Step |
Hyp |
Ref |
Expression |
1 |
|
mamucl.b |
⊢ 𝐵 = ( Base ‘ 𝑅 ) |
2 |
|
mamucl.r |
⊢ ( 𝜑 → 𝑅 ∈ Ring ) |
3 |
|
mamudi.f |
⊢ 𝐹 = ( 𝑅 maMul 〈 𝑀 , 𝑁 , 𝑂 〉 ) |
4 |
|
mamudi.m |
⊢ ( 𝜑 → 𝑀 ∈ Fin ) |
5 |
|
mamudi.n |
⊢ ( 𝜑 → 𝑁 ∈ Fin ) |
6 |
|
mamudi.o |
⊢ ( 𝜑 → 𝑂 ∈ Fin ) |
7 |
|
mamudi.p |
⊢ + = ( +g ‘ 𝑅 ) |
8 |
|
mamudi.x |
⊢ ( 𝜑 → 𝑋 ∈ ( 𝐵 ↑m ( 𝑀 × 𝑁 ) ) ) |
9 |
|
mamudi.y |
⊢ ( 𝜑 → 𝑌 ∈ ( 𝐵 ↑m ( 𝑀 × 𝑁 ) ) ) |
10 |
|
mamudi.z |
⊢ ( 𝜑 → 𝑍 ∈ ( 𝐵 ↑m ( 𝑁 × 𝑂 ) ) ) |
11 |
|
ringcmn |
⊢ ( 𝑅 ∈ Ring → 𝑅 ∈ CMnd ) |
12 |
2 11
|
syl |
⊢ ( 𝜑 → 𝑅 ∈ CMnd ) |
13 |
12
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂 ) ) → 𝑅 ∈ CMnd ) |
14 |
5
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂 ) ) → 𝑁 ∈ Fin ) |
15 |
2
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂 ) ) ∧ 𝑗 ∈ 𝑁 ) → 𝑅 ∈ Ring ) |
16 |
|
elmapi |
⊢ ( 𝑋 ∈ ( 𝐵 ↑m ( 𝑀 × 𝑁 ) ) → 𝑋 : ( 𝑀 × 𝑁 ) ⟶ 𝐵 ) |
17 |
8 16
|
syl |
⊢ ( 𝜑 → 𝑋 : ( 𝑀 × 𝑁 ) ⟶ 𝐵 ) |
18 |
17
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂 ) ) ∧ 𝑗 ∈ 𝑁 ) → 𝑋 : ( 𝑀 × 𝑁 ) ⟶ 𝐵 ) |
19 |
|
simplrl |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂 ) ) ∧ 𝑗 ∈ 𝑁 ) → 𝑖 ∈ 𝑀 ) |
20 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂 ) ) ∧ 𝑗 ∈ 𝑁 ) → 𝑗 ∈ 𝑁 ) |
21 |
18 19 20
|
fovrnd |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂 ) ) ∧ 𝑗 ∈ 𝑁 ) → ( 𝑖 𝑋 𝑗 ) ∈ 𝐵 ) |
22 |
|
elmapi |
⊢ ( 𝑍 ∈ ( 𝐵 ↑m ( 𝑁 × 𝑂 ) ) → 𝑍 : ( 𝑁 × 𝑂 ) ⟶ 𝐵 ) |
23 |
10 22
|
syl |
⊢ ( 𝜑 → 𝑍 : ( 𝑁 × 𝑂 ) ⟶ 𝐵 ) |
24 |
23
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂 ) ) ∧ 𝑗 ∈ 𝑁 ) → 𝑍 : ( 𝑁 × 𝑂 ) ⟶ 𝐵 ) |
25 |
|
simplrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂 ) ) ∧ 𝑗 ∈ 𝑁 ) → 𝑘 ∈ 𝑂 ) |
26 |
24 20 25
|
