Step |
Hyp |
Ref |
Expression |
1 |
|
mhmimalem.f |
⊢ ( 𝜑 → 𝐹 ∈ ( 𝑀 MndHom 𝑁 ) ) |
2 |
|
mhmimalem.s |
⊢ ( 𝜑 → 𝑋 ⊆ ( Base ‘ 𝑀 ) ) |
3 |
|
mhmimalem.a |
⊢ ( 𝜑 → ⊕ = ( +g ‘ 𝑀 ) ) |
4 |
|
mhmimalem.p |
⊢ ( 𝜑 → + = ( +g ‘ 𝑁 ) ) |
5 |
|
mhmimalem.c |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ) → ( 𝑧 ⊕ 𝑥 ) ∈ 𝑋 ) |
6 |
1
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ) ) → 𝐹 ∈ ( 𝑀 MndHom 𝑁 ) ) |
7 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ) ) → 𝑋 ⊆ ( Base ‘ 𝑀 ) ) |
8 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ) ) → 𝑧 ∈ 𝑋 ) |
9 |
7 8
|
sseldd |
⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ) ) → 𝑧 ∈ ( Base ‘ 𝑀 ) ) |
10 |
|
simprr |
⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ) ) → 𝑥 ∈ 𝑋 ) |
11 |
7 10
|
sseldd |
⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ) ) → 𝑥 ∈ ( Base ‘ 𝑀 ) ) |
12 |
|
eqid |
⊢ ( Base ‘ 𝑀 ) = ( Base ‘ 𝑀 ) |
13 |
|
eqid |
⊢ ( +g ‘ 𝑀 ) = ( +g ‘ 𝑀 ) |
14 |
|
eqid |
⊢ ( +g ‘ 𝑁 ) = ( +g ‘ 𝑁 ) |
15 |
12 13 14
|
mhmlin |
⊢ ( ( 𝐹 ∈ ( 𝑀 MndHom 𝑁 ) ∧ 𝑧 ∈ ( Base ‘ 𝑀 ) ∧ 𝑥 ∈ ( Base ‘ 𝑀 ) ) → ( 𝐹 ‘ ( 𝑧 ( +g ‘ 𝑀 ) 𝑥 ) ) = ( ( 𝐹 ‘ 𝑧 ) ( +g ‘ 𝑁 ) ( 𝐹 ‘ 𝑥 ) ) ) |
16 |
6 9 11 15
|
syl3anc |
⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ) ) → ( 𝐹 ‘ ( 𝑧 ( +g ‘ 𝑀 ) 𝑥 ) ) = ( ( 𝐹 ‘ 𝑧 ) ( +g ‘ 𝑁 ) ( 𝐹 ‘ 𝑥 ) ) ) |
17 |
3
|
oveqd |
⊢ ( 𝜑 → ( 𝑧 ⊕ 𝑥 ) = ( 𝑧 ( +g ‘ 𝑀 ) 𝑥 ) ) |
18 |
17
|
fveq2d |
⊢ ( 𝜑 → ( 𝐹 ‘ ( 𝑧 ⊕ 𝑥 ) ) = ( 𝐹 ‘ ( 𝑧 ( +g ‘ 𝑀 ) 𝑥 ) ) ) |
19 |
4
|
oveqd |
⊢ ( 𝜑 → ( ( 𝐹 ‘ 𝑧 ) + ( 𝐹 ‘ 𝑥 ) ) = ( ( 𝐹 ‘ 𝑧 ) ( +g ‘ 𝑁 ) ( 𝐹 ‘ 𝑥 ) ) ) |
20 |
18 19
|
eqeq12d |
