Step |
Hyp |
Ref |
Expression |
1 |
|
smndex1ibas.m |
⊢ 𝑀 = ( EndoFMnd ‘ ℕ0 ) |
2 |
|
smndex1ibas.n |
⊢ 𝑁 ∈ ℕ |
3 |
|
smndex1ibas.i |
⊢ 𝐼 = ( 𝑥 ∈ ℕ0 ↦ ( 𝑥 mod 𝑁 ) ) |
4 |
|
smndex1ibas.g |
⊢ 𝐺 = ( 𝑛 ∈ ( 0 ..^ 𝑁 ) ↦ ( 𝑥 ∈ ℕ0 ↦ 𝑛 ) ) |
5 |
|
smndex1mgm.b |
⊢ 𝐵 = ( { 𝐼 } ∪ ∪ 𝑛 ∈ ( 0 ..^ 𝑁 ) { ( 𝐺 ‘ 𝑛 ) } ) |
6 |
|
smndex1mgm.s |
⊢ 𝑆 = ( 𝑀 ↾s 𝐵 ) |
7 |
1 2 3 4 5
|
smndex1basss |
⊢ 𝐵 ⊆ ( Base ‘ 𝑀 ) |
8 |
|
ssel |
⊢ ( 𝐵 ⊆ ( Base ‘ 𝑀 ) → ( 𝑎 ∈ 𝐵 → 𝑎 ∈ ( Base ‘ 𝑀 ) ) ) |
9 |
|
ssel |
⊢ ( 𝐵 ⊆ ( Base ‘ 𝑀 ) → ( 𝑏 ∈ 𝐵 → 𝑏 ∈ ( Base ‘ 𝑀 ) ) ) |
10 |
8 9
|
anim12d |
⊢ ( 𝐵 ⊆ ( Base ‘ 𝑀 ) → ( ( 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ) → ( 𝑎 ∈ ( Base ‘ 𝑀 ) ∧ 𝑏 ∈ ( Base ‘ 𝑀 ) ) ) ) |
11 |
7 10
|
ax-mp |
⊢ ( ( 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ) → ( 𝑎 ∈ ( Base ‘ 𝑀 ) ∧ 𝑏 ∈ ( Base ‘ 𝑀 ) ) ) |
12 |
|
eqid |
⊢ ( Base ‘ 𝑀 ) = ( Base ‘ 𝑀 ) |
13 |
|
eqid |
⊢ ( +g ‘ 𝑀 ) = ( +g ‘ 𝑀 ) |
14 |
1 12 13
|
efmndov |
⊢ ( ( 𝑎 ∈ ( Base ‘ 𝑀 ) ∧ 𝑏 ∈ ( Base ‘ 𝑀 ) ) → ( 𝑎 ( +g ‘ 𝑀 ) 𝑏 ) = ( 𝑎 ∘ 𝑏 ) ) |
15 |
11 14
|
syl |
⊢ ( ( 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ) → ( 𝑎 ( +g ‘ 𝑀 ) 𝑏 ) = ( 𝑎 ∘ 𝑏 ) ) |
16 |
|
simpl |
⊢ ( ( 𝑎 = 𝐼 ∧ 𝑏 = 𝐼 ) → 𝑎 = 𝐼 ) |
17 |
|
simpr |
⊢ ( ( 𝑎 = 𝐼 ∧ 𝑏 = 𝐼 ) → 𝑏 = 𝐼 ) |
18 |
16 17
|
coeq12d |
⊢ ( ( 𝑎 = 𝐼 ∧ 𝑏 = 𝐼 ) → ( 𝑎 ∘ 𝑏 ) = ( 𝐼 ∘ 𝐼 ) ) |
19 |
1 2 3
|
smndex1iidm |
⊢ ( 𝐼 ∘ 𝐼 ) = 𝐼 |
20 |
18 19
|
eqtrdi |
⊢ ( ( 𝑎 = 𝐼 ∧ 𝑏 = 𝐼 ) → ( 𝑎 ∘ 𝑏 ) = 𝐼 ) |
21 |
20
|
orcd |
