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 |
|
mplsubglem.r |
⊢ ( 𝜑 → 𝑅 ∈ Grp ) |
11 |
|
ssrab2 |
⊢ { 𝑔 ∈ 𝐵 ∣ ( 𝑔 supp 0 ) ∈ 𝐴 } ⊆ 𝐵 |
12 |
9 11
|
eqsstrdi |
⊢ ( 𝜑 → 𝑈 ⊆ 𝐵 ) |
13 |
1 5 10 4 3 2
|
psr0cl |
⊢ ( 𝜑 → ( 𝐷 × { 0 } ) ∈ 𝐵 ) |
14 |
|
eqid |
⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 ) |
15 |
14 3
|
grpidcl |
⊢ ( 𝑅 ∈ Grp → 0 ∈ ( Base ‘ 𝑅 ) ) |
16 |
|
fconst6g |
⊢ ( 0 ∈ ( Base ‘ 𝑅 ) → ( 𝐷 × { 0 } ) : 𝐷 ⟶ ( Base ‘ 𝑅 ) ) |
17 |
10 15 16
|
3syl |
⊢ ( 𝜑 → ( 𝐷 × { 0 } ) : 𝐷 ⟶ ( Base ‘ 𝑅 ) ) |
18 |
|
eldifi |
⊢ ( 𝑢 ∈ ( 𝐷 ∖ ∅ ) → 𝑢 ∈ 𝐷 ) |
19 |
3
|
fvexi |
⊢ 0 ∈ V |
20 |
19
|
fvconst2 |
⊢ ( 𝑢 ∈ 𝐷 → ( ( 𝐷 × { 0 } ) ‘ 𝑢 ) = 0 ) |
21 |
18 20
|
syl |
⊢ ( 𝑢 ∈ ( 𝐷 ∖ ∅ ) → ( ( 𝐷 × { 0 } ) ‘ 𝑢 ) = 0 ) |
22 |
21
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑢 ∈ ( 𝐷 ∖ ∅ ) ) → ( ( 𝐷 × { 0 } ) ‘ 𝑢 ) = 0 ) |
23 |
17 22
|
suppss |
⊢ ( 𝜑 → ( ( 𝐷 × { 0 } ) supp 0 ) ⊆ ∅ ) |
24 |
|
ss0 |
⊢ ( ( ( 𝐷 × { 0 } ) supp 0 ) ⊆ ∅ → ( ( 𝐷 × { 0 } ) supp 0 ) = ∅ ) |
25 |
23 24
|
syl |
⊢ ( 𝜑 → ( ( 𝐷 × { 0 } ) supp 0 ) = ∅ ) |
26 |
25 6
|
eqeltrd |
⊢ ( 𝜑 → ( ( 𝐷 × { 0 } ) supp 0 ) ∈ 𝐴 ) |
27 |
9
|
eleq2d |
⊢ ( 𝜑 → ( ( 𝐷 × { 0 } ) ∈ 𝑈 ↔ ( 𝐷 × { 0 } ) ∈ { 𝑔 ∈ 𝐵 ∣ ( 𝑔 supp 0 ) ∈ 𝐴 } ) ) |
28 |
|
oveq1 |
⊢ ( 𝑔 = ( 𝐷 × { 0 } ) → ( 𝑔 supp 0 ) = ( ( 𝐷 × { 0 } ) supp 0 ) ) |
29 |
28
|
eleq1d |
⊢ ( 𝑔 = ( 𝐷 × { 0 } ) → ( ( 𝑔 supp 0 ) ∈ 𝐴 ↔ ( ( 𝐷 × { 0 } ) supp 0 ) ∈ 𝐴 ) ) |
30 |
29
|
elrab |
⊢ ( ( 𝐷 × { 0 } ) ∈ { 𝑔 ∈ 𝐵 ∣ ( 𝑔 supp 0 ) ∈ 𝐴 } ↔ ( ( 𝐷 × { 0 } ) ∈ 𝐵 ∧ ( ( 𝐷 × { 0 } ) supp 0 ) ∈ 𝐴 ) ) |
31 |
27 30
|
bitrdi |
⊢ ( 𝜑 → ( ( 𝐷 × { 0 } ) ∈ 𝑈 ↔ ( ( 𝐷 × { 0 } ) ∈ 𝐵 ∧ ( ( 𝐷 × { 0 } ) supp 0 ) ∈ 𝐴 ) ) ) |
32 |
13 26 31
|
mpbir2and |
⊢ ( 𝜑 → ( 𝐷 × { 0 } ) ∈ 𝑈 ) |
33 |
32
|
ne0d |
⊢ ( 𝜑 → 𝑈 ≠ ∅ ) |
34 |
|
eqid |
⊢ ( +g ‘ 𝑆 ) = ( +g ‘ 𝑆 ) |
35 |
10
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) ∧ 𝑣 ∈ 𝑈 ) → 𝑅 ∈ Grp ) |
36 |
9
|
eleq2d |
⊢ ( 𝜑 → ( 𝑢 ∈ 𝑈 ↔ 𝑢 ∈ { 𝑔 ∈ 𝐵 ∣ ( 𝑔 supp 0 ) ∈ 𝐴 } ) ) |
37 |
|
oveq1 |
⊢ ( 𝑔 = 𝑢 → ( 𝑔 supp 0 ) = ( 𝑢 supp 0 ) ) |
38 |
37
|
eleq1d |
⊢ ( 𝑔 = 𝑢 → ( ( 𝑔 supp 0 ) ∈ 𝐴 ↔ ( 𝑢 supp 0 ) ∈ 𝐴 ) ) |
39 |
38
|
elrab |
⊢ ( 𝑢 ∈ { 𝑔 ∈ 𝐵 ∣ ( 𝑔 supp 0 ) ∈ 𝐴 } ↔ ( 𝑢 ∈ 𝐵 ∧ ( 𝑢 supp 0 ) ∈ 𝐴 ) ) |
40 |
36 39
