Step |
Hyp |
Ref |
Expression |
1 |
|
subgga.1 |
⊢ 𝑋 = ( Base ‘ 𝐺 ) |
2 |
|
subgga.2 |
⊢ + = ( +g ‘ 𝐺 ) |
3 |
|
subgga.3 |
⊢ 𝐻 = ( 𝐺 ↾s 𝑌 ) |
4 |
|
subgga.4 |
⊢ 𝐹 = ( 𝑥 ∈ 𝑌 , 𝑦 ∈ 𝑋 ↦ ( 𝑥 + 𝑦 ) ) |
5 |
3
|
subggrp |
⊢ ( 𝑌 ∈ ( SubGrp ‘ 𝐺 ) → 𝐻 ∈ Grp ) |
6 |
1
|
fvexi |
⊢ 𝑋 ∈ V |
7 |
5 6
|
jctir |
⊢ ( 𝑌 ∈ ( SubGrp ‘ 𝐺 ) → ( 𝐻 ∈ Grp ∧ 𝑋 ∈ V ) ) |
8 |
|
subgrcl |
⊢ ( 𝑌 ∈ ( SubGrp ‘ 𝐺 ) → 𝐺 ∈ Grp ) |
9 |
8
|
adantr |
⊢ ( ( 𝑌 ∈ ( SubGrp ‘ 𝐺 ) ∧ ( 𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑋 ) ) → 𝐺 ∈ Grp ) |
10 |
1
|
subgss |
⊢ ( 𝑌 ∈ ( SubGrp ‘ 𝐺 ) → 𝑌 ⊆ 𝑋 ) |
11 |
10
|
sselda |
⊢ ( ( 𝑌 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑥 ∈ 𝑌 ) → 𝑥 ∈ 𝑋 ) |
12 |
11
|
adantrr |
⊢ ( ( 𝑌 ∈ ( SubGrp ‘ 𝐺 ) ∧ ( 𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑋 ) ) → 𝑥 ∈ 𝑋 ) |
13 |
|
simprr |
⊢ ( ( 𝑌 ∈ ( SubGrp ‘ 𝐺 ) ∧ ( 𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑋 ) ) → 𝑦 ∈ 𝑋 ) |
14 |
1 2
|
grpcl |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ) → ( 𝑥 + 𝑦 ) ∈ 𝑋 ) |
15 |
9 12 13 14
|
syl3anc |
⊢ ( ( 𝑌 ∈ ( SubGrp ‘ 𝐺 ) ∧ ( 𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑋 ) ) → ( 𝑥 + 𝑦 ) ∈ 𝑋 ) |
16 |
15
|
ralrimivva |
⊢ ( 𝑌 ∈ ( SubGrp ‘ 𝐺 ) → ∀ 𝑥 ∈ 𝑌 ∀ 𝑦 ∈ 𝑋 ( 𝑥 + 𝑦 ) ∈ 𝑋 ) |
17 |
4
|
fmpo |
⊢ ( ∀ 𝑥 ∈ 𝑌 ∀ 𝑦 ∈ 𝑋 ( 𝑥 + 𝑦 ) ∈ 𝑋 ↔ 𝐹 : ( 𝑌 × 𝑋 ) ⟶ 𝑋 ) |
18 |
16 17
|
sylib |
⊢ ( 𝑌 ∈ ( SubGrp ‘ 𝐺 ) → 𝐹 : ( 𝑌 × 𝑋 ) ⟶ 𝑋 ) |
19 |
3
|
subgbas |
⊢ ( 𝑌 ∈ ( SubGrp ‘ 𝐺 ) → 𝑌 = ( Base ‘ 𝐻 ) ) |
20 |
19
|
xpeq1d |
⊢ ( 𝑌 ∈ ( SubGrp ‘ 𝐺 ) → ( 𝑌 × 𝑋 ) = ( ( Base ‘ 𝐻 ) × 𝑋 ) ) |
21 |
20
|
feq2d |
⊢ ( 𝑌 ∈ ( SubGrp ‘ 𝐺 ) → ( 𝐹 : ( 𝑌 × 𝑋 ) ⟶ 𝑋 ↔ 𝐹 : ( ( Base ‘ 𝐻 ) × 𝑋 ) ⟶ 𝑋 ) ) |
22 |
18 21
|
mpbid |
⊢ ( 𝑌 ∈ ( SubGrp ‘ 𝐺 ) → 𝐹 : ( ( Base ‘ 𝐻 ) × 𝑋 ) ⟶ 𝑋 ) |
23 |
|
eqid |
⊢ ( 0g ‘ 𝐺 ) = ( 0g ‘ 𝐺 ) |
24 |
23
|
subg0cl |
⊢ ( 𝑌 ∈ ( SubGrp ‘ 𝐺 ) → ( 0g ‘ 𝐺 ) ∈ 𝑌 ) |
25 |
|
oveq12 |
