Step |
Hyp |
Ref |
Expression |
1 |
|
ocvss.v |
⊢ 𝑉 = ( Base ‘ 𝑊 ) |
2 |
|
ocvss.o |
⊢ ⊥ = ( ocv ‘ 𝑊 ) |
3 |
|
ocvlss.l |
⊢ 𝐿 = ( LSubSp ‘ 𝑊 ) |
4 |
1 2
|
ocvss |
⊢ ( ⊥ ‘ 𝑆 ) ⊆ 𝑉 |
5 |
4
|
a1i |
⊢ ( ( 𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉 ) → ( ⊥ ‘ 𝑆 ) ⊆ 𝑉 ) |
6 |
|
simpr |
⊢ ( ( 𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉 ) → 𝑆 ⊆ 𝑉 ) |
7 |
|
phllmod |
⊢ ( 𝑊 ∈ PreHil → 𝑊 ∈ LMod ) |
8 |
7
|
adantr |
⊢ ( ( 𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉 ) → 𝑊 ∈ LMod ) |
9 |
|
eqid |
⊢ ( 0g ‘ 𝑊 ) = ( 0g ‘ 𝑊 ) |
10 |
1 9
|
lmod0vcl |
⊢ ( 𝑊 ∈ LMod → ( 0g ‘ 𝑊 ) ∈ 𝑉 ) |
11 |
8 10
|
syl |
⊢ ( ( 𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉 ) → ( 0g ‘ 𝑊 ) ∈ 𝑉 ) |
12 |
|
simpll |
⊢ ( ( ( 𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉 ) ∧ 𝑥 ∈ 𝑆 ) → 𝑊 ∈ PreHil ) |
13 |
6
|
sselda |
⊢ ( ( ( 𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉 ) ∧ 𝑥 ∈ 𝑆 ) → 𝑥 ∈ 𝑉 ) |
14 |
|
eqid |
⊢ ( Scalar ‘ 𝑊 ) = ( Scalar ‘ 𝑊 ) |
15 |
|
eqid |
⊢ ( ·𝑖 ‘ 𝑊 ) = ( ·𝑖 ‘ 𝑊 ) |
16 |
|
eqid |
⊢ ( 0g ‘ ( Scalar ‘ 𝑊 ) ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) |
17 |
14 15 1 16 9
|
ip0l |
⊢ ( ( 𝑊 ∈ PreHil ∧ 𝑥 ∈ 𝑉 ) → ( ( 0g ‘ 𝑊 ) ( ·𝑖 ‘ 𝑊 ) 𝑥 ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ) |
18 |
12 13 17
|
syl2anc |
⊢ ( ( ( 𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉 ) ∧ 𝑥 ∈ 𝑆 ) → ( ( 0g ‘ 𝑊 ) ( ·𝑖 ‘ 𝑊 ) 𝑥 ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ) |
19 |
18
|
ralrimiva |
⊢ ( ( 𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉 ) → ∀ 𝑥 ∈ 𝑆 ( ( 0g ‘ 𝑊 ) ( ·𝑖 ‘ 𝑊 ) 𝑥 ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ) |
20 |
1 15 14 16 2
|
elocv |
⊢ ( ( 0g ‘ 𝑊 ) ∈ ( ⊥ ‘ 𝑆 ) ↔ ( 𝑆 ⊆ 𝑉 ∧ ( 0g ‘ 𝑊 ) ∈ 𝑉 ∧ ∀ 𝑥 ∈ 𝑆 ( ( 0g ‘ 𝑊 ) ( ·𝑖 ‘ 𝑊 ) 𝑥 ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ) ) |
21 |
6 11 19 20
|
syl3anbrc |
⊢ ( ( 𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉 ) → ( 0g ‘ 𝑊 ) ∈ ( ⊥ ‘ 𝑆 ) ) |
22 |
21
|
ne0d |
⊢ ( ( 𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉 ) → ( ⊥ ‘ 𝑆 ) ≠ ∅ ) |
23 |
6
|
adantr |
⊢ ( ( ( 𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉 ) ∧ ( 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ 𝑦 ∈ ( ⊥ ‘ 𝑆 ) ∧ 𝑧 ∈ ( ⊥ ‘ 𝑆 ) ) ) → 𝑆 ⊆ 𝑉 ) |
24 |
8
|
adantr |
⊢ ( ( ( 𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉 ) ∧ ( 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ 𝑦 ∈ ( ⊥ ‘ 𝑆 ) ∧ 𝑧 ∈ ( ⊥ ‘ 𝑆 ) ) ) → 𝑊 ∈ LMod ) |
25 |
|
simpr1 |
⊢ ( ( ( 𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉 ) ∧ ( 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ 𝑦 ∈ ( ⊥ ‘ 𝑆 ) ∧ 𝑧 ∈ ( ⊥ ‘ 𝑆 ) ) ) → 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ) |