|
bitrdi |
⊢ ( 𝜑 → ( 𝑢 ∈ 𝑈 ↔ ( 𝑢 ∈ 𝐵 ∧ ( 𝑢 supp 0 ) ∈ 𝐴 ) ) ) |
41 |
40
|
biimpa |
⊢ ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) → ( 𝑢 ∈ 𝐵 ∧ ( 𝑢 supp 0 ) ∈ 𝐴 ) ) |
42 |
41
|
simpld |
⊢ ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) → 𝑢 ∈ 𝐵 ) |
43 |
42
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) ∧ 𝑣 ∈ 𝑈 ) → 𝑢 ∈ 𝐵 ) |
44 |
9
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) → 𝑈 = { 𝑔 ∈ 𝐵 ∣ ( 𝑔 supp 0 ) ∈ 𝐴 } ) |
45 |
44
|
eleq2d |
⊢ ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) → ( 𝑣 ∈ 𝑈 ↔ 𝑣 ∈ { 𝑔 ∈ 𝐵 ∣ ( 𝑔 supp 0 ) ∈ 𝐴 } ) ) |
46 |
|
oveq1 |
⊢ ( 𝑔 = 𝑣 → ( 𝑔 supp 0 ) = ( 𝑣 supp 0 ) ) |
47 |
46
|
eleq1d |
⊢ ( 𝑔 = 𝑣 → ( ( 𝑔 supp 0 ) ∈ 𝐴 ↔ ( 𝑣 supp 0 ) ∈ 𝐴 ) ) |
48 |
47
|
elrab |
⊢ ( 𝑣 ∈ { 𝑔 ∈ 𝐵 ∣ ( 𝑔 supp 0 ) ∈ 𝐴 } ↔ ( 𝑣 ∈ 𝐵 ∧ ( 𝑣 supp 0 ) ∈ 𝐴 ) ) |
49 |
45 48
|
bitrdi |
⊢ ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) → ( 𝑣 ∈ 𝑈 ↔ ( 𝑣 ∈ 𝐵 ∧ ( 𝑣 supp 0 ) ∈ 𝐴 ) ) ) |
50 |
49
|
biimpa |
⊢ ( ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) ∧ 𝑣 ∈ 𝑈 ) → ( 𝑣 ∈ 𝐵 ∧ ( 𝑣 supp 0 ) ∈ 𝐴 ) ) |
51 |
50
|
simpld |
⊢ ( ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) ∧ 𝑣 ∈ 𝑈 ) → 𝑣 ∈ 𝐵 ) |
52 |
1 2 34 35 43 51
|
psraddcl |
⊢ ( ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) ∧ 𝑣 ∈ 𝑈 ) → ( 𝑢 ( +g ‘ 𝑆 ) 𝑣 ) ∈ 𝐵 ) |
53 |
|
ovexd |
⊢ ( ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) ∧ 𝑣 ∈ 𝑈 ) → ( ( 𝑢 ( +g ‘ 𝑆 ) 𝑣 ) supp 0 ) ∈ V ) |
54 |
|
sseq2 |
⊢ ( 𝑥 = ( ( 𝑢 supp 0 ) ∪ ( 𝑣 supp 0 ) ) → ( 𝑦 ⊆ 𝑥 ↔ 𝑦 ⊆ ( ( 𝑢 supp 0 ) ∪ ( 𝑣 supp 0 ) ) ) ) |
55 |
54
|
imbi1d |
⊢ ( 𝑥 = ( ( 𝑢 supp 0 ) ∪ ( 𝑣 supp 0 ) ) → ( ( 𝑦 ⊆ 𝑥 → 𝑦 ∈ 𝐴 ) ↔ ( 𝑦 ⊆ ( ( 𝑢 supp 0 ) ∪ ( 𝑣 supp 0 ) ) → 𝑦 ∈ 𝐴 ) ) ) |
56 |
55
|
albidv |
⊢ ( 𝑥 = ( ( 𝑢 supp 0 ) ∪ ( 𝑣 supp 0 ) ) → ( ∀ 𝑦 ( 𝑦 ⊆ 𝑥 → 𝑦 ∈ 𝐴 ) ↔ ∀ 𝑦 ( 𝑦 ⊆ ( ( 𝑢 supp 0 ) ∪ ( 𝑣 supp 0 ) ) → 𝑦 ∈ 𝐴 ) ) ) |
57 |
8
|
expr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝑦 ⊆ 𝑥 → 𝑦 ∈ 𝐴 ) ) |
58 |
57
|
alrimiv |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ∀ 𝑦 ( 𝑦 ⊆ 𝑥 → 𝑦 ∈ 𝐴 ) ) |
59 |
58
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ( 𝑦 ⊆ 𝑥 → 𝑦 ∈ 𝐴 ) ) |
60 |
59
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) ∧ 𝑣 ∈ 𝑈 ) → ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ( 𝑦 ⊆ 𝑥 → 𝑦 ∈ 𝐴 ) ) |
