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Theorem phlssphl 21677
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 2738 . 2 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → (Base‘𝑋) = (Base‘𝑋))
2 eqidd 2738 . 2 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → (+g𝑋) = (+g𝑋))
3 eqidd 2738 . 2 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → ( ·𝑠𝑋) = ( ·𝑠𝑋))
4 eqidd 2738 . 2 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → (·𝑖𝑋) = (·𝑖𝑋))
5 phllmod 21648 . . . 4 (𝑊 ∈ PreHil → 𝑊 ∈ LMod)
6 phlssphl.x . . . . 5 𝑋 = (𝑊s 𝑈)
7 eqid 2737 . . . . 5 (0g𝑊) = (0g𝑊)
8 eqid 2737 . . . . 5 (0g𝑋) = (0g𝑋)
9 phlssphl.s . . . . 5 𝑆 = (LSubSp‘𝑊)
106, 7, 8, 9lss0v 21015 . . . 4 ((𝑊 ∈ LMod ∧ 𝑈𝑆) → (0g𝑋) = (0g𝑊))
115, 10sylan 580 . . 3 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → (0g𝑋) = (0g𝑊))
1211eqcomd 2743 . 2 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → (0g𝑊) = (0g𝑋))
13 eqidd 2738 . 2 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → (Scalar‘𝑋) = (Scalar‘𝑋))
14 eqidd 2738 . 2 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → (Base‘(Scalar‘𝑋)) = (Base‘(Scalar‘𝑋)))
15 eqidd 2738 . 2 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → (+g‘(Scalar‘𝑋)) = (+g‘(Scalar‘𝑋)))
16 eqidd 2738 . 2 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → (.r‘(Scalar‘𝑋)) = (.r‘(Scalar‘𝑋)))
17 eqidd 2738 . 2 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → (*𝑟‘(Scalar‘𝑋)) = (*𝑟‘(Scalar‘𝑋)))
18 eqidd 2738 . 2 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → (0g‘(Scalar‘𝑋)) = (0g‘(Scalar‘𝑋)))
19 phllvec 21647 . . 3 (𝑊 ∈ PreHil → 𝑊 ∈ LVec)
206, 9lsslvec 21108 . . 3 ((𝑊 ∈ LVec ∧ 𝑈𝑆) → 𝑋 ∈ LVec)
2119, 20sylan 580 . 2 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → 𝑋 ∈ LVec)
22 eqid 2737 . . . . . 6 (Scalar‘𝑊) = (Scalar‘𝑊)
236, 22resssca 17387 . . . . 5 (𝑈𝑆 → (Scalar‘𝑊) = (Scalar‘𝑋))
2423eqcomd 2743 . . . 4 (𝑈𝑆 → (Scalar‘𝑋) = (Scalar‘𝑊))
2524adantl 481 . . 3 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → (Scalar‘𝑋) = (Scalar‘𝑊))
2622phlsrng 21649 . . . 4 (𝑊 ∈ PreHil → (Scalar‘𝑊) ∈ *-Ring)
2726adantr 480 . . 3 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → (Scalar‘𝑊) ∈ *-Ring)
2825, 27eqeltrd 2841 . 2 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → (Scalar‘𝑋) ∈ *-Ring)
29 simpl 482 . . . . 5 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → 𝑊 ∈ PreHil)
30 eqid 2737 . . . . . . 7 (Base‘𝑊) = (Base‘𝑊)
316, 30ressbasss 17284 . . . . . 6 (Base‘𝑋) ⊆ (Base‘𝑊)