⊢ ( 𝜑 → ( ( 𝐹 ‘ ( 𝑧 ⊕ 𝑥 ) ) = ( ( 𝐹 ‘ 𝑧 ) + ( 𝐹 ‘ 𝑥 ) ) ↔ ( 𝐹 ‘ ( 𝑧 ( +g ‘ 𝑀 ) 𝑥 ) ) = ( ( 𝐹 ‘ 𝑧 ) ( +g ‘ 𝑁 ) ( 𝐹 ‘ 𝑥 ) ) ) ) |
21 |
20
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ) ) → ( ( 𝐹 ‘ ( 𝑧 ⊕ 𝑥 ) ) = ( ( 𝐹 ‘ 𝑧 ) + ( 𝐹 ‘ 𝑥 ) ) ↔ ( 𝐹 ‘ ( 𝑧 ( +g ‘ 𝑀 ) 𝑥 ) ) = ( ( 𝐹 ‘ 𝑧 ) ( +g ‘ 𝑁 ) ( 𝐹 ‘ 𝑥 ) ) ) ) |
22 |
16 21
|
mpbird |
⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ) ) → ( 𝐹 ‘ ( 𝑧 ⊕ 𝑥 ) ) = ( ( 𝐹 ‘ 𝑧 ) + ( 𝐹 ‘ 𝑥 ) ) ) |
23 |
|
eqid |
⊢ ( Base ‘ 𝑁 ) = ( Base ‘ 𝑁 ) |
24 |
12 23
|
mhmf |
⊢ ( 𝐹 ∈ ( 𝑀 MndHom 𝑁 ) → 𝐹 : ( Base ‘ 𝑀 ) ⟶ ( Base ‘ 𝑁 ) ) |
25 |
1 24
|
syl |
⊢ ( 𝜑 → 𝐹 : ( Base ‘ 𝑀 ) ⟶ ( Base ‘ 𝑁 ) ) |
26 |
25
|
ffnd |
⊢ ( 𝜑 → 𝐹 Fn ( Base ‘ 𝑀 ) ) |
27 |
26
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ) ) → 𝐹 Fn ( Base ‘ 𝑀 ) ) |
28 |
5
|
3expb |
⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ) ) → ( 𝑧 ⊕ 𝑥 ) ∈ 𝑋 ) |
29 |
|
fnfvima |
⊢ ( ( 𝐹 Fn ( Base ‘ 𝑀 ) ∧ 𝑋 ⊆ ( Base ‘ 𝑀 ) ∧ ( 𝑧 ⊕ 𝑥 ) ∈ 𝑋 ) → ( 𝐹 ‘ ( 𝑧 ⊕ 𝑥 ) ) ∈ ( 𝐹 “ 𝑋 ) ) |
30 |
27 7 28 29
|
syl3anc |
⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ) ) → ( 𝐹 ‘ ( 𝑧 ⊕ 𝑥 ) ) ∈ ( 𝐹 “ 𝑋 ) ) |
31 |
22 30
|
eqeltrrd |
⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ) ) → ( ( 𝐹 ‘ 𝑧 ) + ( 𝐹 ‘ 𝑥 ) ) ∈ ( 𝐹 “ 𝑋 ) ) |
32 |
31
|
anassrs |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑋 ) ∧ 𝑥 ∈ 𝑋 ) → ( ( 𝐹 ‘ 𝑧 ) + ( 𝐹 ‘ 𝑥 ) ) ∈ ( 𝐹 “ 𝑋 ) ) |
33 |
32
|
ralrimiva |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝑋 ) → ∀ 𝑥 ∈ 𝑋 ( ( 𝐹 ‘ 𝑧 ) + ( 𝐹 ‘ 𝑥 ) ) ∈ ( 𝐹 “ 𝑋 ) ) |
34 |
|
oveq2 |
⊢ ( 𝑦 = ( 𝐹 ‘ 𝑥 ) → ( ( 𝐹 ‘ 𝑧 ) + 𝑦 ) = ( ( 𝐹 ‘ 𝑧 ) + ( 𝐹 ‘ 𝑥 ) ) ) |
35 |