61 |
41
|
simprd |
⊢ ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) → ( 𝑢 supp 0 ) ∈ 𝐴 ) |
62 |
61
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) ∧ 𝑣 ∈ 𝑈 ) → ( 𝑢 supp 0 ) ∈ 𝐴 ) |
63 |
50
|
simprd |
⊢ ( ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) ∧ 𝑣 ∈ 𝑈 ) → ( 𝑣 supp 0 ) ∈ 𝐴 ) |
64 |
7
|
ralrimivva |
⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 ∪ 𝑦 ) ∈ 𝐴 ) |
65 |
64
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) ∧ 𝑣 ∈ 𝑈 ) → ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 ∪ 𝑦 ) ∈ 𝐴 ) |
66 |
|
uneq1 |
⊢ ( 𝑥 = ( 𝑢 supp 0 ) → ( 𝑥 ∪ 𝑦 ) = ( ( 𝑢 supp 0 ) ∪ 𝑦 ) ) |
67 |
66
|
eleq1d |
⊢ ( 𝑥 = ( 𝑢 supp 0 ) → ( ( 𝑥 ∪ 𝑦 ) ∈ 𝐴 ↔ ( ( 𝑢 supp 0 ) ∪ 𝑦 ) ∈ 𝐴 ) ) |
68 |
|
uneq2 |
⊢ ( 𝑦 = ( 𝑣 supp 0 ) → ( ( 𝑢 supp 0 ) ∪ 𝑦 ) = ( ( 𝑢 supp 0 ) ∪ ( 𝑣 supp 0 ) ) ) |
69 |
68
|
eleq1d |
⊢ ( 𝑦 = ( 𝑣 supp 0 ) → ( ( ( 𝑢 supp 0 ) ∪ 𝑦 ) ∈ 𝐴 ↔ ( ( 𝑢 supp 0 ) ∪ ( 𝑣 supp 0 ) ) ∈ 𝐴 ) ) |
70 |
67 69
|
rspc2va |
⊢ ( ( ( ( 𝑢 supp 0 ) ∈ 𝐴 ∧ ( 𝑣 supp 0 ) ∈ 𝐴 ) ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 ∪ 𝑦 ) ∈ 𝐴 ) → ( ( 𝑢 supp 0 ) ∪ ( 𝑣 supp 0 ) ) ∈ 𝐴 ) |
71 |
62 63 65 70
|
syl21anc |
⊢ ( ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) ∧ 𝑣 ∈ 𝑈 ) → ( ( 𝑢 supp 0 ) ∪ ( 𝑣 supp 0 ) ) ∈ 𝐴 ) |
72 |
56 60 71
|
rspcdva |
⊢ ( ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) ∧ 𝑣 ∈ 𝑈 ) → ∀ 𝑦 ( 𝑦 ⊆ ( ( 𝑢 supp 0 ) ∪ ( 𝑣 supp 0 ) ) → 𝑦 ∈ 𝐴 ) ) |
73 |
1 14 4 2 52
|
psrelbas |
⊢ ( ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) ∧ 𝑣 ∈ 𝑈 ) → ( 𝑢 ( +g ‘ 𝑆 ) 𝑣 ) : 𝐷 ⟶ ( Base ‘ 𝑅 ) ) |
74 |
|
eqid |
⊢ ( +g ‘ 𝑅 ) = ( +g ‘ 𝑅 ) |
75 |
1 2 74 34 43 51
|
psradd |
⊢ ( ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) ∧ 𝑣 ∈ 𝑈 ) → ( 𝑢 ( +g ‘ 𝑆 ) 𝑣 ) = ( 𝑢 ∘f ( +g ‘ 𝑅 ) 𝑣 ) ) |
76 |
75
|
fveq1d |
⊢ ( ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) ∧ 𝑣 ∈ 𝑈 ) → ( ( 𝑢 ( +g ‘ 𝑆 ) 𝑣 ) ‘ 𝑘 ) = ( ( 𝑢 ∘f ( +g ‘ 𝑅 ) 𝑣 ) ‘ 𝑘 ) ) |
77 |
76
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) ∧ 𝑣 ∈ 𝑈 ) ∧ 𝑘 ∈ ( 𝐷 ∖ ( ( 𝑢 supp 0 ) ∪ ( 𝑣 supp 0 ) ) ) ) → ( ( 𝑢 ( +g ‘ 𝑆 ) 𝑣 ) ‘ 𝑘 ) = ( ( 𝑢 ∘f ( +g ‘ 𝑅 ) 𝑣 ) ‘ 𝑘 ) ) |
78 |
|
eldifi |
