Step |
Hyp |
Ref |
Expression |
1 |
|
natfval.1 |
⊢ 𝑁 = ( 𝐶 Nat 𝐷 ) |
2 |
|
natfval.b |
⊢ 𝐵 = ( Base ‘ 𝐶 ) |
3 |
|
natfval.h |
⊢ 𝐻 = ( Hom ‘ 𝐶 ) |
4 |
|
natfval.j |
⊢ 𝐽 = ( Hom ‘ 𝐷 ) |
5 |
|
natfval.o |
⊢ · = ( comp ‘ 𝐷 ) |
6 |
|
isnat.f |
⊢ ( 𝜑 → 𝐹 ( 𝐶 Func 𝐷 ) 𝐺 ) |
7 |
|
isnat.g |
⊢ ( 𝜑 → 𝐾 ( 𝐶 Func 𝐷 ) 𝐿 ) |
8 |
1 2 3 4 5
|
natfval |
⊢ 𝑁 = ( 𝑓 ∈ ( 𝐶 Func 𝐷 ) , 𝑔 ∈ ( 𝐶 Func 𝐷 ) ↦ ⦋ ( 1st ‘ 𝑓 ) / 𝑟 ⦌ ⦋ ( 1st ‘ 𝑔 ) / 𝑠 ⦌ { 𝑎 ∈ X 𝑥 ∈ 𝐵 ( ( 𝑟 ‘ 𝑥 ) 𝐽 ( 𝑠 ‘ 𝑥 ) ) ∣ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ∀ ℎ ∈ ( 𝑥 𝐻 𝑦 ) ( ( 𝑎 ‘ 𝑦 ) ( 〈 ( 𝑟 ‘ 𝑥 ) , ( 𝑟 ‘ 𝑦 ) 〉 · ( 𝑠 ‘ 𝑦 ) ) ( ( 𝑥 ( 2nd ‘ 𝑓 ) 𝑦 ) ‘ ℎ ) ) = ( ( ( 𝑥 ( 2nd ‘ 𝑔 ) 𝑦 ) ‘ ℎ ) ( 〈 ( 𝑟 ‘ 𝑥 ) , ( 𝑠 ‘ 𝑥 ) 〉 · ( 𝑠 ‘ 𝑦 ) ) ( 𝑎 ‘ 𝑥 ) ) } ) |
9 |
8
|
a1i |
⊢ ( 𝜑 → 𝑁 = ( 𝑓 ∈ ( 𝐶 Func 𝐷 ) , 𝑔 ∈ ( 𝐶 Func 𝐷 ) ↦ ⦋ ( 1st ‘ 𝑓 ) / 𝑟 ⦌ ⦋ ( 1st ‘ 𝑔 ) / 𝑠 ⦌ { 𝑎 ∈ X 𝑥 ∈ 𝐵 ( ( 𝑟 ‘ 𝑥 ) 𝐽 ( 𝑠 ‘ 𝑥 ) ) ∣ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ∀ ℎ ∈ ( 𝑥 𝐻 𝑦 ) ( ( 𝑎 ‘ 𝑦 ) ( 〈 ( 𝑟 ‘ 𝑥 ) , ( 𝑟 ‘ 𝑦 ) 〉 · ( 𝑠 ‘ 𝑦 ) ) ( ( 𝑥 ( 2nd ‘ 𝑓 ) 𝑦 ) ‘ ℎ ) ) = ( ( ( 𝑥 ( 2nd ‘ 𝑔 ) 𝑦 ) ‘ ℎ ) ( 〈 ( 𝑟 ‘ 𝑥 ) , ( 𝑠 ‘ 𝑥 ) 〉 · ( 𝑠 ‘ 𝑦 ) ) ( 𝑎 ‘ 𝑥 ) ) } ) ) |
10 |
|
fvexd |
⊢ ( ( 𝜑 ∧ ( 𝑓 = 〈 𝐹 , 𝐺 〉 ∧ 𝑔 = 〈 𝐾 , 𝐿 〉 ) ) → ( 1st ‘ 𝑓 ) ∈ V ) |
11 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( 𝑓 = 〈 𝐹 , 𝐺 〉 ∧ 𝑔 = 〈 𝐾 , 𝐿 〉 ) ) → 𝑓 = 〈 𝐹 , 𝐺 〉 ) |
12 |
11
|
fveq2d |
⊢ ( ( 𝜑 ∧ ( 𝑓 = 〈 𝐹 , 𝐺 〉 ∧ 𝑔 = 〈 𝐾 , 𝐿 〉 ) ) → ( 1st ‘ 𝑓 ) = ( 1st ‘ 〈 𝐹 , 𝐺 〉 ) ) |