26 |
|
simpr2 |
⊢ ( ( ( 𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉 ) ∧ ( 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ 𝑦 ∈ ( ⊥ ‘ 𝑆 ) ∧ 𝑧 ∈ ( ⊥ ‘ 𝑆 ) ) ) → 𝑦 ∈ ( ⊥ ‘ 𝑆 ) ) |
27 |
4 26
|
sselid |
⊢ ( ( ( 𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉 ) ∧ ( 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ 𝑦 ∈ ( ⊥ ‘ 𝑆 ) ∧ 𝑧 ∈ ( ⊥ ‘ 𝑆 ) ) ) → 𝑦 ∈ 𝑉 ) |
28 |
|
eqid |
⊢ ( ·𝑠 ‘ 𝑊 ) = ( ·𝑠 ‘ 𝑊 ) |
29 |
|
eqid |
⊢ ( Base ‘ ( Scalar ‘ 𝑊 ) ) = ( Base ‘ ( Scalar ‘ 𝑊 ) ) |
30 |
1 14 28 29
|
lmodvscl |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ 𝑦 ∈ 𝑉 ) → ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑦 ) ∈ 𝑉 ) |
31 |
24 25 27 30
|
syl3anc |
⊢ ( ( ( 𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉 ) ∧ ( 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ 𝑦 ∈ ( ⊥ ‘ 𝑆 ) ∧ 𝑧 ∈ ( ⊥ ‘ 𝑆 ) ) ) → ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑦 ) ∈ 𝑉 ) |
32 |
|
simpr3 |
⊢ ( ( ( 𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉 ) ∧ ( 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ 𝑦 ∈ ( ⊥ ‘ 𝑆 ) ∧ 𝑧 ∈ ( ⊥ ‘ 𝑆 ) ) ) → 𝑧 ∈ ( ⊥ ‘ 𝑆 ) ) |
33 |
4 32
|
sselid |
⊢ ( ( ( 𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉 ) ∧ ( 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ 𝑦 ∈ ( ⊥ ‘ 𝑆 ) ∧ 𝑧 ∈ ( ⊥ ‘ 𝑆 ) ) ) → 𝑧 ∈ 𝑉 ) |
34 |
|
eqid |
⊢ ( +g ‘ 𝑊 ) = ( +g ‘ 𝑊 ) |
35 |
1 34
|
lmodvacl |
⊢ ( ( 𝑊 ∈ LMod ∧ ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑦 ) ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) → ( ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑦 ) ( +g ‘ 𝑊 ) 𝑧 ) ∈ 𝑉 ) |
36 |
24 31 33 35
|
syl3anc |
⊢ ( ( ( 𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉 ) ∧ ( 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ 𝑦 ∈ ( ⊥ ‘ 𝑆 ) ∧ 𝑧 ∈ ( ⊥ ‘ 𝑆 ) ) ) → ( ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑦 ) ( +g ‘ 𝑊 ) 𝑧 ) ∈ 𝑉 ) |
37 |
12
|
adantlr |
⊢ ( ( ( ( 𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉 ) ∧ ( 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ 𝑦 ∈ ( ⊥ ‘ 𝑆 ) ∧ 𝑧 ∈ ( ⊥ ‘ 𝑆 ) ) ) ∧ 𝑥 ∈ 𝑆 ) → 𝑊 ∈ PreHil ) |
38 |
31
|
adantr |
⊢ ( ( ( ( 𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉 ) ∧ ( 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ 𝑦 ∈ ( ⊥ ‘ 𝑆 ) ∧ 𝑧 ∈ ( ⊥ ‘ 𝑆 ) ) ) ∧ 𝑥 ∈ 𝑆 ) → ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑦 ) ∈ 𝑉 ) |
39 |
33
|
adantr |
⊢ ( ( ( ( 𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉 ) ∧ ( 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ 𝑦 ∈ ( ⊥ ‘ 𝑆 ) ∧ 𝑧 ∈ ( ⊥ ‘ 𝑆 ) ) ) ∧ 𝑥 ∈ 𝑆 ) → 𝑧 ∈ 𝑉 ) |
40 |
13
|
adantlr |