⊢ ( 𝑘 ∈ ( 𝐷 ∖ ( ( 𝑢 supp 0 ) ∪ ( 𝑣 supp 0 ) ) ) → 𝑘 ∈ 𝐷 ) |
79 |
1 14 4 2 42
|
psrelbas |
⊢ ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) → 𝑢 : 𝐷 ⟶ ( Base ‘ 𝑅 ) ) |
80 |
79
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) ∧ 𝑣 ∈ 𝑈 ) → 𝑢 : 𝐷 ⟶ ( Base ‘ 𝑅 ) ) |
81 |
80
|
ffnd |
⊢ ( ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) ∧ 𝑣 ∈ 𝑈 ) → 𝑢 Fn 𝐷 ) |
82 |
1 14 4 2 51
|
psrelbas |
⊢ ( ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) ∧ 𝑣 ∈ 𝑈 ) → 𝑣 : 𝐷 ⟶ ( Base ‘ 𝑅 ) ) |
83 |
82
|
ffnd |
⊢ ( ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) ∧ 𝑣 ∈ 𝑈 ) → 𝑣 Fn 𝐷 ) |
84 |
|
ovex |
⊢ ( ℕ0 ↑m 𝐼 ) ∈ V |
85 |
4 84
|
rabex2 |
⊢ 𝐷 ∈ V |
86 |
85
|
a1i |
⊢ ( ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) ∧ 𝑣 ∈ 𝑈 ) → 𝐷 ∈ V ) |
87 |
|
inidm |
⊢ ( 𝐷 ∩ 𝐷 ) = 𝐷 |
88 |
|
eqidd |
⊢ ( ( ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) ∧ 𝑣 ∈ 𝑈 ) ∧ 𝑘 ∈ 𝐷 ) → ( 𝑢 ‘ 𝑘 ) = ( 𝑢 ‘ 𝑘 ) ) |
89 |
|
eqidd |
⊢ ( ( ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) ∧ 𝑣 ∈ 𝑈 ) ∧ 𝑘 ∈ 𝐷 ) → ( 𝑣 ‘ 𝑘 ) = ( 𝑣 ‘ 𝑘 ) ) |
90 |
81 83 86 86 87 88 89
|
ofval |
⊢ ( ( ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) ∧ 𝑣 ∈ 𝑈 ) ∧ 𝑘 ∈ 𝐷 ) → ( ( 𝑢 ∘f ( +g ‘ 𝑅 ) 𝑣 ) ‘ 𝑘 ) = ( ( 𝑢 ‘ 𝑘 ) ( +g ‘ 𝑅 ) ( 𝑣 ‘ 𝑘 ) ) ) |
91 |
78 90
|
sylan2 |
⊢ ( ( ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) ∧ 𝑣 ∈ 𝑈 ) ∧ 𝑘 ∈ ( 𝐷 ∖ ( ( 𝑢 supp 0 ) ∪ ( 𝑣 supp 0 ) ) ) ) → ( ( 𝑢 ∘f ( +g ‘ 𝑅 ) 𝑣 ) ‘ 𝑘 ) = ( ( 𝑢 ‘ 𝑘 ) ( +g ‘ 𝑅 ) ( 𝑣 ‘ 𝑘 ) ) ) |
92 |
|
ssun1 |
⊢ ( 𝑢 supp 0 ) ⊆ ( ( 𝑢 supp 0 ) ∪ ( 𝑣 supp 0 ) ) |
93 |
|
sscon |
⊢ ( ( 𝑢 supp 0 ) ⊆ ( ( 𝑢 supp 0 ) ∪ ( 𝑣 supp 0 ) ) → ( 𝐷 ∖ ( ( 𝑢 supp 0 ) ∪ ( 𝑣 supp 0 ) ) ) ⊆ ( 𝐷 ∖ ( 𝑢 supp 0 ) ) ) |
94 |
92 93
|
ax-mp |
⊢ ( 𝐷 ∖ ( ( 𝑢 supp 0 ) ∪ ( 𝑣 supp 0 ) ) ) ⊆ ( 𝐷 ∖ ( 𝑢 supp 0 ) ) |
95 |
94
|
sseli |
⊢ ( 𝑘 ∈ ( 𝐷 ∖ ( ( 𝑢 supp 0 ) ∪ ( 𝑣 supp 0 ) ) ) → 𝑘 ∈ ( 𝐷 ∖ ( 𝑢 supp 0 ) ) ) |
96 |
|
ssidd |
⊢ ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) → ( 𝑢 supp 0 ) ⊆ ( 𝑢 supp 0 ) ) |
97 |
85
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) → 𝐷 ∈ V ) |
98 |
19
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) → 0 ∈ V ) |
99 |
79 96 97 98
|
suppssr |
