Step |
Hyp |
Ref |
Expression |
1 |
|
psrplusg.s |
⊢ 𝑆 = ( 𝐼 mPwSer 𝑅 ) |
2 |
|
psrplusg.b |
⊢ 𝐵 = ( Base ‘ 𝑆 ) |
3 |
|
psrplusg.a |
⊢ + = ( +g ‘ 𝑅 ) |
4 |
|
psrplusg.p |
⊢ ✚ = ( +g ‘ 𝑆 ) |
5 |
|
eqid |
⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 ) |
6 |
|
eqid |
⊢ ( .r ‘ 𝑅 ) = ( .r ‘ 𝑅 ) |
7 |
|
eqid |
⊢ ( TopOpen ‘ 𝑅 ) = ( TopOpen ‘ 𝑅 ) |
8 |
|
eqid |
⊢ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } = { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } |
9 |
|
simpl |
⊢ ( ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → 𝐼 ∈ V ) |
10 |
1 5 8 2 9
|
psrbas |
⊢ ( ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → 𝐵 = ( ( Base ‘ 𝑅 ) ↑m { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ) |
11 |
|
eqid |
⊢ ( ∘f + ↾ ( 𝐵 × 𝐵 ) ) = ( ∘f + ↾ ( 𝐵 × 𝐵 ) ) |
12 |
|
eqid |
⊢ ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ ( 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ↦ ( 𝑅 Σg ( 𝑥 ∈ { 𝑦 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑦 ∘r ≤ 𝑘 } ↦ ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ 𝑅 ) ( 𝑔 ‘ ( 𝑘 ∘f − 𝑥 ) ) ) ) ) ) ) = ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ ( 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ↦ ( 𝑅 Σg ( 𝑥 ∈ { 𝑦 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑦 ∘r ≤ 𝑘 } ↦ ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ 𝑅 ) ( 𝑔 ‘ ( 𝑘 ∘f − 𝑥 ) ) ) ) ) ) ) |
13 |
|
eqid |
⊢ ( 𝑥 ∈ ( Base ‘ 𝑅 ) , 𝑓 ∈ 𝐵 ↦ ( ( { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } × { 𝑥 } ) ∘f ( .r ‘ 𝑅 ) 𝑓 ) ) = ( 𝑥 ∈ ( Base ‘ 𝑅 ) , 𝑓 ∈ 𝐵 ↦ ( ( { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } × { 𝑥 } ) ∘f ( .r ‘ 𝑅 ) 𝑓 ) ) |
14 |
|
eqidd |
⊢ ( ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → ( ∏t ‘ ( { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } × { ( TopOpen ‘ 𝑅 ) } ) ) = ( ∏t ‘ ( { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } × { ( TopOpen ‘ 𝑅 ) } ) ) ) |
15 |
|
simpr |
⊢ ( ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → 𝑅 ∈ V ) |
16 |
1 5 3 6 7 8 10 11 12 13 14 9 15
|
psrval |
⊢ ( ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → 𝑆 = ( { 〈 ( Base ‘ ndx ) , 𝐵 〉 , 〈 ( +g ‘ ndx ) , ( ∘f + ↾ ( 𝐵 × 𝐵 ) ) 〉 , 〈 ( .r ‘ ndx ) , ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ ( 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ↦ ( 𝑅 Σg ( 𝑥 ∈ { 𝑦 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑦 ∘r ≤ 𝑘 } ↦ ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ 𝑅 ) ( 𝑔 ‘ ( 𝑘 ∘f − 𝑥 ) ) ) ) ) ) ) 〉 } ∪ { 〈 ( Scalar ‘ ndx ) , 𝑅 〉 , 〈 ( ·𝑠 ‘ ndx ) , ( 𝑥 ∈ ( Base ‘ 𝑅 ) , 𝑓 ∈ 𝐵 ↦ ( ( { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } × { 𝑥 } ) ∘f ( .r ‘ 𝑅 ) 𝑓 ) ) 〉 , 〈 ( TopSet ‘ ndx ) , ( ∏t ‘ ( { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } × { ( TopOpen ‘ 𝑅 ) } ) ) 〉 } ) ) |
17 |
16
|
fveq2d |
⊢ ( ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → ( +g ‘ 𝑆 ) = ( +g ‘ ( { 〈 ( Base ‘ ndx ) , 𝐵 〉 , 〈 ( +g ‘ ndx ) , ( ∘f + ↾ ( 𝐵 × 𝐵 ) ) 〉 , 〈 ( .r ‘ ndx ) , ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ ( 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ↦ ( 𝑅 Σg ( 𝑥 ∈ { 𝑦 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑦 ∘r ≤ 𝑘 } ↦ ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ 𝑅 ) ( 𝑔 ‘ ( 𝑘 ∘f − 𝑥 ) ) ) ) ) ) ) 〉 } ∪ { 〈 ( Scalar ‘ ndx ) , 𝑅 〉 , 〈 ( ·𝑠 ‘ ndx ) , ( 𝑥 ∈ ( Base ‘ 𝑅 ) , 𝑓 ∈ 𝐵 ↦ ( ( { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } × { 𝑥 } ) ∘f ( .r ‘ 𝑅 ) 𝑓 ) ) 〉 , 〈 ( TopSet ‘ ndx ) , ( ∏t ‘ ( { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } × { ( TopOpen ‘ 𝑅 ) } ) ) 〉 } ) ) ) |
18 |
2
|
fvexi |
⊢ 𝐵 ∈ V |
19 |
18 18
|
xpex |
⊢ ( 𝐵 × 𝐵 ) ∈ V |
20 |
|
ofexg |
⊢ ( ( 𝐵 × 𝐵 ) ∈ V → ( ∘f + ↾ ( 𝐵 × 𝐵 ) ) ∈ V ) |
21 |
|
psrvalstr |
⊢ ( { 〈 ( Base ‘ ndx ) , 𝐵 〉 , 〈 ( +g ‘ ndx ) , ( ∘f + ↾ ( 𝐵 × 𝐵 ) ) 〉 , 〈 ( .r ‘ ndx ) , ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ ( 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ↦ ( 𝑅 Σg ( 𝑥 ∈ { 𝑦 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑦 ∘r ≤ 𝑘 } ↦ ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ 𝑅 ) ( 𝑔 ‘ ( 𝑘 ∘f − 𝑥 ) ) ) ) ) ) ) 〉 } ∪ { 〈 ( Scalar ‘ ndx ) , 𝑅 〉 , 〈 ( ·𝑠 ‘ ndx ) , ( 𝑥 ∈ ( Base ‘ 𝑅 ) , 𝑓 ∈ 𝐵 ↦ ( ( { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } × { 𝑥 } ) ∘f ( .r ‘ 𝑅 ) 𝑓 ) ) 〉 , 〈 ( TopSet ‘ ndx ) , ( ∏t ‘ ( { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } × { ( TopOpen ‘ 𝑅 ) } ) ) 〉 } ) Struct 〈 1 , 9 〉 |
22 |
|
plusgid |
⊢ +g = Slot ( +g ‘ ndx ) |
23 |
|
snsstp2 |
⊢ { 〈 ( +g ‘ ndx ) , ( ∘f + ↾ ( 𝐵 × 𝐵 ) ) 〉 } ⊆ { 〈 ( Base ‘ ndx ) , 𝐵 〉 , 〈 ( +g ‘ ndx ) , ( ∘f + ↾ ( 𝐵 × 𝐵 ) ) 〉 , 〈 ( .r ‘ ndx ) , ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ ( 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ↦ ( 𝑅 Σg ( 𝑥 ∈ { 𝑦 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑦 ∘r ≤ 𝑘 } ↦ ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ 𝑅 ) ( 𝑔 ‘ ( 𝑘 ∘f − 𝑥 ) ) ) ) ) ) ) 〉 } |