⊢ ( ( ( ( 𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉 ) ∧ ( 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ 𝑦 ∈ ( ⊥ ‘ 𝑆 ) ∧ 𝑧 ∈ ( ⊥ ‘ 𝑆 ) ) ) ∧ 𝑥 ∈ 𝑆 ) → 𝑥 ∈ 𝑉 ) |
41 |
|
eqid |
⊢ ( +g ‘ ( Scalar ‘ 𝑊 ) ) = ( +g ‘ ( Scalar ‘ 𝑊 ) ) |
42 |
14 15 1 34 41
|
ipdir |
⊢ ( ( 𝑊 ∈ PreHil ∧ ( ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑦 ) ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ∧ 𝑥 ∈ 𝑉 ) ) → ( ( ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑦 ) ( +g ‘ 𝑊 ) 𝑧 ) ( ·𝑖 ‘ 𝑊 ) 𝑥 ) = ( ( ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑦 ) ( ·𝑖 ‘ 𝑊 ) 𝑥 ) ( +g ‘ ( Scalar ‘ 𝑊 ) ) ( 𝑧 ( ·𝑖 ‘ 𝑊 ) 𝑥 ) ) ) |
43 |
37 38 39 40 42
|
syl13anc |
⊢ ( ( ( ( 𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉 ) ∧ ( 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ 𝑦 ∈ ( ⊥ ‘ 𝑆 ) ∧ 𝑧 ∈ ( ⊥ ‘ 𝑆 ) ) ) ∧ 𝑥 ∈ 𝑆 ) → ( ( ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑦 ) ( +g ‘ 𝑊 ) 𝑧 ) ( ·𝑖 ‘ 𝑊 ) 𝑥 ) = ( ( ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑦 ) ( ·𝑖 ‘ 𝑊 ) 𝑥 ) ( +g ‘ ( Scalar ‘ 𝑊 ) ) ( 𝑧 ( ·𝑖 ‘ 𝑊 ) 𝑥 ) ) ) |
44 |
25
|
adantr |
⊢ ( ( ( ( 𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉 ) ∧ ( 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ 𝑦 ∈ ( ⊥ ‘ 𝑆 ) ∧ 𝑧 ∈ ( ⊥ ‘ 𝑆 ) ) ) ∧ 𝑥 ∈ 𝑆 ) → 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ) |
45 |
27
|
adantr |
⊢ ( ( ( ( 𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉 ) ∧ ( 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ 𝑦 ∈ ( ⊥ ‘ 𝑆 ) ∧ 𝑧 ∈ ( ⊥ ‘ 𝑆 ) ) ) ∧ 𝑥 ∈ 𝑆 ) → 𝑦 ∈ 𝑉 ) |
46 |
|
eqid |
⊢ ( .r ‘ ( Scalar ‘ 𝑊 ) ) = ( .r ‘ ( Scalar ‘ 𝑊 ) ) |
47 |
14 15 1 29 28 46
|
ipass |
⊢ ( ( 𝑊 ∈ PreHil ∧ ( 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ 𝑦 ∈ 𝑉 ∧ 𝑥 ∈ 𝑉 ) ) → ( ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑦 ) ( ·𝑖 ‘ 𝑊 ) 𝑥 ) = ( 𝑟 ( .r ‘ ( Scalar ‘ 𝑊 ) ) ( 𝑦 ( ·𝑖 ‘ 𝑊 ) 𝑥 ) ) ) |
48 |
37 44 45 40 47
|
syl13anc |
⊢ ( ( ( ( 𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉 ) ∧ ( 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ 𝑦 ∈ ( ⊥ ‘ 𝑆 ) ∧ 𝑧 ∈ ( ⊥ ‘ 𝑆 ) ) ) ∧ 𝑥 ∈ 𝑆 ) → ( ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑦 ) ( ·𝑖 ‘ 𝑊 ) 𝑥 ) = ( 𝑟 ( .r ‘ ( Scalar ‘ 𝑊 ) ) ( 𝑦 ( ·𝑖 ‘ 𝑊 ) 𝑥 ) ) ) |
49 |
1 15 14 16 2
|
ocvi |
⊢ ( ( 𝑦 ∈ ( ⊥ ‘ 𝑆 ) ∧ 𝑥 ∈ 𝑆 ) → ( 𝑦 ( ·𝑖 ‘ 𝑊 ) 𝑥 ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ) |
50 |
26 49
|
sylan |
⊢ ( ( ( ( 𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉 ) ∧ ( 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ 𝑦 ∈ ( ⊥ ‘ 𝑆 ) ∧ 𝑧 ∈ ( ⊥ ‘ 𝑆 ) ) ) ∧ 𝑥 ∈ 𝑆 ) → ( 𝑦 ( ·𝑖 ‘ 𝑊 ) 𝑥 ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ) |
51 |
50
|
oveq2d |
