Step |
Hyp |
Ref |
Expression |
1 |
|
mplsubglem.s |
⊢ 𝑆 = ( 𝐼 mPwSer 𝑅 ) |
2 |
|
mplsubglem.b |
⊢ 𝐵 = ( Base ‘ 𝑆 ) |
3 |
|
mplsubglem.z |
⊢ 0 = ( 0g ‘ 𝑅 ) |
4 |
|
mplsubglem.d |
⊢ 𝐷 = { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } |
5 |
|
mplsubglem.i |
⊢ ( 𝜑 → 𝐼 ∈ 𝑊 ) |
6 |
|
mplsubglem.0 |
⊢ ( 𝜑 → ∅ ∈ 𝐴 ) |
7 |
|
mplsubglem.a |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ) → ( 𝑥 ∪ 𝑦 ) ∈ 𝐴 ) |
8 |
|
mplsubglem.y |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ⊆ 𝑥 ) ) → 𝑦 ∈ 𝐴 ) |
9 |
|
mplsubglem.u |
⊢ ( 𝜑 → 𝑈 = { 𝑔 ∈ 𝐵 ∣ ( 𝑔 supp 0 ) ∈ 𝐴 } ) |
10 |
|
mpllsslem.r |
⊢ ( 𝜑 → 𝑅 ∈ Ring ) |
11 |
1 5 10
|
psrsca |
⊢ ( 𝜑 → 𝑅 = ( Scalar ‘ 𝑆 ) ) |
12 |
|
eqidd |
⊢ ( 𝜑 → ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 ) ) |
13 |
2
|
a1i |
⊢ ( 𝜑 → 𝐵 = ( Base ‘ 𝑆 ) ) |
14 |
|
eqidd |
⊢ ( 𝜑 → ( +g ‘ 𝑆 ) = ( +g ‘ 𝑆 ) ) |
15 |
|
eqidd |
⊢ ( 𝜑 → ( ·𝑠 ‘ 𝑆 ) = ( ·𝑠 ‘ 𝑆 ) ) |
16 |
|
eqidd |
⊢ ( 𝜑 → ( LSubSp ‘ 𝑆 ) = ( LSubSp ‘ 𝑆 ) ) |
17 |
|
ringgrp |
⊢ ( 𝑅 ∈ Ring → 𝑅 ∈ Grp ) |
18 |
10 17
|
syl |
⊢ ( 𝜑 → 𝑅 ∈ Grp ) |
19 |
1 2 3 4 5 6 7 8 9 18
|
mplsubglem |
⊢ ( 𝜑 → 𝑈 ∈ ( SubGrp ‘ 𝑆 ) ) |
20 |
2
|
subgss |
⊢ ( 𝑈 ∈ ( SubGrp ‘ 𝑆 ) → 𝑈 ⊆ 𝐵 ) |
21 |
19 20
|
syl |
⊢ ( 𝜑 → 𝑈 ⊆ 𝐵 ) |
22 |
|
eqid |
⊢ ( 0g ‘ 𝑆 ) = ( 0g ‘ 𝑆 ) |
23 |
22
|
subg0cl |
⊢ ( 𝑈 ∈ ( SubGrp ‘ 𝑆 ) → ( 0g ‘ 𝑆 ) ∈ 𝑈 ) |
