Step |
Hyp |
Ref |
Expression |
1 |
|
pwssplit1.y |
⊢ 𝑌 = ( 𝑊 ↑s 𝑈 ) |
2 |
|
pwssplit1.z |
⊢ 𝑍 = ( 𝑊 ↑s 𝑉 ) |
3 |
|
pwssplit1.b |
⊢ 𝐵 = ( Base ‘ 𝑌 ) |
4 |
|
pwssplit1.c |
⊢ 𝐶 = ( Base ‘ 𝑍 ) |
5 |
|
pwssplit1.f |
⊢ 𝐹 = ( 𝑥 ∈ 𝐵 ↦ ( 𝑥 ↾ 𝑉 ) ) |
6 |
|
eqid |
⊢ ( +g ‘ 𝑌 ) = ( +g ‘ 𝑌 ) |
7 |
|
eqid |
⊢ ( +g ‘ 𝑍 ) = ( +g ‘ 𝑍 ) |
8 |
|
simp1 |
⊢ ( ( 𝑊 ∈ Grp ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈 ) → 𝑊 ∈ Grp ) |
9 |
|
simp2 |
⊢ ( ( 𝑊 ∈ Grp ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈 ) → 𝑈 ∈ 𝑋 ) |
10 |
1
|
pwsgrp |
⊢ ( ( 𝑊 ∈ Grp ∧ 𝑈 ∈ 𝑋 ) → 𝑌 ∈ Grp ) |
11 |
8 9 10
|
syl2anc |
⊢ ( ( 𝑊 ∈ Grp ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈 ) → 𝑌 ∈ Grp ) |
12 |
|
simp3 |
⊢ ( ( 𝑊 ∈ Grp ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈 ) → 𝑉 ⊆ 𝑈 ) |
13 |
9 12
|
ssexd |
⊢ ( ( 𝑊 ∈ Grp ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈 ) → 𝑉 ∈ V ) |
14 |
2
|
pwsgrp |
⊢ ( ( 𝑊 ∈ Grp ∧ 𝑉 ∈ V ) → 𝑍 ∈ Grp ) |
15 |
8 13 14
|
syl2anc |
⊢ ( ( 𝑊 ∈ Grp ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈 ) → 𝑍 ∈ Grp ) |
16 |
1 2 3 4 5
|
pwssplit0 |
⊢ ( ( 𝑊 ∈ Grp ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈 ) → 𝐹 : 𝐵 ⟶ 𝐶 ) |
17 |
|
offres |
⊢ ( ( 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ) → ( ( 𝑎 ∘f ( +g ‘ 𝑊 ) 𝑏 ) ↾ 𝑉 ) = ( ( 𝑎 ↾ 𝑉 ) ∘f ( +g ‘ 𝑊 ) ( 𝑏 ↾ 𝑉 ) ) ) |
18 |
17
|
adantl |
⊢ ( ( ( 𝑊 ∈ Grp ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈 ) ∧ ( 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ) ) → ( ( 𝑎 ∘f ( +g ‘ 𝑊 ) 𝑏 ) ↾ 𝑉 ) = ( ( 𝑎 ↾ 𝑉 ) ∘f ( +g ‘ 𝑊 ) ( 𝑏 ↾ 𝑉 ) ) ) |
19 |
8
|
adantr |
⊢ ( ( ( 𝑊 ∈ Grp ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈 ) ∧ ( 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ) ) → 𝑊 ∈ Grp ) |
20 |
|
simpl2 |
⊢ ( ( ( 𝑊 ∈ Grp ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈 ) ∧ ( 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ) ) → 𝑈 ∈ 𝑋 ) |
21 |
|
simprl |
⊢ ( ( ( 𝑊 ∈ Grp ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈 ) ∧ ( 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ) ) → 𝑎 ∈ 𝐵 ) |
22 |
|
simprr |
⊢ ( ( ( 𝑊 ∈ Grp ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈 ) ∧ ( 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ) ) → 𝑏 ∈ 𝐵 ) |
23 |
|
eqid |
⊢ ( +g ‘ 𝑊 ) = ( +g ‘ 𝑊 ) |
24 |
1 3 19 20 21 22 23 6
|
pwsplusgval |
⊢ ( ( ( 𝑊 ∈ Grp ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈 ) ∧ ( 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ) ) → ( 𝑎 ( +g ‘ 𝑌 ) 𝑏 ) = ( 𝑎 ∘f ( +g ‘ 𝑊 ) 𝑏 ) ) |
25 |
24
|
reseq1d |
⊢ ( ( ( 𝑊 ∈ Grp ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈 ) ∧ ( 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ) ) → ( ( 𝑎 ( +g ‘ 𝑌 ) 𝑏 ) ↾ 𝑉 ) = ( ( 𝑎 ∘f ( +g ‘ 𝑊 ) 𝑏 ) ↾ 𝑉 ) ) |
26 |
5
|
fvtresfn |
⊢ ( 𝑎 ∈ 𝐵 → ( 𝐹 ‘ 𝑎 ) = ( 𝑎 ↾ 𝑉 ) ) |
27 |
5
|
fvtresfn |
⊢ ( 𝑏 ∈ 𝐵 → ( 𝐹 ‘ 𝑏 ) = ( 𝑏 ↾ 𝑉 ) ) |