⊢ ( ( 𝑎 = 𝐼 ∧ 𝑏 = 𝐼 ) → ( ( 𝑎 ∘ 𝑏 ) = 𝐼 ∨ ∃ 𝑘 ∈ ( 0 ..^ 𝑁 ) ( 𝑎 ∘ 𝑏 ) = ( 𝐺 ‘ 𝑘 ) ) ) |
22 |
21
|
ex |
⊢ ( 𝑎 = 𝐼 → ( 𝑏 = 𝐼 → ( ( 𝑎 ∘ 𝑏 ) = 𝐼 ∨ ∃ 𝑘 ∈ ( 0 ..^ 𝑁 ) ( 𝑎 ∘ 𝑏 ) = ( 𝐺 ‘ 𝑘 ) ) ) ) |
23 |
|
simpll |
⊢ ( ( ( 𝑎 = 𝐼 ∧ 𝑘 ∈ ( 0 ..^ 𝑁 ) ) ∧ 𝑏 = ( 𝐺 ‘ 𝑘 ) ) → 𝑎 = 𝐼 ) |
24 |
|
simpr |
⊢ ( ( ( 𝑎 = 𝐼 ∧ 𝑘 ∈ ( 0 ..^ 𝑁 ) ) ∧ 𝑏 = ( 𝐺 ‘ 𝑘 ) ) → 𝑏 = ( 𝐺 ‘ 𝑘 ) ) |
25 |
23 24
|
coeq12d |
⊢ ( ( ( 𝑎 = 𝐼 ∧ 𝑘 ∈ ( 0 ..^ 𝑁 ) ) ∧ 𝑏 = ( 𝐺 ‘ 𝑘 ) ) → ( 𝑎 ∘ 𝑏 ) = ( 𝐼 ∘ ( 𝐺 ‘ 𝑘 ) ) ) |
26 |
1 2 3 4
|
smndex1igid |
⊢ ( 𝑘 ∈ ( 0 ..^ 𝑁 ) → ( 𝐼 ∘ ( 𝐺 ‘ 𝑘 ) ) = ( 𝐺 ‘ 𝑘 ) ) |
27 |
26
|
ad2antlr |
⊢ ( ( ( 𝑎 = 𝐼 ∧ 𝑘 ∈ ( 0 ..^ 𝑁 ) ) ∧ 𝑏 = ( 𝐺 ‘ 𝑘 ) ) → ( 𝐼 ∘ ( 𝐺 ‘ 𝑘 ) ) = ( 𝐺 ‘ 𝑘 ) ) |
28 |
25 27
|
eqtrd |
⊢ ( ( ( 𝑎 = 𝐼 ∧ 𝑘 ∈ ( 0 ..^ 𝑁 ) ) ∧ 𝑏 = ( 𝐺 ‘ 𝑘 ) ) → ( 𝑎 ∘ 𝑏 ) = ( 𝐺 ‘ 𝑘 ) ) |
29 |
28
|
ex |
⊢ ( ( 𝑎 = 𝐼 ∧ 𝑘 ∈ ( 0 ..^ 𝑁 ) ) → ( 𝑏 = ( 𝐺 ‘ 𝑘 ) → ( 𝑎 ∘ 𝑏 ) = ( 𝐺 ‘ 𝑘 ) ) ) |
30 |
29
|
reximdva |
⊢ ( 𝑎 = 𝐼 → ( ∃ 𝑘 ∈ ( 0 ..^ 𝑁 ) 𝑏 = ( 𝐺 ‘ 𝑘 ) → ∃ 𝑘 ∈ ( 0 ..^ 𝑁 ) ( 𝑎 ∘ 𝑏 ) = ( 𝐺 ‘ 𝑘 ) ) ) |
31 |
30
|
imp |
⊢ ( ( 𝑎 = 𝐼 ∧ ∃ 𝑘 ∈ ( 0 ..^ 𝑁 ) 𝑏 = ( 𝐺 ‘ 𝑘 ) ) → ∃ 𝑘 ∈ ( 0 ..^ 𝑁 ) ( 𝑎 ∘ 𝑏 ) = ( 𝐺 ‘ 𝑘 ) ) |
32 |
31
|
olcd |
⊢ ( ( 𝑎 = 𝐼 ∧ ∃ 𝑘 ∈ ( 0 ..^ 𝑁 ) 𝑏 = ( 𝐺 ‘ 𝑘 ) ) → ( ( 𝑎 ∘ 𝑏 ) = 𝐼 ∨ ∃ 𝑘 ∈ ( 0 ..^ 𝑁 ) ( 𝑎 ∘ 𝑏 ) = ( 𝐺 ‘ 𝑘 ) ) ) |
33 |
32
|
ex |
⊢ ( 𝑎 = 𝐼 → ( ∃ 𝑘 ∈ ( 0 ..^ 𝑁 ) 𝑏 = ( 𝐺 ‘ 𝑘 ) → ( ( 𝑎 ∘ 𝑏 ) = 𝐼 ∨ ∃ 𝑘 ∈ ( 0 ..^ 𝑁 ) ( 𝑎 ∘ 𝑏 ) = ( 𝐺 ‘ 𝑘 ) ) ) ) |
34 |
22 33
|
jaod |
⊢ ( 𝑎 = 𝐼 → ( ( 𝑏 = 𝐼 ∨ ∃ 𝑘 ∈ ( 0 ..^ 𝑁 ) 𝑏 = ( 𝐺 ‘ 𝑘 ) ) → ( ( 𝑎 ∘ 𝑏 ) = 𝐼 ∨ ∃ 𝑘 ∈ ( 0 ..^ 𝑁 ) ( 𝑎 ∘ 𝑏 ) = ( 𝐺 ‘ 𝑘 ) ) ) ) |