24 |
|
ne0i |
⊢ ( ( 0g ‘ 𝑆 ) ∈ 𝑈 → 𝑈 ≠ ∅ ) |
25 |
19 23 24
|
3syl |
⊢ ( 𝜑 → 𝑈 ≠ ∅ ) |
26 |
19
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑢 ∈ ( Base ‘ 𝑅 ) ∧ 𝑣 ∈ 𝑈 ∧ 𝑤 ∈ 𝑈 ) ) → 𝑈 ∈ ( SubGrp ‘ 𝑆 ) ) |
27 |
|
eqid |
⊢ ( ·𝑠 ‘ 𝑆 ) = ( ·𝑠 ‘ 𝑆 ) |
28 |
|
eqid |
⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 ) |
29 |
10
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑢 ∈ ( Base ‘ 𝑅 ) ∧ 𝑣 ∈ 𝑈 ) ) → 𝑅 ∈ Ring ) |
30 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( 𝑢 ∈ ( Base ‘ 𝑅 ) ∧ 𝑣 ∈ 𝑈 ) ) → 𝑢 ∈ ( Base ‘ 𝑅 ) ) |
31 |
|
simprr |
⊢ ( ( 𝜑 ∧ ( 𝑢 ∈ ( Base ‘ 𝑅 ) ∧ 𝑣 ∈ 𝑈 ) ) → 𝑣 ∈ 𝑈 ) |
32 |
9
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑢 ∈ ( Base ‘ 𝑅 ) ∧ 𝑣 ∈ 𝑈 ) ) → 𝑈 = { 𝑔 ∈ 𝐵 ∣ ( 𝑔 supp 0 ) ∈ 𝐴 } ) |
33 |
32
|
eleq2d |
⊢ ( ( 𝜑 ∧ ( 𝑢 ∈ ( Base ‘ 𝑅 ) ∧ 𝑣 ∈ 𝑈 ) ) → ( 𝑣 ∈ 𝑈 ↔ 𝑣 ∈ { 𝑔 ∈ 𝐵 ∣ ( 𝑔 supp 0 ) ∈ 𝐴 } ) ) |
34 |
|
oveq1 |
⊢ ( 𝑔 = 𝑣 → ( 𝑔 supp 0 ) = ( 𝑣 supp 0 ) ) |
35 |
34
|
eleq1d |
⊢ ( 𝑔 = 𝑣 → ( ( 𝑔 supp 0 ) ∈ 𝐴 ↔ ( 𝑣 supp 0 ) ∈ 𝐴 ) ) |
36 |
35
|
elrab |
⊢ ( 𝑣 ∈ { 𝑔 ∈ 𝐵 ∣ ( 𝑔 supp 0 ) ∈ 𝐴 } ↔ ( 𝑣 ∈ 𝐵 ∧ ( 𝑣 supp 0 ) ∈ 𝐴 ) ) |
37 |
33 36
|
bitrdi |
⊢ ( ( 𝜑 ∧ ( 𝑢 ∈ ( Base ‘ 𝑅 ) ∧ 𝑣 ∈ 𝑈 ) ) → ( 𝑣 ∈ 𝑈 ↔ ( 𝑣 ∈ 𝐵 ∧ ( 𝑣 supp 0 ) ∈ 𝐴 ) ) ) |
38 |
31 37
|
mpbid |
⊢ ( ( 𝜑 ∧ ( 𝑢 ∈ ( Base ‘ 𝑅 ) ∧ 𝑣 ∈ 𝑈 ) ) → ( 𝑣 ∈ 𝐵 ∧ ( 𝑣 supp 0 ) ∈ 𝐴 ) ) |
39 |
38
|
