Step |
Hyp |
Ref |
Expression |
1 |
|
isassa.v |
⊢ 𝑉 = ( Base ‘ 𝑊 ) |
2 |
|
isassa.f |
⊢ 𝐹 = ( Scalar ‘ 𝑊 ) |
3 |
|
isassa.b |
⊢ 𝐵 = ( Base ‘ 𝐹 ) |
4 |
|
isassa.s |
⊢ · = ( ·𝑠 ‘ 𝑊 ) |
5 |
|
isassa.t |
⊢ × = ( .r ‘ 𝑊 ) |
6 |
|
fvexd |
⊢ ( 𝑤 = 𝑊 → ( Scalar ‘ 𝑤 ) ∈ V ) |
7 |
|
fveq2 |
⊢ ( 𝑤 = 𝑊 → ( Scalar ‘ 𝑤 ) = ( Scalar ‘ 𝑊 ) ) |
8 |
7 2
|
eqtr4di |
⊢ ( 𝑤 = 𝑊 → ( Scalar ‘ 𝑤 ) = 𝐹 ) |
9 |
|
fveq2 |
⊢ ( 𝑓 = 𝐹 → ( Base ‘ 𝑓 ) = ( Base ‘ 𝐹 ) ) |
10 |
9 3
|
eqtr4di |
⊢ ( 𝑓 = 𝐹 → ( Base ‘ 𝑓 ) = 𝐵 ) |
11 |
10
|
adantl |
⊢ ( ( 𝑤 = 𝑊 ∧ 𝑓 = 𝐹 ) → ( Base ‘ 𝑓 ) = 𝐵 ) |
12 |
|
fveq2 |
⊢ ( 𝑤 = 𝑊 → ( Base ‘ 𝑤 ) = ( Base ‘ 𝑊 ) ) |
13 |
12 1
|
eqtr4di |
⊢ ( 𝑤 = 𝑊 → ( Base ‘ 𝑤 ) = 𝑉 ) |
14 |
|
simpr |
⊢ ( ( 𝑠 = · ∧ 𝑡 = × ) → 𝑡 = × ) |
15 |
|
simpl |
⊢ ( ( 𝑠 = · ∧ 𝑡 = × ) → 𝑠 = · ) |
16 |
15
|
oveqd |
⊢ ( ( 𝑠 = · ∧ 𝑡 = × ) → ( 𝑟 𝑠 𝑥 ) = ( 𝑟 · 𝑥 ) ) |
17 |
|
eqidd |
⊢ ( ( 𝑠 = · ∧ 𝑡 = × ) → 𝑦 = 𝑦 ) |
18 |
14 16 17
|
oveq123d |
⊢ ( ( 𝑠 = · ∧ 𝑡 = × ) → ( ( 𝑟 𝑠 𝑥 ) 𝑡 𝑦 ) = ( ( 𝑟 · 𝑥 ) × 𝑦 ) ) |
19 |
|
eqidd |
⊢ ( ( 𝑠 = · ∧ 𝑡 = × ) → 𝑟 = 𝑟 ) |
20 |
14
|
oveqd |
⊢ ( ( 𝑠 = · ∧ 𝑡 = × ) → ( 𝑥 𝑡 𝑦 ) = ( 𝑥 × 𝑦 ) ) |
21 |
15 19 20
|
oveq123d |
⊢ ( ( 𝑠 = · ∧ 𝑡 = × ) → ( 𝑟 𝑠 ( 𝑥 𝑡 𝑦 ) ) = ( 𝑟 · ( 𝑥 × 𝑦 ) ) ) |