13 |
|
relfunc |
⊢ Rel ( 𝐶 Func 𝐷 ) |
14 |
|
brrelex12 |
⊢ ( ( Rel ( 𝐶 Func 𝐷 ) ∧ 𝐹 ( 𝐶 Func 𝐷 ) 𝐺 ) → ( 𝐹 ∈ V ∧ 𝐺 ∈ V ) ) |
15 |
13 6 14
|
sylancr |
⊢ ( 𝜑 → ( 𝐹 ∈ V ∧ 𝐺 ∈ V ) ) |
16 |
|
op1stg |
⊢ ( ( 𝐹 ∈ V ∧ 𝐺 ∈ V ) → ( 1st ‘ 〈 𝐹 , 𝐺 〉 ) = 𝐹 ) |
17 |
15 16
|
syl |
⊢ ( 𝜑 → ( 1st ‘ 〈 𝐹 , 𝐺 〉 ) = 𝐹 ) |
18 |
17
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑓 = 〈 𝐹 , 𝐺 〉 ∧ 𝑔 = 〈 𝐾 , 𝐿 〉 ) ) → ( 1st ‘ 〈 𝐹 , 𝐺 〉 ) = 𝐹 ) |
19 |
12 18
|
eqtrd |
⊢ ( ( 𝜑 ∧ ( 𝑓 = 〈 𝐹 , 𝐺 〉 ∧ 𝑔 = 〈 𝐾 , 𝐿 〉 ) ) → ( 1st ‘ 𝑓 ) = 𝐹 ) |
20 |
|
fvexd |
⊢ ( ( ( 𝜑 ∧ ( 𝑓 = 〈 𝐹 , 𝐺 〉 ∧ 𝑔 = 〈 𝐾 , 𝐿 〉 ) ) ∧ 𝑟 = 𝐹 ) → ( 1st ‘ 𝑔 ) ∈ V ) |
21 |
|
simplrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑓 = 〈 𝐹 , 𝐺 〉 ∧ 𝑔 = 〈 𝐾 , 𝐿 〉 ) ) ∧ 𝑟 = 𝐹 ) → 𝑔 = 〈 𝐾 , 𝐿 〉 ) |
22 |
21
|
fveq2d |
⊢ ( ( ( 𝜑 ∧ ( 𝑓 = 〈 𝐹 , 𝐺 〉 ∧ 𝑔 = 〈 𝐾 , 𝐿 〉 ) ) ∧ 𝑟 = 𝐹 ) → ( 1st ‘ 𝑔 ) = ( 1st ‘ 〈 𝐾 , 𝐿 〉 ) ) |
23 |
|
brrelex12 |
⊢ ( ( Rel ( 𝐶 Func 𝐷 ) ∧ 𝐾 ( 𝐶 Func 𝐷 ) 𝐿 ) → ( 𝐾 ∈ V ∧ 𝐿 ∈ V ) ) |
24 |
13 7 23
|
sylancr |
⊢ ( 𝜑 → ( 𝐾 ∈ V ∧ 𝐿 ∈ V ) ) |
25 |
|
op1stg |
⊢ ( ( 𝐾 ∈ V ∧ 𝐿 ∈ V ) → ( 1st ‘ 〈 𝐾 , 𝐿 〉 ) = 𝐾 ) |
26 |
24 25
|
syl |
⊢ ( 𝜑 → ( 1st ‘ 〈 𝐾 , 𝐿 〉 ) = 𝐾 ) |
27 |
26
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑓 = 〈 𝐹 , 𝐺 〉 ∧ 𝑔 = 〈 𝐾 , 𝐿 〉 ) ) ∧ 𝑟 = 𝐹 ) → ( 1st ‘ 〈 𝐾 , 𝐿 〉 ) = 𝐾 ) |
28 |
22 27
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ ( 𝑓 = 〈 𝐹 , 𝐺 〉 ∧ 𝑔 = 〈 𝐾 , 𝐿 〉 ) ) ∧ 𝑟 = 𝐹 ) → ( 1st ‘ 𝑔 ) = 𝐾 ) |
29 |
|