simpld |
⊢ ( ( 𝜑 ∧ ( 𝑢 ∈ ( Base ‘ 𝑅 ) ∧ 𝑣 ∈ 𝑈 ) ) → 𝑣 ∈ 𝐵 ) |
40 |
1 27 28 2 29 30 39
|
psrvscacl |
⊢ ( ( 𝜑 ∧ ( 𝑢 ∈ ( Base ‘ 𝑅 ) ∧ 𝑣 ∈ 𝑈 ) ) → ( 𝑢 ( ·𝑠 ‘ 𝑆 ) 𝑣 ) ∈ 𝐵 ) |
41 |
|
ovex |
⊢ ( ( 𝑢 ( ·𝑠 ‘ 𝑆 ) 𝑣 ) supp 0 ) ∈ V |
42 |
41
|
a1i |
⊢ ( ( 𝜑 ∧ ( 𝑢 ∈ ( Base ‘ 𝑅 ) ∧ 𝑣 ∈ 𝑈 ) ) → ( ( 𝑢 ( ·𝑠 ‘ 𝑆 ) 𝑣 ) supp 0 ) ∈ V ) |
43 |
|
sseq2 |
⊢ ( 𝑥 = ( 𝑣 supp 0 ) → ( 𝑦 ⊆ 𝑥 ↔ 𝑦 ⊆ ( 𝑣 supp 0 ) ) ) |
44 |
43
|
imbi1d |
⊢ ( 𝑥 = ( 𝑣 supp 0 ) → ( ( 𝑦 ⊆ 𝑥 → 𝑦 ∈ 𝐴 ) ↔ ( 𝑦 ⊆ ( 𝑣 supp 0 ) → 𝑦 ∈ 𝐴 ) ) ) |
45 |
44
|
albidv |
⊢ ( 𝑥 = ( 𝑣 supp 0 ) → ( ∀ 𝑦 ( 𝑦 ⊆ 𝑥 → 𝑦 ∈ 𝐴 ) ↔ ∀ 𝑦 ( 𝑦 ⊆ ( 𝑣 supp 0 ) → 𝑦 ∈ 𝐴 ) ) ) |
46 |
8
|
expr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝑦 ⊆ 𝑥 → 𝑦 ∈ 𝐴 ) ) |
47 |
46
|
alrimiv |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ∀ 𝑦 ( 𝑦 ⊆ 𝑥 → 𝑦 ∈ 𝐴 ) ) |
48 |
47
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ( 𝑦 ⊆ 𝑥 → 𝑦 ∈ 𝐴 ) ) |
49 |
48
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑢 ∈ ( Base ‘ 𝑅 ) ∧ 𝑣 ∈ 𝑈 ) ) → ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ( 𝑦 ⊆ 𝑥 → 𝑦 ∈ 𝐴 ) ) |
50 |
38
|
simprd |
⊢ ( ( 𝜑 ∧ ( 𝑢 ∈ ( Base ‘ 𝑅 ) ∧ 𝑣 ∈ 𝑈 ) ) → ( 𝑣 supp 0 ) ∈ 𝐴 ) |
51 |
45 49 50
|
rspcdva |
⊢ ( ( 𝜑 ∧ ( 𝑢 ∈ ( Base ‘ 𝑅 ) ∧ 𝑣 ∈ 𝑈 ) ) → ∀ 𝑦 ( 𝑦 ⊆ ( 𝑣 supp 0 ) → 𝑦 ∈ 𝐴 ) ) |
52 |
1 28 4 2 40
|
psrelbas |
⊢ ( ( 𝜑 ∧ ( 𝑢 ∈ ( Base ‘ 𝑅 ) ∧ 𝑣 ∈ 𝑈 ) ) → ( 𝑢 ( ·𝑠 ‘ 𝑆 ) 𝑣 ) : 𝐷 ⟶ ( Base ‘ 𝑅 ) ) |