3231sseli 3979 . . . . 5 (𝑥 ∈ (Base‘𝑋) → 𝑥 ∈ (Base‘𝑊))
3331sseli 3979 . . . . 5 (𝑦 ∈ (Base‘𝑋) → 𝑦 ∈ (Base‘𝑊))
34 eqid 2737 . . . . . 6 (·𝑖𝑊) = (·𝑖𝑊)
35 eqid 2737 . . . . . 6 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
3622, 34, 30, 35ipcl 21651 . . . . 5 ((𝑊 ∈ PreHil ∧ 𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊)) → (𝑥(·𝑖𝑊)𝑦) ∈ (Base‘(Scalar‘𝑊)))
3729, 32, 33, 36syl3an 1161 . . . 4 (((𝑊 ∈ PreHil ∧ 𝑈𝑆) ∧ 𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋)) → (𝑥(·𝑖𝑊)𝑦) ∈ (Base‘(Scalar‘𝑊)))
3824fveq2d 6910 . . . . . . 7 (𝑈𝑆 → (Base‘(Scalar‘𝑋)) = (Base‘(Scalar‘𝑊)))
3938eleq2d 2827 . . . . . 6 (𝑈𝑆 → ((𝑥(·𝑖𝑊)𝑦) ∈ (Base‘(Scalar‘𝑋)) ↔ (𝑥(·𝑖𝑊)𝑦) ∈ (Base‘(Scalar‘𝑊))))
4039adantl 481 . . . . 5 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → ((𝑥(·𝑖𝑊)𝑦) ∈ (Base‘(Scalar‘𝑋)) ↔ (𝑥(·𝑖𝑊)𝑦) ∈ (Base‘(Scalar‘𝑊))))
41403ad2ant1 1134 . . . 4 (((𝑊 ∈ PreHil ∧ 𝑈𝑆) ∧ 𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋)) → ((𝑥(·𝑖𝑊)𝑦) ∈ (Base‘(Scalar‘𝑋)) ↔ (𝑥(·𝑖𝑊)𝑦) ∈ (Base‘(Scalar‘𝑊))))
4237, 41mpbird 257 . . 3 (((𝑊 ∈ PreHil ∧ 𝑈𝑆) ∧ 𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋)) → (𝑥(·𝑖𝑊)𝑦) ∈ (Base‘(Scalar‘𝑋)))
43 eqid 2737 . . . . . . . 8 (·𝑖𝑋) = (·𝑖𝑋)
446, 34, 43ssipeq 21674 . . . . . . 7 (𝑈𝑆 → (·𝑖𝑋) = (·𝑖𝑊))
4544oveqd 7448 . . . . . 6 (𝑈𝑆 → (𝑥(·𝑖𝑋)𝑦) = (𝑥(·𝑖𝑊)𝑦))
4645eleq1d 2826 . . . . 5 (𝑈𝑆 → ((𝑥(·𝑖𝑋)𝑦) ∈ (Base‘(Scalar‘𝑋)) ↔ (𝑥(·𝑖𝑊)𝑦) ∈ (Base‘(Scalar‘𝑋))))
4746adantl 481 . . . 4 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → ((𝑥(·𝑖𝑋)𝑦) ∈ (Base‘(Scalar‘𝑋)) ↔ (𝑥(·𝑖𝑊)𝑦) ∈ (Base‘(Scalar‘𝑋))))
48473ad2ant1 1134 . . 3 (((𝑊 ∈ PreHil ∧ 𝑈𝑆) ∧ 𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋)) → ((𝑥(·𝑖𝑋)𝑦) ∈ (Base‘(Scalar‘𝑋)) ↔ (𝑥(·𝑖𝑊)𝑦) ∈ (Base‘(Scalar‘𝑋))))
4942, 48mpbird 257 . 2 (((𝑊 ∈ PreHil ∧ 𝑈𝑆) ∧ 𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋)) → (𝑥(·𝑖𝑋)𝑦) ∈ (Base‘(Scalar‘𝑋)))
50293ad2ant1 1134 . . . . 5 (((𝑊 ∈ PreHil ∧ 𝑈𝑆) ∧ 𝑞 ∈ (Base‘(Scalar‘𝑋)) ∧ (𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋) ∧ 𝑧 ∈ (Base‘𝑋))) → 𝑊 ∈ PreHil)
515adantr 480 . . . . . . 7 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → 𝑊 ∈ LMod)
52513ad2ant1 1134 . . . . . 6 (((𝑊 ∈ PreHil ∧ 𝑈𝑆) ∧ 𝑞 ∈ (Base‘(Scalar‘𝑋)) ∧ (𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋) ∧ 𝑧 ∈ (Base‘𝑋))) → 𝑊 ∈ LMod)
5325fveq2d 6910 . . . . . . . . 9 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → (Base‘(Scalar‘𝑋)) = (Base‘(Scalar‘𝑊)))