fovrnd |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂 ) ) ∧ 𝑗 ∈ 𝑁 ) → ( 𝑗 𝑍 𝑘 ) ∈ 𝐵 ) |
27 |
|
eqid |
⊢ ( .r ‘ 𝑅 ) = ( .r ‘ 𝑅 ) |
28 |
1 27
|
ringcl |
⊢ ( ( 𝑅 ∈ Ring ∧ ( 𝑖 𝑋 𝑗 ) ∈ 𝐵 ∧ ( 𝑗 𝑍 𝑘 ) ∈ 𝐵 ) → ( ( 𝑖 𝑋 𝑗 ) ( .r ‘ 𝑅 ) ( 𝑗 𝑍 𝑘 ) ) ∈ 𝐵 ) |
29 |
15 21 26 28
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂 ) ) ∧ 𝑗 ∈ 𝑁 ) → ( ( 𝑖 𝑋 𝑗 ) ( .r ‘ 𝑅 ) ( 𝑗 𝑍 𝑘 ) ) ∈ 𝐵 ) |
30 |
|
elmapi |
⊢ ( 𝑌 ∈ ( 𝐵 ↑m ( 𝑀 × 𝑁 ) ) → 𝑌 : ( 𝑀 × 𝑁 ) ⟶ 𝐵 ) |
31 |
9 30
|
syl |
⊢ ( 𝜑 → 𝑌 : ( 𝑀 × 𝑁 ) ⟶ 𝐵 ) |
32 |
31
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂 ) ) ∧ 𝑗 ∈ 𝑁 ) → 𝑌 : ( 𝑀 × 𝑁 ) ⟶ 𝐵 ) |
33 |
32 19 20
|
fovrnd |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂 ) ) ∧ 𝑗 ∈ 𝑁 ) → ( 𝑖 𝑌 𝑗 ) ∈ 𝐵 ) |
34 |
1 27
|
ringcl |
⊢ ( ( 𝑅 ∈ Ring ∧ ( 𝑖 𝑌 𝑗 ) ∈ 𝐵 ∧ ( 𝑗 𝑍 𝑘 ) ∈ 𝐵 ) → ( ( 𝑖 𝑌 𝑗 ) ( .r ‘ 𝑅 ) ( 𝑗 𝑍 𝑘 ) ) ∈ 𝐵 ) |
35 |
15 33 26 34
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂 ) ) ∧ 𝑗 ∈ 𝑁 ) → ( ( 𝑖 𝑌 𝑗 ) ( .r ‘ 𝑅 ) ( 𝑗 𝑍 𝑘 ) ) ∈ 𝐵 ) |
36 |
|
eqid |
⊢ ( 𝑗 ∈ 𝑁 ↦ ( ( 𝑖 𝑋 𝑗 ) ( .r ‘ 𝑅 ) ( 𝑗 𝑍 𝑘 ) ) ) = ( 𝑗 ∈ 𝑁 ↦ ( ( 𝑖 𝑋 𝑗 ) ( .r ‘ 𝑅 ) ( 𝑗 𝑍 𝑘 ) ) ) |
37 |
|
eqid |
⊢ ( 𝑗 ∈ 𝑁 ↦ ( ( 𝑖 𝑌 𝑗 ) ( .r ‘ 𝑅 ) ( 𝑗 𝑍 𝑘 ) ) ) = ( 𝑗 ∈ 𝑁 ↦ ( ( 𝑖 𝑌 𝑗 ) ( .r ‘ 𝑅 ) ( 𝑗 𝑍 𝑘 ) ) ) |
38 |
1 7 13 14 29 35 36 37
|
gsummptfidmadd2 |
⊢ ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂 ) ) → ( 𝑅 Σg ( ( 𝑗 ∈ 𝑁 ↦ ( ( 𝑖 𝑋 𝑗 ) ( .r ‘ 𝑅 ) ( 𝑗 𝑍 𝑘 ) ) ) ∘f + ( 𝑗 ∈ 𝑁 ↦ ( ( 𝑖 𝑌 𝑗 ) ( .r ‘ 𝑅 ) ( 𝑗 𝑍 𝑘 ) ) ) ) ) = ( ( 𝑅 Σg ( 𝑗 ∈ 𝑁 ↦ ( ( 𝑖 𝑋 𝑗 ) ( .r ‘ 𝑅 ) ( 𝑗 𝑍 𝑘 ) ) ) ) + ( 𝑅 Σg ( 𝑗 ∈ 𝑁 ↦ ( ( 𝑖 𝑌 𝑗 ) ( .r ‘ 𝑅 ) ( 𝑗 𝑍 𝑘 ) ) ) ) ) ) |
39 |
8
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂 ) ) ∧ 𝑗 ∈ 𝑁 ) → 𝑋 ∈ ( 𝐵 ↑m ( 𝑀 × 𝑁 ) ) ) |
40 |
|
ffn |
⊢ ( 𝑋 : ( 𝑀 × 𝑁 ) ⟶ 𝐵 → 𝑋 Fn ( 𝑀 × 𝑁 ) ) |
