖𝑊)𝑦))
4645eleq1d 2902 . . . . 5 (𝑈𝑆 → ((𝑥(·𝑖𝑋)𝑦) ∈ (Base‘(Scalar‘𝑋)) ↔ (𝑥(·𝑖𝑊)𝑦) ∈ (Base‘(Scalar‘𝑋))))
4746adantl 482 . . . 4 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → ((𝑥(·𝑖𝑋)𝑦) ∈ (Base‘(Scalar‘𝑋)) ↔ (𝑥(·𝑖𝑊)𝑦) ∈ (Base‘(Scalar‘𝑋))))
48473ad2ant1 1127 . . 3 (((𝑊 ∈ PreHil ∧ 𝑈𝑆) ∧ 𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋)) → ((𝑥(·𝑖𝑋)𝑦) ∈ (Base‘(Scalar‘𝑋)) ↔ (𝑥(·𝑖𝑊)𝑦) ∈ (Base‘(Scalar‘𝑋))))
4942, 48mpbird 258 . 2 (((𝑊 ∈ PreHil ∧ 𝑈𝑆) ∧ 𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋)) → (𝑥(·𝑖𝑋)𝑦) ∈ (Base‘(Scalar‘𝑋)))
50293ad2ant1 1127 . . . . 5 (((𝑊 ∈ PreHil ∧ 𝑈𝑆) ∧ 𝑞 ∈ (Base‘(Scalar‘𝑋)) ∧ (𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋) ∧ 𝑧 ∈ (Base‘𝑋))) → 𝑊 ∈ PreHil)
515adantr 481 . . . . . . 7 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → 𝑊 ∈ LMod)
52513ad2ant1 1127 . . . . . 6 (((𝑊 ∈ PreHil ∧ 𝑈𝑆) ∧ 𝑞 ∈ (Base‘(Scalar‘𝑋)) ∧ (𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋) ∧ 𝑧 ∈ (Base‘𝑋))) → 𝑊 ∈ LMod)
5325fveq2d 6671 . . . . . . . . 9 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → (Base‘(Scalar‘𝑋)) = (Base‘(Scalar‘𝑊)))
5453eleq2d 2903 . . . . . . . 8 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → (𝑞 ∈ (Base‘(Scalar‘𝑋)) ↔ 𝑞 ∈ (Base‘(Scalar‘𝑊))))
5554biimpa 477 . . . . . . 7 (((𝑊 ∈ PreHil ∧ 𝑈𝑆) ∧ 𝑞 ∈ (Base‘(Scalar‘𝑋))) → 𝑞 ∈ (Base‘(Scalar‘𝑊)))
56553adant3 1126 . . . . . 6 (((𝑊 ∈ PreHil ∧ 𝑈𝑆) ∧ 𝑞 ∈ (Base‘(Scalar‘𝑋)) ∧ (𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋) ∧ 𝑧 ∈ (Base‘𝑋))) → 𝑞 ∈ (Base‘(Scalar‘𝑊)))
57323ad2ant1 1127 . . . . . . 7 ((𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋) ∧ 𝑧 ∈ (Base‘𝑋)) → 𝑥 ∈ (Base‘𝑊))
58573ad2ant3 1129 . . . . . 6 (((𝑊 ∈ PreHil ∧ 𝑈𝑆) ∧ 𝑞 ∈ (Base‘(Scalar‘𝑋)) ∧ (𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋) ∧ 𝑧 ∈ (Base‘𝑋))) → 𝑥 ∈ (Base‘𝑊))
59 eqid 2826 . . . . . . 7 ( ·𝑠𝑊) = ( ·𝑠𝑊)
6030, 22, 59, 35lmodvscl 19571 . . . . . 6 ((𝑊 ∈ LMod ∧ 𝑞 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑥 ∈ (Base‘𝑊)) → (𝑞( ·𝑠𝑊)𝑥) ∈ (Base‘𝑊))
6152, 56, 58, 60syl3anc 1365 . . . . 5 (((𝑊 ∈ PreHil ∧ 𝑈𝑆) ∧ 𝑞 ∈ (Base‘(Scalar‘𝑋)) ∧ (𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋) ∧ 𝑧 ∈ (Base‘𝑋))) → (𝑞( ·𝑠𝑊)𝑥) ∈ (Base‘𝑊))
62333ad2ant2 1128 . . . . . 6 ((𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋) ∧ 𝑧 ∈ (Base‘𝑋)) → 𝑦 ∈ (Base‘𝑊))