24 |
|
ssun1 |
⊢ { 〈 ( Base ‘ ndx ) , 𝐵 〉 , 〈 ( +g ‘ ndx ) , ( ∘f + ↾ ( 𝐵 × 𝐵 ) ) 〉 , 〈 ( .r ‘ ndx ) , ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ ( 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ↦ ( 𝑅 Σg ( 𝑥 ∈ { 𝑦 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑦 ∘r ≤ 𝑘 } ↦ ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ 𝑅 ) ( 𝑔 ‘ ( 𝑘 ∘f − 𝑥 ) ) ) ) ) ) ) 〉 } ⊆ ( { 〈 ( Base ‘ ndx ) , 𝐵 〉 , 〈 ( +g ‘ ndx ) , ( ∘f + ↾ ( 𝐵 × 𝐵 ) ) 〉 , 〈 ( .r ‘ ndx ) , ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ ( 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ↦ ( 𝑅 Σg ( 𝑥 ∈ { 𝑦 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑦 ∘r ≤ 𝑘 } ↦ ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ 𝑅 ) ( 𝑔 ‘ ( 𝑘 ∘f − 𝑥 ) ) ) ) ) ) ) 〉 } ∪ { 〈 ( Scalar ‘ ndx ) , 𝑅 〉 , 〈 ( ·𝑠 ‘ ndx ) , ( 𝑥 ∈ ( Base ‘ 𝑅 ) , 𝑓 ∈ 𝐵 ↦ ( ( { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } × { 𝑥 } ) ∘f ( .r ‘ 𝑅 ) 𝑓 ) ) 〉 , 〈 ( TopSet ‘ ndx ) , ( ∏t ‘ ( { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } × { ( TopOpen ‘ 𝑅 ) } ) ) 〉 } ) |
25 |
23 24
|
sstri |
⊢ { 〈 ( +g ‘ ndx ) , ( ∘f + ↾ ( 𝐵 × 𝐵 ) ) 〉 } ⊆ ( { 〈 ( Base ‘ ndx ) , 𝐵 〉 , 〈 ( +g ‘ ndx ) , ( ∘f + ↾ ( 𝐵 × 𝐵 ) ) 〉 , 〈 ( .r ‘ ndx ) , ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ ( 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ↦ ( 𝑅 Σg ( 𝑥 ∈ { 𝑦 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑦 ∘r ≤ 𝑘 } ↦ ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ 𝑅 ) ( 𝑔 ‘ ( 𝑘 ∘f − 𝑥 ) ) ) ) ) ) ) 〉 } ∪ { 〈 ( Scalar ‘ ndx ) , 𝑅 〉 , 〈 ( ·𝑠 ‘ ndx ) , ( 𝑥 ∈ ( Base ‘ 𝑅 ) , 𝑓 ∈ 𝐵 ↦ ( ( { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } × { 𝑥 } ) ∘f ( .r ‘ 𝑅 ) 𝑓 ) ) 〉 , 〈 ( TopSet ‘ ndx ) , ( ∏t ‘ ( { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } × { ( TopOpen ‘ 𝑅 ) } ) ) 〉 } ) |
26 |
21 22 25
|
strfv |
⊢ ( ( ∘f + ↾ ( 𝐵 × 𝐵 ) ) ∈ V → ( ∘f + ↾ ( 𝐵 × 𝐵 ) ) = ( +g ‘ ( { 〈 ( Base ‘ ndx ) , 𝐵 〉 , 〈 ( +g ‘ ndx ) , ( ∘f + ↾ ( 𝐵 × 𝐵 ) ) 〉 , 〈 ( .r ‘ ndx ) , ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ ( 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ↦ ( 𝑅 Σg ( 𝑥 ∈ { 𝑦 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑦 ∘r ≤ 𝑘 } ↦ ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ 𝑅 ) ( 𝑔 ‘ ( 𝑘 ∘f − 𝑥 ) ) ) ) ) ) ) 〉 } ∪ { 〈 ( Scalar ‘ ndx ) , 𝑅 〉 , 〈 ( ·𝑠 ‘ ndx ) , ( 𝑥 ∈ ( Base ‘ 𝑅 ) , 𝑓 ∈ 𝐵 ↦ ( ( { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } × { 𝑥 } ) ∘f ( .r ‘ 𝑅 ) 𝑓 ) ) 〉 , 〈 ( TopSet ‘ ndx ) , ( ∏t ‘ ( { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } × { ( TopOpen ‘ 𝑅 ) } ) ) 〉 } ) ) ) |