41 |
39 16 40
|
3syl |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂 ) ) ∧ 𝑗 ∈ 𝑁 ) → 𝑋 Fn ( 𝑀 × 𝑁 ) ) |
42 |
9
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂 ) ) ∧ 𝑗 ∈ 𝑁 ) → 𝑌 ∈ ( 𝐵 ↑m ( 𝑀 × 𝑁 ) ) ) |
43 |
|
ffn |
⊢ ( 𝑌 : ( 𝑀 × 𝑁 ) ⟶ 𝐵 → 𝑌 Fn ( 𝑀 × 𝑁 ) ) |
44 |
42 30 43
|
3syl |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂 ) ) ∧ 𝑗 ∈ 𝑁 ) → 𝑌 Fn ( 𝑀 × 𝑁 ) ) |
45 |
|
xpfi |
⊢ ( ( 𝑀 ∈ Fin ∧ 𝑁 ∈ Fin ) → ( 𝑀 × 𝑁 ) ∈ Fin ) |
46 |
4 5 45
|
syl2anc |
⊢ ( 𝜑 → ( 𝑀 × 𝑁 ) ∈ Fin ) |
47 |
46
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂 ) ) ∧ 𝑗 ∈ 𝑁 ) → ( 𝑀 × 𝑁 ) ∈ Fin ) |
48 |
|
opelxpi |
⊢ ( ( 𝑖 ∈ 𝑀 ∧ 𝑗 ∈ 𝑁 ) → 〈 𝑖 , 𝑗 〉 ∈ ( 𝑀 × 𝑁 ) ) |
49 |
48
|
adantlr |
⊢ ( ( ( 𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂 ) ∧ 𝑗 ∈ 𝑁 ) → 〈 𝑖 , 𝑗 〉 ∈ ( 𝑀 × 𝑁 ) ) |
50 |
49
|
adantll |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂 ) ) ∧ 𝑗 ∈ 𝑁 ) → 〈 𝑖 , 𝑗 〉 ∈ ( 𝑀 × 𝑁 ) ) |
51 |
|
fnfvof |
⊢ ( ( ( 𝑋 Fn ( 𝑀 × 𝑁 ) ∧ 𝑌 Fn ( 𝑀 × 𝑁 ) ) ∧ ( ( 𝑀 × 𝑁 ) ∈ Fin ∧ 〈 𝑖 , 𝑗 〉 ∈ ( 𝑀 × 𝑁 ) ) ) → ( ( 𝑋 ∘f + 𝑌 ) ‘ 〈 𝑖 , 𝑗 〉 ) = ( ( 𝑋 ‘ 〈 𝑖 , 𝑗 〉 ) + ( 𝑌 ‘ 〈 𝑖 , 𝑗 〉 ) ) ) |
52 |
41 44 47 50 51
|
syl22anc |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂 ) ) ∧ 𝑗 ∈ 𝑁 ) → ( ( 𝑋 ∘f + 𝑌 ) ‘ 〈 𝑖 , 𝑗 〉 ) = ( ( 𝑋 ‘ 〈 𝑖 , 𝑗 〉 ) + ( 𝑌 ‘ 〈 𝑖 , 𝑗 〉 ) ) ) |
53 |
|
df-ov |
⊢ ( 𝑖 ( 𝑋 ∘f + 𝑌 ) 𝑗 ) = ( ( 𝑋 ∘f + 𝑌 ) ‘ 〈 𝑖 , 𝑗 〉 ) |
54 |
|
df-ov |
⊢ ( 𝑖 𝑋 𝑗 ) = ( 𝑋 ‘ 〈 𝑖 , 𝑗 〉 ) |
55 |
|
df-ov |
⊢ ( 𝑖 𝑌 𝑗 ) = ( 𝑌 ‘ 〈 𝑖 , 𝑗 〉 ) |
56 |
54 55
|
oveq12i |
⊢ ( ( 𝑖 𝑋 𝑗 ) + ( 𝑖 𝑌 𝑗 ) ) = ( ( 𝑋 ‘ 〈 𝑖 , 𝑗 〉 ) + ( 𝑌 ‘ 〈 𝑖 , 𝑗 〉 ) ) |
57 |
52 53 56
|
3eqtr4g |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂 ) ) ∧ 𝑗 ∈ 𝑁 ) → ( 𝑖 ( 𝑋 ∘f + 𝑌 ) 𝑗 ) = ( ( 𝑖 𝑋 𝑗 ) + ( 𝑖 𝑌 𝑗 ) ) ) |