simplr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑓 = 〈 𝐹 , 𝐺 〉 ∧ 𝑔 = 〈 𝐾 , 𝐿 〉 ) ) ∧ 𝑟 = 𝐹 ) ∧ 𝑠 = 𝐾 ) → 𝑟 = 𝐹 ) |
30 |
29
|
fveq1d |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑓 = 〈 𝐹 , 𝐺 〉 ∧ 𝑔 = 〈 𝐾 , 𝐿 〉 ) ) ∧ 𝑟 = 𝐹 ) ∧ 𝑠 = 𝐾 ) → ( 𝑟 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑥 ) ) |
31 |
|
simpr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑓 = 〈 𝐹 , 𝐺 〉 ∧ 𝑔 = 〈 𝐾 , 𝐿 〉 ) ) ∧ 𝑟 = 𝐹 ) ∧ 𝑠 = 𝐾 ) → 𝑠 = 𝐾 ) |
32 |
31
|
fveq1d |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑓 = 〈 𝐹 , 𝐺 〉 ∧ 𝑔 = 〈 𝐾 , 𝐿 〉 ) ) ∧ 𝑟 = 𝐹 ) ∧ 𝑠 = 𝐾 ) → ( 𝑠 ‘ 𝑥 ) = ( 𝐾 ‘ 𝑥 ) ) |
33 |
30 32
|
oveq12d |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑓 = 〈 𝐹 , 𝐺 〉 ∧ 𝑔 = 〈 𝐾 , 𝐿 〉 ) ) ∧ 𝑟 = 𝐹 ) ∧ 𝑠 = 𝐾 ) → ( ( 𝑟 ‘ 𝑥 ) 𝐽 ( 𝑠 ‘ 𝑥 ) ) = ( ( 𝐹 ‘ 𝑥 ) 𝐽 ( 𝐾 ‘ 𝑥 ) ) ) |
34 |
33
|
ixpeq2dv |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑓 = 〈 𝐹 , 𝐺 〉 ∧ 𝑔 = 〈 𝐾 , 𝐿 〉 ) ) ∧ 𝑟 = 𝐹 ) ∧ 𝑠 = 𝐾 ) → X 𝑥 ∈ 𝐵 ( ( 𝑟 ‘ 𝑥 ) 𝐽 ( 𝑠 ‘ 𝑥 ) ) = X 𝑥 ∈ 𝐵 ( ( 𝐹 ‘ 𝑥 ) 𝐽 ( 𝐾 ‘ 𝑥 ) ) ) |
35 |
29
|
fveq1d |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑓 = 〈 𝐹 , 𝐺 〉 ∧ 𝑔 = 〈 𝐾 , 𝐿 〉 ) ) ∧ 𝑟 = 𝐹 ) ∧ 𝑠 = 𝐾 ) → ( 𝑟 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑦 ) ) |
36 |
30 35
|
opeq12d |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑓 = 〈 𝐹 , 𝐺 〉 ∧ 𝑔 = 〈 𝐾 , 𝐿 〉 ) ) ∧ 𝑟 = 𝐹 ) ∧ 𝑠 = 𝐾 ) → 〈 ( 𝑟 ‘ 𝑥 ) , ( 𝑟 ‘ 𝑦 ) 〉 = 〈 ( 𝐹 ‘ 𝑥 ) , ( 𝐹 ‘ 𝑦 ) 〉 ) |
37 |
31
|