63623ad2ant3 1129 . . . . 5 (((𝑊 ∈ PreHil ∧ 𝑈𝑆) ∧ 𝑞 ∈ (Base‘(Scalar‘𝑋)) ∧ (𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋) ∧ 𝑧 ∈ (Base‘𝑋))) → 𝑦 ∈ (Base‘𝑊))
6431sseli 3967 . . . . . . 7 (𝑧 ∈ (Base‘𝑋) → 𝑧 ∈ (Base‘𝑊))
65643ad2ant3 1129 . . . . . 6 ((𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋) ∧ 𝑧 ∈ (Base‘𝑋)) → 𝑧 ∈ (Base‘𝑊))
66653ad2ant3 1129 . . . . 5 (((𝑊 ∈ PreHil ∧ 𝑈𝑆) ∧ 𝑞 ∈ (Base‘(Scalar‘𝑋)) ∧ (𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋) ∧ 𝑧 ∈ (Base‘𝑋))) → 𝑧 ∈ (Base‘𝑊))
67 eqid 2826 . . . . . 6 (+g𝑊) = (+g𝑊)
68 eqid 2826 . . . . . 6 (+g‘(Scalar‘𝑊)) = (+g‘(Scalar‘𝑊))
6922, 34, 30, 67, 68ipdir 20699 . . . . 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 1366 . . . 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 2826 . . . . . . 7 (.r‘(Scalar‘𝑊)) = (.r‘(Scalar‘𝑊))
7222, 34, 30, 35, 59, 71ipass 20705 . . . . . 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 1366 . . . . 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 7163 . . . 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 2861 . . 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 16602 . . . . . . . . 9 (𝑈𝑆 → (+g𝑊) = (+g𝑋))
7776eqcomd 2832 . . . . . . . 8 (𝑈𝑆 → (+g𝑋) = (+g𝑊))
786, 59ressvsca 16641 . . . . . . . . . 10 (𝑈𝑆 → ( ·𝑠𝑊) = ( ·𝑠𝑋))
7978eqcomd 2832 . . . . . . . . 9 (𝑈𝑆 → ( ·𝑠𝑋) = ( ·𝑠𝑊))
8079oveqd 7165 . . . . . . . 8 (𝑈𝑆 → (𝑞( ·𝑠𝑋)𝑥) = (𝑞( ·𝑠𝑊)𝑥))
81 eqidd 2827 . . . . . . . 8 (𝑈𝑆𝑦 = 𝑦)
8277, 80, 81oveq123d 7169 . . . . . . 7 (𝑈𝑆 → ((𝑞( ·𝑠𝑋)𝑥)(+g𝑋)𝑦) = ((𝑞( ·𝑠𝑊)𝑥)(+g𝑊)𝑦))
83 eqidd 2827 . . . . . . 7 (𝑈𝑆𝑧 = 𝑧)
8444, 82, 83oveq123d 7169 . . . . . 6 (𝑈𝑆 → (((𝑞( ·𝑠𝑋)𝑥)(+g𝑋)𝑦)(·𝑖𝑋)𝑧) = (((𝑞( ·𝑠𝑊)𝑥)(+g𝑊)𝑦)(·𝑖𝑊)𝑧))
8524fveq2d 6671 . . . . . . 7 (𝑈𝑆 → (+g‘(Scalar‘𝑋)) = (+g‘(Scalar‘𝑊)))
8624fveq2d 6671 . . . . . . . 8 (𝑈𝑆 → (.r‘(Scalar‘𝑋)) = (.r‘(Scalar‘𝑊)))
87 eqidd 2827 . . . . . . . 8 (𝑈𝑆𝑞 = 𝑞)
8844oveqd 7165 . . . . . . . 8 (𝑈𝑆 → (𝑥(·𝑖𝑋)𝑧) = (𝑥(·𝑖𝑊)𝑧))
8986, 87, 88oveq123d 7169 . . . . . . 7 (𝑈𝑆 → (𝑞(.r‘(Scalar‘𝑋))(𝑥(·𝑖𝑋)𝑧)) = (𝑞(.r‘(Scalar‘𝑊))(𝑥(·𝑖𝑊)𝑧)))
9044oveqd 7165 . . . . . . 7 (𝑈𝑆 → (𝑦(·𝑖𝑋)𝑧) = (𝑦(·𝑖𝑊)𝑧))
