Step |
Hyp |
Ref |
Expression |
1 |
|
ascldimul.a |
⊢ 𝐴 = ( algSc ‘ 𝑊 ) |
2 |
|
ascldimul.f |
⊢ 𝐹 = ( Scalar ‘ 𝑊 ) |
3 |
|
ascldimul.k |
⊢ 𝐾 = ( Base ‘ 𝐹 ) |
4 |
|
ascldimul.t |
⊢ × = ( .r ‘ 𝑊 ) |
5 |
|
ascldimul.s |
⊢ · = ( .r ‘ 𝐹 ) |
6 |
|
assaring |
⊢ ( 𝑊 ∈ AssAlg → 𝑊 ∈ Ring ) |
7 |
|
eqid |
⊢ ( Base ‘ 𝑊 ) = ( Base ‘ 𝑊 ) |
8 |
|
eqid |
⊢ ( 1r ‘ 𝑊 ) = ( 1r ‘ 𝑊 ) |
9 |
7 8
|
ringidcl |
⊢ ( 𝑊 ∈ Ring → ( 1r ‘ 𝑊 ) ∈ ( Base ‘ 𝑊 ) ) |
10 |
6 9
|
syl |
⊢ ( 𝑊 ∈ AssAlg → ( 1r ‘ 𝑊 ) ∈ ( Base ‘ 𝑊 ) ) |
11 |
7 4 8
|
ringlidm |
⊢ ( ( 𝑊 ∈ Ring ∧ ( 1r ‘ 𝑊 ) ∈ ( Base ‘ 𝑊 ) ) → ( ( 1r ‘ 𝑊 ) × ( 1r ‘ 𝑊 ) ) = ( 1r ‘ 𝑊 ) ) |
12 |
6 10 11
|
syl2anc |
⊢ ( 𝑊 ∈ AssAlg → ( ( 1r ‘ 𝑊 ) × ( 1r ‘ 𝑊 ) ) = ( 1r ‘ 𝑊 ) ) |
13 |
12
|
oveq2d |
⊢ ( 𝑊 ∈ AssAlg → ( 𝑆 ( ·𝑠 ‘ 𝑊 ) ( ( 1r ‘ 𝑊 ) × ( 1r ‘ 𝑊 ) ) ) = ( 𝑆 ( ·𝑠 ‘ 𝑊 ) ( 1r ‘ 𝑊 ) ) ) |
14 |
13
|
oveq2d |
⊢ ( 𝑊 ∈ AssAlg → ( 𝑅 ( ·𝑠 ‘ 𝑊 ) ( 𝑆 ( ·𝑠 ‘ 𝑊 ) ( ( 1r ‘ 𝑊 ) × ( 1r ‘ 𝑊 ) ) ) ) = ( 𝑅 ( ·𝑠 ‘ 𝑊 ) ( 𝑆 ( ·𝑠 ‘ 𝑊 ) ( 1r ‘ 𝑊 ) ) ) ) |
15 |
14
|
3ad2ant1 |
⊢ ( ( 𝑊 ∈ AssAlg ∧ 𝑅 ∈ 𝐾 ∧ 𝑆 ∈ 𝐾 ) → ( 𝑅 ( ·𝑠 ‘ 𝑊 ) ( 𝑆 ( ·𝑠 ‘ 𝑊 ) ( ( 1r ‘ 𝑊 ) × ( 1r ‘ 𝑊 ) ) ) ) = ( 𝑅 ( ·𝑠 ‘ 𝑊 ) ( 𝑆 ( ·𝑠 ‘ 𝑊 ) ( 1r ‘ 𝑊 ) ) ) ) |
16 |
|
simp1 |
⊢ ( ( 𝑊 ∈ AssAlg ∧ 𝑅 ∈ 𝐾 ∧ 𝑆 ∈ 𝐾 ) → 𝑊 ∈ AssAlg ) |