⊢ ( ( 𝑥 = ( 0g ‘ 𝐺 ) ∧ 𝑦 = 𝑢 ) → ( 𝑥 + 𝑦 ) = ( ( 0g ‘ 𝐺 ) + 𝑢 ) ) |
26 |
|
ovex |
⊢ ( ( 0g ‘ 𝐺 ) + 𝑢 ) ∈ V |
27 |
25 4 26
|
ovmpoa |
⊢ ( ( ( 0g ‘ 𝐺 ) ∈ 𝑌 ∧ 𝑢 ∈ 𝑋 ) → ( ( 0g ‘ 𝐺 ) 𝐹 𝑢 ) = ( ( 0g ‘ 𝐺 ) + 𝑢 ) ) |
28 |
24 27
|
sylan |
⊢ ( ( 𝑌 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑢 ∈ 𝑋 ) → ( ( 0g ‘ 𝐺 ) 𝐹 𝑢 ) = ( ( 0g ‘ 𝐺 ) + 𝑢 ) ) |
29 |
3 23
|
subg0 |
⊢ ( 𝑌 ∈ ( SubGrp ‘ 𝐺 ) → ( 0g ‘ 𝐺 ) = ( 0g ‘ 𝐻 ) ) |
30 |
29
|
oveq1d |
⊢ ( 𝑌 ∈ ( SubGrp ‘ 𝐺 ) → ( ( 0g ‘ 𝐺 ) 𝐹 𝑢 ) = ( ( 0g ‘ 𝐻 ) 𝐹 𝑢 ) ) |
31 |
30
|
adantr |
⊢ ( ( 𝑌 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑢 ∈ 𝑋 ) → ( ( 0g ‘ 𝐺 ) 𝐹 𝑢 ) = ( ( 0g ‘ 𝐻 ) 𝐹 𝑢 ) ) |
32 |
1 2 23
|
grplid |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑢 ∈ 𝑋 ) → ( ( 0g ‘ 𝐺 ) + 𝑢 ) = 𝑢 ) |
33 |
8 32
|
sylan |
⊢ ( ( 𝑌 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑢 ∈ 𝑋 ) → ( ( 0g ‘ 𝐺 ) + 𝑢 ) = 𝑢 ) |
34 |
28 31 33
|
3eqtr3d |
⊢ ( ( 𝑌 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑢 ∈ 𝑋 ) → ( ( 0g ‘ 𝐻 ) 𝐹 𝑢 ) = 𝑢 ) |
35 |
8
|
ad2antrr |
⊢ ( ( ( 𝑌 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑢 ∈ 𝑋 ) ∧ ( 𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌 ) ) → 𝐺 ∈ Grp ) |
36 |
10
|
ad2antrr |
⊢ ( ( ( 𝑌 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑢 ∈ 𝑋 ) ∧ ( 𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌 ) ) → 𝑌 ⊆ 𝑋 ) |
37 |
|
simprl |
⊢ ( ( ( 𝑌 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑢 ∈ 𝑋 ) ∧ ( 𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌 ) ) → 𝑣 ∈ 𝑌 ) |
38 |
36 37
|
sseldd |
⊢ ( ( ( 𝑌 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑢 ∈ 𝑋 ) ∧ ( 𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌 ) ) → 𝑣 ∈ 𝑋 ) |
39 |
|
simprr |
⊢ ( ( ( 𝑌 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑢 ∈ 𝑋 ) ∧ ( 𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌 ) ) → 𝑤 ∈ 𝑌 ) |
40 |
36 39
|
sseldd |