35 |
|
simpr |
⊢ ( ( ( 𝑏 = 𝐼 ∧ 𝑘 ∈ ( 0 ..^ 𝑁 ) ) ∧ 𝑎 = ( 𝐺 ‘ 𝑘 ) ) → 𝑎 = ( 𝐺 ‘ 𝑘 ) ) |
36 |
|
simpll |
⊢ ( ( ( 𝑏 = 𝐼 ∧ 𝑘 ∈ ( 0 ..^ 𝑁 ) ) ∧ 𝑎 = ( 𝐺 ‘ 𝑘 ) ) → 𝑏 = 𝐼 ) |
37 |
35 36
|
coeq12d |
⊢ ( ( ( 𝑏 = 𝐼 ∧ 𝑘 ∈ ( 0 ..^ 𝑁 ) ) ∧ 𝑎 = ( 𝐺 ‘ 𝑘 ) ) → ( 𝑎 ∘ 𝑏 ) = ( ( 𝐺 ‘ 𝑘 ) ∘ 𝐼 ) ) |
38 |
1 2 3
|
smndex1ibas |
⊢ 𝐼 ∈ ( Base ‘ 𝑀 ) |
39 |
1 2 3 4
|
smndex1gid |
⊢ ( ( 𝐼 ∈ ( Base ‘ 𝑀 ) ∧ 𝑘 ∈ ( 0 ..^ 𝑁 ) ) → ( ( 𝐺 ‘ 𝑘 ) ∘ 𝐼 ) = ( 𝐺 ‘ 𝑘 ) ) |
40 |
38 39
|
mpan |
⊢ ( 𝑘 ∈ ( 0 ..^ 𝑁 ) → ( ( 𝐺 ‘ 𝑘 ) ∘ 𝐼 ) = ( 𝐺 ‘ 𝑘 ) ) |
41 |
40
|
ad2antlr |
⊢ ( ( ( 𝑏 = 𝐼 ∧ 𝑘 ∈ ( 0 ..^ 𝑁 ) ) ∧ 𝑎 = ( 𝐺 ‘ 𝑘 ) ) → ( ( 𝐺 ‘ 𝑘 ) ∘ 𝐼 ) = ( 𝐺 ‘ 𝑘 ) ) |
42 |
37 41
|
eqtrd |
⊢ ( ( ( 𝑏 = 𝐼 ∧ 𝑘 ∈ ( 0 ..^ 𝑁 ) ) ∧ 𝑎 = ( 𝐺 ‘ 𝑘 ) ) → ( 𝑎 ∘ 𝑏 ) = ( 𝐺 ‘ 𝑘 ) ) |
43 |
42
|
ex |
⊢ ( ( 𝑏 = 𝐼 ∧ 𝑘 ∈ ( 0 ..^ 𝑁 ) ) → ( 𝑎 = ( 𝐺 ‘ 𝑘 ) → ( 𝑎 ∘ 𝑏 ) = ( 𝐺 ‘ 𝑘 ) ) ) |
44 |
43
|
reximdva |
⊢ ( 𝑏 = 𝐼 → ( ∃ 𝑘 ∈ ( 0 ..^ 𝑁 ) 𝑎 = ( 𝐺 ‘ 𝑘 ) → ∃ 𝑘 ∈ ( 0 ..^ 𝑁 ) ( 𝑎 ∘ 𝑏 ) = ( 𝐺 ‘ 𝑘 ) ) ) |
45 |
44
|
imp |
⊢ ( ( 𝑏 = 𝐼 ∧ ∃ 𝑘 ∈ ( 0 ..^ 𝑁 ) 𝑎 = ( 𝐺 ‘ 𝑘 ) ) → ∃ 𝑘 ∈ ( 0 ..^ 𝑁 ) ( 𝑎 ∘ 𝑏 ) = ( 𝐺 ‘ 𝑘 ) ) |
46 |
45
|
olcd |
⊢ ( ( 𝑏 = 𝐼 ∧ ∃ 𝑘 ∈ ( 0 ..^ 𝑁 ) 𝑎 = ( 𝐺 ‘ 𝑘 ) ) → ( ( 𝑎 ∘ 𝑏 ) = 𝐼 ∨ ∃ 𝑘 ∈ ( 0 ..^ 𝑁 ) ( 𝑎 ∘ 𝑏 ) = ( 𝐺 ‘ 𝑘 ) ) ) |
47 |
46
|
expcom |
⊢ ( ∃ 𝑘 ∈ ( 0 ..^ 𝑁 ) 𝑎 = ( 𝐺 ‘ 𝑘 ) → ( 𝑏 = 𝐼 → ( ( 𝑎 ∘ 𝑏 ) = 𝐼 ∨ ∃ 𝑘 ∈ ( 0 ..^ 𝑁 ) ( 𝑎 ∘ 𝑏 ) = ( 𝐺 ‘ 𝑘 ) ) ) ) |
48 |
|
fveq2 |
⊢ ( 𝑘 = 𝑚 → ( 𝐺 ‘ 𝑘 ) = ( 𝐺 ‘ 𝑚 ) ) |
49 |
48
|
eqeq2d |
⊢ ( 𝑘 = 𝑚 → ( 𝑏 = ( 𝐺 ‘ 𝑘 ) ↔ 𝑏 = ( 𝐺 ‘ 𝑚 ) ) ) |
50 |
49
|
cbvrexvw |