58 |
57
|
oveq1d |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂 ) ) ∧ 𝑗 ∈ 𝑁 ) → ( ( 𝑖 ( 𝑋 ∘f + 𝑌 ) 𝑗 ) ( .r ‘ 𝑅 ) ( 𝑗 𝑍 𝑘 ) ) = ( ( ( 𝑖 𝑋 𝑗 ) + ( 𝑖 𝑌 𝑗 ) ) ( .r ‘ 𝑅 ) ( 𝑗 𝑍 𝑘 ) ) ) |
59 |
1 7 27
|
ringdir |
⊢ ( ( 𝑅 ∈ Ring ∧ ( ( 𝑖 𝑋 𝑗 ) ∈ 𝐵 ∧ ( 𝑖 𝑌 𝑗 ) ∈ 𝐵 ∧ ( 𝑗 𝑍 𝑘 ) ∈ 𝐵 ) ) → ( ( ( 𝑖 𝑋 𝑗 ) + ( 𝑖 𝑌 𝑗 ) ) ( .r ‘ 𝑅 ) ( 𝑗 𝑍 𝑘 ) ) = ( ( ( 𝑖 𝑋 𝑗 ) ( .r ‘ 𝑅 ) ( 𝑗 𝑍 𝑘 ) ) + ( ( 𝑖 𝑌 𝑗 ) ( .r ‘ 𝑅 ) ( 𝑗 𝑍 𝑘 ) ) ) ) |
60 |
15 21 33 26 59
|
syl13anc |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂 ) ) ∧ 𝑗 ∈ 𝑁 ) → ( ( ( 𝑖 𝑋 𝑗 ) + ( 𝑖 𝑌 𝑗 ) ) ( .r ‘ 𝑅 ) ( 𝑗 𝑍 𝑘 ) ) = ( ( ( 𝑖 𝑋 𝑗 ) ( .r ‘ 𝑅 ) ( 𝑗 𝑍 𝑘 ) ) + ( ( 𝑖 𝑌 𝑗 ) ( .r ‘ 𝑅 ) ( 𝑗 𝑍 𝑘 ) ) ) ) |
61 |
58 60
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂 ) ) ∧ 𝑗 ∈ 𝑁 ) → ( ( 𝑖 ( 𝑋 ∘f + 𝑌 ) 𝑗 ) ( .r ‘ 𝑅 ) ( 𝑗 𝑍 𝑘 ) ) = ( ( ( 𝑖 𝑋 𝑗 ) ( .r ‘ 𝑅 ) ( 𝑗 𝑍 𝑘 ) ) + ( ( 𝑖 𝑌 𝑗 ) ( .r ‘ 𝑅 ) ( 𝑗 𝑍 𝑘 ) ) ) ) |
62 |
61
|
mpteq2dva |
⊢ ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂 ) ) → ( 𝑗 ∈ 𝑁 ↦ ( ( 𝑖 ( 𝑋 ∘f + 𝑌 ) 𝑗 ) ( .r ‘ 𝑅 ) ( 𝑗 𝑍 𝑘 ) ) ) = ( 𝑗 ∈ 𝑁 ↦ ( ( ( 𝑖 𝑋 𝑗 ) ( .r ‘ 𝑅 ) ( 𝑗 𝑍 𝑘 ) ) + ( ( 𝑖 𝑌 𝑗 ) ( .r ‘ 𝑅 ) ( 𝑗 𝑍 𝑘 ) ) ) ) ) |
63 |
|
eqidd |
⊢ ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂 ) ) → ( 𝑗 ∈ 𝑁 ↦ ( ( 𝑖 𝑋 𝑗 ) ( .r ‘ 𝑅 ) ( 𝑗 𝑍 𝑘 ) ) ) = ( 𝑗 ∈ 𝑁 ↦ ( ( 𝑖 𝑋 𝑗 ) ( .r ‘ 𝑅 ) ( 𝑗 𝑍 𝑘 ) ) ) ) |
64 |
|
eqidd |
⊢ ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂 ) ) → ( 𝑗 ∈ 𝑁 ↦ ( ( 𝑖 𝑌 𝑗 ) ( .r ‘ 𝑅 ) ( 𝑗 𝑍 𝑘 ) ) ) = ( 𝑗 ∈ 𝑁 ↦ ( ( 𝑖 𝑌 𝑗 ) ( .r ‘ 𝑅 ) ( 𝑗 𝑍 𝑘 ) ) ) ) |
65 |
14 29 35 63 64
|
offval2 |
⊢ ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂 ) ) → ( ( 𝑗 ∈ 𝑁 ↦ ( ( 𝑖 𝑋 𝑗 ) ( .r ‘ 𝑅 ) ( 𝑗 𝑍 𝑘 ) ) ) ∘f + ( 𝑗 ∈ 𝑁 ↦ ( ( 𝑖 𝑌 𝑗 ) ( .r ‘ 𝑅 ) ( 𝑗 𝑍 𝑘 ) ) ) ) = ( 𝑗 ∈ 𝑁 ↦ ( ( ( 𝑖 𝑋 𝑗 ) ( .r ‘ 𝑅 ) ( 𝑗 𝑍 𝑘 ) ) + ( ( 𝑖 𝑌 𝑗 ) ( .r ‘ 𝑅 ) ( 𝑗 𝑍 𝑘 ) ) ) ) ) |