27 |
19 20 26
|
mp2b |
⊢ ( ∘f + ↾ ( 𝐵 × 𝐵 ) ) = ( +g ‘ ( { 〈 ( Base ‘ ndx ) , 𝐵 〉 , 〈 ( +g ‘ ndx ) , ( ∘f + ↾ ( 𝐵 × 𝐵 ) ) 〉 , 〈 ( .r ‘ ndx ) , ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ ( 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ↦ ( 𝑅 Σg ( 𝑥 ∈ { 𝑦 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑦 ∘r ≤ 𝑘 } ↦ ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ 𝑅 ) ( 𝑔 ‘ ( 𝑘 ∘f − 𝑥 ) ) ) ) ) ) ) 〉 } ∪ { 〈 ( Scalar ‘ ndx ) , 𝑅 〉 , 〈 ( ·𝑠 ‘ ndx ) , ( 𝑥 ∈ ( Base ‘ 𝑅 ) , 𝑓 ∈ 𝐵 ↦ ( ( { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } × { 𝑥 } ) ∘f ( .r ‘ 𝑅 ) 𝑓 ) ) 〉 , 〈 ( TopSet ‘ ndx ) , ( ∏t ‘ ( { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } × { ( TopOpen ‘ 𝑅 ) } ) ) 〉 } ) ) |
28 |
17 4 27
|
3eqtr4g |
⊢ ( ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → ✚ = ( ∘f + ↾ ( 𝐵 × 𝐵 ) ) ) |
29 |
|
reldmpsr |
⊢ Rel dom mPwSer |
30 |
29
|
ovprc |
⊢ ( ¬ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → ( 𝐼 mPwSer 𝑅 ) = ∅ ) |
31 |
1 30
|
syl5eq |
⊢ ( ¬ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → 𝑆 = ∅ ) |
32 |
31
|
fveq2d |
⊢ ( ¬ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → ( +g ‘ 𝑆 ) = ( +g ‘ ∅ ) ) |
33 |
22
|
str0 |
⊢ ∅ = ( +g ‘ ∅ ) |
34 |
32 4 33
|
3eqtr4g |
⊢ ( ¬ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → ✚ = ∅ ) |
35 |
31
|
fveq2d |
⊢ ( ¬ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → ( Base ‘ 𝑆 ) = ( Base ‘ ∅ ) ) |
36 |
|
base0 |
⊢ ∅ = ( Base ‘ ∅ ) |
37 |
35 2 36
|
3eqtr4g |
⊢ ( ¬ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → 𝐵 = ∅ ) |
38 |
37
|
xpeq2d |
⊢ ( ¬ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → ( 𝐵 × 𝐵 ) = ( 𝐵 × ∅ ) ) |
39 |
|
xp0 |
⊢ ( 𝐵 × ∅ ) = ∅ |
40 |
38 39
|
eqtrdi |
⊢ ( ¬ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → ( 𝐵 × 𝐵 ) = ∅ ) |
41 |
40
|
reseq2d |
⊢ ( ¬ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → ( ∘f + ↾ ( 𝐵 × 𝐵 ) ) = ( ∘f + ↾ ∅ ) ) |
42 |
|
res0 |
⊢ ( ∘f + ↾ ∅ ) = ∅ |
43 |
41 42
|
eqtrdi |
⊢ ( ¬ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → ( ∘f + ↾ ( 𝐵 × 𝐵 ) ) = ∅ ) |
44 |
34 43
|
eqtr4d |
⊢ ( ¬ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → ✚ = ( ∘f + ↾ ( 𝐵 × 𝐵 ) ) ) |
45 |
28 44
|
pm2.61i |
⊢ ✚ = ( ∘f + ↾ ( 𝐵 × 𝐵 ) ) |