Step |
Hyp |
Ref |
Expression |
1 |
|
issubmnd.b |
⊢ 𝐵 = ( Base ‘ 𝐺 ) |
2 |
|
issubmnd.p |
⊢ + = ( +g ‘ 𝐺 ) |
3 |
|
issubmnd.z |
⊢ 0 = ( 0g ‘ 𝐺 ) |
4 |
|
issubmnd.h |
⊢ 𝐻 = ( 𝐺 ↾s 𝑆 ) |
5 |
|
simplr |
⊢ ( ( ( ( 𝐺 ∈ Mnd ∧ 𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ) ∧ 𝐻 ∈ Mnd ) ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) ) → 𝐻 ∈ Mnd ) |
6 |
|
simprl |
⊢ ( ( ( ( 𝐺 ∈ Mnd ∧ 𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ) ∧ 𝐻 ∈ Mnd ) ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) ) → 𝑥 ∈ 𝑆 ) |
7 |
|
simpll2 |
⊢ ( ( ( ( 𝐺 ∈ Mnd ∧ 𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ) ∧ 𝐻 ∈ Mnd ) ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) ) → 𝑆 ⊆ 𝐵 ) |
8 |
4 1
|
ressbas2 |
⊢ ( 𝑆 ⊆ 𝐵 → 𝑆 = ( Base ‘ 𝐻 ) ) |
9 |
7 8
|
syl |
⊢ ( ( ( ( 𝐺 ∈ Mnd ∧ 𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ) ∧ 𝐻 ∈ Mnd ) ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) ) → 𝑆 = ( Base ‘ 𝐻 ) ) |
10 |
6 9
|
eleqtrd |
⊢ ( ( ( ( 𝐺 ∈ Mnd ∧ 𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ) ∧ 𝐻 ∈ Mnd ) ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) ) → 𝑥 ∈ ( Base ‘ 𝐻 ) ) |
11 |
|
simprr |
⊢ ( ( ( ( 𝐺 ∈ Mnd ∧ 𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ) ∧ 𝐻 ∈ Mnd ) ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) ) → 𝑦 ∈ 𝑆 ) |
12 |
11 9
|
eleqtrd |
⊢ ( ( ( ( 𝐺 ∈ Mnd ∧ 𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ) ∧ 𝐻 ∈ Mnd ) ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) ) → 𝑦 ∈ ( Base ‘ 𝐻 ) ) |
13 |
|
eqid |
⊢ ( Base ‘ 𝐻 ) = ( Base ‘ 𝐻 ) |
14 |
|
eqid |
⊢ ( +g ‘ 𝐻 ) = ( +g ‘ 𝐻 ) |
15 |
13 14
|
mndcl |
⊢ ( ( 𝐻 ∈ Mnd ∧ 𝑥 ∈ ( Base ‘ 𝐻 ) ∧ 𝑦 ∈ ( Base ‘ 𝐻 ) ) → ( 𝑥 ( +g ‘ 𝐻 ) 𝑦 ) ∈ ( Base ‘ 𝐻 ) ) |
16 |
5 10 12 15
|
syl3anc |
⊢ ( ( ( ( 𝐺 ∈ Mnd ∧ 𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ) ∧ 𝐻 ∈ Mnd ) ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) ) → ( 𝑥 ( +g ‘ 𝐻 ) 𝑦 ) ∈ ( Base ‘ 𝐻 ) ) |
17 |
1
|
fvexi |
⊢ 𝐵 ∈ V |
18 |
17
|
ssex |
⊢ ( 𝑆 ⊆ 𝐵 → 𝑆 ∈ V ) |
19 |
18
|
3ad2ant2 |
⊢ ( ( 𝐺 ∈ Mnd ∧ 𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ) → 𝑆 ∈ V ) |
20 |
4 2
|
ressplusg |
⊢ ( 𝑆 ∈ V → + = ( +g ‘ 𝐻 ) ) |
21 |
19 20
|
syl |
⊢ ( ( 𝐺 ∈ Mnd ∧ 𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ) → + = ( +g ‘ 𝐻 ) ) |
22 |
21
|
ad2antrr |
⊢ ( ( ( ( 𝐺 ∈ Mnd ∧ 𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ) ∧ 𝐻 ∈ Mnd ) ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) ) → + = ( +g ‘ 𝐻 ) ) |
23 |
22
|
oveqd |
⊢ ( ( ( ( 𝐺 ∈ Mnd ∧ 𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ) ∧ 𝐻 ∈ Mnd ) ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) ) → ( 𝑥 + 𝑦 ) = ( 𝑥 ( +g ‘ 𝐻 ) 𝑦 ) ) |
24 |
16 23 9
|
3eltr4d |
⊢ ( ( ( ( 𝐺 ∈ Mnd ∧ 𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ) ∧ 𝐻 ∈ Mnd ) ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) ) → ( 𝑥 + 𝑦 ) ∈ 𝑆 ) |
25 |
24
|
ralrimivva |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ 𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ) ∧ 𝐻 ∈ Mnd ) → ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 + 𝑦 ) ∈ 𝑆 ) |
26 |
|
simpl2 |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ 𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ) ∧ ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 + 𝑦 ) ∈ 𝑆 ) → 𝑆 ⊆ 𝐵 ) |
27 |
26 8
|
syl |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ 𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ) ∧ ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 + 𝑦 ) ∈ 𝑆 ) → 𝑆 = ( Base ‘ 𝐻 ) ) |
28 |
21
|
adantr |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ 𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ) ∧ ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 + 𝑦 ) ∈ 𝑆 ) → + = ( +g ‘ 𝐻 ) ) |
29 |
|
ovrspc2v |
⊢ ( ( ( 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ) ∧ ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 + 𝑦 ) ∈ 𝑆 ) → ( 𝑢 + 𝑣 ) ∈ 𝑆 ) |