66 |
62 65
|
eqtr4d |
⊢ ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂 ) ) → ( 𝑗 ∈ 𝑁 ↦ ( ( 𝑖 ( 𝑋 ∘f + 𝑌 ) 𝑗 ) ( .r ‘ 𝑅 ) ( 𝑗 𝑍 𝑘 ) ) ) = ( ( 𝑗 ∈ 𝑁 ↦ ( ( 𝑖 𝑋 𝑗 ) ( .r ‘ 𝑅 ) ( 𝑗 𝑍 𝑘 ) ) ) ∘f + ( 𝑗 ∈ 𝑁 ↦ ( ( 𝑖 𝑌 𝑗 ) ( .r ‘ 𝑅 ) ( 𝑗 𝑍 𝑘 ) ) ) ) ) |
67 |
66
|
oveq2d |
⊢ ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂 ) ) → ( 𝑅 Σg ( 𝑗 ∈ 𝑁 ↦ ( ( 𝑖 ( 𝑋 ∘f + 𝑌 ) 𝑗 ) ( .r ‘ 𝑅 ) ( 𝑗 𝑍 𝑘 ) ) ) ) = ( 𝑅 Σg ( ( 𝑗 ∈ 𝑁 ↦ ( ( 𝑖 𝑋 𝑗 ) ( .r ‘ 𝑅 ) ( 𝑗 𝑍 𝑘 ) ) ) ∘f + ( 𝑗 ∈ 𝑁 ↦ ( ( 𝑖 𝑌 𝑗 ) ( .r ‘ 𝑅 ) ( 𝑗 𝑍 𝑘 ) ) ) ) ) ) |
68 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂 ) ) → 𝑅 ∈ Ring ) |
69 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂 ) ) → 𝑀 ∈ Fin ) |
70 |
6
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂 ) ) → 𝑂 ∈ Fin ) |
71 |
8
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂 ) ) → 𝑋 ∈ ( 𝐵 ↑m ( 𝑀 × 𝑁 ) ) ) |
72 |
10
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂 ) ) → 𝑍 ∈ ( 𝐵 ↑m ( 𝑁 × 𝑂 ) ) ) |
73 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂 ) ) → 𝑖 ∈ 𝑀 ) |
74 |
|
simprr |
⊢ ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂 ) ) → 𝑘 ∈ 𝑂 ) |
75 |
3 1 27 68 69 14 70 71 72 73 74
|
mamufv |
⊢ ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂 ) ) → ( 𝑖 ( 𝑋 𝐹 𝑍 ) 𝑘 ) = ( 𝑅 Σg ( 𝑗 ∈ 𝑁 ↦ ( ( 𝑖 𝑋 𝑗 ) ( .r ‘ 𝑅 ) ( 𝑗 𝑍 𝑘 ) ) ) ) ) |
76 |
9
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂 ) ) → 𝑌 ∈ ( 𝐵 ↑m ( 𝑀 × 𝑁 ) ) ) |
77 |
3 1 27 68 69 14 70 76 72 73 74
|
mamufv |
⊢ ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂 ) ) → ( 𝑖 ( 𝑌 𝐹 𝑍 ) 𝑘 ) = ( 𝑅 Σg ( 𝑗 ∈ 𝑁 ↦ ( ( 𝑖 𝑌 𝑗 ) ( .r ‘ 𝑅 ) ( 𝑗 𝑍 𝑘 ) ) ) ) ) |