⊢ ( ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) ∧ 𝑘 ∈ ( 𝐷 ∖ ( 𝑢 supp 0 ) ) ) → ( 𝑢 ‘ 𝑘 ) = 0 ) |
100 |
99
|
adantlr |
⊢ ( ( ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) ∧ 𝑣 ∈ 𝑈 ) ∧ 𝑘 ∈ ( 𝐷 ∖ ( 𝑢 supp 0 ) ) ) → ( 𝑢 ‘ 𝑘 ) = 0 ) |
101 |
95 100
|
sylan2 |
⊢ ( ( ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) ∧ 𝑣 ∈ 𝑈 ) ∧ 𝑘 ∈ ( 𝐷 ∖ ( ( 𝑢 supp 0 ) ∪ ( 𝑣 supp 0 ) ) ) ) → ( 𝑢 ‘ 𝑘 ) = 0 ) |
102 |
|
ssun2 |
⊢ ( 𝑣 supp 0 ) ⊆ ( ( 𝑢 supp 0 ) ∪ ( 𝑣 supp 0 ) ) |
103 |
|
sscon |
⊢ ( ( 𝑣 supp 0 ) ⊆ ( ( 𝑢 supp 0 ) ∪ ( 𝑣 supp 0 ) ) → ( 𝐷 ∖ ( ( 𝑢 supp 0 ) ∪ ( 𝑣 supp 0 ) ) ) ⊆ ( 𝐷 ∖ ( 𝑣 supp 0 ) ) ) |
104 |
102 103
|
ax-mp |
⊢ ( 𝐷 ∖ ( ( 𝑢 supp 0 ) ∪ ( 𝑣 supp 0 ) ) ) ⊆ ( 𝐷 ∖ ( 𝑣 supp 0 ) ) |
105 |
104
|
sseli |
⊢ ( 𝑘 ∈ ( 𝐷 ∖ ( ( 𝑢 supp 0 ) ∪ ( 𝑣 supp 0 ) ) ) → 𝑘 ∈ ( 𝐷 ∖ ( 𝑣 supp 0 ) ) ) |
106 |
|
ssidd |
⊢ ( ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) ∧ 𝑣 ∈ 𝑈 ) → ( 𝑣 supp 0 ) ⊆ ( 𝑣 supp 0 ) ) |
107 |
19
|
a1i |
⊢ ( ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) ∧ 𝑣 ∈ 𝑈 ) → 0 ∈ V ) |
108 |
82 106 86 107
|
suppssr |
⊢ ( ( ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) ∧ 𝑣 ∈ 𝑈 ) ∧ 𝑘 ∈ ( 𝐷 ∖ ( 𝑣 supp 0 ) ) ) → ( 𝑣 ‘ 𝑘 ) = 0 ) |
109 |
105 108
|
sylan2 |
⊢ ( ( ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) ∧ 𝑣 ∈ 𝑈 ) ∧ 𝑘 ∈ ( 𝐷 ∖ ( ( 𝑢 supp 0 ) ∪ ( 𝑣 supp 0 ) ) ) ) → ( 𝑣 ‘ 𝑘 ) = 0 ) |
110 |
101 109
|
oveq12d |
⊢ ( ( ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) ∧ 𝑣 ∈ 𝑈 ) ∧ 𝑘 ∈ ( 𝐷 ∖ ( ( 𝑢 supp 0 ) ∪ ( 𝑣 supp 0 ) ) ) ) → ( ( 𝑢 ‘ 𝑘 ) ( +g ‘ 𝑅 ) ( 𝑣 ‘ 𝑘 ) ) = ( 0 ( +g ‘ 𝑅 ) 0 ) ) |
111 |
14 74 3
|
grplid |
⊢ ( ( 𝑅 ∈ Grp ∧ 0 ∈ ( Base ‘ 𝑅 ) ) → ( 0 ( +g ‘ 𝑅 ) 0 ) = 0 ) |
112 |
35 15 111
|
syl2anc2 |
⊢ ( ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) ∧ 𝑣 ∈ 𝑈 ) → ( 0 ( +g ‘ 𝑅 ) 0 ) = 0 ) |
113 |
112
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) ∧ 𝑣 ∈ 𝑈 ) ∧ 𝑘 ∈ ( 𝐷 ∖ ( ( 𝑢 supp 0 ) ∪ ( 𝑣 supp 0 ) ) ) ) → ( 0 ( +g ‘ 𝑅 ) 0 ) = 0 ) |
114 |
110 113
|
eqtrd |
⊢ ( ( ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) ∧ 𝑣 ∈ 𝑈 ) ∧ 𝑘 ∈ ( 𝐷 ∖ ( ( 𝑢 supp 0 ) ∪ ( 𝑣 supp 0 ) ) ) ) → ( ( 𝑢 ‘ 𝑘 ) ( +g ‘ 𝑅 ) ( 𝑣 ‘ 𝑘 ) ) = 0 ) |
115 |
77 91 114
|
3eqtrd |
⊢ ( ( ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) ∧ 𝑣 ∈ 𝑈 ) ∧ 𝑘 ∈ ( 𝐷 ∖ ( ( 𝑢 supp 0 ) ∪ ( 𝑣 supp 0 ) ) ) ) → ( ( 𝑢 ( +g ‘ 𝑆 ) 𝑣 ) ‘ 𝑘 ) = 0 ) |
