Step |
Hyp |
Ref |
Expression |
1 |
|
frlmplusgvalb.f |
⊢ 𝐹 = ( 𝑅 freeLMod 𝐼 ) |
2 |
|
frlmplusgvalb.b |
⊢ 𝐵 = ( Base ‘ 𝐹 ) |
3 |
|
frlmplusgvalb.i |
⊢ ( 𝜑 → 𝐼 ∈ 𝑊 ) |
4 |
|
frlmplusgvalb.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝐵 ) |
5 |
|
frlmplusgvalb.z |
⊢ ( 𝜑 → 𝑍 ∈ 𝐵 ) |
6 |
|
frlmplusgvalb.r |
⊢ ( 𝜑 → 𝑅 ∈ Ring ) |
7 |
|
frlmplusgvalb.y |
⊢ ( 𝜑 → 𝑌 ∈ 𝐵 ) |
8 |
|
frlmplusgvalb.a |
⊢ + = ( +g ‘ 𝑅 ) |
9 |
|
frlmplusgvalb.p |
⊢ ✚ = ( +g ‘ 𝐹 ) |
10 |
|
eqid |
⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 ) |
11 |
1 10 2
|
frlmbasmap |
⊢ ( ( 𝐼 ∈ 𝑊 ∧ 𝑍 ∈ 𝐵 ) → 𝑍 ∈ ( ( Base ‘ 𝑅 ) ↑m 𝐼 ) ) |
12 |
3 5 11
|
syl2anc |
⊢ ( 𝜑 → 𝑍 ∈ ( ( Base ‘ 𝑅 ) ↑m 𝐼 ) ) |
13 |
|
fvexd |
⊢ ( 𝜑 → ( Base ‘ 𝑅 ) ∈ V ) |
14 |
13 3
|
elmapd |
⊢ ( 𝜑 → ( 𝑍 ∈ ( ( Base ‘ 𝑅 ) ↑m 𝐼 ) ↔ 𝑍 : 𝐼 ⟶ ( Base ‘ 𝑅 ) ) ) |
15 |
12 14
|
mpbid |
⊢ ( 𝜑 → 𝑍 : 𝐼 ⟶ ( Base ‘ 𝑅 ) ) |
16 |
15
|
ffnd |
⊢ ( 𝜑 → 𝑍 Fn 𝐼 ) |
17 |
1
|
frlmlmod |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊 ) → 𝐹 ∈ LMod ) |
18 |
6 3 17
|
syl2anc |
⊢ ( 𝜑 → 𝐹 ∈ LMod ) |
19 |
|
lmodgrp |
⊢ ( 𝐹 ∈ LMod → 𝐹 ∈ Grp ) |
20 |
18 19
|
syl |
⊢ ( 𝜑 → 𝐹 ∈ Grp ) |
21 |
2 9
|
grpcl |
⊢ ( ( 𝐹 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ✚ 𝑌 ) ∈ 𝐵 ) |
22 |
20 4 7 21
|
syl3anc |
⊢ ( 𝜑 → ( 𝑋 ✚ 𝑌 ) ∈ 𝐵 ) |
23 |
1 10 2
|
frlmbasmap |
⊢ ( ( 𝐼 ∈ 𝑊 ∧ ( 𝑋 ✚ 𝑌 ) ∈ 𝐵 ) → ( 𝑋 ✚ 𝑌 ) ∈ ( ( Base ‘ 𝑅 ) ↑m 𝐼 ) ) |
24 |
3 22 23
|
syl2anc |
⊢ ( 𝜑 → ( 𝑋 ✚ 𝑌 ) ∈ ( ( Base ‘ 𝑅 ) ↑m 𝐼 ) ) |
25 |
13 3
|
elmapd |
⊢ ( 𝜑 → ( ( 𝑋 ✚ 𝑌 ) ∈ ( ( Base ‘ 𝑅 ) ↑m 𝐼 ) ↔ ( 𝑋 ✚ 𝑌 ) : 𝐼 ⟶ ( Base ‘ 𝑅 ) ) ) |
26 |
24 25
|
mpbid |
⊢ ( 𝜑 → ( 𝑋 ✚ 𝑌 ) : 𝐼 ⟶ ( Base ‘ 𝑅 ) ) |
27 |
26
|
ffnd |
⊢ ( 𝜑 → ( 𝑋 ✚ 𝑌 ) Fn 𝐼 ) |
28 |
|
eqfnfv |
⊢ ( ( 𝑍 Fn 𝐼 ∧ ( 𝑋 ✚ 𝑌 ) Fn 𝐼 ) → ( 𝑍 = ( 𝑋 ✚ 𝑌 ) ↔ ∀ 𝑖 ∈ 𝐼 ( 𝑍 ‘ 𝑖 ) = ( ( 𝑋 ✚ 𝑌 ) ‘ 𝑖 ) ) ) |
29 |
16 27 28
|
syl2anc |
⊢ ( 𝜑 → ( 𝑍 = ( 𝑋 ✚ 𝑌 ) ↔ ∀ 𝑖 ∈ 𝐼 ( 𝑍 ‘ 𝑖 ) = ( ( 𝑋 ✚ 𝑌 ) ‘ 𝑖 ) ) ) |
30 |
6
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝐼 ) → 𝑅 ∈ Ring ) |
31 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝐼 ) → 𝐼 ∈ 𝑊 ) |
32 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝐼 ) → 𝑋 ∈ 𝐵 ) |
33 |
7
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝐼 ) → 𝑌 ∈ 𝐵 ) |
34 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝐼 ) → 𝑖 ∈ 𝐼 ) |
35 |
1 2 30 31 32 33 34 8 9
|
frlmvplusgvalc |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝐼 ) → ( ( 𝑋 ✚ 𝑌 ) ‘ 𝑖 ) = ( ( 𝑋 ‘ 𝑖 ) + ( 𝑌 ‘ 𝑖 ) ) ) |
36 |
35
|
eqeq2d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝐼 ) → ( ( 𝑍 ‘ 𝑖 ) = ( ( 𝑋 ✚ 𝑌 ) ‘ 𝑖 ) ↔ ( 𝑍 ‘ 𝑖 ) = ( ( 𝑋 ‘ 𝑖 ) + ( 𝑌 ‘ 𝑖 ) ) ) ) |
37 |
36
|
ralbidva |
⊢ ( 𝜑 → ( ∀ 𝑖 ∈ 𝐼 ( 𝑍 ‘ 𝑖 ) = ( ( 𝑋 ✚ 𝑌 ) ‘ 𝑖 ) ↔ ∀ 𝑖 ∈ 𝐼 ( 𝑍 ‘ 𝑖 ) = ( ( 𝑋 ‘ 𝑖 ) + ( 𝑌 ‘ 𝑖 ) ) ) ) |
38 |
29 37
|
bitrd |
⊢ ( 𝜑 → ( 𝑍 = ( 𝑋 ✚ 𝑌 ) ↔ ∀ 𝑖 ∈ 𝐼 ( 𝑍 ‘ 𝑖 ) = ( ( 𝑋 ‘ 𝑖 ) + ( 𝑌 ‘ 𝑖 ) ) ) ) |