78 |
75 77
|
oveq12d |
⊢ ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂 ) ) → ( ( 𝑖 ( 𝑋 𝐹 𝑍 ) 𝑘 ) + ( 𝑖 ( 𝑌 𝐹 𝑍 ) 𝑘 ) ) = ( ( 𝑅 Σg ( 𝑗 ∈ 𝑁 ↦ ( ( 𝑖 𝑋 𝑗 ) ( .r ‘ 𝑅 ) ( 𝑗 𝑍 𝑘 ) ) ) ) + ( 𝑅 Σg ( 𝑗 ∈ 𝑁 ↦ ( ( 𝑖 𝑌 𝑗 ) ( .r ‘ 𝑅 ) ( 𝑗 𝑍 𝑘 ) ) ) ) ) ) |
79 |
38 67 78
|
3eqtr4d |
⊢ ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂 ) ) → ( 𝑅 Σg ( 𝑗 ∈ 𝑁 ↦ ( ( 𝑖 ( 𝑋 ∘f + 𝑌 ) 𝑗 ) ( .r ‘ 𝑅 ) ( 𝑗 𝑍 𝑘 ) ) ) ) = ( ( 𝑖 ( 𝑋 𝐹 𝑍 ) 𝑘 ) + ( 𝑖 ( 𝑌 𝐹 𝑍 ) 𝑘 ) ) ) |
80 |
|
ringmnd |
⊢ ( 𝑅 ∈ Ring → 𝑅 ∈ Mnd ) |
81 |
2 80
|
syl |
⊢ ( 𝜑 → 𝑅 ∈ Mnd ) |
82 |
1 7
|
mndvcl |
⊢ ( ( 𝑅 ∈ Mnd ∧ 𝑋 ∈ ( 𝐵 ↑m ( 𝑀 × 𝑁 ) ) ∧ 𝑌 ∈ ( 𝐵 ↑m ( 𝑀 × 𝑁 ) ) ) → ( 𝑋 ∘f + 𝑌 ) ∈ ( 𝐵 ↑m ( 𝑀 × 𝑁 ) ) ) |
83 |
81 8 9 82
|
syl3anc |
⊢ ( 𝜑 → ( 𝑋 ∘f + 𝑌 ) ∈ ( 𝐵 ↑m ( 𝑀 × 𝑁 ) ) ) |
84 |
83
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂 ) ) → ( 𝑋 ∘f + 𝑌 ) ∈ ( 𝐵 ↑m ( 𝑀 × 𝑁 ) ) ) |
85 |
3 1 27 68 69 14 70 84 72 73 74
|
mamufv |
⊢ ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂 ) ) → ( 𝑖 ( ( 𝑋 ∘f + 𝑌 ) 𝐹 𝑍 ) 𝑘 ) = ( 𝑅 Σg ( 𝑗 ∈ 𝑁 ↦ ( ( 𝑖 ( 𝑋 ∘f + 𝑌 ) 𝑗 ) ( .r ‘ 𝑅 ) ( 𝑗 𝑍 𝑘 ) ) ) ) ) |
86 |
1 2 3 4 5 6 8 10
|
mamucl |
⊢ ( 𝜑 → ( 𝑋 𝐹 𝑍 ) ∈ ( 𝐵 ↑m ( 𝑀 × 𝑂 ) ) ) |
87 |
|
elmapi |
⊢ ( ( 𝑋 𝐹 𝑍 ) ∈ ( 𝐵 ↑m ( 𝑀 × 𝑂 ) ) → ( 𝑋 𝐹 𝑍 ) : ( 𝑀 × 𝑂 ) ⟶ 𝐵 ) |
88 |
|
ffn |
⊢ ( ( 𝑋 𝐹 𝑍 ) : ( 𝑀 × 𝑂 ) ⟶ 𝐵 → ( 𝑋 𝐹 𝑍 ) Fn ( 𝑀 × 𝑂 ) ) |
89 |
86 87 88
|
3syl |
⊢ ( 𝜑 → ( 𝑋 𝐹 𝑍 ) Fn ( 𝑀 × 𝑂 ) ) |
90 |
89
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂 ) ) → ( 𝑋 𝐹 𝑍 ) Fn ( 𝑀 × 𝑂 ) ) |
91 |
1 2 3 4 5 6 9 10
|
mamucl |
⊢ ( 𝜑 → ( 𝑌 𝐹 𝑍 ) ∈ ( 𝐵 ↑m ( 𝑀 × 𝑂 ) ) ) |
92 |
|
elmapi |
⊢ ( ( 𝑌 𝐹 𝑍 ) ∈ ( 𝐵 ↑m ( 𝑀 × 𝑂 ) ) → ( 𝑌 𝐹 𝑍 ) : ( 𝑀 × 𝑂 ) ⟶ 𝐵 ) |
93 |
|
ffn |