30 |
29
|
ancoms |
⊢ ( ( ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 + 𝑦 ) ∈ 𝑆 ∧ ( 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ) ) → ( 𝑢 + 𝑣 ) ∈ 𝑆 ) |
31 |
30
|
3impb |
⊢ ( ( ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 + 𝑦 ) ∈ 𝑆 ∧ 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ) → ( 𝑢 + 𝑣 ) ∈ 𝑆 ) |
32 |
31
|
3adant1l |
⊢ ( ( ( ( 𝐺 ∈ Mnd ∧ 𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ) ∧ ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 + 𝑦 ) ∈ 𝑆 ) ∧ 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ) → ( 𝑢 + 𝑣 ) ∈ 𝑆 ) |
33 |
|
simpl1 |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ 𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ) ∧ ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 + 𝑦 ) ∈ 𝑆 ) → 𝐺 ∈ Mnd ) |
34 |
26
|
sseld |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ 𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ) ∧ ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 + 𝑦 ) ∈ 𝑆 ) → ( 𝑢 ∈ 𝑆 → 𝑢 ∈ 𝐵 ) ) |
35 |
26
|
sseld |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ 𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ) ∧ ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 + 𝑦 ) ∈ 𝑆 ) → ( 𝑣 ∈ 𝑆 → 𝑣 ∈ 𝐵 ) ) |
36 |
26
|
sseld |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ 𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ) ∧ ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 + 𝑦 ) ∈ 𝑆 ) → ( 𝑤 ∈ 𝑆 → 𝑤 ∈ 𝐵 ) ) |
37 |
34 35 36
|
3anim123d |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ 𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ) ∧ ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 + 𝑦 ) ∈ 𝑆 ) → ( ( 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆 ) → ( 𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ) ) |
38 |
37
|
imp |
⊢ ( ( ( ( 𝐺 ∈ Mnd ∧ 𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ) ∧ ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 + 𝑦 ) ∈ 𝑆 ) ∧ ( 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆 ) ) → ( 𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ) |
39 |
1 2
|
mndass |
⊢ ( ( 𝐺 ∈ Mnd ∧ ( 𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ) → ( ( 𝑢 + 𝑣 ) + 𝑤 ) = ( 𝑢 + ( 𝑣 + 𝑤 ) ) ) |
40 |
33 38 39
|
syl2an2r |
⊢ ( ( ( ( 𝐺 ∈ Mnd ∧ 𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ) ∧ ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 + 𝑦 ) ∈ 𝑆 ) ∧ ( 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆 ) ) → ( ( 𝑢 + 𝑣 ) + 𝑤 ) = ( 𝑢 + ( 𝑣 + 𝑤 ) ) ) |
41 |
|
simpl3 |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ 𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ) ∧ ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 + 𝑦 ) ∈ 𝑆 ) → 0 ∈ 𝑆 ) |
42 |
26
|
sselda |
⊢ ( ( ( ( 𝐺 ∈ Mnd ∧ 𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ) ∧ ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 + 𝑦 ) ∈ 𝑆 ) ∧ 𝑢 ∈ 𝑆 ) → 𝑢 ∈ 𝐵 ) |
43 |
1 2 3
|
mndlid |
⊢ ( ( 𝐺 ∈ Mnd ∧ 𝑢 ∈ 𝐵 ) → ( 0 + 𝑢 ) = 𝑢 ) |
44 |
33 42 43
|
syl2an2r |
⊢ ( ( ( ( 𝐺 ∈ Mnd ∧ 𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ) ∧ ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 + 𝑦 ) ∈ 𝑆 ) ∧ 𝑢 ∈ 𝑆 ) → ( 0 + 𝑢 ) = 𝑢 ) |
45 |
1 2 3
|
mndrid |
⊢ ( ( 𝐺 ∈ Mnd ∧ 𝑢 ∈ 𝐵 ) → ( 𝑢 + 0 ) = 𝑢 ) |
46 |
33 42 45
|
syl2an2r |
⊢ ( ( ( ( 𝐺 ∈ Mnd ∧ 𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ) ∧ ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 + 𝑦 ) ∈ 𝑆 ) ∧ 𝑢 ∈ 𝑆 ) → ( 𝑢 + 0 ) = 𝑢 ) |
47 |
27 28 32 40 41 44 46
|
ismndd |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ 𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ) ∧ ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 + 𝑦 ) ∈ 𝑆 ) → 𝐻 ∈ Mnd ) |
48 |
25 47
|
impbida |
⊢ ( ( 𝐺 ∈ Mnd ∧ 𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ) → ( 𝐻 ∈ Mnd ↔ ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 + 𝑦 ) ∈ 𝑆 ) ) |