116 |
73 115
|
suppss |
⊢ ( ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) ∧ 𝑣 ∈ 𝑈 ) → ( ( 𝑢 ( +g ‘ 𝑆 ) 𝑣 ) supp 0 ) ⊆ ( ( 𝑢 supp 0 ) ∪ ( 𝑣 supp 0 ) ) ) |
117 |
|
sseq1 |
⊢ ( 𝑦 = ( ( 𝑢 ( +g ‘ 𝑆 ) 𝑣 ) supp 0 ) → ( 𝑦 ⊆ ( ( 𝑢 supp 0 ) ∪ ( 𝑣 supp 0 ) ) ↔ ( ( 𝑢 ( +g ‘ 𝑆 ) 𝑣 ) supp 0 ) ⊆ ( ( 𝑢 supp 0 ) ∪ ( 𝑣 supp 0 ) ) ) ) |
118 |
|
eleq1 |
⊢ ( 𝑦 = ( ( 𝑢 ( +g ‘ 𝑆 ) 𝑣 ) supp 0 ) → ( 𝑦 ∈ 𝐴 ↔ ( ( 𝑢 ( +g ‘ 𝑆 ) 𝑣 ) supp 0 ) ∈ 𝐴 ) ) |
119 |
117 118
|
imbi12d |
⊢ ( 𝑦 = ( ( 𝑢 ( +g ‘ 𝑆 ) 𝑣 ) supp 0 ) → ( ( 𝑦 ⊆ ( ( 𝑢 supp 0 ) ∪ ( 𝑣 supp 0 ) ) → 𝑦 ∈ 𝐴 ) ↔ ( ( ( 𝑢 ( +g ‘ 𝑆 ) 𝑣 ) supp 0 ) ⊆ ( ( 𝑢 supp 0 ) ∪ ( 𝑣 supp 0 ) ) → ( ( 𝑢 ( +g ‘ 𝑆 ) 𝑣 ) supp 0 ) ∈ 𝐴 ) ) ) |
120 |
119
|
spcgv |
⊢ ( ( ( 𝑢 ( +g ‘ 𝑆 ) 𝑣 ) supp 0 ) ∈ V → ( ∀ 𝑦 ( 𝑦 ⊆ ( ( 𝑢 supp 0 ) ∪ ( 𝑣 supp 0 ) ) → 𝑦 ∈ 𝐴 ) → ( ( ( 𝑢 ( +g ‘ 𝑆 ) 𝑣 ) supp 0 ) ⊆ ( ( 𝑢 supp 0 ) ∪ ( 𝑣 supp 0 ) ) → ( ( 𝑢 ( +g ‘ 𝑆 ) 𝑣 ) supp 0 ) ∈ 𝐴 ) ) ) |
121 |
53 72 116 120
|
syl3c |
⊢ ( ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) ∧ 𝑣 ∈ 𝑈 ) → ( ( 𝑢 ( +g ‘ 𝑆 ) 𝑣 ) supp 0 ) ∈ 𝐴 ) |
122 |
9
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) ∧ 𝑣 ∈ 𝑈 ) → 𝑈 = { 𝑔 ∈ 𝐵 ∣ ( 𝑔 supp 0 ) ∈ 𝐴 } ) |
123 |
122
|
eleq2d |
⊢ ( ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) ∧ 𝑣 ∈ 𝑈 ) → ( ( 𝑢 ( +g ‘ 𝑆 ) 𝑣 ) ∈ 𝑈 ↔ ( 𝑢 ( +g ‘ 𝑆 ) 𝑣 ) ∈ { 𝑔 ∈ 𝐵 ∣ ( 𝑔 supp 0 ) ∈ 𝐴 } ) ) |
124 |
|
oveq1 |
⊢ ( 𝑔 = ( 𝑢 ( +g ‘ 𝑆 ) 𝑣 ) → ( 𝑔 supp 0 ) = ( ( 𝑢 ( +g ‘ 𝑆 ) 𝑣 ) supp 0 ) ) |
125 |
124
|
eleq1d |
⊢ ( 𝑔 = ( 𝑢 ( +g ‘ 𝑆 ) 𝑣 ) → ( ( 𝑔 supp 0 ) ∈ 𝐴 ↔ ( ( 𝑢 ( +g ‘ 𝑆 ) 𝑣 ) supp 0 ) ∈ 𝐴 ) ) |
126 |
125
|
elrab |
⊢ ( ( 𝑢 ( +g ‘ 𝑆 ) 𝑣 ) ∈ { 𝑔 ∈ 𝐵 ∣ ( 𝑔 supp 0 ) ∈ 𝐴 } ↔ ( ( 𝑢 ( +g ‘ 𝑆 ) 𝑣 ) ∈ 𝐵 ∧ ( ( 𝑢 ( +g ‘ 𝑆 ) 𝑣 ) supp 0 ) ∈ 𝐴 ) ) |
127 |
123 126
|
bitrdi |
⊢ ( ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) ∧ 𝑣 ∈ 𝑈 ) → ( ( 𝑢 ( +g ‘ 𝑆 ) 𝑣 ) ∈ 𝑈 ↔ ( ( 𝑢 ( +g ‘ 𝑆 ) 𝑣 ) ∈ 𝐵 ∧ ( ( 𝑢 ( +g ‘ 𝑆 ) 𝑣 ) supp 0 ) ∈ 𝐴 ) ) ) |
128 |
52 121 127
|
mpbir2and |
⊢ ( ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) ∧ 𝑣 ∈ 𝑈 ) → ( 𝑢 ( +g ‘ 𝑆 ) 𝑣 ) ∈ 𝑈 ) |
129 |
128
|
ralrimiva |
⊢ ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) → ∀ 𝑣 ∈ 𝑈 ( 𝑢 ( +g ‘ 𝑆 ) 𝑣 ) ∈ 𝑈 ) |