22 |
18 21
|
eqeq12d |
⊢ ( ( 𝑠 = · ∧ 𝑡 = × ) → ( ( ( 𝑟 𝑠 𝑥 ) 𝑡 𝑦 ) = ( 𝑟 𝑠 ( 𝑥 𝑡 𝑦 ) ) ↔ ( ( 𝑟 · 𝑥 ) × 𝑦 ) = ( 𝑟 · ( 𝑥 × 𝑦 ) ) ) ) |
23 |
|
eqidd |
⊢ ( ( 𝑠 = · ∧ 𝑡 = × ) → 𝑥 = 𝑥 ) |
24 |
15
|
oveqd |
⊢ ( ( 𝑠 = · ∧ 𝑡 = × ) → ( 𝑟 𝑠 𝑦 ) = ( 𝑟 · 𝑦 ) ) |
25 |
14 23 24
|
oveq123d |
⊢ ( ( 𝑠 = · ∧ 𝑡 = × ) → ( 𝑥 𝑡 ( 𝑟 𝑠 𝑦 ) ) = ( 𝑥 × ( 𝑟 · 𝑦 ) ) ) |
26 |
25 21
|
eqeq12d |
⊢ ( ( 𝑠 = · ∧ 𝑡 = × ) → ( ( 𝑥 𝑡 ( 𝑟 𝑠 𝑦 ) ) = ( 𝑟 𝑠 ( 𝑥 𝑡 𝑦 ) ) ↔ ( 𝑥 × ( 𝑟 · 𝑦 ) ) = ( 𝑟 · ( 𝑥 × 𝑦 ) ) ) ) |
27 |
22 26
|
anbi12d |
⊢ ( ( 𝑠 = · ∧ 𝑡 = × ) → ( ( ( ( 𝑟 𝑠 𝑥 ) 𝑡 𝑦 ) = ( 𝑟 𝑠 ( 𝑥 𝑡 𝑦 ) ) ∧ ( 𝑥 𝑡 ( 𝑟 𝑠 𝑦 ) ) = ( 𝑟 𝑠 ( 𝑥 𝑡 𝑦 ) ) ) ↔ ( ( ( 𝑟 · 𝑥 ) × 𝑦 ) = ( 𝑟 · ( 𝑥 × 𝑦 ) ) ∧ ( 𝑥 × ( 𝑟 · 𝑦 ) ) = ( 𝑟 · ( 𝑥 × 𝑦 ) ) ) ) ) |
28 |
4 5 27
|
sbcie2s |
⊢ ( 𝑤 = 𝑊 → ( [ ( ·𝑠 ‘ 𝑤 ) / 𝑠 ] [ ( .r ‘ 𝑤 ) / 𝑡 ] ( ( ( 𝑟 𝑠 𝑥 ) 𝑡 𝑦 ) = ( 𝑟 𝑠 ( 𝑥 𝑡 𝑦 ) ) ∧ ( 𝑥 𝑡 ( 𝑟 𝑠 𝑦 ) ) = ( 𝑟 𝑠 ( 𝑥 𝑡 𝑦 ) ) ) ↔ ( ( ( 𝑟 · 𝑥 ) × 𝑦 ) = ( 𝑟 · ( 𝑥 × 𝑦 ) ) ∧ ( 𝑥 × ( 𝑟 · 𝑦 ) ) = ( 𝑟 · ( 𝑥 × 𝑦 ) ) ) ) ) |
29 |
13 28
|
raleqbidv |
⊢ ( 𝑤 = 𝑊 → ( ∀ 𝑦 ∈ ( Base ‘ 𝑤 ) [ ( ·𝑠 ‘ 𝑤 ) / 𝑠 ] [ ( .r ‘ 𝑤 ) / 𝑡 ] ( ( ( 𝑟 𝑠 𝑥 ) 𝑡 𝑦 ) = ( 𝑟 𝑠 ( 𝑥 𝑡 𝑦 ) ) ∧ ( 𝑥 𝑡 ( 𝑟 𝑠 𝑦 ) ) = ( 𝑟 𝑠 ( 𝑥 𝑡 𝑦 ) ) ) ↔ ∀ 𝑦 ∈ 𝑉 ( ( ( 𝑟 · 𝑥 ) × 𝑦 ) = ( 𝑟 · ( 𝑥 × 𝑦 ) ) ∧ ( 𝑥 × ( 𝑟 · 𝑦 ) ) = ( 𝑟 · ( 𝑥 × 𝑦 ) ) ) ) ) |