28 |
26 27
|
oveqan12d |
⊢ ( ( 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ) → ( ( 𝐹 ‘ 𝑎 ) ∘f ( +g ‘ 𝑊 ) ( 𝐹 ‘ 𝑏 ) ) = ( ( 𝑎 ↾ 𝑉 ) ∘f ( +g ‘ 𝑊 ) ( 𝑏 ↾ 𝑉 ) ) ) |
29 |
28
|
adantl |
⊢ ( ( ( 𝑊 ∈ Grp ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈 ) ∧ ( 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ) ) → ( ( 𝐹 ‘ 𝑎 ) ∘f ( +g ‘ 𝑊 ) ( 𝐹 ‘ 𝑏 ) ) = ( ( 𝑎 ↾ 𝑉 ) ∘f ( +g ‘ 𝑊 ) ( 𝑏 ↾ 𝑉 ) ) ) |
30 |
18 25 29
|
3eqtr4d |
⊢ ( ( ( 𝑊 ∈ Grp ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈 ) ∧ ( 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ) ) → ( ( 𝑎 ( +g ‘ 𝑌 ) 𝑏 ) ↾ 𝑉 ) = ( ( 𝐹 ‘ 𝑎 ) ∘f ( +g ‘ 𝑊 ) ( 𝐹 ‘ 𝑏 ) ) ) |
31 |
3 6
|
grpcl |
⊢ ( ( 𝑌 ∈ Grp ∧ 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ) → ( 𝑎 ( +g ‘ 𝑌 ) 𝑏 ) ∈ 𝐵 ) |
32 |
31
|
3expb |
⊢ ( ( 𝑌 ∈ Grp ∧ ( 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ) ) → ( 𝑎 ( +g ‘ 𝑌 ) 𝑏 ) ∈ 𝐵 ) |
33 |
11 32
|
sylan |
⊢ ( ( ( 𝑊 ∈ Grp ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈 ) ∧ ( 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ) ) → ( 𝑎 ( +g ‘ 𝑌 ) 𝑏 ) ∈ 𝐵 ) |
34 |
5
|
fvtresfn |
⊢ ( ( 𝑎 ( +g ‘ 𝑌 ) 𝑏 ) ∈ 𝐵 → ( 𝐹 ‘ ( 𝑎 ( +g ‘ 𝑌 ) 𝑏 ) ) = ( ( 𝑎 ( +g ‘ 𝑌 ) 𝑏 ) ↾ 𝑉 ) ) |
35 |
33 34
|
syl |
⊢ ( ( ( 𝑊 ∈ Grp ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈 ) ∧ ( 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ) ) → ( 𝐹 ‘ ( 𝑎 ( +g ‘ 𝑌 ) 𝑏 ) ) = ( ( 𝑎 ( +g ‘ 𝑌 ) 𝑏 ) ↾ 𝑉 ) ) |
36 |
13
|
adantr |
⊢ ( ( ( 𝑊 ∈ Grp ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈 ) ∧ ( 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ) ) → 𝑉 ∈ V ) |
37 |
16
|
ffvelrnda |
⊢ ( ( ( 𝑊 ∈ Grp ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈 ) ∧ 𝑎 ∈ 𝐵 ) → ( 𝐹 ‘ 𝑎 ) ∈ 𝐶 ) |
38 |
37
|
adantrr |
⊢ ( ( ( 𝑊 ∈ Grp ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈 ) ∧ ( 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ) ) → ( 𝐹 ‘ 𝑎 ) ∈ 𝐶 ) |
39 |
16
|
ffvelrnda |
⊢ ( ( ( 𝑊 ∈ Grp ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈 ) ∧ 𝑏 ∈ 𝐵 ) → ( 𝐹 ‘ 𝑏 ) ∈ 𝐶 ) |
40 |
39
|
adantrl |
⊢ ( ( ( 𝑊 ∈ Grp ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈 ) ∧ ( 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ) ) → ( 𝐹 ‘ 𝑏 ) ∈ 𝐶 ) |
41 |
2 4 19 36 38 40 23 7
|
pwsplusgval |
⊢ ( ( ( 𝑊 ∈ Grp ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈 ) ∧ ( 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ) ) → ( ( 𝐹 ‘ 𝑎 ) ( +g ‘ 𝑍 ) ( 𝐹 ‘ 𝑏 ) ) = ( ( 𝐹 ‘ 𝑎 ) ∘f ( +g ‘ 𝑊 ) ( 𝐹 ‘ 𝑏 ) ) ) |
42 |
30 35 41
|
3eqtr4d |
⊢ ( ( ( 𝑊 ∈ Grp ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈 ) ∧ ( 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ) ) → ( 𝐹 ‘ ( 𝑎 ( +g ‘ 𝑌 ) 𝑏 ) ) = ( ( 𝐹 ‘ 𝑎 ) ( +g ‘ 𝑍 ) ( 𝐹 ‘ 𝑏 ) ) ) |
43 |
3 4 6 7 11 15 16 42
|
isghmd |
⊢ ( ( 𝑊 ∈ Grp ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈 ) → 𝐹 ∈ ( 𝑌 GrpHom 𝑍 ) ) |