⊢ ( ∃ 𝑘 ∈ ( 0 ..^ 𝑁 ) 𝑏 = ( 𝐺 ‘ 𝑘 ) ↔ ∃ 𝑚 ∈ ( 0 ..^ 𝑁 ) 𝑏 = ( 𝐺 ‘ 𝑚 ) ) |
51 |
|
simpr |
⊢ ( ( ( ( 𝑚 ∈ ( 0 ..^ 𝑁 ) ∧ 𝑏 = ( 𝐺 ‘ 𝑚 ) ) ∧ 𝑘 ∈ ( 0 ..^ 𝑁 ) ) ∧ 𝑎 = ( 𝐺 ‘ 𝑘 ) ) → 𝑎 = ( 𝐺 ‘ 𝑘 ) ) |
52 |
|
simpllr |
⊢ ( ( ( ( 𝑚 ∈ ( 0 ..^ 𝑁 ) ∧ 𝑏 = ( 𝐺 ‘ 𝑚 ) ) ∧ 𝑘 ∈ ( 0 ..^ 𝑁 ) ) ∧ 𝑎 = ( 𝐺 ‘ 𝑘 ) ) → 𝑏 = ( 𝐺 ‘ 𝑚 ) ) |
53 |
51 52
|
coeq12d |
⊢ ( ( ( ( 𝑚 ∈ ( 0 ..^ 𝑁 ) ∧ 𝑏 = ( 𝐺 ‘ 𝑚 ) ) ∧ 𝑘 ∈ ( 0 ..^ 𝑁 ) ) ∧ 𝑎 = ( 𝐺 ‘ 𝑘 ) ) → ( 𝑎 ∘ 𝑏 ) = ( ( 𝐺 ‘ 𝑘 ) ∘ ( 𝐺 ‘ 𝑚 ) ) ) |
54 |
1 2 3 4
|
smndex1gbas |
⊢ ( 𝑚 ∈ ( 0 ..^ 𝑁 ) → ( 𝐺 ‘ 𝑚 ) ∈ ( Base ‘ 𝑀 ) ) |
55 |
1 2 3 4
|
smndex1gid |
⊢ ( ( ( 𝐺 ‘ 𝑚 ) ∈ ( Base ‘ 𝑀 ) ∧ 𝑘 ∈ ( 0 ..^ 𝑁 ) ) → ( ( 𝐺 ‘ 𝑘 ) ∘ ( 𝐺 ‘ 𝑚 ) ) = ( 𝐺 ‘ 𝑘 ) ) |
56 |
54 55
|
sylan |
⊢ ( ( 𝑚 ∈ ( 0 ..^ 𝑁 ) ∧ 𝑘 ∈ ( 0 ..^ 𝑁 ) ) → ( ( 𝐺 ‘ 𝑘 ) ∘ ( 𝐺 ‘ 𝑚 ) ) = ( 𝐺 ‘ 𝑘 ) ) |
57 |
56
|
ad4ant13 |
⊢ ( ( ( ( 𝑚 ∈ ( 0 ..^ 𝑁 ) ∧ 𝑏 = ( 𝐺 ‘ 𝑚 ) ) ∧ 𝑘 ∈ ( 0 ..^ 𝑁 ) ) ∧ 𝑎 = ( 𝐺 ‘ 𝑘 ) ) → ( ( 𝐺 ‘ 𝑘 ) ∘ ( 𝐺 ‘ 𝑚 ) ) = ( 𝐺 ‘ 𝑘 ) ) |
58 |
53 57
|
eqtrd |
⊢ ( ( ( ( 𝑚 ∈ ( 0 ..^ 𝑁 ) ∧ 𝑏 = ( 𝐺 ‘ 𝑚 ) ) ∧ 𝑘 ∈ ( 0 ..^ 𝑁 ) ) ∧ 𝑎 = ( 𝐺 ‘ 𝑘 ) ) → ( 𝑎 ∘ 𝑏 ) = ( 𝐺 ‘ 𝑘 ) ) |
59 |
58
|
ex |
⊢ ( ( ( 𝑚 ∈ ( 0 ..^ 𝑁 ) ∧ 𝑏 = ( 𝐺 ‘ 𝑚 ) ) ∧ 𝑘 ∈ ( 0 ..^ 𝑁 ) ) → ( 𝑎 = ( 𝐺 ‘ 𝑘 ) → ( 𝑎 ∘ 𝑏 ) = ( 𝐺 ‘ 𝑘 ) ) ) |
60 |
59
|
reximdva |
⊢ ( ( 𝑚 ∈ ( 0 ..^ 𝑁 ) ∧ 𝑏 = ( 𝐺 ‘ 𝑚 ) ) → ( ∃ 𝑘 ∈ ( 0 ..^ 𝑁 ) 𝑎 = ( 𝐺 ‘ 𝑘 ) → ∃ 𝑘 ∈ ( 0 ..^ 𝑁 ) ( 𝑎 ∘ 𝑏 ) = ( 𝐺 ‘ 𝑘 ) ) ) |
61 |
60
|
rexlimiva |
⊢ ( ∃ 𝑚 ∈ ( 0 ..^ 𝑁 ) 𝑏 = ( 𝐺 ‘ 𝑚 ) → ( ∃ 𝑘 ∈ ( 0 ..^ 𝑁 ) 𝑎 = ( 𝐺 ‘ 𝑘 ) → ∃ 𝑘 ∈ ( 0 ..^ 𝑁 ) ( 𝑎 ∘ 𝑏 ) = ( 𝐺 ‘ 𝑘 ) ) ) |