9185, 89, 90oveq123d 7169 . . . . . 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 2842 . . . . 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 482 . . . 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 1127 . . 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 258 . 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 482 . . . . . 6 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → (·𝑖𝑋) = (·𝑖𝑊))
9796oveqdr 7176 . . . . 5 (((𝑊 ∈ PreHil ∧ 𝑈𝑆) ∧ 𝑥 ∈ (Base‘𝑋)) → (𝑥(·𝑖𝑋)𝑥) = (𝑥(·𝑖𝑊)𝑥))
9824fveq2d 6671 . . . . . . 7 (𝑈𝑆 → (0g‘(Scalar‘𝑋)) = (0g‘(Scalar‘𝑊)))
9998adantl 482 . . . . . 6 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → (0g‘(Scalar‘𝑋)) = (0g‘(Scalar‘𝑊)))
10099adantr 481 . . . . 5 (((𝑊 ∈ PreHil ∧ 𝑈𝑆) ∧ 𝑥 ∈ (Base‘𝑋)) → (0g‘(Scalar‘𝑋)) = (0g‘(Scalar‘𝑊)))
10197, 100eqeq12d 2842 . . . 4 (((𝑊 ∈ PreHil ∧ 𝑈𝑆) ∧ 𝑥 ∈ (Base‘𝑋)) → ((𝑥(·𝑖𝑋)𝑥) = (0g‘(Scalar‘𝑋)) ↔ (𝑥(·𝑖𝑊)𝑥) = (0g‘(Scalar‘𝑊))))
102 eqid 2826 . . . . . . 7 (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
10322, 34, 30, 102, 7ipeq0 20698 . . . . . 6 ((𝑊 ∈ PreHil ∧ 𝑥 ∈ (Base‘𝑊)) → ((𝑥(·𝑖𝑊)𝑥) = (0g‘(Scalar‘𝑊)) ↔ 𝑥 = (0g𝑊)))
10429, 32, 103syl2an 595 . . . . 5 (((𝑊 ∈ PreHil ∧ 𝑈𝑆) ∧ 𝑥 ∈ (Base‘𝑋)) → ((𝑥(·𝑖𝑊)𝑥) = (0g‘(Scalar‘𝑊)) ↔ 𝑥 = (0g𝑊)))
105104biimpd 230 . . . 4 (((𝑊 ∈ PreHil ∧ 𝑈𝑆) ∧ 𝑥 ∈ (Base‘𝑋)) → ((𝑥(·𝑖𝑊)𝑥) = (0g‘(Scalar‘𝑊)) → 𝑥 = (0g𝑊)))
106101, 105sylbid 241 . . 3 (((𝑊 ∈ PreHil ∧ 𝑈𝑆) ∧ 𝑥 ∈ (Base‘𝑋)) → ((𝑥(·𝑖𝑋)𝑥) = (0g‘(Scalar‘𝑋)) → 𝑥 = (0g𝑊)))
1071063impia 1111 . 2 (((𝑊 ∈ PreHil ∧ 𝑈𝑆) ∧ 𝑥 ∈ (Base‘𝑋) ∧ (𝑥(·𝑖𝑋)𝑥) = (0g‘(Scalar‘𝑋))) → 𝑥 = (0g𝑊))
10824fveq2d 6671 . . . . . . 7 (𝑈𝑆 → (*𝑟‘(Scalar‘𝑋)) = (*𝑟‘(Scalar‘𝑊)))
109108fveq1d 6669 . . . . . 6 (𝑈𝑆 → ((*𝑟‘(Scalar‘𝑋))‘(𝑥(·𝑖𝑊)𝑦)) = ((*𝑟‘(Scalar‘𝑊))‘(𝑥(·𝑖𝑊)𝑦)))
110109adantl 482 . . . . 5 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → ((*𝑟‘(Scalar‘𝑋))‘(𝑥(·𝑖𝑊)𝑦)) = ((*𝑟‘(Scalar‘𝑊))‘(𝑥(·𝑖𝑊)𝑦)))
1111103ad2ant1 1127 . . . 4 (((𝑊 ∈ PreHil ∧ 𝑈𝑆) ∧ 𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋)) → ((*𝑟‘(Scalar‘𝑋))‘(𝑥(·𝑖𝑊)𝑦)) = ((*𝑟‘(Scalar‘𝑊))‘(𝑥(·𝑖𝑊)𝑦)))
112 eqid 2826 . . . . . 6 (*𝑟‘(Scalar‘𝑊)) = (*𝑟‘(Scalar‘𝑊))
11322, 34, 30, 112ipcj 20694 . . . . 5 ((𝑊 ∈ PreHil ∧ 𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊)) → ((*𝑟‘(Scalar‘𝑊))‘(𝑥(·𝑖𝑊)𝑦)) = (𝑦(·𝑖𝑊)𝑥))
11429, 32, 33, 113syl3an 1154 . . . 4 (((𝑊 ∈ PreHil ∧ 𝑈𝑆) ∧ 𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋)) → ((*𝑟‘(Scalar‘𝑊))‘(𝑥(·𝑖𝑊)𝑦)) = (𝑦(·𝑖𝑊)𝑥))