⊢ ( ( ( ( 𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉 ) ∧ ( 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ 𝑦 ∈ ( ⊥ ‘ 𝑆 ) ∧ 𝑧 ∈ ( ⊥ ‘ 𝑆 ) ) ) ∧ 𝑥 ∈ 𝑆 ) → ( 𝑟 ( .r ‘ ( Scalar ‘ 𝑊 ) ) ( 𝑦 ( ·𝑖 ‘ 𝑊 ) 𝑥 ) ) = ( 𝑟 ( .r ‘ ( Scalar ‘ 𝑊 ) ) ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ) ) |
52 |
24
|
adantr |
⊢ ( ( ( ( 𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉 ) ∧ ( 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ 𝑦 ∈ ( ⊥ ‘ 𝑆 ) ∧ 𝑧 ∈ ( ⊥ ‘ 𝑆 ) ) ) ∧ 𝑥 ∈ 𝑆 ) → 𝑊 ∈ LMod ) |
53 |
14
|
lmodring |
⊢ ( 𝑊 ∈ LMod → ( Scalar ‘ 𝑊 ) ∈ Ring ) |
54 |
52 53
|
syl |
⊢ ( ( ( ( 𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉 ) ∧ ( 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ 𝑦 ∈ ( ⊥ ‘ 𝑆 ) ∧ 𝑧 ∈ ( ⊥ ‘ 𝑆 ) ) ) ∧ 𝑥 ∈ 𝑆 ) → ( Scalar ‘ 𝑊 ) ∈ Ring ) |
55 |
29 46 16
|
ringrz |
⊢ ( ( ( Scalar ‘ 𝑊 ) ∈ Ring ∧ 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ) → ( 𝑟 ( .r ‘ ( Scalar ‘ 𝑊 ) ) ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ) |
56 |
54 44 55
|
syl2anc |
⊢ ( ( ( ( 𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉 ) ∧ ( 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ 𝑦 ∈ ( ⊥ ‘ 𝑆 ) ∧ 𝑧 ∈ ( ⊥ ‘ 𝑆 ) ) ) ∧ 𝑥 ∈ 𝑆 ) → ( 𝑟 ( .r ‘ ( Scalar ‘ 𝑊 ) ) ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ) |
57 |
48 51 56
|
3eqtrd |
⊢ ( ( ( ( 𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉 ) ∧ ( 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ 𝑦 ∈ ( ⊥ ‘ 𝑆 ) ∧ 𝑧 ∈ ( ⊥ ‘ 𝑆 ) ) ) ∧ 𝑥 ∈ 𝑆 ) → ( ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑦 ) ( ·𝑖 ‘ 𝑊 ) 𝑥 ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ) |
58 |
1 15 14 16 2
|
ocvi |
⊢ ( ( 𝑧 ∈ ( ⊥ ‘ 𝑆 ) ∧ 𝑥 ∈ 𝑆 ) → ( 𝑧 ( ·𝑖 ‘ 𝑊 ) 𝑥 ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ) |
59 |
32 58
|
sylan |
⊢ ( ( ( ( 𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉 ) ∧ ( 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ 𝑦 ∈ ( ⊥ ‘ 𝑆 ) ∧ 𝑧 ∈ ( ⊥ ‘ 𝑆 ) ) ) ∧ 𝑥 ∈ 𝑆 ) → ( 𝑧 ( ·𝑖 ‘ 𝑊 ) 𝑥 ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ) |
60 |
57 59
|
oveq12d |
⊢ ( ( ( ( 𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉 ) ∧ ( 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ 𝑦 ∈ ( ⊥ ‘ 𝑆 ) ∧ 𝑧 ∈ ( ⊥ ‘ 𝑆 ) ) ) ∧ 𝑥 ∈ 𝑆 ) → ( ( ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑦 ) ( ·𝑖 ‘ 𝑊 ) 𝑥 ) ( +g ‘ ( Scalar ‘ 𝑊 ) ) ( 𝑧 ( ·𝑖 ‘ 𝑊 ) 𝑥 ) ) = ( ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ( +g ‘ ( Scalar ‘ 𝑊 ) ) ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ) ) |
61 |
14
|
lmodfgrp |
⊢ ( 𝑊 ∈ LMod → ( Scalar ‘ 𝑊 ) ∈ Grp ) |
62 |
29 16
|
grpidcl |
⊢ ( ( Scalar ‘ 𝑊 ) ∈ Grp → ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ) |
63 |
29 41 16
|
grplid |