53 |
|
eqid |
⊢ ( .r ‘ 𝑅 ) = ( .r ‘ 𝑅 ) |
54 |
30
|
adantr |
⊢ ( ( ( 𝜑 ∧ ( 𝑢 ∈ ( Base ‘ 𝑅 ) ∧ 𝑣 ∈ 𝑈 ) ) ∧ 𝑘 ∈ ( 𝐷 ∖ ( 𝑣 supp 0 ) ) ) → 𝑢 ∈ ( Base ‘ 𝑅 ) ) |
55 |
39
|
adantr |
⊢ ( ( ( 𝜑 ∧ ( 𝑢 ∈ ( Base ‘ 𝑅 ) ∧ 𝑣 ∈ 𝑈 ) ) ∧ 𝑘 ∈ ( 𝐷 ∖ ( 𝑣 supp 0 ) ) ) → 𝑣 ∈ 𝐵 ) |
56 |
|
eldifi |
⊢ ( 𝑘 ∈ ( 𝐷 ∖ ( 𝑣 supp 0 ) ) → 𝑘 ∈ 𝐷 ) |
57 |
56
|
adantl |
⊢ ( ( ( 𝜑 ∧ ( 𝑢 ∈ ( Base ‘ 𝑅 ) ∧ 𝑣 ∈ 𝑈 ) ) ∧ 𝑘 ∈ ( 𝐷 ∖ ( 𝑣 supp 0 ) ) ) → 𝑘 ∈ 𝐷 ) |
58 |
1 27 28 2 53 4 54 55 57
|
psrvscaval |
⊢ ( ( ( 𝜑 ∧ ( 𝑢 ∈ ( Base ‘ 𝑅 ) ∧ 𝑣 ∈ 𝑈 ) ) ∧ 𝑘 ∈ ( 𝐷 ∖ ( 𝑣 supp 0 ) ) ) → ( ( 𝑢 ( ·𝑠 ‘ 𝑆 ) 𝑣 ) ‘ 𝑘 ) = ( 𝑢 ( .r ‘ 𝑅 ) ( 𝑣 ‘ 𝑘 ) ) ) |
59 |
1 28 4 2 39
|
psrelbas |
⊢ ( ( 𝜑 ∧ ( 𝑢 ∈ ( Base ‘ 𝑅 ) ∧ 𝑣 ∈ 𝑈 ) ) → 𝑣 : 𝐷 ⟶ ( Base ‘ 𝑅 ) ) |
60 |
|
ssidd |
⊢ ( ( 𝜑 ∧ ( 𝑢 ∈ ( Base ‘ 𝑅 ) ∧ 𝑣 ∈ 𝑈 ) ) → ( 𝑣 supp 0 ) ⊆ ( 𝑣 supp 0 ) ) |
61 |
|
ovex |
⊢ ( ℕ0 ↑m 𝐼 ) ∈ V |
62 |
4 61
|
rabex2 |
⊢ 𝐷 ∈ V |
63 |
62
|
a1i |
⊢ ( ( 𝜑 ∧ ( 𝑢 ∈ ( Base ‘ 𝑅 ) ∧ 𝑣 ∈ 𝑈 ) ) → 𝐷 ∈ V ) |
64 |
3
|
fvexi |
⊢ 0 ∈ V |
65 |
64
|
a1i |
⊢ ( ( 𝜑 ∧ ( 𝑢 ∈ ( Base ‘ 𝑅 ) ∧ 𝑣 ∈ 𝑈 ) ) → 0 ∈ V ) |
66 |
59 60 63 65
|
suppssr |
⊢ ( ( ( 𝜑 ∧ ( 𝑢 ∈ ( Base ‘ 𝑅 ) ∧ 𝑣 ∈ 𝑈 ) ) ∧ 𝑘 ∈ ( 𝐷 ∖ ( 𝑣 supp 0 ) ) ) → ( 𝑣 ‘ 𝑘 ) = 0 ) |
67 |
66
|
oveq2d |
⊢ ( ( ( 𝜑 ∧ ( 𝑢 ∈ ( Base ‘ 𝑅 ) ∧ 𝑣 ∈ 𝑈 ) ) ∧ 𝑘 ∈ ( 𝐷 ∖ ( 𝑣 supp 0 ) ) ) → ( 𝑢 ( .r ‘ 𝑅 ) ( 𝑣 ‘ 𝑘 ) ) = ( 𝑢 ( .r ‘ 𝑅 ) 0 ) ) |
