Step |
Hyp |
Ref |
Expression |
1 |
|
oppgval.2 |
⊢ + = ( +g ‘ 𝑅 ) |
2 |
|
oppgval.3 |
⊢ 𝑂 = ( oppg ‘ 𝑅 ) |
3 |
|
oppgplusfval.4 |
⊢ ✚ = ( +g ‘ 𝑂 ) |
4 |
1 2
|
oppgval |
⊢ 𝑂 = ( 𝑅 sSet 〈 ( +g ‘ ndx ) , tpos + 〉 ) |
5 |
4
|
fveq2i |
⊢ ( +g ‘ 𝑂 ) = ( +g ‘ ( 𝑅 sSet 〈 ( +g ‘ ndx ) , tpos + 〉 ) ) |
6 |
1
|
fvexi |
⊢ + ∈ V |
7 |
6
|
tposex |
⊢ tpos + ∈ V |
8 |
|
plusgid |
⊢ +g = Slot ( +g ‘ ndx ) |
9 |
8
|
setsid |
⊢ ( ( 𝑅 ∈ V ∧ tpos + ∈ V ) → tpos + = ( +g ‘ ( 𝑅 sSet 〈 ( +g ‘ ndx ) , tpos + 〉 ) ) ) |
10 |
7 9
|
mpan2 |
⊢ ( 𝑅 ∈ V → tpos + = ( +g ‘ ( 𝑅 sSet 〈 ( +g ‘ ndx ) , tpos + 〉 ) ) ) |
11 |
5 10
|
eqtr4id |
⊢ ( 𝑅 ∈ V → ( +g ‘ 𝑂 ) = tpos + ) |
12 |
|
tpos0 |
⊢ tpos ∅ = ∅ |
13 |
8
|
str0 |
⊢ ∅ = ( +g ‘ ∅ ) |
14 |
12 13
|
eqtr2i |
⊢ ( +g ‘ ∅ ) = tpos ∅ |
15 |
|
reldmsets |
⊢ Rel dom sSet |
16 |
15
|
ovprc1 |
⊢ ( ¬ 𝑅 ∈ V → ( 𝑅 sSet 〈 ( +g ‘ ndx ) , tpos + 〉 ) = ∅ ) |
17 |
4 16
|
syl5eq |
⊢ ( ¬ 𝑅 ∈ V → 𝑂 = ∅ ) |
18 |
17
|
fveq2d |
⊢ ( ¬ 𝑅 ∈ V → ( +g ‘ 𝑂 ) = ( +g ‘ ∅ ) ) |
19 |
|
fvprc |
⊢ ( ¬ 𝑅 ∈ V → ( +g ‘ 𝑅 ) = ∅ ) |
20 |
1 19
|
syl5eq |
⊢ ( ¬ 𝑅 ∈ V → + = ∅ ) |
21 |
20
|
tposeqd |
⊢ ( ¬ 𝑅 ∈ V → tpos + = tpos ∅ ) |
22 |
14 18 21
|
3eqtr4a |
⊢ ( ¬ 𝑅 ∈ V → ( +g ‘ 𝑂 ) = tpos + ) |
23 |
11 22
|
pm2.61i |
⊢ ( +g ‘ 𝑂 ) = tpos + |
24 |
3 23
|
eqtri |
⊢ ✚ = tpos + |