⊢ ( ( ( 𝑌 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑢 ∈ 𝑋 ) ∧ ( 𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌 ) ) → 𝑤 ∈ 𝑋 ) |
41 |
|
simplr |
⊢ ( ( ( 𝑌 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑢 ∈ 𝑋 ) ∧ ( 𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌 ) ) → 𝑢 ∈ 𝑋 ) |
42 |
1 2
|
grpass |
⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑣 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋 ∧ 𝑢 ∈ 𝑋 ) ) → ( ( 𝑣 + 𝑤 ) + 𝑢 ) = ( 𝑣 + ( 𝑤 + 𝑢 ) ) ) |
43 |
35 38 40 41 42
|
syl13anc |
⊢ ( ( ( 𝑌 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑢 ∈ 𝑋 ) ∧ ( 𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌 ) ) → ( ( 𝑣 + 𝑤 ) + 𝑢 ) = ( 𝑣 + ( 𝑤 + 𝑢 ) ) ) |
44 |
1 2
|
grpcl |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑤 ∈ 𝑋 ∧ 𝑢 ∈ 𝑋 ) → ( 𝑤 + 𝑢 ) ∈ 𝑋 ) |
45 |
35 40 41 44
|
syl3anc |
⊢ ( ( ( 𝑌 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑢 ∈ 𝑋 ) ∧ ( 𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌 ) ) → ( 𝑤 + 𝑢 ) ∈ 𝑋 ) |
46 |
|
oveq12 |
⊢ ( ( 𝑥 = 𝑣 ∧ 𝑦 = ( 𝑤 + 𝑢 ) ) → ( 𝑥 + 𝑦 ) = ( 𝑣 + ( 𝑤 + 𝑢 ) ) ) |
47 |
|
ovex |
⊢ ( 𝑣 + ( 𝑤 + 𝑢 ) ) ∈ V |
48 |
46 4 47
|
ovmpoa |
⊢ ( ( 𝑣 ∈ 𝑌 ∧ ( 𝑤 + 𝑢 ) ∈ 𝑋 ) → ( 𝑣 𝐹 ( 𝑤 + 𝑢 ) ) = ( 𝑣 + ( 𝑤 + 𝑢 ) ) ) |
49 |
37 45 48
|
syl2anc |
⊢ ( ( ( 𝑌 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑢 ∈ 𝑋 ) ∧ ( 𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌 ) ) → ( 𝑣 𝐹 ( 𝑤 + 𝑢 ) ) = ( 𝑣 + ( 𝑤 + 𝑢 ) ) ) |
50 |
43 49
|
eqtr4d |
⊢ ( ( ( 𝑌 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑢 ∈ 𝑋 ) ∧ ( 𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌 ) ) → ( ( 𝑣 + 𝑤 ) + 𝑢 ) = ( 𝑣 𝐹 ( 𝑤 + 𝑢 ) ) ) |
51 |
2
|
subgcl |
⊢ ( ( 𝑌 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌 ) → ( 𝑣 + 𝑤 ) ∈ 𝑌 ) |
52 |
51
|
3expb |
⊢ ( ( 𝑌 ∈ ( SubGrp ‘ 𝐺 ) ∧ ( 𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌 ) ) → ( 𝑣 + 𝑤 ) ∈ 𝑌 ) |