⊢ ( ( 𝑌 𝐹 𝑍 ) : ( 𝑀 × 𝑂 ) ⟶ 𝐵 → ( 𝑌 𝐹 𝑍 ) Fn ( 𝑀 × 𝑂 ) ) |
94 |
91 92 93
|
3syl |
⊢ ( 𝜑 → ( 𝑌 𝐹 𝑍 ) Fn ( 𝑀 × 𝑂 ) ) |
95 |
94
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂 ) ) → ( 𝑌 𝐹 𝑍 ) Fn ( 𝑀 × 𝑂 ) ) |
96 |
|
xpfi |
⊢ ( ( 𝑀 ∈ Fin ∧ 𝑂 ∈ Fin ) → ( 𝑀 × 𝑂 ) ∈ Fin ) |
97 |
4 6 96
|
syl2anc |
⊢ ( 𝜑 → ( 𝑀 × 𝑂 ) ∈ Fin ) |
98 |
97
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂 ) ) → ( 𝑀 × 𝑂 ) ∈ Fin ) |
99 |
|
opelxpi |
⊢ ( ( 𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂 ) → 〈 𝑖 , 𝑘 〉 ∈ ( 𝑀 × 𝑂 ) ) |
100 |
99
|
adantl |
⊢ ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂 ) ) → 〈 𝑖 , 𝑘 〉 ∈ ( 𝑀 × 𝑂 ) ) |
101 |
|
fnfvof |
⊢ ( ( ( ( 𝑋 𝐹 𝑍 ) Fn ( 𝑀 × 𝑂 ) ∧ ( 𝑌 𝐹 𝑍 ) Fn ( 𝑀 × 𝑂 ) ) ∧ ( ( 𝑀 × 𝑂 ) ∈ Fin ∧ 〈 𝑖 , 𝑘 〉 ∈ ( 𝑀 × 𝑂 ) ) ) → ( ( ( 𝑋 𝐹 𝑍 ) ∘f + ( 𝑌 𝐹 𝑍 ) ) ‘ 〈 𝑖 , 𝑘 〉 ) = ( ( ( 𝑋 𝐹 𝑍 ) ‘ 〈 𝑖 , 𝑘 〉 ) + ( ( 𝑌 𝐹 𝑍 ) ‘ 〈 𝑖 , 𝑘 〉 ) ) ) |
102 |
90 95 98 100 101
|
syl22anc |
⊢ ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂 ) ) → ( ( ( 𝑋 𝐹 𝑍 ) ∘f + ( 𝑌 𝐹 𝑍 ) ) ‘ 〈 𝑖 , 𝑘 〉 ) = ( ( ( 𝑋 𝐹 𝑍 ) ‘ 〈 𝑖 , 𝑘 〉 ) + ( ( 𝑌 𝐹 𝑍 ) ‘ 〈 𝑖 , 𝑘 〉 ) ) ) |
103 |
|
df-ov |
⊢ ( 𝑖 ( ( 𝑋 𝐹 𝑍 ) ∘f + ( 𝑌 𝐹 𝑍 ) ) 𝑘 ) = ( ( ( 𝑋 𝐹 𝑍 ) ∘f + ( 𝑌 𝐹 𝑍 ) ) ‘ 〈 𝑖 , 𝑘 〉 ) |
104 |
|
df-ov |
⊢ ( 𝑖 ( 𝑋 𝐹 𝑍 ) 𝑘 ) = ( ( 𝑋 𝐹 𝑍 ) ‘ 〈 𝑖 , 𝑘 〉 ) |
105 |
|
df-ov |
⊢ ( 𝑖 ( 𝑌 𝐹 𝑍 ) 𝑘 ) = ( ( 𝑌 𝐹 𝑍 ) ‘ 〈 𝑖 , 𝑘 〉 ) |
106 |
104 105
|
oveq12i |
⊢ ( ( 𝑖 ( 𝑋 𝐹 𝑍 ) 𝑘 ) + ( 𝑖 ( 𝑌 𝐹 𝑍 ) 𝑘 ) ) = ( ( ( 𝑋 𝐹 𝑍 ) ‘ 〈 𝑖 , 𝑘 〉 ) + ( ( 𝑌 𝐹 𝑍 ) ‘ 〈 𝑖 , 𝑘 〉 ) ) |
107 |
102 103 106
|
3eqtr4g |
⊢ ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂 ) ) → ( 𝑖 ( ( 𝑋 𝐹 𝑍 ) ∘f + ( 𝑌 𝐹 𝑍 ) ) 𝑘 ) = ( ( 𝑖 ( 𝑋 𝐹 𝑍 ) 𝑘 ) + ( 𝑖 ( 𝑌 𝐹 𝑍 ) 𝑘 ) ) ) |
108 |
79 85 107
|
3eqtr4d |