115111, 114eqtrd 2861 . . 3 (((𝑊 ∈ PreHil ∧ 𝑈𝑆) ∧ 𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋)) → ((*𝑟‘(Scalar‘𝑋))‘(𝑥(·𝑖𝑊)𝑦)) = (𝑦(·𝑖𝑊)𝑥))
11645fveq2d 6671 . . . . . 6 (𝑈𝑆 → ((*𝑟‘(Scalar‘𝑋))‘(𝑥(·𝑖𝑋)𝑦)) = ((*𝑟‘(Scalar‘𝑋))‘(𝑥(·𝑖𝑊)𝑦)))
11744oveqd 7165 . . . . . 6 (𝑈𝑆 → (𝑦(·𝑖𝑋)𝑥) = (𝑦(·𝑖𝑊)𝑥))
118116, 117eqeq12d 2842 . . . . 5 (𝑈𝑆 → (((*𝑟‘(Scalar‘𝑋))‘(𝑥(·𝑖𝑋)𝑦)) = (𝑦(·𝑖𝑋)𝑥) ↔ ((*𝑟‘(Scalar‘𝑋))‘(𝑥(·𝑖𝑊)𝑦)) = (𝑦(·𝑖𝑊)𝑥)))
119118adantl 482 . . . 4 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → (((*𝑟‘(Scalar‘𝑋))‘(𝑥(·𝑖𝑋)𝑦)) = (𝑦(·𝑖𝑋)𝑥) ↔ ((*𝑟‘(Scalar‘𝑋))‘(𝑥(·𝑖𝑊)𝑦)) = (𝑦(·𝑖𝑊)𝑥)))
1201193ad2ant1 1127 . . 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 258 . 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 20714 1 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → 𝑋 ∈ PreHil)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1081   = wceq 1530  wcel 2107  cfv 6352  (class class class)co 7148  Basecbs 16473  s cress 16474  +gcplusg 16555  .rcmulr 16556  *𝑟cstv 16557  Scalarcsca 16558   ·𝑠 cvsca 16559  ·𝑖cip 16560  0gc0g 16703  *-Ringcsr 19535  LModclmod 19554  LSubSpclss 19623  LVecclvec 19794  PreHilcphl 20684
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-nel 3129  df-ral 3148  df-rex 3149  df-reu 3150  df-rmo 3151  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6146  df-ord 6192  df-on 6193  df-lim 6194  df-suc 6195  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7569  df-1st 7680  df-2nd 7681  df-wrecs 7938  df-recs 7999  df-rdg 8037  df-er 8279  df-en 8499  df-dom 8500  df-sdom 8501  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11628  df-2 11689  df-3 11690  df-4 11691  df-5 11692  df-6 11693  df-7 11694  df-8 11695  df-ndx 16476  df-slot 16477  df-base 16479  df-sets 16480  df-ress 16481  df-plusg 16568  df-mulr 16569  df-sca 16571  df-vsca 16572  df-ip 16573  df-0g 16705  df-mgm 17842  df-sgrp 17890  df-mnd 17901  df-grp 18036  df-minusg 18037  df-sbg 18038  df-subg 18206  df-ghm 18286  df-mgp 19160  df-ur 19172  df-ring 19219  df-subrg 19453  df-lmod 19556  df-lss 19624  df-lsp 19664  df-lmhm 19714  df-lvec 19795  df-sra 19864  df-rgmod 19865  df-phl 20686
This theorem is referenced by:  cphsscph  23769
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