17 |
|
simp2 |
⊢ ( ( 𝑊 ∈ AssAlg ∧ 𝑅 ∈ 𝐾 ∧ 𝑆 ∈ 𝐾 ) → 𝑅 ∈ 𝐾 ) |
18 |
10
|
3ad2ant1 |
⊢ ( ( 𝑊 ∈ AssAlg ∧ 𝑅 ∈ 𝐾 ∧ 𝑆 ∈ 𝐾 ) → ( 1r ‘ 𝑊 ) ∈ ( Base ‘ 𝑊 ) ) |
19 |
|
assalmod |
⊢ ( 𝑊 ∈ AssAlg → 𝑊 ∈ LMod ) |
20 |
19
|
3ad2ant1 |
⊢ ( ( 𝑊 ∈ AssAlg ∧ 𝑅 ∈ 𝐾 ∧ 𝑆 ∈ 𝐾 ) → 𝑊 ∈ LMod ) |
21 |
|
simp3 |
⊢ ( ( 𝑊 ∈ AssAlg ∧ 𝑅 ∈ 𝐾 ∧ 𝑆 ∈ 𝐾 ) → 𝑆 ∈ 𝐾 ) |
22 |
|
eqid |
⊢ ( ·𝑠 ‘ 𝑊 ) = ( ·𝑠 ‘ 𝑊 ) |
23 |
7 2 22 3
|
lmodvscl |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑆 ∈ 𝐾 ∧ ( 1r ‘ 𝑊 ) ∈ ( Base ‘ 𝑊 ) ) → ( 𝑆 ( ·𝑠 ‘ 𝑊 ) ( 1r ‘ 𝑊 ) ) ∈ ( Base ‘ 𝑊 ) ) |
24 |
20 21 18 23
|
syl3anc |
⊢ ( ( 𝑊 ∈ AssAlg ∧ 𝑅 ∈ 𝐾 ∧ 𝑆 ∈ 𝐾 ) → ( 𝑆 ( ·𝑠 ‘ 𝑊 ) ( 1r ‘ 𝑊 ) ) ∈ ( Base ‘ 𝑊 ) ) |
25 |
7 2 3 22 4
|
assaass |
⊢ ( ( 𝑊 ∈ AssAlg ∧ ( 𝑅 ∈ 𝐾 ∧ ( 1r ‘ 𝑊 ) ∈ ( Base ‘ 𝑊 ) ∧ ( 𝑆 ( ·𝑠 ‘ 𝑊 ) ( 1r ‘ 𝑊 ) ) ∈ ( Base ‘ 𝑊 ) ) ) → ( ( 𝑅 ( ·𝑠 ‘ 𝑊 ) ( 1r ‘ 𝑊 ) ) × ( 𝑆 ( ·𝑠 ‘ 𝑊 ) ( 1r ‘ 𝑊 ) ) ) = ( 𝑅 ( ·𝑠 ‘ 𝑊 ) ( ( 1r ‘ 𝑊 ) × ( 𝑆 ( ·𝑠 ‘ 𝑊 ) ( 1r ‘ 𝑊 ) ) ) ) ) |
26 |
16 17 18 24 25
|
syl13anc |
⊢ ( ( 𝑊 ∈ AssAlg ∧ 𝑅 ∈ 𝐾 ∧ 𝑆 ∈ 𝐾 ) → ( ( 𝑅 ( ·𝑠 ‘ 𝑊 ) ( 1r ‘ 𝑊 ) ) × ( 𝑆 ( ·𝑠 ‘ 𝑊 ) ( 1r ‘ 𝑊 ) ) ) = ( 𝑅 ( ·𝑠 ‘ 𝑊 ) ( ( 1r ‘ 𝑊 ) × ( 𝑆 ( ·𝑠 ‘ 𝑊 ) ( 1r ‘ 𝑊 ) ) ) ) ) |
27 |
7 2 3 22 4
|
assaassr |