30 |
13 29
|
raleqbidv |
⊢ ( 𝑤 = 𝑊 → ( ∀ 𝑥 ∈ ( Base ‘ 𝑤 ) ∀ 𝑦 ∈ ( Base ‘ 𝑤 ) [ ( ·𝑠 ‘ 𝑤 ) / 𝑠 ] [ ( .r ‘ 𝑤 ) / 𝑡 ] ( ( ( 𝑟 𝑠 𝑥 ) 𝑡 𝑦 ) = ( 𝑟 𝑠 ( 𝑥 𝑡 𝑦 ) ) ∧ ( 𝑥 𝑡 ( 𝑟 𝑠 𝑦 ) ) = ( 𝑟 𝑠 ( 𝑥 𝑡 𝑦 ) ) ) ↔ ∀ 𝑥 ∈ 𝑉 ∀ 𝑦 ∈ 𝑉 ( ( ( 𝑟 · 𝑥 ) × 𝑦 ) = ( 𝑟 · ( 𝑥 × 𝑦 ) ) ∧ ( 𝑥 × ( 𝑟 · 𝑦 ) ) = ( 𝑟 · ( 𝑥 × 𝑦 ) ) ) ) ) |
31 |
30
|
adantr |
⊢ ( ( 𝑤 = 𝑊 ∧ 𝑓 = 𝐹 ) → ( ∀ 𝑥 ∈ ( Base ‘ 𝑤 ) ∀ 𝑦 ∈ ( Base ‘ 𝑤 ) [ ( ·𝑠 ‘ 𝑤 ) / 𝑠 ] [ ( .r ‘ 𝑤 ) / 𝑡 ] ( ( ( 𝑟 𝑠 𝑥 ) 𝑡 𝑦 ) = ( 𝑟 𝑠 ( 𝑥 𝑡 𝑦 ) ) ∧ ( 𝑥 𝑡 ( 𝑟 𝑠 𝑦 ) ) = ( 𝑟 𝑠 ( 𝑥 𝑡 𝑦 ) ) ) ↔ ∀ 𝑥 ∈ 𝑉 ∀ 𝑦 ∈ 𝑉 ( ( ( 𝑟 · 𝑥 ) × 𝑦 ) = ( 𝑟 · ( 𝑥 × 𝑦 ) ) ∧ ( 𝑥 × ( 𝑟 · 𝑦 ) ) = ( 𝑟 · ( 𝑥 × 𝑦 ) ) ) ) ) |
32 |
11 31
|
raleqbidv |
⊢ ( ( 𝑤 = 𝑊 ∧ 𝑓 = 𝐹 ) → ( ∀ 𝑟 ∈ ( Base ‘ 𝑓 ) ∀ 𝑥 ∈ ( Base ‘ 𝑤 ) ∀ 𝑦 ∈ ( Base ‘ 𝑤 ) [ ( ·𝑠 ‘ 𝑤 ) / 𝑠 ] [ ( .r ‘ 𝑤 ) / 𝑡 ] ( ( ( 𝑟 𝑠 𝑥 ) 𝑡 𝑦 ) = ( 𝑟 𝑠 ( 𝑥 𝑡 𝑦 ) ) ∧ ( 𝑥 𝑡 ( 𝑟 𝑠 𝑦 ) ) = ( 𝑟 𝑠 ( 𝑥 𝑡 𝑦 ) ) ) ↔ ∀ 𝑟 ∈ 𝐵 ∀ 𝑥 ∈ 𝑉 ∀ 𝑦 ∈ 𝑉 ( ( ( 𝑟 · 𝑥 ) × 𝑦 ) = ( 𝑟 · ( 𝑥 × 𝑦 ) ) ∧ ( 𝑥 × ( 𝑟 · 𝑦 ) ) = ( 𝑟 · ( 𝑥 × 𝑦 ) ) ) ) ) |
33 |
6 8 32
|
sbcied2 |