62 |
61
|
imp |
⊢ ( ( ∃ 𝑚 ∈ ( 0 ..^ 𝑁 ) 𝑏 = ( 𝐺 ‘ 𝑚 ) ∧ ∃ 𝑘 ∈ ( 0 ..^ 𝑁 ) 𝑎 = ( 𝐺 ‘ 𝑘 ) ) → ∃ 𝑘 ∈ ( 0 ..^ 𝑁 ) ( 𝑎 ∘ 𝑏 ) = ( 𝐺 ‘ 𝑘 ) ) |
63 |
62
|
olcd |
⊢ ( ( ∃ 𝑚 ∈ ( 0 ..^ 𝑁 ) 𝑏 = ( 𝐺 ‘ 𝑚 ) ∧ ∃ 𝑘 ∈ ( 0 ..^ 𝑁 ) 𝑎 = ( 𝐺 ‘ 𝑘 ) ) → ( ( 𝑎 ∘ 𝑏 ) = 𝐼 ∨ ∃ 𝑘 ∈ ( 0 ..^ 𝑁 ) ( 𝑎 ∘ 𝑏 ) = ( 𝐺 ‘ 𝑘 ) ) ) |
64 |
63
|
expcom |
⊢ ( ∃ 𝑘 ∈ ( 0 ..^ 𝑁 ) 𝑎 = ( 𝐺 ‘ 𝑘 ) → ( ∃ 𝑚 ∈ ( 0 ..^ 𝑁 ) 𝑏 = ( 𝐺 ‘ 𝑚 ) → ( ( 𝑎 ∘ 𝑏 ) = 𝐼 ∨ ∃ 𝑘 ∈ ( 0 ..^ 𝑁 ) ( 𝑎 ∘ 𝑏 ) = ( 𝐺 ‘ 𝑘 ) ) ) ) |
65 |
50 64
|
syl5bi |
⊢ ( ∃ 𝑘 ∈ ( 0 ..^ 𝑁 ) 𝑎 = ( 𝐺 ‘ 𝑘 ) → ( ∃ 𝑘 ∈ ( 0 ..^ 𝑁 ) 𝑏 = ( 𝐺 ‘ 𝑘 ) → ( ( 𝑎 ∘ 𝑏 ) = 𝐼 ∨ ∃ 𝑘 ∈ ( 0 ..^ 𝑁 ) ( 𝑎 ∘ 𝑏 ) = ( 𝐺 ‘ 𝑘 ) ) ) ) |
66 |
47 65
|
jaod |
⊢ ( ∃ 𝑘 ∈ ( 0 ..^ 𝑁 ) 𝑎 = ( 𝐺 ‘ 𝑘 ) → ( ( 𝑏 = 𝐼 ∨ ∃ 𝑘 ∈ ( 0 ..^ 𝑁 ) 𝑏 = ( 𝐺 ‘ 𝑘 ) ) → ( ( 𝑎 ∘ 𝑏 ) = 𝐼 ∨ ∃ 𝑘 ∈ ( 0 ..^ 𝑁 ) ( 𝑎 ∘ 𝑏 ) = ( 𝐺 ‘ 𝑘 ) ) ) ) |
67 |
34 66
|
jaoi |
⊢ ( ( 𝑎 = 𝐼 ∨ ∃ 𝑘 ∈ ( 0 ..^ 𝑁 ) 𝑎 = ( 𝐺 ‘ 𝑘 ) ) → ( ( 𝑏 = 𝐼 ∨ ∃ 𝑘 ∈ ( 0 ..^ 𝑁 ) 𝑏 = ( 𝐺 ‘ 𝑘 ) ) → ( ( 𝑎 ∘ 𝑏 ) = 𝐼 ∨ ∃ 𝑘 ∈ ( 0 ..^ 𝑁 ) ( 𝑎 ∘ 𝑏 ) = ( 𝐺 ‘ 𝑘 ) ) ) ) |
68 |
67
|
imp |
⊢ ( ( ( 𝑎 = 𝐼 ∨ ∃ 𝑘 ∈ ( 0 ..^ 𝑁 ) 𝑎 = ( 𝐺 ‘ 𝑘 ) ) ∧ ( 𝑏 = 𝐼 ∨ ∃ 𝑘 ∈ ( 0 ..^ 𝑁 ) 𝑏 = ( 𝐺 ‘ 𝑘 ) ) ) → ( ( 𝑎 ∘ 𝑏 ) = 𝐼 ∨ ∃ 𝑘 ∈ ( 0 ..^ 𝑁 ) ( 𝑎 ∘ 𝑏 ) = ( 𝐺 ‘ 𝑘 ) ) ) |
69 |
5
|
eleq2i |
⊢ ( 𝑎 ∈ 𝐵 ↔ 𝑎 ∈ ( { 𝐼 } ∪ ∪ 𝑛 ∈ ( 0 ..^ 𝑁 ) { ( 𝐺 ‘ 𝑛 ) } ) ) |
70 |
|
fveq2 |
⊢ ( 𝑛 = 𝑘 → ( 𝐺 ‘ 𝑛 ) = ( 𝐺 ‘ 𝑘 ) ) |
71 |
70
|
sneqd |
⊢ ( 𝑛 = 𝑘 → { ( 𝐺 ‘ 𝑛 ) } = { ( 𝐺 ‘ 𝑘 ) } ) |
72 |
71
|
cbviunv |
⊢ ∪ 𝑛 ∈ ( 0 ..^ 𝑁 ) { ( 𝐺 ‘ 𝑛 ) } = ∪ 𝑘 ∈ ( 0 ..^ 𝑁 ) { ( 𝐺 ‘ 𝑘 ) } |
73 |