68 |
28 53 3
|
ringrz |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑢 ∈ ( Base ‘ 𝑅 ) ) → ( 𝑢 ( .r ‘ 𝑅 ) 0 ) = 0 ) |
69 |
10 30 68
|
syl2an2r |
⊢ ( ( 𝜑 ∧ ( 𝑢 ∈ ( Base ‘ 𝑅 ) ∧ 𝑣 ∈ 𝑈 ) ) → ( 𝑢 ( .r ‘ 𝑅 ) 0 ) = 0 ) |
70 |
69
|
adantr |
⊢ ( ( ( 𝜑 ∧ ( 𝑢 ∈ ( Base ‘ 𝑅 ) ∧ 𝑣 ∈ 𝑈 ) ) ∧ 𝑘 ∈ ( 𝐷 ∖ ( 𝑣 supp 0 ) ) ) → ( 𝑢 ( .r ‘ 𝑅 ) 0 ) = 0 ) |
71 |
58 67 70
|
3eqtrd |
⊢ ( ( ( 𝜑 ∧ ( 𝑢 ∈ ( Base ‘ 𝑅 ) ∧ 𝑣 ∈ 𝑈 ) ) ∧ 𝑘 ∈ ( 𝐷 ∖ ( 𝑣 supp 0 ) ) ) → ( ( 𝑢 ( ·𝑠 ‘ 𝑆 ) 𝑣 ) ‘ 𝑘 ) = 0 ) |
72 |
52 71
|
suppss |
⊢ ( ( 𝜑 ∧ ( 𝑢 ∈ ( Base ‘ 𝑅 ) ∧ 𝑣 ∈ 𝑈 ) ) → ( ( 𝑢 ( ·𝑠 ‘ 𝑆 ) 𝑣 ) supp 0 ) ⊆ ( 𝑣 supp 0 ) ) |
73 |
|
sseq1 |
⊢ ( 𝑦 = ( ( 𝑢 ( ·𝑠 ‘ 𝑆 ) 𝑣 ) supp 0 ) → ( 𝑦 ⊆ ( 𝑣 supp 0 ) ↔ ( ( 𝑢 ( ·𝑠 ‘ 𝑆 ) 𝑣 ) supp 0 ) ⊆ ( 𝑣 supp 0 ) ) ) |
74 |
|
eleq1 |
⊢ ( 𝑦 = ( ( 𝑢 ( ·𝑠 ‘ 𝑆 ) 𝑣 ) supp 0 ) → ( 𝑦 ∈ 𝐴 ↔ ( ( 𝑢 ( ·𝑠 ‘ 𝑆 ) 𝑣 ) supp 0 ) ∈ 𝐴 ) ) |
75 |
73 74
|
imbi12d |
⊢ ( 𝑦 = ( ( 𝑢 ( ·𝑠 ‘ 𝑆 ) 𝑣 ) supp 0 ) → ( ( 𝑦 ⊆ ( 𝑣 supp 0 ) → 𝑦 ∈ 𝐴 ) ↔ ( ( ( 𝑢 ( ·𝑠 ‘ 𝑆 ) 𝑣 ) supp 0 ) ⊆ ( 𝑣 supp 0 ) → ( ( 𝑢 ( ·𝑠 ‘ 𝑆 ) 𝑣 ) supp 0 ) ∈ 𝐴 ) ) ) |
76 |
75
|
spcgv |
⊢ ( ( ( 𝑢 ( ·𝑠 ‘ 𝑆 ) 𝑣 ) supp 0 ) ∈ V → ( ∀ 𝑦 ( 𝑦 ⊆ ( 𝑣 supp 0 ) → 𝑦 ∈ 𝐴 ) → ( ( ( 𝑢 ( ·𝑠 ‘ 𝑆 ) 𝑣 ) supp 0 ) ⊆ ( 𝑣 supp 0 ) → ( ( 𝑢 ( ·𝑠 ‘ 𝑆 ) 𝑣 ) supp 0 ) ∈ 𝐴 ) ) ) |
77 |
42 51 72 76
|
syl3c |
⊢ ( ( 𝜑 ∧ ( 𝑢 ∈ ( Base ‘ 𝑅 ) ∧ 𝑣 ∈ 𝑈 ) ) → ( ( 𝑢 ( ·𝑠 ‘ 𝑆 ) 𝑣 ) supp 0 ) ∈ 𝐴 ) |
