Step |
Hyp |
Ref |
Expression |
1 |
|
imasmnd.u |
⊢ ( 𝜑 → 𝑈 = ( 𝐹 “s 𝑅 ) ) |
2 |
|
imasmnd.v |
⊢ ( 𝜑 → 𝑉 = ( Base ‘ 𝑅 ) ) |
3 |
|
imasmnd.p |
⊢ + = ( +g ‘ 𝑅 ) |
4 |
|
imasmnd.f |
⊢ ( 𝜑 → 𝐹 : 𝑉 –onto→ 𝐵 ) |
5 |
|
imasmnd.e |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ∧ ( 𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉 ) ) → ( ( ( 𝐹 ‘ 𝑎 ) = ( 𝐹 ‘ 𝑝 ) ∧ ( 𝐹 ‘ 𝑏 ) = ( 𝐹 ‘ 𝑞 ) ) → ( 𝐹 ‘ ( 𝑎 + 𝑏 ) ) = ( 𝐹 ‘ ( 𝑝 + 𝑞 ) ) ) ) |
6 |
|
imasmnd.r |
⊢ ( 𝜑 → 𝑅 ∈ Mnd ) |
7 |
|
imasmnd.z |
⊢ 0 = ( 0g ‘ 𝑅 ) |
8 |
6
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) → 𝑅 ∈ Mnd ) |
9 |
|
simp2 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) → 𝑥 ∈ 𝑉 ) |
10 |
2
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) → 𝑉 = ( Base ‘ 𝑅 ) ) |
11 |
9 10
|
eleqtrd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) → 𝑥 ∈ ( Base ‘ 𝑅 ) ) |
12 |
|
simp3 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) → 𝑦 ∈ 𝑉 ) |
13 |
12 10
|
eleqtrd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) → 𝑦 ∈ ( Base ‘ 𝑅 ) ) |
14 |
|
eqid |
⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 ) |
15 |
14 3
|
mndcl |
⊢ ( ( 𝑅 ∈ Mnd ∧ 𝑥 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ ( Base ‘ 𝑅 ) ) → ( 𝑥 + 𝑦 ) ∈ ( Base ‘ 𝑅 ) ) |
16 |
8 11 13 15
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) → ( 𝑥 + 𝑦 ) ∈ ( Base ‘ 𝑅 ) ) |
17 |
16 10
|
eleqtrrd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) → ( 𝑥 + 𝑦 ) ∈ 𝑉 ) |
18 |
6
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) → 𝑅 ∈ Mnd ) |
19 |
11
|
3adant3r3 |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) → 𝑥 ∈ ( Base ‘ 𝑅 ) ) |
20 |
13
|
3adant3r3 |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) → 𝑦 ∈ ( Base ‘ 𝑅 ) ) |
21 |
|
simpr3 |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) → 𝑧 ∈ 𝑉 ) |
22 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) → 𝑉 = ( Base ‘ 𝑅 ) ) |
23 |
21 22
|
eleqtrd |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) → 𝑧 ∈ ( Base ‘ 𝑅 ) ) |
24 |
14 3
|
mndass |
⊢ ( ( 𝑅 ∈ Mnd ∧ ( 𝑥 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ ( Base ‘ 𝑅 ) ∧ 𝑧 ∈ ( Base ‘ 𝑅 ) ) ) → ( ( 𝑥 + 𝑦 ) + 𝑧 ) = ( 𝑥 + ( 𝑦 + 𝑧 ) ) ) |
25 |
18 19 20 23 24
|
syl13anc |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) → ( ( 𝑥 + 𝑦 ) + 𝑧 ) = ( 𝑥 + ( 𝑦 + 𝑧 ) ) ) |
26 |
25
|
fveq2d |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) → ( 𝐹 ‘ ( ( 𝑥 + 𝑦 ) + 𝑧 ) ) = ( 𝐹 ‘ ( 𝑥 + ( 𝑦 + 𝑧 ) ) ) ) |
27 |
14 7
|
mndidcl |
⊢ ( 𝑅 ∈ Mnd → 0 ∈ ( Base ‘ 𝑅 ) ) |
28 |
6 27
|
syl |
⊢ ( 𝜑 → 0 ∈ ( Base ‘ 𝑅 ) ) |
29 |
28 2
|
eleqtrrd |
⊢ ( 𝜑 → 0 ∈ 𝑉 ) |
30 |
2
|
eleq2d |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝑉 ↔ 𝑥 ∈ ( Base ‘ 𝑅 ) ) ) |
31 |
30
|
biimpa |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑉 ) → 𝑥 ∈ ( Base ‘ 𝑅 ) ) |
32 |
14 3 7
|
mndlid |
⊢ ( ( 𝑅 ∈ Mnd ∧ 𝑥 ∈ ( Base ‘ 𝑅 ) ) → ( 0 + 𝑥 ) = 𝑥 ) |
33 |
6 31 32
|
syl2an2r |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑉 ) → ( 0 + 𝑥 ) = 𝑥 ) |
34 |
33
|
fveq2d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑉 ) → ( 𝐹 ‘ ( 0 + 𝑥 ) ) = ( 𝐹 ‘ 𝑥 ) ) |
35 |
14 3 7
|
mndrid |
⊢ ( ( 𝑅 ∈ Mnd ∧ 𝑥 ∈ ( Base ‘ 𝑅 ) ) → ( 𝑥 + 0 ) = 𝑥 ) |
36 |
6 31 35
|
syl2an2r |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑉 ) → ( 𝑥 + 0 ) = 𝑥 ) |
37 |
36
|
fveq2d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑉 ) → ( 𝐹 ‘ ( 𝑥 + 0 ) ) = ( 𝐹 ‘ 𝑥 ) ) |
38 |
1 2 3 4 5 6 17 26 29 34 37
|
imasmnd2 |
⊢ ( 𝜑 → ( 𝑈 ∈ Mnd ∧ ( 𝐹 ‘ 0 ) = ( 0g ‘ 𝑈 ) ) ) |