⊢ ( ( ( Scalar ‘ 𝑊 ) ∈ Grp ∧ ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ) → ( ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ( +g ‘ ( Scalar ‘ 𝑊 ) ) ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ) |
64 |
62 63
|
mpdan |
⊢ ( ( Scalar ‘ 𝑊 ) ∈ Grp → ( ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ( +g ‘ ( Scalar ‘ 𝑊 ) ) ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ) |
65 |
52 61 64
|
3syl |
⊢ ( ( ( ( 𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉 ) ∧ ( 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ 𝑦 ∈ ( ⊥ ‘ 𝑆 ) ∧ 𝑧 ∈ ( ⊥ ‘ 𝑆 ) ) ) ∧ 𝑥 ∈ 𝑆 ) → ( ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ( +g ‘ ( Scalar ‘ 𝑊 ) ) ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ) |
66 |
43 60 65
|
3eqtrd |
⊢ ( ( ( ( 𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉 ) ∧ ( 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ 𝑦 ∈ ( ⊥ ‘ 𝑆 ) ∧ 𝑧 ∈ ( ⊥ ‘ 𝑆 ) ) ) ∧ 𝑥 ∈ 𝑆 ) → ( ( ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑦 ) ( +g ‘ 𝑊 ) 𝑧 ) ( ·𝑖 ‘ 𝑊 ) 𝑥 ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ) |
67 |
66
|
ralrimiva |
⊢ ( ( ( 𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉 ) ∧ ( 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ 𝑦 ∈ ( ⊥ ‘ 𝑆 ) ∧ 𝑧 ∈ ( ⊥ ‘ 𝑆 ) ) ) → ∀ 𝑥 ∈ 𝑆 ( ( ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑦 ) ( +g ‘ 𝑊 ) 𝑧 ) ( ·𝑖 ‘ 𝑊 ) 𝑥 ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ) |
68 |
1 15 14 16 2
|
elocv |
⊢ ( ( ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑦 ) ( +g ‘ 𝑊 ) 𝑧 ) ∈ ( ⊥ ‘ 𝑆 ) ↔ ( 𝑆 ⊆ 𝑉 ∧ ( ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑦 ) ( +g ‘ 𝑊 ) 𝑧 ) ∈ 𝑉 ∧ ∀ 𝑥 ∈ 𝑆 ( ( ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑦 ) ( +g ‘ 𝑊 ) 𝑧 ) ( ·𝑖 ‘ 𝑊 ) 𝑥 ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ) ) |
69 |
23 36 67 68
|
syl3anbrc |
⊢ ( ( ( 𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉 ) ∧ ( 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ 𝑦 ∈ ( ⊥ ‘ 𝑆 ) ∧ 𝑧 ∈ ( ⊥ ‘ 𝑆 ) ) ) → ( ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑦 ) ( +g ‘ 𝑊 ) 𝑧 ) ∈ ( ⊥ ‘ 𝑆 ) ) |
70 |
69
|
ralrimivvva |
⊢ ( ( 𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉 ) → ∀ 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∀ 𝑦 ∈ ( ⊥ ‘ 𝑆 ) ∀ 𝑧 ∈ ( ⊥ ‘ 𝑆 ) ( ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑦 ) ( +g ‘ 𝑊 ) 𝑧 ) ∈ ( ⊥ ‘ 𝑆 ) ) |
71 |
14 29 1 34 28 3
|
islss |
⊢ ( ( ⊥ ‘ 𝑆 ) ∈ 𝐿 ↔ ( ( ⊥ ‘ 𝑆 ) ⊆ 𝑉 ∧ ( ⊥ ‘ 𝑆 ) ≠ ∅ ∧ ∀ 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∀ 𝑦 ∈ ( ⊥ ‘ 𝑆 ) ∀ 𝑧 ∈ ( ⊥ ‘ 𝑆 ) ( ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑦 ) ( +g ‘ 𝑊 ) 𝑧 ) ∈ ( ⊥ ‘ 𝑆 ) ) ) |
72 |
5 22 70 71
|
syl3anbrc |
⊢ ( ( 𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉 ) → ( ⊥ ‘ 𝑆 ) ∈ 𝐿 ) |