72
|
uneq2i |
⊢ ( { 𝐼 } ∪ ∪ 𝑛 ∈ ( 0 ..^ 𝑁 ) { ( 𝐺 ‘ 𝑛 ) } ) = ( { 𝐼 } ∪ ∪ 𝑘 ∈ ( 0 ..^ 𝑁 ) { ( 𝐺 ‘ 𝑘 ) } ) |
74 |
73
|
eleq2i |
⊢ ( 𝑎 ∈ ( { 𝐼 } ∪ ∪ 𝑛 ∈ ( 0 ..^ 𝑁 ) { ( 𝐺 ‘ 𝑛 ) } ) ↔ 𝑎 ∈ ( { 𝐼 } ∪ ∪ 𝑘 ∈ ( 0 ..^ 𝑁 ) { ( 𝐺 ‘ 𝑘 ) } ) ) |
75 |
69 74
|
bitri |
⊢ ( 𝑎 ∈ 𝐵 ↔ 𝑎 ∈ ( { 𝐼 } ∪ ∪ 𝑘 ∈ ( 0 ..^ 𝑁 ) { ( 𝐺 ‘ 𝑘 ) } ) ) |
76 |
|
elun |
⊢ ( 𝑎 ∈ ( { 𝐼 } ∪ ∪ 𝑘 ∈ ( 0 ..^ 𝑁 ) { ( 𝐺 ‘ 𝑘 ) } ) ↔ ( 𝑎 ∈ { 𝐼 } ∨ 𝑎 ∈ ∪ 𝑘 ∈ ( 0 ..^ 𝑁 ) { ( 𝐺 ‘ 𝑘 ) } ) ) |
77 |
|
velsn |
⊢ ( 𝑎 ∈ { 𝐼 } ↔ 𝑎 = 𝐼 ) |
78 |
|
eliun |
⊢ ( 𝑎 ∈ ∪ 𝑘 ∈ ( 0 ..^ 𝑁 ) { ( 𝐺 ‘ 𝑘 ) } ↔ ∃ 𝑘 ∈ ( 0 ..^ 𝑁 ) 𝑎 ∈ { ( 𝐺 ‘ 𝑘 ) } ) |
79 |
|
velsn |
⊢ ( 𝑎 ∈ { ( 𝐺 ‘ 𝑘 ) } ↔ 𝑎 = ( 𝐺 ‘ 𝑘 ) ) |
80 |
79
|
rexbii |
⊢ ( ∃ 𝑘 ∈ ( 0 ..^ 𝑁 ) 𝑎 ∈ { ( 𝐺 ‘ 𝑘 ) } ↔ ∃ 𝑘 ∈ ( 0 ..^ 𝑁 ) 𝑎 = ( 𝐺 ‘ 𝑘 ) ) |
81 |
78 80
|
bitri |
⊢ ( 𝑎 ∈ ∪ 𝑘 ∈ ( 0 ..^ 𝑁 ) { ( 𝐺 ‘ 𝑘 ) } ↔ ∃ 𝑘 ∈ ( 0 ..^ 𝑁 ) 𝑎 = ( 𝐺 ‘ 𝑘 ) ) |
82 |
77 81
|
orbi12i |
⊢ ( ( 𝑎 ∈ { 𝐼 } ∨ 𝑎 ∈ ∪ 𝑘 ∈ ( 0 ..^ 𝑁 ) { ( 𝐺 ‘ 𝑘 ) } ) ↔ ( 𝑎 = 𝐼 ∨ ∃ 𝑘 ∈ ( 0 ..^ 𝑁 ) 𝑎 = ( 𝐺 ‘ 𝑘 ) ) ) |
83 |
75 76 82
|
3bitri |
⊢ ( 𝑎 ∈ 𝐵 ↔ ( 𝑎 = 𝐼 ∨ ∃ 𝑘 ∈ ( 0 ..^ 𝑁 ) 𝑎 = ( 𝐺 ‘ 𝑘 ) ) ) |
84 |
5
|
eleq2i |
⊢ ( 𝑏 ∈ 𝐵 ↔ 𝑏 ∈ ( { 𝐼 } ∪ ∪ 𝑛 ∈ ( 0 ..^ 𝑁 ) { ( 𝐺 ‘ 𝑛 ) } ) ) |
85 |
73
|
eleq2i |
⊢ ( 𝑏 ∈ ( { 𝐼 } ∪ ∪ 𝑛 ∈ ( 0 ..^ 𝑁 ) { ( 𝐺 ‘ 𝑛 ) } ) ↔ 𝑏 ∈ ( { 𝐼 } ∪ ∪ 𝑘 ∈ ( 0 ..^ 𝑁 ) { ( 𝐺 ‘ 𝑘 ) } ) ) |
86 |
84 85
|
bitri |
⊢ ( 𝑏 ∈ 𝐵 ↔ 𝑏 ∈ ( { 𝐼 } ∪ ∪ 𝑘 ∈ ( 0 ..^ 𝑁 ) { ( 𝐺 ‘ 𝑘 ) } ) ) |
87 |
|
elun |
⊢ ( 𝑏 ∈ ( { 𝐼 } ∪ ∪ 𝑘 ∈ ( 0 ..^ 𝑁 ) { ( 𝐺 ‘ 𝑘 ) } ) ↔ ( 𝑏 ∈ { 𝐼 } ∨ 𝑏 ∈ ∪ 𝑘 ∈ ( 0 ..^ 𝑁 ) { ( 𝐺 ‘ 𝑘 ) } ) ) |