5453eleq2d 2827 . . . . . . . 8 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → (𝑞 ∈ (Base‘(Scalar‘𝑋)) ↔ 𝑞 ∈ (Base‘(Scalar‘𝑊))))
5554biimpa 476 . . . . . . 7 (((𝑊 ∈ PreHil ∧ 𝑈𝑆) ∧ 𝑞 ∈ (Base‘(Scalar‘𝑋))) → 𝑞 ∈ (Base‘(Scalar‘𝑊)))
56553adant3 1133 . . . . . 6 (((𝑊 ∈ PreHil ∧ 𝑈𝑆) ∧ 𝑞 ∈ (Base‘(Scalar‘𝑋)) ∧ (𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋) ∧ 𝑧 ∈ (Base‘𝑋))) → 𝑞 ∈ (Base‘(Scalar‘𝑊)))
57323ad2ant1 1134 . . . . . . 7 ((𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋) ∧ 𝑧 ∈ (Base‘𝑋)) → 𝑥 ∈ (Base‘𝑊))
58573ad2ant3 1136 . . . . . 6 (((𝑊 ∈ PreHil ∧ 𝑈𝑆) ∧ 𝑞 ∈ (Base‘(Scalar‘𝑋)) ∧ (𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋) ∧ 𝑧 ∈ (Base‘𝑋))) → 𝑥 ∈ (Base‘𝑊))
59 eqid 2737 . . . . . . 7 ( ·𝑠𝑊) = ( ·𝑠𝑊)
6030, 22, 59, 35lmodvscl 20876 . . . . . 6 ((𝑊 ∈ LMod ∧ 𝑞 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑥 ∈ (Base‘𝑊)) → (𝑞( ·𝑠𝑊)𝑥) ∈ (Base‘𝑊))
6152, 56, 58, 60syl3anc 1373 . . . . 5 (((𝑊 ∈ PreHil ∧ 𝑈𝑆) ∧ 𝑞 ∈ (Base‘(Scalar‘𝑋)) ∧ (𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋) ∧ 𝑧 ∈ (Base‘𝑋))) → (𝑞( ·𝑠𝑊)𝑥) ∈ (Base‘𝑊))
62333ad2ant2 1135 . . . . . 6 ((𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋) ∧ 𝑧 ∈ (Base‘𝑋)) → 𝑦 ∈ (Base‘𝑊))
63623ad2ant3 1136 . . . . 5 (((𝑊 ∈ PreHil ∧ 𝑈𝑆) ∧ 𝑞 ∈ (Base‘(Scalar‘𝑋)) ∧ (𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋) ∧ 𝑧 ∈ (Base‘𝑋))) → 𝑦 ∈ (Base‘𝑊))
6431sseli 3979 . . . . . . 7 (𝑧 ∈ (Base‘𝑋) → 𝑧 ∈ (Base‘𝑊))
65643ad2ant3 1136 . . . . . 6 ((𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋) ∧ 𝑧 ∈ (Base‘𝑋)) → 𝑧 ∈ (Base‘𝑊))
66653ad2ant3 1136 . . . . 5 (((𝑊 ∈ PreHil ∧ 𝑈𝑆) ∧ 𝑞 ∈ (Base‘(Scalar‘𝑋)) ∧ (𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋) ∧ 𝑧 ∈ (Base‘𝑋))) → 𝑧 ∈ (Base‘𝑊))
67 eqid 2737 . . . . . 6 (+g𝑊) = (+g𝑊)
68 eqid 2737 . . . . . 6 (+g‘(Scalar‘𝑊)) = (+g‘(Scalar‘𝑊))
6922, 34, 30, 67, 68ipdir 21657 . . . . 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 1374 . . . 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 2737 . . . . . . 7 (.r‘(Scalar‘𝑊)) = (.r‘(Scalar‘𝑊))
7222, 34, 30, 35, 59, 71ipass 21663 . . . . . 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 1374 . . . . 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 7446 . . . 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 2777 . . 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 17334 . . . . . . . . 9 (𝑈𝑆 → (+g𝑊) = (+g𝑋))
7776eqcomd 2743 . . . . . . . 8 (𝑈𝑆 → (+g𝑋) = (+g𝑊))
786, 59ressvsca 17388 . . . . . . . . . 10 (𝑈𝑆 → ( ·𝑠𝑊) = ( ·𝑠𝑋))
7978eqcomd 2743 . . . . . . . . 9 (𝑈𝑆 → ( ·𝑠𝑋) = ( ·𝑠𝑊))