34
|
eleq1d |
⊢ ( 𝑦 = ( 𝐹 ‘ 𝑥 ) → ( ( ( 𝐹 ‘ 𝑧 ) + 𝑦 ) ∈ ( 𝐹 “ 𝑋 ) ↔ ( ( 𝐹 ‘ 𝑧 ) + ( 𝐹 ‘ 𝑥 ) ) ∈ ( 𝐹 “ 𝑋 ) ) ) |
36 |
35
|
ralima |
⊢ ( ( 𝐹 Fn ( Base ‘ 𝑀 ) ∧ 𝑋 ⊆ ( Base ‘ 𝑀 ) ) → ( ∀ 𝑦 ∈ ( 𝐹 “ 𝑋 ) ( ( 𝐹 ‘ 𝑧 ) + 𝑦 ) ∈ ( 𝐹 “ 𝑋 ) ↔ ∀ 𝑥 ∈ 𝑋 ( ( 𝐹 ‘ 𝑧 ) + ( 𝐹 ‘ 𝑥 ) ) ∈ ( 𝐹 “ 𝑋 ) ) ) |
37 |
26 2 36
|
syl2anc |
⊢ ( 𝜑 → ( ∀ 𝑦 ∈ ( 𝐹 “ 𝑋 ) ( ( 𝐹 ‘ 𝑧 ) + 𝑦 ) ∈ ( 𝐹 “ 𝑋 ) ↔ ∀ 𝑥 ∈ 𝑋 ( ( 𝐹 ‘ 𝑧 ) + ( 𝐹 ‘ 𝑥 ) ) ∈ ( 𝐹 “ 𝑋 ) ) ) |
38 |
37
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝑋 ) → ( ∀ 𝑦 ∈ ( 𝐹 “ 𝑋 ) ( ( 𝐹 ‘ 𝑧 ) + 𝑦 ) ∈ ( 𝐹 “ 𝑋 ) ↔ ∀ 𝑥 ∈ 𝑋 ( ( 𝐹 ‘ 𝑧 ) + ( 𝐹 ‘ 𝑥 ) ) ∈ ( 𝐹 “ 𝑋 ) ) ) |
39 |
33 38
|
mpbird |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝑋 ) → ∀ 𝑦 ∈ ( 𝐹 “ 𝑋 ) ( ( 𝐹 ‘ 𝑧 ) + 𝑦 ) ∈ ( 𝐹 “ 𝑋 ) ) |
40 |
39
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑧 ∈ 𝑋 ∀ 𝑦 ∈ ( 𝐹 “ 𝑋 ) ( ( 𝐹 ‘ 𝑧 ) + 𝑦 ) ∈ ( 𝐹 “ 𝑋 ) ) |
41 |
|
oveq1 |
⊢ ( 𝑥 = ( 𝐹 ‘ 𝑧 ) → ( 𝑥 + 𝑦 ) = ( ( 𝐹 ‘ 𝑧 ) + 𝑦 ) ) |
42 |
41
|
eleq1d |
⊢ ( 𝑥 = ( 𝐹 ‘ 𝑧 ) → ( ( 𝑥 + 𝑦 ) ∈ ( 𝐹 “ 𝑋 ) ↔ ( ( 𝐹 ‘ 𝑧 ) + 𝑦 ) ∈ ( 𝐹 “ 𝑋 ) ) ) |
43 |
42
|
ralbidv |
⊢ ( 𝑥 = ( 𝐹 ‘ 𝑧 ) → ( ∀ 𝑦 ∈ ( 𝐹 “ 𝑋 ) ( 𝑥 + 𝑦 ) ∈ ( 𝐹 “ 𝑋 ) ↔ ∀ 𝑦 ∈ ( 𝐹 “ 𝑋 ) ( ( 𝐹 ‘ 𝑧 ) + 𝑦 ) ∈ ( 𝐹 “ 𝑋 ) ) ) |
44 |
43
|
ralima |
⊢ ( ( 𝐹 Fn ( Base ‘ 𝑀 ) ∧ 𝑋 ⊆ ( Base ‘ 𝑀 ) ) → ( ∀ 𝑥 ∈ ( 𝐹 “ 𝑋 ) ∀ 𝑦 ∈ ( 𝐹 “ 𝑋 ) ( 𝑥 + 𝑦 ) ∈ ( 𝐹 “ 𝑋 ) ↔ ∀ 𝑧 ∈ 𝑋 ∀ 𝑦 ∈ ( 𝐹 “ 𝑋 ) ( ( 𝐹 ‘ 𝑧 ) + 𝑦 ) ∈ ( 𝐹 “ 𝑋 ) ) ) |
45 |
26 2 44
|
syl2anc |
⊢ ( 𝜑 → ( ∀ 𝑥 ∈ ( 𝐹 “ 𝑋 ) ∀ 𝑦 ∈ ( 𝐹 “ 𝑋 ) ( 𝑥 + 𝑦 ) ∈ ( 𝐹 “ 𝑋 ) ↔ ∀ 𝑧 ∈ 𝑋 ∀ 𝑦 ∈ ( 𝐹 “ 𝑋 ) ( ( 𝐹 ‘ 𝑧 ) + 𝑦 ) ∈ ( 𝐹 “ 𝑋 ) ) ) |
46 |
40 45
|
mpbird |
⊢ ( 𝜑 → ∀ 𝑥 ∈ ( 𝐹 “ 𝑋 ) ∀ 𝑦 ∈ ( 𝐹 “ 𝑋 ) ( 𝑥 + 𝑦 ) ∈ ( 𝐹 “ 𝑋 ) ) |