53 |
52
|
adantlr |
⊢ ( ( ( 𝑌 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑢 ∈ 𝑋 ) ∧ ( 𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌 ) ) → ( 𝑣 + 𝑤 ) ∈ 𝑌 ) |
54 |
|
oveq12 |
⊢ ( ( 𝑥 = ( 𝑣 + 𝑤 ) ∧ 𝑦 = 𝑢 ) → ( 𝑥 + 𝑦 ) = ( ( 𝑣 + 𝑤 ) + 𝑢 ) ) |
55 |
|
ovex |
⊢ ( ( 𝑣 + 𝑤 ) + 𝑢 ) ∈ V |
56 |
54 4 55
|
ovmpoa |
⊢ ( ( ( 𝑣 + 𝑤 ) ∈ 𝑌 ∧ 𝑢 ∈ 𝑋 ) → ( ( 𝑣 + 𝑤 ) 𝐹 𝑢 ) = ( ( 𝑣 + 𝑤 ) + 𝑢 ) ) |
57 |
53 41 56
|
syl2anc |
⊢ ( ( ( 𝑌 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑢 ∈ 𝑋 ) ∧ ( 𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌 ) ) → ( ( 𝑣 + 𝑤 ) 𝐹 𝑢 ) = ( ( 𝑣 + 𝑤 ) + 𝑢 ) ) |
58 |
|
oveq12 |
⊢ ( ( 𝑥 = 𝑤 ∧ 𝑦 = 𝑢 ) → ( 𝑥 + 𝑦 ) = ( 𝑤 + 𝑢 ) ) |
59 |
|
ovex |
⊢ ( 𝑤 + 𝑢 ) ∈ V |
60 |
58 4 59
|
ovmpoa |
⊢ ( ( 𝑤 ∈ 𝑌 ∧ 𝑢 ∈ 𝑋 ) → ( 𝑤 𝐹 𝑢 ) = ( 𝑤 + 𝑢 ) ) |
61 |
39 41 60
|
syl2anc |
⊢ ( ( ( 𝑌 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑢 ∈ 𝑋 ) ∧ ( 𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌 ) ) → ( 𝑤 𝐹 𝑢 ) = ( 𝑤 + 𝑢 ) ) |
62 |
61
|
oveq2d |
⊢ ( ( ( 𝑌 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑢 ∈ 𝑋 ) ∧ ( 𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌 ) ) → ( 𝑣 𝐹 ( 𝑤 𝐹 𝑢 ) ) = ( 𝑣 𝐹 ( 𝑤 + 𝑢 ) ) ) |
63 |
50 57 62
|
3eqtr4d |
⊢ ( ( ( 𝑌 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑢 ∈ 𝑋 ) ∧ ( 𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌 ) ) → ( ( 𝑣 + 𝑤 ) 𝐹 𝑢 ) = ( 𝑣 𝐹 ( 𝑤 𝐹 𝑢 ) ) ) |
64 |
63
|
ralrimivva |
⊢ ( ( 𝑌 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑢 ∈ 𝑋 ) → ∀ 𝑣 ∈ 𝑌 ∀ 𝑤 ∈ 𝑌 ( ( 𝑣 + 𝑤 ) 𝐹 𝑢 ) = ( 𝑣 𝐹 ( 𝑤 𝐹 𝑢 ) ) ) |
65 |
3 2
|
ressplusg |
⊢ ( 𝑌 ∈ ( SubGrp ‘ 𝐺 ) → + = ( +g ‘ 𝐻 ) ) |
66 |
65
|
oveqd |
⊢ ( 𝑌 ∈ ( SubGrp ‘ 𝐺 ) → ( 𝑣 + 𝑤 ) = ( 𝑣 ( +g ‘ 𝐻 ) 𝑤 ) ) |
67 |
66
|
oveq1d |
⊢ ( 𝑌 ∈ ( SubGrp ‘ 𝐺 ) → ( ( 𝑣 + 𝑤 ) 𝐹 𝑢 ) = ( ( 𝑣 ( +g ‘ 𝐻 ) 𝑤 ) 𝐹 𝑢 ) ) |
68 |
67
|
eqeq1d |