8079oveqd 7448 . . . . . . . 8 (𝑈𝑆 → (𝑞( ·𝑠𝑋)𝑥) = (𝑞( ·𝑠𝑊)𝑥))
81 eqidd 2738 . . . . . . . 8 (𝑈𝑆𝑦 = 𝑦)
8277, 80, 81oveq123d 7452 . . . . . . 7 (𝑈𝑆 → ((𝑞( ·𝑠𝑋)𝑥)(+g𝑋)𝑦) = ((𝑞( ·𝑠𝑊)𝑥)(+g𝑊)𝑦))
83 eqidd 2738 . . . . . . 7 (𝑈𝑆𝑧 = 𝑧)
8444, 82, 83oveq123d 7452 . . . . . 6 (𝑈𝑆 → (((𝑞( ·𝑠𝑋)𝑥)(+g𝑋)𝑦)(·𝑖𝑋)𝑧) = (((𝑞( ·𝑠𝑊)𝑥)(+g𝑊)𝑦)(·𝑖𝑊)𝑧))
8524fveq2d 6910 . . . . . . 7 (𝑈𝑆 → (+g‘(Scalar‘𝑋)) = (+g‘(Scalar‘𝑊)))
8624fveq2d 6910 . . . . . . . 8 (𝑈𝑆 → (.r‘(Scalar‘𝑋)) = (.r‘(Scalar‘𝑊)))
87 eqidd 2738 . . . . . . . 8 (𝑈𝑆𝑞 = 𝑞)
8844oveqd 7448 . . . . . . . 8 (𝑈𝑆 → (𝑥(·𝑖𝑋)𝑧) = (𝑥(·𝑖𝑊)𝑧))
8986, 87, 88oveq123d 7452 . . . . . . 7 (𝑈𝑆 → (𝑞(.r‘(Scalar‘𝑋))(𝑥(·𝑖𝑋)𝑧)) = (𝑞(.r‘(Scalar‘𝑊))(𝑥(·𝑖𝑊)𝑧)))
9044oveqd 7448 . . . . . . 7 (𝑈𝑆 → (𝑦(·𝑖𝑋)𝑧) = (𝑦(·𝑖𝑊)𝑧))
9185, 89, 90oveq123d 7452 . . . . . 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 2753 . . . . 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 481 . . . 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 1134 . . 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 257 . 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 481 . . . . . 6 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → (·𝑖𝑋) = (·𝑖𝑊))
9796oveqdr 7459 . . . . 5 (((𝑊 ∈ PreHil ∧ 𝑈𝑆) ∧ 𝑥 ∈ (Base‘𝑋)) → (𝑥(·𝑖𝑋)𝑥) = (𝑥(·𝑖𝑊)𝑥))
9824fveq2d 6910 . . . . . . 7 (𝑈𝑆 → (0g‘(Scalar‘𝑋)) = (0g‘(Scalar‘𝑊)))
9998adantl 481 . . . . . 6 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → (0g‘(Scalar‘𝑋)) = (0g‘(Scalar‘𝑊)))
10099adantr 480 . . . . 5 (((𝑊 ∈ PreHil ∧ 𝑈𝑆) ∧ 𝑥 ∈ (Base‘𝑋)) → (0g‘(Scalar‘𝑋)) = (0g‘(Scalar‘𝑊)))
10197, 100eqeq12d 2753 . . . 4 (((𝑊 ∈ PreHil ∧ 𝑈𝑆) ∧ 𝑥 ∈ (Base‘𝑋)) → ((𝑥(·𝑖𝑋)𝑥) = (0g‘(Scalar‘𝑋)) ↔ (𝑥(·𝑖𝑊)𝑥) = (0g‘(Scalar‘𝑊))))
102 eqid 2737 . . . . . . 7 (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
10322, 34, 30, 102, 7ipeq0 21656 . . . . . 6 ((𝑊 ∈ PreHil ∧ 𝑥 ∈ (Base‘𝑊)) → ((𝑥(·𝑖𝑊)𝑥) = (0g‘(Scalar‘𝑊)) ↔ 𝑥 = (0g𝑊)))
10429, 32, 103syl2an 596 . . . . 5 (((𝑊 ∈ PreHil ∧ 𝑈𝑆) ∧ 𝑥 ∈ (Base‘𝑋)) → ((𝑥(·𝑖𝑊)𝑥) = (0g‘(Scalar‘𝑊)) ↔ 𝑥 = (0g𝑊)))
105104biimpd 229 . . . 4 (((𝑊 ∈ PreHil ∧ 𝑈𝑆) ∧ 𝑥 ∈ (Base‘𝑋)) → ((𝑥(·𝑖𝑊)𝑥) = (0g‘(Scalar‘𝑊)) → 𝑥 = (0g𝑊)))
106101, 105sylbid 240 . . 3 (((𝑊 ∈ PreHil ∧ 𝑈𝑆) ∧ 𝑥 ∈ (Base‘𝑋)) → ((𝑥(·𝑖𝑋)𝑥) = (0g‘(Scalar‘𝑋)) → 𝑥 = (0g𝑊)))
1071063impia 1118 . 2 (((𝑊 ∈ PreHil ∧ 𝑈𝑆) ∧ 𝑥 ∈ (Base‘𝑋) ∧ (𝑥(·𝑖𝑋)𝑥) = (0g‘(Scalar‘𝑋))) → 𝑥 = (0g𝑊))
10824fveq2d 6910 . . . . . . 7 (𝑈𝑆 → (*𝑟‘(Scalar‘𝑋)) = (*𝑟‘(Scalar‘𝑊)))
109108fveq1d 6908 . . . . . 6 (𝑈𝑆 → ((*𝑟‘(Scalar‘𝑋))‘(𝑥(·𝑖𝑊)𝑦)) = ((*𝑟‘(Scalar‘𝑊))‘(𝑥(·𝑖𝑊)𝑦)))