88 |
|
velsn |
⊢ ( 𝑏 ∈ { 𝐼 } ↔ 𝑏 = 𝐼 ) |
89 |
|
eliun |
⊢ ( 𝑏 ∈ ∪ 𝑘 ∈ ( 0 ..^ 𝑁 ) { ( 𝐺 ‘ 𝑘 ) } ↔ ∃ 𝑘 ∈ ( 0 ..^ 𝑁 ) 𝑏 ∈ { ( 𝐺 ‘ 𝑘 ) } ) |
90 |
|
velsn |
⊢ ( 𝑏 ∈ { ( 𝐺 ‘ 𝑘 ) } ↔ 𝑏 = ( 𝐺 ‘ 𝑘 ) ) |
91 |
90
|
rexbii |
⊢ ( ∃ 𝑘 ∈ ( 0 ..^ 𝑁 ) 𝑏 ∈ { ( 𝐺 ‘ 𝑘 ) } ↔ ∃ 𝑘 ∈ ( 0 ..^ 𝑁 ) 𝑏 = ( 𝐺 ‘ 𝑘 ) ) |
92 |
89 91
|
bitri |
⊢ ( 𝑏 ∈ ∪ 𝑘 ∈ ( 0 ..^ 𝑁 ) { ( 𝐺 ‘ 𝑘 ) } ↔ ∃ 𝑘 ∈ ( 0 ..^ 𝑁 ) 𝑏 = ( 𝐺 ‘ 𝑘 ) ) |
93 |
88 92
|
orbi12i |
⊢ ( ( 𝑏 ∈ { 𝐼 } ∨ 𝑏 ∈ ∪ 𝑘 ∈ ( 0 ..^ 𝑁 ) { ( 𝐺 ‘ 𝑘 ) } ) ↔ ( 𝑏 = 𝐼 ∨ ∃ 𝑘 ∈ ( 0 ..^ 𝑁 ) 𝑏 = ( 𝐺 ‘ 𝑘 ) ) ) |
94 |
86 87 93
|
3bitri |
⊢ ( 𝑏 ∈ 𝐵 ↔ ( 𝑏 = 𝐼 ∨ ∃ 𝑘 ∈ ( 0 ..^ 𝑁 ) 𝑏 = ( 𝐺 ‘ 𝑘 ) ) ) |
95 |
83 94
|
anbi12i |
⊢ ( ( 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ) ↔ ( ( 𝑎 = 𝐼 ∨ ∃ 𝑘 ∈ ( 0 ..^ 𝑁 ) 𝑎 = ( 𝐺 ‘ 𝑘 ) ) ∧ ( 𝑏 = 𝐼 ∨ ∃ 𝑘 ∈ ( 0 ..^ 𝑁 ) 𝑏 = ( 𝐺 ‘ 𝑘 ) ) ) ) |
96 |
5
|
eleq2i |
⊢ ( ( 𝑎 ∘ 𝑏 ) ∈ 𝐵 ↔ ( 𝑎 ∘ 𝑏 ) ∈ ( { 𝐼 } ∪ ∪ 𝑛 ∈ ( 0 ..^ 𝑁 ) { ( 𝐺 ‘ 𝑛 ) } ) ) |
97 |
73
|
eleq2i |
⊢ ( ( 𝑎 ∘ 𝑏 ) ∈ ( { 𝐼 } ∪ ∪ 𝑛 ∈ ( 0 ..^ 𝑁 ) { ( 𝐺 ‘ 𝑛 ) } ) ↔ ( 𝑎 ∘ 𝑏 ) ∈ ( { 𝐼 } ∪ ∪ 𝑘 ∈ ( 0 ..^ 𝑁 ) { ( 𝐺 ‘ 𝑘 ) } ) ) |
98 |
96 97
|
bitri |
⊢ ( ( 𝑎 ∘ 𝑏 ) ∈ 𝐵 ↔ ( 𝑎 ∘ 𝑏 ) ∈ ( { 𝐼 } ∪ ∪ 𝑘 ∈ ( 0 ..^ 𝑁 ) { ( 𝐺 ‘ 𝑘 ) } ) ) |
99 |
|
elun |
⊢ ( ( 𝑎 ∘ 𝑏 ) ∈ ( { 𝐼 } ∪ ∪ 𝑘 ∈ ( 0 ..^ 𝑁 ) { ( 𝐺 ‘ 𝑘 ) } ) ↔ ( ( 𝑎 ∘ 𝑏 ) ∈ { 𝐼 } ∨ ( 𝑎 ∘ 𝑏 ) ∈ ∪ 𝑘 ∈ ( 0 ..^ 𝑁 ) { ( 𝐺 ‘ 𝑘 ) } ) ) |
100 |
|
vex |
⊢ 𝑎 ∈ V |
101 |
|
vex |
⊢ 𝑏 ∈ V |
102 |
100 101
|
coex |
⊢ ( 𝑎 ∘ 𝑏 ) ∈ V |
103 |
102
|
elsn |
⊢ ( ( 𝑎 ∘ 𝑏 ) ∈ { 𝐼 } ↔ ( 𝑎 ∘ 𝑏 ) = 𝐼 ) |
104 |
|
eliun |