78 |
32
|
eleq2d |
⊢ ( ( 𝜑 ∧ ( 𝑢 ∈ ( Base ‘ 𝑅 ) ∧ 𝑣 ∈ 𝑈 ) ) → ( ( 𝑢 ( ·𝑠 ‘ 𝑆 ) 𝑣 ) ∈ 𝑈 ↔ ( 𝑢 ( ·𝑠 ‘ 𝑆 ) 𝑣 ) ∈ { 𝑔 ∈ 𝐵 ∣ ( 𝑔 supp 0 ) ∈ 𝐴 } ) ) |
79 |
|
oveq1 |
⊢ ( 𝑔 = ( 𝑢 ( ·𝑠 ‘ 𝑆 ) 𝑣 ) → ( 𝑔 supp 0 ) = ( ( 𝑢 ( ·𝑠 ‘ 𝑆 ) 𝑣 ) supp 0 ) ) |
80 |
79
|
eleq1d |
⊢ ( 𝑔 = ( 𝑢 ( ·𝑠 ‘ 𝑆 ) 𝑣 ) → ( ( 𝑔 supp 0 ) ∈ 𝐴 ↔ ( ( 𝑢 ( ·𝑠 ‘ 𝑆 ) 𝑣 ) supp 0 ) ∈ 𝐴 ) ) |
81 |
80
|
elrab |
⊢ ( ( 𝑢 ( ·𝑠 ‘ 𝑆 ) 𝑣 ) ∈ { 𝑔 ∈ 𝐵 ∣ ( 𝑔 supp 0 ) ∈ 𝐴 } ↔ ( ( 𝑢 ( ·𝑠 ‘ 𝑆 ) 𝑣 ) ∈ 𝐵 ∧ ( ( 𝑢 ( ·𝑠 ‘ 𝑆 ) 𝑣 ) supp 0 ) ∈ 𝐴 ) ) |
82 |
78 81
|
bitrdi |
⊢ ( ( 𝜑 ∧ ( 𝑢 ∈ ( Base ‘ 𝑅 ) ∧ 𝑣 ∈ 𝑈 ) ) → ( ( 𝑢 ( ·𝑠 ‘ 𝑆 ) 𝑣 ) ∈ 𝑈 ↔ ( ( 𝑢 ( ·𝑠 ‘ 𝑆 ) 𝑣 ) ∈ 𝐵 ∧ ( ( 𝑢 ( ·𝑠 ‘ 𝑆 ) 𝑣 ) supp 0 ) ∈ 𝐴 ) ) ) |
83 |
40 77 82
|
mpbir2and |
⊢ ( ( 𝜑 ∧ ( 𝑢 ∈ ( Base ‘ 𝑅 ) ∧ 𝑣 ∈ 𝑈 ) ) → ( 𝑢 ( ·𝑠 ‘ 𝑆 ) 𝑣 ) ∈ 𝑈 ) |
84 |
83
|
3adantr3 |
⊢ ( ( 𝜑 ∧ ( 𝑢 ∈ ( Base ‘ 𝑅 ) ∧ 𝑣 ∈ 𝑈 ∧ 𝑤 ∈ 𝑈 ) ) → ( 𝑢 ( ·𝑠 ‘ 𝑆 ) 𝑣 ) ∈ 𝑈 ) |
85 |
|
simpr3 |
⊢ ( ( 𝜑 ∧ ( 𝑢 ∈ ( Base ‘ 𝑅 ) ∧ 𝑣 ∈ 𝑈 ∧ 𝑤 ∈ 𝑈 ) ) → 𝑤 ∈ 𝑈 ) |
86 |
|
eqid |
⊢ ( +g ‘ 𝑆 ) = ( +g ‘ 𝑆 ) |
87 |
86
|
subgcl |
⊢ ( ( 𝑈 ∈ ( SubGrp ‘ 𝑆 ) ∧ ( 𝑢 ( ·𝑠 ‘ 𝑆 ) 𝑣 ) ∈ 𝑈 ∧ 𝑤 ∈ 𝑈 ) → ( ( 𝑢 ( ·𝑠 ‘ 𝑆 ) 𝑣 ) ( +g ‘ 𝑆 ) 𝑤 ) ∈ 𝑈 ) |
88 |
26 84 85 87
|
syl3anc |
⊢ ( ( 𝜑 ∧ ( 𝑢 ∈ ( Base ‘ 𝑅 ) ∧ 𝑣 ∈ 𝑈 ∧ 𝑤 ∈ 𝑈 ) ) → ( ( 𝑢 ( ·𝑠 ‘ 𝑆 ) 𝑣 ) ( +g ‘ 𝑆 ) 𝑤 ) ∈ 𝑈 ) |
89 |
11 12 13 14 15 16 21 25 88
|
islssd |
⊢ ( 𝜑 → 𝑈 ∈ ( LSubSp ‘ 𝑆 ) ) |