110109adantl 481 . . . . 5 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → ((*𝑟‘(Scalar‘𝑋))‘(𝑥(·𝑖𝑊)𝑦)) = ((*𝑟‘(Scalar‘𝑊))‘(𝑥(·𝑖𝑊)𝑦)))
1111103ad2ant1 1134 . . . 4 (((𝑊 ∈ PreHil ∧ 𝑈𝑆) ∧ 𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋)) → ((*𝑟‘(Scalar‘𝑋))‘(𝑥(·𝑖𝑊)𝑦)) = ((*𝑟‘(Scalar‘𝑊))‘(𝑥(·𝑖𝑊)𝑦)))
112 eqid 2737 . . . . . 6 (*𝑟‘(Scalar‘𝑊)) = (*𝑟‘(Scalar‘𝑊))
11322, 34, 30, 112ipcj 21652 . . . . 5 ((𝑊 ∈ PreHil ∧ 𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊)) → ((*𝑟‘(Scalar‘𝑊))‘(𝑥(·𝑖𝑊)𝑦)) = (𝑦(·𝑖𝑊)𝑥))
11429, 32, 33, 113syl3an 1161 . . . 4 (((𝑊 ∈ PreHil ∧ 𝑈𝑆) ∧ 𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋)) → ((*𝑟‘(Scalar‘𝑊))‘(𝑥(·𝑖𝑊)𝑦)) = (𝑦(·𝑖𝑊)𝑥))
115111, 114eqtrd 2777 . . 3 (((𝑊 ∈ PreHil ∧ 𝑈𝑆) ∧ 𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋)) → ((*𝑟‘(Scalar‘𝑋))‘(𝑥(·𝑖𝑊)𝑦)) = (𝑦(·𝑖𝑊)𝑥))
11645fveq2d 6910 . . . . . 6 (𝑈𝑆 → ((*𝑟‘(Scalar‘𝑋))‘(𝑥(·𝑖𝑋)𝑦)) = ((*𝑟‘(Scalar‘𝑋))‘(𝑥(·𝑖𝑊)𝑦)))
11744oveqd 7448 . . . . . 6 (𝑈𝑆 → (𝑦(·𝑖𝑋)𝑥) = (𝑦(·𝑖𝑊)𝑥))
118116, 117eqeq12d 2753 . . . . 5 (𝑈𝑆 → (((*𝑟‘(Scalar‘𝑋))‘(𝑥(·𝑖𝑋)𝑦)) = (𝑦(·𝑖𝑋)𝑥) ↔ ((*𝑟‘(Scalar‘𝑋))‘(𝑥(·𝑖𝑊)𝑦)) = (𝑦(·𝑖𝑊)𝑥)))
119118adantl 481 . . . 4 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → (((*𝑟‘(Scalar‘𝑋))‘(𝑥(·𝑖𝑋)𝑦)) = (𝑦(·𝑖𝑋)𝑥) ↔ ((*𝑟‘(Scalar‘𝑋))‘(𝑥(·𝑖𝑊)𝑦)) = (𝑦(·𝑖𝑊)𝑥)))
1201193ad2ant1 1134 . . 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 257 . 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 21672 1 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → 𝑋 ∈ PreHil)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  cfv 6561  (class class class)co 7431  Basecbs 17247  s cress 17274  +gcplusg 17297  .rcmulr 17298  *𝑟cstv 17299  Scalarcsca 17300   ·𝑠 cvsca 17301  ·𝑖cip 17302  0gc0g 17484  *-Ringcsr 20839  LModclmod 20858  LSubSpclss 20929  LVecclvec 21101  PreHilcphl 21642
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 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-mulr 17311  df-sca 17313  df-vsca 17314  df-ip 17315  df-0g 17486  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-grp 18954  df-minusg 18955  df-sbg 18956  df-subg 19141  df-ghm 19231  df-cmn 19800  df-abl 19801  df-mgp 20138  df-rng 20150  df-ur 20179  df-ring 20232  df-subrg 20570  df-lmod 20860  df-lss 20930  df-lsp 20970  df-lmhm 21021  df-lvec 21102  df-sra 21172  df-rgmod 21173  df-phl 21644
This theorem is referenced by:  cphsscph  25285
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