⊢ ( ( 𝑎 ∘ 𝑏 ) ∈ ∪ 𝑘 ∈ ( 0 ..^ 𝑁 ) { ( 𝐺 ‘ 𝑘 ) } ↔ ∃ 𝑘 ∈ ( 0 ..^ 𝑁 ) ( 𝑎 ∘ 𝑏 ) ∈ { ( 𝐺 ‘ 𝑘 ) } ) |
105 |
102
|
elsn |
⊢ ( ( 𝑎 ∘ 𝑏 ) ∈ { ( 𝐺 ‘ 𝑘 ) } ↔ ( 𝑎 ∘ 𝑏 ) = ( 𝐺 ‘ 𝑘 ) ) |
106 |
105
|
rexbii |
⊢ ( ∃ 𝑘 ∈ ( 0 ..^ 𝑁 ) ( 𝑎 ∘ 𝑏 ) ∈ { ( 𝐺 ‘ 𝑘 ) } ↔ ∃ 𝑘 ∈ ( 0 ..^ 𝑁 ) ( 𝑎 ∘ 𝑏 ) = ( 𝐺 ‘ 𝑘 ) ) |
107 |
104 106
|
bitri |
⊢ ( ( 𝑎 ∘ 𝑏 ) ∈ ∪ 𝑘 ∈ ( 0 ..^ 𝑁 ) { ( 𝐺 ‘ 𝑘 ) } ↔ ∃ 𝑘 ∈ ( 0 ..^ 𝑁 ) ( 𝑎 ∘ 𝑏 ) = ( 𝐺 ‘ 𝑘 ) ) |
108 |
103 107
|
orbi12i |
⊢ ( ( ( 𝑎 ∘ 𝑏 ) ∈ { 𝐼 } ∨ ( 𝑎 ∘ 𝑏 ) ∈ ∪ 𝑘 ∈ ( 0 ..^ 𝑁 ) { ( 𝐺 ‘ 𝑘 ) } ) ↔ ( ( 𝑎 ∘ 𝑏 ) = 𝐼 ∨ ∃ 𝑘 ∈ ( 0 ..^ 𝑁 ) ( 𝑎 ∘ 𝑏 ) = ( 𝐺 ‘ 𝑘 ) ) ) |
109 |
98 99 108
|
3bitri |
⊢ ( ( 𝑎 ∘ 𝑏 ) ∈ 𝐵 ↔ ( ( 𝑎 ∘ 𝑏 ) = 𝐼 ∨ ∃ 𝑘 ∈ ( 0 ..^ 𝑁 ) ( 𝑎 ∘ 𝑏 ) = ( 𝐺 ‘ 𝑘 ) ) ) |
110 |
68 95 109
|
3imtr4i |
⊢ ( ( 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ) → ( 𝑎 ∘ 𝑏 ) ∈ 𝐵 ) |
111 |
15 110
|
eqeltrd |
⊢ ( ( 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ) → ( 𝑎 ( +g ‘ 𝑀 ) 𝑏 ) ∈ 𝐵 ) |
112 |
111
|
rgen2 |
⊢ ∀ 𝑎 ∈ 𝐵 ∀ 𝑏 ∈ 𝐵 ( 𝑎 ( +g ‘ 𝑀 ) 𝑏 ) ∈ 𝐵 |
113 |
6
|
ovexi |
⊢ 𝑆 ∈ V |
114 |
1 2 3 4 5 6
|
smndex1bas |
⊢ ( Base ‘ 𝑆 ) = 𝐵 |
115 |
114
|
eqcomi |
⊢ 𝐵 = ( Base ‘ 𝑆 ) |
116 |
115
|
fvexi |
⊢ 𝐵 ∈ V |
117 |
6 13
|
ressplusg |
⊢ ( 𝐵 ∈ V → ( +g ‘ 𝑀 ) = ( +g ‘ 𝑆 ) ) |
118 |
116 117
|
ax-mp |
⊢ ( +g ‘ 𝑀 ) = ( +g ‘ 𝑆 ) |
119 |
115 118
|
ismgm |
⊢ ( 𝑆 ∈ V → ( 𝑆 ∈ Mgm ↔ ∀ 𝑎 ∈ 𝐵 ∀ 𝑏 ∈ 𝐵 ( 𝑎 ( +g ‘ 𝑀 ) 𝑏 ) ∈ 𝐵 ) ) |
120 |
113 119
|
ax-mp |
⊢ ( 𝑆 ∈ Mgm ↔ ∀ 𝑎 ∈ 𝐵 ∀ 𝑏 ∈ 𝐵 ( 𝑎 ( +g ‘ 𝑀 ) 𝑏 ) ∈ 𝐵 ) |
121 |
112 120
|
mpbir |
⊢ 𝑆 ∈ Mgm |