Step |
Hyp |
Ref |
Expression |
1 |
|
frlmphl.y |
⊢ 𝑌 = ( 𝑅 freeLMod 𝐼 ) |
2 |
|
frlmphl.b |
⊢ 𝐵 = ( Base ‘ 𝑅 ) |
3 |
|
frlmphl.t |
⊢ · = ( .r ‘ 𝑅 ) |
4 |
|
frlmphl.v |
⊢ 𝑉 = ( Base ‘ 𝑌 ) |
5 |
|
frlmphl.j |
⊢ , = ( ·𝑖 ‘ 𝑌 ) |
6 |
1 2 4
|
frlmbasmap |
⊢ ( ( 𝐼 ∈ 𝑊 ∧ 𝐹 ∈ 𝑉 ) → 𝐹 ∈ ( 𝐵 ↑m 𝐼 ) ) |
7 |
6
|
ad2ant2r |
⊢ ( ( ( 𝐼 ∈ 𝑊 ∧ 𝑅 ∈ 𝑋 ) ∧ ( 𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑉 ) ) → 𝐹 ∈ ( 𝐵 ↑m 𝐼 ) ) |
8 |
|
elmapi |
⊢ ( 𝐹 ∈ ( 𝐵 ↑m 𝐼 ) → 𝐹 : 𝐼 ⟶ 𝐵 ) |
9 |
|
ffn |
⊢ ( 𝐹 : 𝐼 ⟶ 𝐵 → 𝐹 Fn 𝐼 ) |
10 |
7 8 9
|
3syl |
⊢ ( ( ( 𝐼 ∈ 𝑊 ∧ 𝑅 ∈ 𝑋 ) ∧ ( 𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑉 ) ) → 𝐹 Fn 𝐼 ) |
11 |
1 2 4
|
frlmbasmap |
⊢ ( ( 𝐼 ∈ 𝑊 ∧ 𝐺 ∈ 𝑉 ) → 𝐺 ∈ ( 𝐵 ↑m 𝐼 ) ) |
12 |
11
|
ad2ant2rl |
⊢ ( ( ( 𝐼 ∈ 𝑊 ∧ 𝑅 ∈ 𝑋 ) ∧ ( 𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑉 ) ) → 𝐺 ∈ ( 𝐵 ↑m 𝐼 ) ) |
13 |
|
elmapi |
⊢ ( 𝐺 ∈ ( 𝐵 ↑m 𝐼 ) → 𝐺 : 𝐼 ⟶ 𝐵 ) |
14 |
|
ffn |
⊢ ( 𝐺 : 𝐼 ⟶ 𝐵 → 𝐺 Fn 𝐼 ) |
15 |
12 13 14
|
3syl |
⊢ ( ( ( 𝐼 ∈ 𝑊 ∧ 𝑅 ∈ 𝑋 ) ∧ ( 𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑉 ) ) → 𝐺 Fn 𝐼 ) |
16 |
|
simpll |
⊢ ( ( ( 𝐼 ∈ 𝑊 ∧ 𝑅 ∈ 𝑋 ) ∧ ( 𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑉 ) ) → 𝐼 ∈ 𝑊 ) |
17 |
|
inidm |
⊢ ( 𝐼 ∩ 𝐼 ) = 𝐼 |
18 |
|
eqidd |
⊢ ( ( ( ( 𝐼 ∈ 𝑊 ∧ 𝑅 ∈ 𝑋 ) ∧ ( 𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑉 ) ) ∧ 𝑥 ∈ 𝐼 ) → ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑥 ) ) |
19 |
|
eqidd |
⊢ ( ( ( ( 𝐼 ∈ 𝑊 ∧ 𝑅 ∈ 𝑋 ) ∧ ( 𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑉 ) ) ∧ 𝑥 ∈ 𝐼 ) → ( 𝐺 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑥 ) ) |
20 |
10 15 16 16 17 18 19
|
offval |