⊢ ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂 ) ) → ( 𝑖 ( ( 𝑋 ∘f + 𝑌 ) 𝐹 𝑍 ) 𝑘 ) = ( 𝑖 ( ( 𝑋 𝐹 𝑍 ) ∘f + ( 𝑌 𝐹 𝑍 ) ) 𝑘 ) ) |
109 |
108
|
ralrimivva |
⊢ ( 𝜑 → ∀ 𝑖 ∈ 𝑀 ∀ 𝑘 ∈ 𝑂 ( 𝑖 ( ( 𝑋 ∘f + 𝑌 ) 𝐹 𝑍 ) 𝑘 ) = ( 𝑖 ( ( 𝑋 𝐹 𝑍 ) ∘f + ( 𝑌 𝐹 𝑍 ) ) 𝑘 ) ) |
110 |
1 2 3 4 5 6 83 10
|
mamucl |
⊢ ( 𝜑 → ( ( 𝑋 ∘f + 𝑌 ) 𝐹 𝑍 ) ∈ ( 𝐵 ↑m ( 𝑀 × 𝑂 ) ) ) |
111 |
|
elmapi |
⊢ ( ( ( 𝑋 ∘f + 𝑌 ) 𝐹 𝑍 ) ∈ ( 𝐵 ↑m ( 𝑀 × 𝑂 ) ) → ( ( 𝑋 ∘f + 𝑌 ) 𝐹 𝑍 ) : ( 𝑀 × 𝑂 ) ⟶ 𝐵 ) |
112 |
|
ffn |
⊢ ( ( ( 𝑋 ∘f + 𝑌 ) 𝐹 𝑍 ) : ( 𝑀 × 𝑂 ) ⟶ 𝐵 → ( ( 𝑋 ∘f + 𝑌 ) 𝐹 𝑍 ) Fn ( 𝑀 × 𝑂 ) ) |
113 |
110 111 112
|
3syl |
⊢ ( 𝜑 → ( ( 𝑋 ∘f + 𝑌 ) 𝐹 𝑍 ) Fn ( 𝑀 × 𝑂 ) ) |
114 |
1 7
|
mndvcl |
⊢ ( ( 𝑅 ∈ Mnd ∧ ( 𝑋 𝐹 𝑍 ) ∈ ( 𝐵 ↑m ( 𝑀 × 𝑂 ) ) ∧ ( 𝑌 𝐹 𝑍 ) ∈ ( 𝐵 ↑m ( 𝑀 × 𝑂 ) ) ) → ( ( 𝑋 𝐹 𝑍 ) ∘f + ( 𝑌 𝐹 𝑍 ) ) ∈ ( 𝐵 ↑m ( 𝑀 × 𝑂 ) ) ) |
115 |
81 86 91 114
|
syl3anc |
⊢ ( 𝜑 → ( ( 𝑋 𝐹 𝑍 ) ∘f + ( 𝑌 𝐹 𝑍 ) ) ∈ ( 𝐵 ↑m ( 𝑀 × 𝑂 ) ) ) |
116 |
|
elmapi |
⊢ ( ( ( 𝑋 𝐹 𝑍 ) ∘f + ( 𝑌 𝐹 𝑍 ) ) ∈ ( 𝐵 ↑m ( 𝑀 × 𝑂 ) ) → ( ( 𝑋 𝐹 𝑍 ) ∘f + ( 𝑌 𝐹 𝑍 ) ) : ( 𝑀 × 𝑂 ) ⟶ 𝐵 ) |
117 |
|
ffn |
⊢ ( ( ( 𝑋 𝐹 𝑍 ) ∘f + ( 𝑌 𝐹 𝑍 ) ) : ( 𝑀 × 𝑂 ) ⟶ 𝐵 → ( ( 𝑋 𝐹 𝑍 ) ∘f + ( 𝑌 𝐹 𝑍 ) ) Fn ( 𝑀 × 𝑂 ) ) |
118 |
115 116 117
|
3syl |
⊢ ( 𝜑 → ( ( 𝑋 𝐹 𝑍 ) ∘f + ( 𝑌 𝐹 𝑍 ) ) Fn ( 𝑀 × 𝑂 ) ) |
119 |
|
eqfnov2 |
⊢ ( ( ( ( 𝑋 ∘f + 𝑌 ) 𝐹 𝑍 ) Fn ( 𝑀 × 𝑂 ) ∧ ( ( 𝑋 𝐹 𝑍 ) ∘f + ( 𝑌 𝐹 𝑍 ) ) Fn ( 𝑀 × 𝑂 ) ) → ( ( ( 𝑋 ∘f + 𝑌 ) 𝐹 𝑍 ) = ( ( 𝑋 𝐹 𝑍 ) ∘f + ( 𝑌 𝐹 𝑍 ) ) ↔ ∀ 𝑖 ∈ 𝑀 ∀ 𝑘 ∈ 𝑂 ( 𝑖 ( ( 𝑋 ∘f + 𝑌 ) 𝐹 𝑍 ) 𝑘 ) = ( 𝑖 ( ( 𝑋 𝐹 𝑍 ) ∘f + ( 𝑌 𝐹 𝑍 ) ) 𝑘 ) ) ) |
120 |
113 118 119
|
syl2anc |
⊢ ( 𝜑 → ( ( ( 𝑋 ∘f + 𝑌 ) 𝐹 𝑍 ) = ( ( 𝑋 𝐹 𝑍 ) ∘f + ( 𝑌 𝐹 𝑍 ) ) ↔ ∀ 𝑖 ∈ 𝑀 ∀ 𝑘 ∈ 𝑂 ( 𝑖 ( ( 𝑋 ∘f + 𝑌 ) 𝐹 𝑍 ) 𝑘 ) = ( 𝑖 ( ( 𝑋 𝐹 𝑍 ) ∘f + ( 𝑌 𝐹 𝑍 ) ) 𝑘 ) ) ) |
121 |
109 120
|
mpbird |
⊢ ( 𝜑 → ( ( 𝑋 ∘f + 𝑌 ) 𝐹 𝑍 ) = ( ( 𝑋 𝐹 𝑍 ) ∘f + ( 𝑌 𝐹 𝑍 ) ) ) |