Step |
Hyp |
Ref |
Expression |
1 |
|
opprval.1 |
⊢ 𝐵 = ( Base ‘ 𝑅 ) |
2 |
|
opprval.2 |
⊢ · = ( .r ‘ 𝑅 ) |
3 |
|
opprval.3 |
⊢ 𝑂 = ( oppr ‘ 𝑅 ) |
4 |
|
opprmulfval.4 |
⊢ ∙ = ( .r ‘ 𝑂 ) |
5 |
1 2 3
|
opprval |
⊢ 𝑂 = ( 𝑅 sSet 〈 ( .r ‘ ndx ) , tpos · 〉 ) |
6 |
5
|
fveq2i |
⊢ ( .r ‘ 𝑂 ) = ( .r ‘ ( 𝑅 sSet 〈 ( .r ‘ ndx ) , tpos · 〉 ) ) |
7 |
2
|
fvexi |
⊢ · ∈ V |
8 |
7
|
tposex |
⊢ tpos · ∈ V |
9 |
|
mulrid |
⊢ .r = Slot ( .r ‘ ndx ) |
10 |
9
|
setsid |
⊢ ( ( 𝑅 ∈ V ∧ tpos · ∈ V ) → tpos · = ( .r ‘ ( 𝑅 sSet 〈 ( .r ‘ ndx ) , tpos · 〉 ) ) ) |
11 |
8 10
|
mpan2 |
⊢ ( 𝑅 ∈ V → tpos · = ( .r ‘ ( 𝑅 sSet 〈 ( .r ‘ ndx ) , tpos · 〉 ) ) ) |
12 |
6 11
|
eqtr4id |
⊢ ( 𝑅 ∈ V → ( .r ‘ 𝑂 ) = tpos · ) |
13 |
|
tpos0 |
⊢ tpos ∅ = ∅ |
14 |
9
|
str0 |
⊢ ∅ = ( .r ‘ ∅ ) |
15 |
13 14
|
eqtr2i |
⊢ ( .r ‘ ∅ ) = tpos ∅ |
16 |
|
fvprc |
⊢ ( ¬ 𝑅 ∈ V → ( oppr ‘ 𝑅 ) = ∅ ) |
17 |
3 16
|
eqtrid |
⊢ ( ¬ 𝑅 ∈ V → 𝑂 = ∅ ) |
18 |
17
|
fveq2d |
⊢ ( ¬ 𝑅 ∈ V → ( .r ‘ 𝑂 ) = ( .r ‘ ∅ ) ) |
19 |
|
fvprc |
⊢ ( ¬ 𝑅 ∈ V → ( .r ‘ 𝑅 ) = ∅ ) |
20 |
2 19
|
eqtrid |
⊢ ( ¬ 𝑅 ∈ V → · = ∅ ) |
21 |
20
|
tposeqd |
⊢ ( ¬ 𝑅 ∈ V → tpos · = tpos ∅ ) |
22 |
15 18 21
|
3eqtr4a |
⊢ ( ¬ 𝑅 ∈ V → ( .r ‘ 𝑂 ) = tpos · ) |
23 |
12 22
|
pm2.61i |
⊢ ( .r ‘ 𝑂 ) = tpos · |
24 |
4 23
|
eqtri |
⊢ ∙ = tpos · |