| Step |
Hyp |
Ref |
Expression |
| 1 |
|
opsrbas.s |
⊢ 𝑆 = ( 𝐼 mPwSer 𝑅 ) |
| 2 |
|
opsrbas.o |
⊢ 𝑂 = ( ( 𝐼 ordPwSer 𝑅 ) ‘ 𝑇 ) |
| 3 |
|
opsrbas.t |
⊢ ( 𝜑 → 𝑇 ⊆ ( 𝐼 × 𝐼 ) ) |
| 4 |
|
opsrbaslem.1 |
⊢ 𝐸 = Slot 𝑁 |
| 5 |
|
opsrbaslem.2 |
⊢ 𝑁 ∈ ℕ |
| 6 |
|
opsrbaslem.3 |
⊢ 𝑁 < ; 1 0 |
| 7 |
|
eqid |
⊢ ( le ‘ 𝑂 ) = ( le ‘ 𝑂 ) |
| 8 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) ) → 𝐼 ∈ V ) |
| 9 |
|
simprr |
⊢ ( ( 𝜑 ∧ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) ) → 𝑅 ∈ V ) |
| 10 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) ) → 𝑇 ⊆ ( 𝐼 × 𝐼 ) ) |
| 11 |
1 2 7 8 9 10
|
opsrval2 |
⊢ ( ( 𝜑 ∧ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) ) → 𝑂 = ( 𝑆 sSet ⟨ ( le ‘ ndx ) , ( le ‘ 𝑂 ) ⟩ ) ) |
| 12 |
11
|
fveq2d |
⊢ ( ( 𝜑 ∧ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) ) → ( 𝐸 ‘ 𝑂 ) = ( 𝐸 ‘ ( 𝑆 sSet ⟨ ( le ‘ ndx ) , ( le ‘ 𝑂 ) ⟩ ) ) ) |
| 13 |
4 5
|
ndxid |
⊢ 𝐸 = Slot ( 𝐸 ‘ ndx ) |
| 14 |
5
|
nnrei |
⊢ 𝑁 ∈ ℝ |
| 15 |
14 6
|
ltneii |
⊢ 𝑁 ≠ ; 1 0 |
| 16 |
4 5
|
ndxarg |
⊢ ( 𝐸 ‘ ndx ) = 𝑁 |
| 17 |
|
plendx |
⊢ ( le ‘ ndx ) = ; 1 0 |
| 18 |
16 17
|
neeq12i |
⊢ ( ( 𝐸 ‘ ndx ) ≠ ( le ‘ ndx ) ↔ 𝑁 ≠ ; 1 0 ) |
| 19 |
15 18
|
mpbir |
⊢ ( 𝐸 ‘ ndx ) ≠ ( le ‘ ndx ) |
| 20 |
13 19
|
setsnid |
⊢ ( 𝐸 ‘ 𝑆 ) = ( 𝐸 ‘ ( 𝑆 sSet ⟨ ( le ‘ ndx ) , ( le ‘ 𝑂 ) ⟩ ) ) |
| 21 |
12 20
|
syl6reqr |
⊢ ( ( 𝜑 ∧ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) ) → ( 𝐸 ‘ 𝑆 ) = ( 𝐸 ‘ 𝑂 ) ) |
| 22 |
|
0fv |
⊢ ( ∅ ‘ 𝑇 ) = ∅ |
| 23 |
22
|
eqcomi |
⊢ ∅ = ( ∅ ‘ 𝑇 ) |
| 24 |
|
reldmpsr |
⊢ Rel dom mPwSer |
| 25 |
24
|
ovprc |
⊢ ( ¬ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → ( 𝐼 mPwSer 𝑅 ) = ∅ ) |
| 26 |
|
reldmopsr |
⊢ Rel dom ordPwSer |
| 27 |
26
|
ovprc |
⊢ ( ¬ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → ( 𝐼 ordPwSer 𝑅 ) = ∅ ) |
| 28 |
27
|
fveq1d |
⊢ ( ¬ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → ( ( 𝐼 ordPwSer 𝑅 ) ‘ 𝑇 ) = ( ∅ ‘ 𝑇 ) ) |
| 29 |
23 25 28
|
3eqtr4a |
⊢ ( ¬ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → ( 𝐼 mPwSer 𝑅 ) = ( ( 𝐼 ordPwSer 𝑅 ) ‘ 𝑇 ) ) |
| 30 |
29
|
adantl |
⊢ ( ( 𝜑 ∧ ¬ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) ) → ( 𝐼 mPwSer 𝑅 ) = ( ( 𝐼 ordPwSer 𝑅 ) ‘ 𝑇 ) ) |
| 31 |
30 1 2
|
3eqtr4g |
⊢ ( ( 𝜑 ∧ ¬ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) ) → 𝑆 = 𝑂 ) |
| 32 |
31
|
fveq2d |
⊢ ( ( 𝜑 ∧ ¬ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) ) → ( 𝐸 ‘ 𝑆 ) = ( 𝐸 ‘ 𝑂 ) ) |
| 33 |
21 32
|
pm2.61dan |
⊢ ( 𝜑 → ( 𝐸 ‘ 𝑆 ) = ( 𝐸 ‘ 𝑂 ) ) |