130 |
1 5 10
|
psrgrp |
⊢ ( 𝜑 → 𝑆 ∈ Grp ) |
131 |
|
eqid |
⊢ ( invg ‘ 𝑆 ) = ( invg ‘ 𝑆 ) |
132 |
2 131
|
grpinvcl |
⊢ ( ( 𝑆 ∈ Grp ∧ 𝑢 ∈ 𝐵 ) → ( ( invg ‘ 𝑆 ) ‘ 𝑢 ) ∈ 𝐵 ) |
133 |
130 42 132
|
syl2an2r |
⊢ ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) → ( ( invg ‘ 𝑆 ) ‘ 𝑢 ) ∈ 𝐵 ) |
134 |
|
ovexd |
⊢ ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) → ( ( ( invg ‘ 𝑆 ) ‘ 𝑢 ) supp 0 ) ∈ V ) |
135 |
|
sseq2 |
⊢ ( 𝑥 = ( 𝑢 supp 0 ) → ( 𝑦 ⊆ 𝑥 ↔ 𝑦 ⊆ ( 𝑢 supp 0 ) ) ) |
136 |
135
|
imbi1d |
⊢ ( 𝑥 = ( 𝑢 supp 0 ) → ( ( 𝑦 ⊆ 𝑥 → 𝑦 ∈ 𝐴 ) ↔ ( 𝑦 ⊆ ( 𝑢 supp 0 ) → 𝑦 ∈ 𝐴 ) ) ) |
137 |
136
|
albidv |
⊢ ( 𝑥 = ( 𝑢 supp 0 ) → ( ∀ 𝑦 ( 𝑦 ⊆ 𝑥 → 𝑦 ∈ 𝐴 ) ↔ ∀ 𝑦 ( 𝑦 ⊆ ( 𝑢 supp 0 ) → 𝑦 ∈ 𝐴 ) ) ) |
138 |
59
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) → ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ( 𝑦 ⊆ 𝑥 → 𝑦 ∈ 𝐴 ) ) |
139 |
137 138 61
|
rspcdva |
⊢ ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) → ∀ 𝑦 ( 𝑦 ⊆ ( 𝑢 supp 0 ) → 𝑦 ∈ 𝐴 ) ) |
140 |
5
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) → 𝐼 ∈ 𝑊 ) |
141 |
10
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) → 𝑅 ∈ Grp ) |
142 |
|
eqid |
⊢ ( invg ‘ 𝑅 ) = ( invg ‘ 𝑅 ) |
143 |
1 140 141 4 142 2 131 42
|
psrneg |
⊢ ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) → ( ( invg ‘ 𝑆 ) ‘ 𝑢 ) = ( ( invg ‘ 𝑅 ) ∘ 𝑢 ) ) |
144 |
143
|
oveq1d |
⊢ ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) → ( ( ( invg ‘ 𝑆 ) ‘ 𝑢 ) supp 0 ) = ( ( ( invg ‘ 𝑅 ) ∘ 𝑢 ) supp 0 ) ) |
145 |
14 142
|
grpinvfn |
⊢ ( invg ‘ 𝑅 ) Fn ( Base ‘ 𝑅 ) |
146 |
145
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) → ( invg ‘ 𝑅 ) Fn ( Base ‘ 𝑅 ) ) |
147 |
3 142
|
grpinvid |
⊢ ( 𝑅 ∈ Grp → ( ( invg ‘ 𝑅 ) ‘ 0 ) = 0 ) |
148 |
141 147
|
syl |
⊢ ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) → ( ( invg ‘ 𝑅 ) ‘ 0 ) = 0 ) |
149 |
146 79 97 98 148
|
suppcoss |
⊢ ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) → ( ( ( invg ‘ 𝑅 ) ∘ 𝑢 ) supp 0 ) ⊆ ( 𝑢 supp 0 ) ) |
150 |
144 149
|
eqsstrd |
⊢ ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) → ( ( ( invg ‘ 𝑆 ) ‘ 𝑢 ) supp 0 ) ⊆ ( 𝑢 supp 0 ) ) |
151 |
|
sseq1 |