⊢ ( 𝑤 = 𝑊 → ( [ ( Scalar ‘ 𝑤 ) / 𝑓 ] ∀ 𝑟 ∈ ( Base ‘ 𝑓 ) ∀ 𝑥 ∈ ( Base ‘ 𝑤 ) ∀ 𝑦 ∈ ( Base ‘ 𝑤 ) [ ( ·𝑠 ‘ 𝑤 ) / 𝑠 ] [ ( .r ‘ 𝑤 ) / 𝑡 ] ( ( ( 𝑟 𝑠 𝑥 ) 𝑡 𝑦 ) = ( 𝑟 𝑠 ( 𝑥 𝑡 𝑦 ) ) ∧ ( 𝑥 𝑡 ( 𝑟 𝑠 𝑦 ) ) = ( 𝑟 𝑠 ( 𝑥 𝑡 𝑦 ) ) ) ↔ ∀ 𝑟 ∈ 𝐵 ∀ 𝑥 ∈ 𝑉 ∀ 𝑦 ∈ 𝑉 ( ( ( 𝑟 · 𝑥 ) × 𝑦 ) = ( 𝑟 · ( 𝑥 × 𝑦 ) ) ∧ ( 𝑥 × ( 𝑟 · 𝑦 ) ) = ( 𝑟 · ( 𝑥 × 𝑦 ) ) ) ) ) |
34 |
|
df-assa |
⊢ AssAlg = { 𝑤 ∈ ( LMod ∩ Ring ) ∣ [ ( Scalar ‘ 𝑤 ) / 𝑓 ] ∀ 𝑟 ∈ ( Base ‘ 𝑓 ) ∀ 𝑥 ∈ ( Base ‘ 𝑤 ) ∀ 𝑦 ∈ ( Base ‘ 𝑤 ) [ ( ·𝑠 ‘ 𝑤 ) / 𝑠 ] [ ( .r ‘ 𝑤 ) / 𝑡 ] ( ( ( 𝑟 𝑠 𝑥 ) 𝑡 𝑦 ) = ( 𝑟 𝑠 ( 𝑥 𝑡 𝑦 ) ) ∧ ( 𝑥 𝑡 ( 𝑟 𝑠 𝑦 ) ) = ( 𝑟 𝑠 ( 𝑥 𝑡 𝑦 ) ) ) } |
35 |
33 34
|
elrab2 |
⊢ ( 𝑊 ∈ AssAlg ↔ ( 𝑊 ∈ ( LMod ∩ Ring ) ∧ ∀ 𝑟 ∈ 𝐵 ∀ 𝑥 ∈ 𝑉 ∀ 𝑦 ∈ 𝑉 ( ( ( 𝑟 · 𝑥 ) × 𝑦 ) = ( 𝑟 · ( 𝑥 × 𝑦 ) ) ∧ ( 𝑥 × ( 𝑟 · 𝑦 ) ) = ( 𝑟 · ( 𝑥 × 𝑦 ) ) ) ) ) |
36 |
|
elin |
⊢ ( 𝑊 ∈ ( LMod ∩ Ring ) ↔ ( 𝑊 ∈ LMod ∧ 𝑊 ∈ Ring ) ) |
37 |
36
|
anbi1i |
⊢ ( ( 𝑊 ∈ ( LMod ∩ Ring ) ∧ ∀ 𝑟 ∈ 𝐵 ∀ 𝑥 ∈ 𝑉 ∀ 𝑦 ∈ 𝑉 ( ( ( 𝑟 · 𝑥 ) × 𝑦 ) = ( 𝑟 · ( 𝑥 × 𝑦 ) ) ∧ ( 𝑥 × ( 𝑟 · 𝑦 ) ) = ( 𝑟 · ( 𝑥 × 𝑦 ) ) ) ) ↔ ( ( 𝑊 ∈ LMod ∧ 𝑊 ∈ Ring ) ∧ ∀ 𝑟 ∈ 𝐵 ∀ 𝑥 ∈ 𝑉 ∀ 𝑦 ∈ 𝑉 ( ( ( 𝑟 · 𝑥 ) × 𝑦 ) = ( 𝑟 · ( 𝑥 × 𝑦 ) ) ∧ ( 𝑥 × ( 𝑟 · 𝑦 ) ) = ( 𝑟 · ( 𝑥 × 𝑦 ) ) ) ) ) |
38 |
35 37
|
bitri |
⊢ ( 𝑊 ∈ AssAlg ↔ ( ( 𝑊 ∈ LMod ∧ 𝑊 ∈ Ring ) ∧ ∀ 𝑟 ∈ 𝐵 ∀ 𝑥 ∈ 𝑉 ∀ 𝑦 ∈ 𝑉 ( ( ( 𝑟 · 𝑥 ) × 𝑦 ) = ( 𝑟 · ( 𝑥 × 𝑦 ) ) ∧ ( 𝑥 × ( 𝑟 · 𝑦 ) ) = ( 𝑟 · ( 𝑥 × 𝑦 ) ) ) ) ) |