⊢ ( 𝑌 ∈ ( SubGrp ‘ 𝐺 ) → ( ( ( 𝑣 + 𝑤 ) 𝐹 𝑢 ) = ( 𝑣 𝐹 ( 𝑤 𝐹 𝑢 ) ) ↔ ( ( 𝑣 ( +g ‘ 𝐻 ) 𝑤 ) 𝐹 𝑢 ) = ( 𝑣 𝐹 ( 𝑤 𝐹 𝑢 ) ) ) ) |
69 |
19 68
|
raleqbidv |
⊢ ( 𝑌 ∈ ( SubGrp ‘ 𝐺 ) → ( ∀ 𝑤 ∈ 𝑌 ( ( 𝑣 + 𝑤 ) 𝐹 𝑢 ) = ( 𝑣 𝐹 ( 𝑤 𝐹 𝑢 ) ) ↔ ∀ 𝑤 ∈ ( Base ‘ 𝐻 ) ( ( 𝑣 ( +g ‘ 𝐻 ) 𝑤 ) 𝐹 𝑢 ) = ( 𝑣 𝐹 ( 𝑤 𝐹 𝑢 ) ) ) ) |
70 |
19 69
|
raleqbidv |
⊢ ( 𝑌 ∈ ( SubGrp ‘ 𝐺 ) → ( ∀ 𝑣 ∈ 𝑌 ∀ 𝑤 ∈ 𝑌 ( ( 𝑣 + 𝑤 ) 𝐹 𝑢 ) = ( 𝑣 𝐹 ( 𝑤 𝐹 𝑢 ) ) ↔ ∀ 𝑣 ∈ ( Base ‘ 𝐻 ) ∀ 𝑤 ∈ ( Base ‘ 𝐻 ) ( ( 𝑣 ( +g ‘ 𝐻 ) 𝑤 ) 𝐹 𝑢 ) = ( 𝑣 𝐹 ( 𝑤 𝐹 𝑢 ) ) ) ) |
71 |
70
|
biimpa |
⊢ ( ( 𝑌 ∈ ( SubGrp ‘ 𝐺 ) ∧ ∀ 𝑣 ∈ 𝑌 ∀ 𝑤 ∈ 𝑌 ( ( 𝑣 + 𝑤 ) 𝐹 𝑢 ) = ( 𝑣 𝐹 ( 𝑤 𝐹 𝑢 ) ) ) → ∀ 𝑣 ∈ ( Base ‘ 𝐻 ) ∀ 𝑤 ∈ ( Base ‘ 𝐻 ) ( ( 𝑣 ( +g ‘ 𝐻 ) 𝑤 ) 𝐹 𝑢 ) = ( 𝑣 𝐹 ( 𝑤 𝐹 𝑢 ) ) ) |
72 |
64 71
|
syldan |
⊢ ( ( 𝑌 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑢 ∈ 𝑋 ) → ∀ 𝑣 ∈ ( Base ‘ 𝐻 ) ∀ 𝑤 ∈ ( Base ‘ 𝐻 ) ( ( 𝑣 ( +g ‘ 𝐻 ) 𝑤 ) 𝐹 𝑢 ) = ( 𝑣 𝐹 ( 𝑤 𝐹 𝑢 ) ) ) |
73 |
34 72
|
jca |
⊢ ( ( 𝑌 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑢 ∈ 𝑋 ) → ( ( ( 0g ‘ 𝐻 ) 𝐹 𝑢 ) = 𝑢 ∧ ∀ 𝑣 ∈ ( Base ‘ 𝐻 ) ∀ 𝑤 ∈ ( Base ‘ 𝐻 ) ( ( 𝑣 ( +g ‘ 𝐻 ) 𝑤 ) 𝐹 𝑢 ) = ( 𝑣 𝐹 ( 𝑤 𝐹 𝑢 ) ) ) ) |
74 |
73
|
ralrimiva |
⊢ ( 𝑌 ∈ ( SubGrp ‘ 𝐺 ) → ∀ 𝑢 ∈ 𝑋 ( ( ( 0g ‘ 𝐻 ) 𝐹 𝑢 ) = 𝑢 ∧ ∀ 𝑣 ∈ ( Base ‘ 𝐻 ) ∀ 𝑤 ∈ ( Base ‘ 𝐻 ) ( ( 𝑣 ( +g ‘ 𝐻 ) 𝑤 ) 𝐹 𝑢 ) = ( 𝑣 𝐹 ( 𝑤 𝐹 𝑢 ) ) ) ) |
75 |
22 74
|
jca |
⊢ ( 𝑌 ∈ ( SubGrp ‘ 𝐺 ) → ( 𝐹 : ( ( Base ‘ 𝐻 ) × 𝑋 ) ⟶ 𝑋 ∧ ∀ 𝑢 ∈ 𝑋 ( ( ( 0g ‘ 𝐻 ) 𝐹 𝑢 ) = 𝑢 ∧ ∀ 𝑣 ∈ ( Base ‘ 𝐻 ) ∀ 𝑤 ∈ ( Base ‘ 𝐻 ) ( ( 𝑣 ( +g ‘ 𝐻 ) 𝑤 ) 𝐹 𝑢 ) = ( 𝑣 𝐹 ( 𝑤 𝐹 𝑢 ) ) ) ) ) |
76 |
|
eqid |
⊢ ( Base ‘ 𝐻 ) = ( Base ‘ 𝐻 ) |
77 |
|
eqid |
⊢ ( +g ‘ 𝐻 ) = ( +g ‘ 𝐻 ) |
78 |
|
eqid |
⊢ ( 0g ‘ 𝐻 ) = ( 0g ‘ 𝐻 ) |
79 |
76 77 78
|
isga |
⊢ ( 𝐹 ∈ ( 𝐻 GrpAct 𝑋 ) ↔ ( ( 𝐻 ∈ Grp ∧ 𝑋 ∈ V ) ∧ ( 𝐹 : ( ( Base ‘ 𝐻 ) × 𝑋 ) ⟶ 𝑋 ∧ ∀ 𝑢 ∈ 𝑋 ( ( ( 0g ‘ 𝐻 ) 𝐹 𝑢 ) = 𝑢 ∧ ∀ 𝑣 ∈ ( Base ‘ 𝐻 ) ∀ 𝑤 ∈ ( Base ‘ 𝐻 ) ( ( 𝑣 ( +g ‘ 𝐻 ) 𝑤 ) 𝐹 𝑢 ) = ( 𝑣 𝐹 ( 𝑤 𝐹 𝑢 ) ) ) ) ) ) |
80 |
7 75 79
|
sylanbrc |
⊢ ( 𝑌 ∈ ( SubGrp ‘ 𝐺 ) → 𝐹 ∈ ( 𝐻 GrpAct 𝑋 ) ) |