⊢ ( ( ( 𝐼 ∈ 𝑊 ∧ 𝑅 ∈ 𝑋 ) ∧ ( 𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑉 ) ) → ( 𝐹 ∘f · 𝐺 ) = ( 𝑥 ∈ 𝐼 ↦ ( ( 𝐹 ‘ 𝑥 ) · ( 𝐺 ‘ 𝑥 ) ) ) ) |
21 |
20
|
oveq2d |
⊢ ( ( ( 𝐼 ∈ 𝑊 ∧ 𝑅 ∈ 𝑋 ) ∧ ( 𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑉 ) ) → ( 𝑅 Σg ( 𝐹 ∘f · 𝐺 ) ) = ( 𝑅 Σg ( 𝑥 ∈ 𝐼 ↦ ( ( 𝐹 ‘ 𝑥 ) · ( 𝐺 ‘ 𝑥 ) ) ) ) ) |
22 |
|
ovexd |
⊢ ( ( ( 𝐼 ∈ 𝑊 ∧ 𝑅 ∈ 𝑋 ) ∧ ( 𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑉 ) ) → ( 𝑅 Σg ( 𝑥 ∈ 𝐼 ↦ ( ( 𝐹 ‘ 𝑥 ) · ( 𝐺 ‘ 𝑥 ) ) ) ) ∈ V ) |
23 |
|
fveq1 |
⊢ ( 𝑓 = 𝐹 → ( 𝑓 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑥 ) ) |
24 |
23
|
oveq1d |
⊢ ( 𝑓 = 𝐹 → ( ( 𝑓 ‘ 𝑥 ) · ( 𝑔 ‘ 𝑥 ) ) = ( ( 𝐹 ‘ 𝑥 ) · ( 𝑔 ‘ 𝑥 ) ) ) |
25 |
24
|
mpteq2dv |
⊢ ( 𝑓 = 𝐹 → ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑓 ‘ 𝑥 ) · ( 𝑔 ‘ 𝑥 ) ) ) = ( 𝑥 ∈ 𝐼 ↦ ( ( 𝐹 ‘ 𝑥 ) · ( 𝑔 ‘ 𝑥 ) ) ) ) |
26 |
25
|
oveq2d |
⊢ ( 𝑓 = 𝐹 → ( 𝑅 Σg ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑓 ‘ 𝑥 ) · ( 𝑔 ‘ 𝑥 ) ) ) ) = ( 𝑅 Σg ( 𝑥 ∈ 𝐼 ↦ ( ( 𝐹 ‘ 𝑥 ) · ( 𝑔 ‘ 𝑥 ) ) ) ) ) |
27 |
|
fveq1 |
⊢ ( 𝑔 = 𝐺 → ( 𝑔 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑥 ) ) |
28 |
27
|
oveq2d |
⊢ ( 𝑔 = 𝐺 → ( ( 𝐹 ‘ 𝑥 ) · ( 𝑔 ‘ 𝑥 ) ) = ( ( 𝐹 ‘ 𝑥 ) · ( 𝐺 ‘ 𝑥 ) ) ) |
29 |
28
|
mpteq2dv |
⊢ ( 𝑔 = 𝐺 → ( 𝑥 ∈ 𝐼 ↦ ( ( 𝐹 ‘ 𝑥 ) · ( 𝑔 ‘ 𝑥 ) ) ) = ( 𝑥 ∈ 𝐼 ↦ ( ( 𝐹 ‘ 𝑥 ) · ( 𝐺 ‘ 𝑥 ) ) ) ) |
30 |
29
|
oveq2d |
⊢ ( 𝑔 = 𝐺 → ( 𝑅 Σg ( 𝑥 ∈ 𝐼 ↦ ( ( 𝐹 ‘ 𝑥 ) · ( 𝑔 ‘ 𝑥 ) ) ) ) = ( 𝑅 Σg ( 𝑥 ∈ 𝐼 ↦ ( ( 𝐹 ‘ 𝑥 ) · ( 𝐺 ‘ 𝑥 ) ) ) ) ) |
31 |
|
eqid |
⊢ ( 𝑓 ∈ ( 𝐵 ↑m 𝐼 ) , 𝑔 ∈ ( 𝐵 ↑m 𝐼 ) ↦ ( 𝑅 Σg ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑓 ‘ 𝑥 ) · ( 𝑔 ‘ 𝑥 ) ) ) ) ) = ( 𝑓 ∈ ( 𝐵 ↑m 𝐼 ) , 𝑔 ∈ ( 𝐵 ↑m 𝐼 ) ↦ ( 𝑅 Σg ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑓 ‘ 𝑥 ) · ( 𝑔 ‘ 𝑥 ) ) ) ) ) |
32 |
26 30 31
|
ovmpog |