⊢ ( ( 𝑊 ∈ AssAlg ∧ ( 𝑆 ∈ 𝐾 ∧ ( 1r ‘ 𝑊 ) ∈ ( Base ‘ 𝑊 ) ∧ ( 1r ‘ 𝑊 ) ∈ ( Base ‘ 𝑊 ) ) ) → ( ( 1r ‘ 𝑊 ) × ( 𝑆 ( ·𝑠 ‘ 𝑊 ) ( 1r ‘ 𝑊 ) ) ) = ( 𝑆 ( ·𝑠 ‘ 𝑊 ) ( ( 1r ‘ 𝑊 ) × ( 1r ‘ 𝑊 ) ) ) ) |
28 |
16 21 18 18 27
|
syl13anc |
⊢ ( ( 𝑊 ∈ AssAlg ∧ 𝑅 ∈ 𝐾 ∧ 𝑆 ∈ 𝐾 ) → ( ( 1r ‘ 𝑊 ) × ( 𝑆 ( ·𝑠 ‘ 𝑊 ) ( 1r ‘ 𝑊 ) ) ) = ( 𝑆 ( ·𝑠 ‘ 𝑊 ) ( ( 1r ‘ 𝑊 ) × ( 1r ‘ 𝑊 ) ) ) ) |
29 |
28
|
oveq2d |
⊢ ( ( 𝑊 ∈ AssAlg ∧ 𝑅 ∈ 𝐾 ∧ 𝑆 ∈ 𝐾 ) → ( 𝑅 ( ·𝑠 ‘ 𝑊 ) ( ( 1r ‘ 𝑊 ) × ( 𝑆 ( ·𝑠 ‘ 𝑊 ) ( 1r ‘ 𝑊 ) ) ) ) = ( 𝑅 ( ·𝑠 ‘ 𝑊 ) ( 𝑆 ( ·𝑠 ‘ 𝑊 ) ( ( 1r ‘ 𝑊 ) × ( 1r ‘ 𝑊 ) ) ) ) ) |
30 |
26 29
|
eqtrd |
⊢ ( ( 𝑊 ∈ AssAlg ∧ 𝑅 ∈ 𝐾 ∧ 𝑆 ∈ 𝐾 ) → ( ( 𝑅 ( ·𝑠 ‘ 𝑊 ) ( 1r ‘ 𝑊 ) ) × ( 𝑆 ( ·𝑠 ‘ 𝑊 ) ( 1r ‘ 𝑊 ) ) ) = ( 𝑅 ( ·𝑠 ‘ 𝑊 ) ( 𝑆 ( ·𝑠 ‘ 𝑊 ) ( ( 1r ‘ 𝑊 ) × ( 1r ‘ 𝑊 ) ) ) ) ) |
31 |
7 2 22 3 5
|
lmodvsass |
⊢ ( ( 𝑊 ∈ LMod ∧ ( 𝑅 ∈ 𝐾 ∧ 𝑆 ∈ 𝐾 ∧ ( 1r ‘ 𝑊 ) ∈ ( Base ‘ 𝑊 ) ) ) → ( ( 𝑅 · 𝑆 ) ( ·𝑠 ‘ 𝑊 ) ( 1r ‘ 𝑊 ) ) = ( 𝑅 ( ·𝑠 ‘ 𝑊 ) ( 𝑆 ( ·𝑠 ‘ 𝑊 ) ( 1r ‘ 𝑊 ) ) ) ) |
32 |
20 17 21 18 31
|
syl13anc |
⊢ ( ( 𝑊 ∈ AssAlg ∧ 𝑅 ∈ 𝐾 ∧ 𝑆 ∈ 𝐾 ) → ( ( 𝑅 · 𝑆 ) ( ·𝑠 ‘ 𝑊 ) ( 1r ‘ 𝑊 ) ) = ( 𝑅 ( ·𝑠 ‘ 𝑊 ) ( 𝑆 ( ·𝑠 ‘ 𝑊 ) ( 1r ‘ 𝑊 ) ) ) ) |
33 |
15 30 32
|
3eqtr4rd |
⊢ ( ( 𝑊 ∈ AssAlg ∧ 𝑅 ∈ 𝐾 ∧ 𝑆 ∈ 𝐾 ) → ( ( 𝑅 · 𝑆 ) ( ·𝑠 ‘ 𝑊 ) ( 1r ‘ 𝑊 ) ) = ( ( 𝑅 ( ·𝑠 ‘ 𝑊 ) ( 1r ‘ 𝑊 ) ) × ( 𝑆 ( ·𝑠 ‘ 𝑊 ) ( 1r ‘ 𝑊 ) ) ) ) |