⊢ ( 𝑦 = ( ( ( invg ‘ 𝑆 ) ‘ 𝑢 ) supp 0 ) → ( 𝑦 ⊆ ( 𝑢 supp 0 ) ↔ ( ( ( invg ‘ 𝑆 ) ‘ 𝑢 ) supp 0 ) ⊆ ( 𝑢 supp 0 ) ) ) |
152 |
|
eleq1 |
⊢ ( 𝑦 = ( ( ( invg ‘ 𝑆 ) ‘ 𝑢 ) supp 0 ) → ( 𝑦 ∈ 𝐴 ↔ ( ( ( invg ‘ 𝑆 ) ‘ 𝑢 ) supp 0 ) ∈ 𝐴 ) ) |
153 |
151 152
|
imbi12d |
⊢ ( 𝑦 = ( ( ( invg ‘ 𝑆 ) ‘ 𝑢 ) supp 0 ) → ( ( 𝑦 ⊆ ( 𝑢 supp 0 ) → 𝑦 ∈ 𝐴 ) ↔ ( ( ( ( invg ‘ 𝑆 ) ‘ 𝑢 ) supp 0 ) ⊆ ( 𝑢 supp 0 ) → ( ( ( invg ‘ 𝑆 ) ‘ 𝑢 ) supp 0 ) ∈ 𝐴 ) ) ) |
154 |
153
|
spcgv |
⊢ ( ( ( ( invg ‘ 𝑆 ) ‘ 𝑢 ) supp 0 ) ∈ V → ( ∀ 𝑦 ( 𝑦 ⊆ ( 𝑢 supp 0 ) → 𝑦 ∈ 𝐴 ) → ( ( ( ( invg ‘ 𝑆 ) ‘ 𝑢 ) supp 0 ) ⊆ ( 𝑢 supp 0 ) → ( ( ( invg ‘ 𝑆 ) ‘ 𝑢 ) supp 0 ) ∈ 𝐴 ) ) ) |
155 |
134 139 150 154
|
syl3c |
⊢ ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) → ( ( ( invg ‘ 𝑆 ) ‘ 𝑢 ) supp 0 ) ∈ 𝐴 ) |
156 |
44
|
eleq2d |
⊢ ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) → ( ( ( invg ‘ 𝑆 ) ‘ 𝑢 ) ∈ 𝑈 ↔ ( ( invg ‘ 𝑆 ) ‘ 𝑢 ) ∈ { 𝑔 ∈ 𝐵 ∣ ( 𝑔 supp 0 ) ∈ 𝐴 } ) ) |
157 |
|
oveq1 |
⊢ ( 𝑔 = ( ( invg ‘ 𝑆 ) ‘ 𝑢 ) → ( 𝑔 supp 0 ) = ( ( ( invg ‘ 𝑆 ) ‘ 𝑢 ) supp 0 ) ) |
158 |
157
|
eleq1d |
⊢ ( 𝑔 = ( ( invg ‘ 𝑆 ) ‘ 𝑢 ) → ( ( 𝑔 supp 0 ) ∈ 𝐴 ↔ ( ( ( invg ‘ 𝑆 ) ‘ 𝑢 ) supp 0 ) ∈ 𝐴 ) ) |
159 |
158
|
elrab |
⊢ ( ( ( invg ‘ 𝑆 ) ‘ 𝑢 ) ∈ { 𝑔 ∈ 𝐵 ∣ ( 𝑔 supp 0 ) ∈ 𝐴 } ↔ ( ( ( invg ‘ 𝑆 ) ‘ 𝑢 ) ∈ 𝐵 ∧ ( ( ( invg ‘ 𝑆 ) ‘ 𝑢 ) supp 0 ) ∈ 𝐴 ) ) |
160 |
156 159
|
bitrdi |
⊢ ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) → ( ( ( invg ‘ 𝑆 ) ‘ 𝑢 ) ∈ 𝑈 ↔ ( ( ( invg ‘ 𝑆 ) ‘ 𝑢 ) ∈ 𝐵 ∧ ( ( ( invg ‘ 𝑆 ) ‘ 𝑢 ) supp 0 ) ∈ 𝐴 ) ) ) |
161 |
133 155 160
|
mpbir2and |
⊢ ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) → ( ( invg ‘ 𝑆 ) ‘ 𝑢 ) ∈ 𝑈 ) |
162 |
129 161
|
jca |
⊢ ( ( 𝜑 ∧ 𝑢 ∈ 𝑈 ) → ( ∀ 𝑣 ∈ 𝑈 ( 𝑢 ( +g ‘ 𝑆 ) 𝑣 ) ∈ 𝑈 ∧ ( ( invg ‘ 𝑆 ) ‘ 𝑢 ) ∈ 𝑈 ) ) |
163 |
162
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑢 ∈ 𝑈 ( ∀ 𝑣 ∈ 𝑈 ( 𝑢 ( +g ‘ 𝑆 ) 𝑣 ) ∈ 𝑈 ∧ ( ( invg ‘ 𝑆 ) ‘ 𝑢 ) ∈ 𝑈 ) ) |
164 |
2 34 131
|
issubg2 |
⊢ ( 𝑆 ∈ Grp → ( 𝑈 ∈ ( SubGrp ‘ 𝑆 ) ↔ ( 𝑈 ⊆ 𝐵 ∧ 𝑈 ≠ ∅ ∧ ∀ 𝑢 ∈ 𝑈 ( ∀ 𝑣 ∈ 𝑈 ( 𝑢 ( +g ‘ 𝑆 ) 𝑣 ) ∈ 𝑈 ∧ ( ( invg ‘ 𝑆 ) ‘ 𝑢 ) ∈ 𝑈 ) ) ) ) |
165 |
130 164
|
syl |
⊢ ( 𝜑 → ( 𝑈 ∈ ( SubGrp ‘ 𝑆 ) ↔ ( 𝑈 ⊆ 𝐵 ∧ 𝑈 ≠ ∅ ∧ ∀ 𝑢 ∈ 𝑈 ( ∀ 𝑣 ∈ 𝑈 ( 𝑢 ( +g ‘ 𝑆 ) 𝑣 ) ∈ 𝑈 ∧ ( ( invg ‘ 𝑆 ) ‘ 𝑢 ) ∈ 𝑈 ) ) ) ) |
166 |
12 33 163 165
|
mpbir3and |
⊢ ( 𝜑 → 𝑈 ∈ ( SubGrp ‘ 𝑆 ) ) |