⊢ ( ( 𝐹 ∈ ( 𝐵 ↑m 𝐼 ) ∧ 𝐺 ∈ ( 𝐵 ↑m 𝐼 ) ∧ ( 𝑅 Σg ( 𝑥 ∈ 𝐼 ↦ ( ( 𝐹 ‘ 𝑥 ) · ( 𝐺 ‘ 𝑥 ) ) ) ) ∈ V ) → ( 𝐹 ( 𝑓 ∈ ( 𝐵 ↑m 𝐼 ) , 𝑔 ∈ ( 𝐵 ↑m 𝐼 ) ↦ ( 𝑅 Σg ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑓 ‘ 𝑥 ) · ( 𝑔 ‘ 𝑥 ) ) ) ) ) 𝐺 ) = ( 𝑅 Σg ( 𝑥 ∈ 𝐼 ↦ ( ( 𝐹 ‘ 𝑥 ) · ( 𝐺 ‘ 𝑥 ) ) ) ) ) |
33 |
7 12 22 32
|
syl3anc |
⊢ ( ( ( 𝐼 ∈ 𝑊 ∧ 𝑅 ∈ 𝑋 ) ∧ ( 𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑉 ) ) → ( 𝐹 ( 𝑓 ∈ ( 𝐵 ↑m 𝐼 ) , 𝑔 ∈ ( 𝐵 ↑m 𝐼 ) ↦ ( 𝑅 Σg ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑓 ‘ 𝑥 ) · ( 𝑔 ‘ 𝑥 ) ) ) ) ) 𝐺 ) = ( 𝑅 Σg ( 𝑥 ∈ 𝐼 ↦ ( ( 𝐹 ‘ 𝑥 ) · ( 𝐺 ‘ 𝑥 ) ) ) ) ) |
34 |
1 2 3
|
frlmip |
⊢ ( ( 𝐼 ∈ 𝑊 ∧ 𝑅 ∈ 𝑋 ) → ( 𝑓 ∈ ( 𝐵 ↑m 𝐼 ) , 𝑔 ∈ ( 𝐵 ↑m 𝐼 ) ↦ ( 𝑅 Σg ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑓 ‘ 𝑥 ) · ( 𝑔 ‘ 𝑥 ) ) ) ) ) = ( ·𝑖 ‘ 𝑌 ) ) |
35 |
34
|
adantr |
⊢ ( ( ( 𝐼 ∈ 𝑊 ∧ 𝑅 ∈ 𝑋 ) ∧ ( 𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑉 ) ) → ( 𝑓 ∈ ( 𝐵 ↑m 𝐼 ) , 𝑔 ∈ ( 𝐵 ↑m 𝐼 ) ↦ ( 𝑅 Σg ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑓 ‘ 𝑥 ) · ( 𝑔 ‘ 𝑥 ) ) ) ) ) = ( ·𝑖 ‘ 𝑌 ) ) |
36 |
35 5
|
eqtr4di |
⊢ ( ( ( 𝐼 ∈ 𝑊 ∧ 𝑅 ∈ 𝑋 ) ∧ ( 𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑉 ) ) → ( 𝑓 ∈ ( 𝐵 ↑m 𝐼 ) , 𝑔 ∈ ( 𝐵 ↑m 𝐼 ) ↦ ( 𝑅 Σg ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑓 ‘ 𝑥 ) · ( 𝑔 ‘ 𝑥 ) ) ) ) ) = , ) |
37 |
36
|
oveqd |
⊢ ( ( ( 𝐼 ∈ 𝑊 ∧ 𝑅 ∈ 𝑋 ) ∧ ( 𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑉 ) ) → ( 𝐹 ( 𝑓 ∈ ( 𝐵 ↑m 𝐼 ) , 𝑔 ∈ ( 𝐵 ↑m 𝐼 ) ↦ ( 𝑅 Σg ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑓 ‘ 𝑥 ) · ( 𝑔 ‘ 𝑥 ) ) ) ) ) 𝐺 ) = ( 𝐹 , 𝐺 ) ) |
38 |
21 33 37
|
3eqtr2rd |
⊢ ( ( ( 𝐼 ∈ 𝑊 ∧ 𝑅 ∈ 𝑋 ) ∧ ( 𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑉 ) ) → ( 𝐹 , 𝐺 ) = ( 𝑅 Σg ( 𝐹 ∘f · 𝐺 ) ) ) |