34 |
2
|
assasca |
⊢ ( 𝑊 ∈ AssAlg → 𝐹 ∈ CRing ) |
35 |
|
crngring |
⊢ ( 𝐹 ∈ CRing → 𝐹 ∈ Ring ) |
36 |
34 35
|
syl |
⊢ ( 𝑊 ∈ AssAlg → 𝐹 ∈ Ring ) |
37 |
3 5
|
ringcl |
⊢ ( ( 𝐹 ∈ Ring ∧ 𝑅 ∈ 𝐾 ∧ 𝑆 ∈ 𝐾 ) → ( 𝑅 · 𝑆 ) ∈ 𝐾 ) |
38 |
36 37
|
syl3an1 |
⊢ ( ( 𝑊 ∈ AssAlg ∧ 𝑅 ∈ 𝐾 ∧ 𝑆 ∈ 𝐾 ) → ( 𝑅 · 𝑆 ) ∈ 𝐾 ) |
39 |
1 2 3 22 8
|
asclval |
⊢ ( ( 𝑅 · 𝑆 ) ∈ 𝐾 → ( 𝐴 ‘ ( 𝑅 · 𝑆 ) ) = ( ( 𝑅 · 𝑆 ) ( ·𝑠 ‘ 𝑊 ) ( 1r ‘ 𝑊 ) ) ) |
40 |
38 39
|
syl |
⊢ ( ( 𝑊 ∈ AssAlg ∧ 𝑅 ∈ 𝐾 ∧ 𝑆 ∈ 𝐾 ) → ( 𝐴 ‘ ( 𝑅 · 𝑆 ) ) = ( ( 𝑅 · 𝑆 ) ( ·𝑠 ‘ 𝑊 ) ( 1r ‘ 𝑊 ) ) ) |
41 |
1 2 3 22 8
|
asclval |
⊢ ( 𝑅 ∈ 𝐾 → ( 𝐴 ‘ 𝑅 ) = ( 𝑅 ( ·𝑠 ‘ 𝑊 ) ( 1r ‘ 𝑊 ) ) ) |
42 |
17 41
|
syl |
⊢ ( ( 𝑊 ∈ AssAlg ∧ 𝑅 ∈ 𝐾 ∧ 𝑆 ∈ 𝐾 ) → ( 𝐴 ‘ 𝑅 ) = ( 𝑅 ( ·𝑠 ‘ 𝑊 ) ( 1r ‘ 𝑊 ) ) ) |
43 |
1 2 3 22 8
|
asclval |
⊢ ( 𝑆 ∈ 𝐾 → ( 𝐴 ‘ 𝑆 ) = ( 𝑆 ( ·𝑠 ‘ 𝑊 ) ( 1r ‘ 𝑊 ) ) ) |
44 |
21 43
|
syl |
⊢ ( ( 𝑊 ∈ AssAlg ∧ 𝑅 ∈ 𝐾 ∧ 𝑆 ∈ 𝐾 ) → ( 𝐴 ‘ 𝑆 ) = ( 𝑆 ( ·𝑠 ‘ 𝑊 ) ( 1r ‘ 𝑊 ) ) ) |
45 |
42 44
|
oveq12d |
⊢ ( ( 𝑊 ∈ AssAlg ∧ 𝑅 ∈ 𝐾 ∧ 𝑆 ∈ 𝐾 ) → ( ( 𝐴 ‘ 𝑅 ) × ( 𝐴 ‘ 𝑆 ) ) = ( ( 𝑅 ( ·𝑠 ‘ 𝑊 ) ( 1r ‘ 𝑊 ) ) × ( 𝑆 ( ·𝑠 ‘ 𝑊 ) ( 1r ‘ 𝑊 ) ) ) ) |
46 |
33 40 45
|
3eqtr4d |
⊢ ( ( 𝑊 ∈ AssAlg ∧ 𝑅 ∈ 𝐾 ∧ 𝑆 ∈ 𝐾 ) → ( 𝐴 ‘ ( 𝑅 · 𝑆 ) ) = ( ( 𝐴 ‘ 𝑅 ) × ( 𝐴 ‘ 𝑆 ) ) ) |