fveq1d |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑓 = 〈 𝐹 , 𝐺 〉 ∧ 𝑔 = 〈 𝐾 , 𝐿 〉 ) ) ∧ 𝑟 = 𝐹 ) ∧ 𝑠 = 𝐾 ) → ( 𝑠 ‘ 𝑦 ) = ( 𝐾 ‘ 𝑦 ) ) |
38 |
36 37
|
oveq12d |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑓 = 〈 𝐹 , 𝐺 〉 ∧ 𝑔 = 〈 𝐾 , 𝐿 〉 ) ) ∧ 𝑟 = 𝐹 ) ∧ 𝑠 = 𝐾 ) → ( 〈 ( 𝑟 ‘ 𝑥 ) , ( 𝑟 ‘ 𝑦 ) 〉 · ( 𝑠 ‘ 𝑦 ) ) = ( 〈 ( 𝐹 ‘ 𝑥 ) , ( 𝐹 ‘ 𝑦 ) 〉 · ( 𝐾 ‘ 𝑦 ) ) ) |
39 |
|
eqidd |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑓 = 〈 𝐹 , 𝐺 〉 ∧ 𝑔 = 〈 𝐾 , 𝐿 〉 ) ) ∧ 𝑟 = 𝐹 ) ∧ 𝑠 = 𝐾 ) → ( 𝑎 ‘ 𝑦 ) = ( 𝑎 ‘ 𝑦 ) ) |
40 |
11
|
ad2antrr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑓 = 〈 𝐹 , 𝐺 〉 ∧ 𝑔 = 〈 𝐾 , 𝐿 〉 ) ) ∧ 𝑟 = 𝐹 ) ∧ 𝑠 = 𝐾 ) → 𝑓 = 〈 𝐹 , 𝐺 〉 ) |
41 |
40
|
fveq2d |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑓 = 〈 𝐹 , 𝐺 〉 ∧ 𝑔 = 〈 𝐾 , 𝐿 〉 ) ) ∧ 𝑟 = 𝐹 ) ∧ 𝑠 = 𝐾 ) → ( 2nd ‘ 𝑓 ) = ( 2nd ‘ 〈 𝐹 , 𝐺 〉 ) ) |
42 |
|
op2ndg |
⊢ ( ( 𝐹 ∈ V ∧ 𝐺 ∈ V ) → ( 2nd ‘ 〈 𝐹 , 𝐺 〉 ) = 𝐺 ) |
43 |
15 42
|
syl |
⊢ ( 𝜑 → ( 2nd ‘ 〈 𝐹 , 𝐺 〉 ) = 𝐺 ) |
44 |
43
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑓 = 〈 𝐹 , 𝐺 〉 ∧ 𝑔 = 〈 𝐾 , 𝐿 〉 ) ) ∧ 𝑟 = 𝐹 ) ∧ 𝑠 = 𝐾 ) → ( 2nd ‘ 〈 𝐹 , 𝐺 〉 ) = 𝐺 ) |
45 |
41 44
|
eqtrd |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑓 = 〈 𝐹 , 𝐺 〉 ∧ 𝑔 = 〈 𝐾 , 𝐿 〉 ) ) ∧ 𝑟 = 𝐹 ) ∧ 𝑠 = 𝐾 ) → ( 2nd ‘ 𝑓 ) = 𝐺 ) |
46 |
45
|
oveqd |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑓 = 〈 𝐹 , 𝐺 〉 ∧ 𝑔 = 〈 𝐾 , 𝐿 〉 ) ) ∧ 𝑟 = 𝐹 ) ∧ 𝑠 = 𝐾 ) → ( 𝑥 ( 2nd ‘ 𝑓 ) 𝑦 ) = ( 𝑥 𝐺 𝑦 ) ) |
47 |
46
|
fveq1d |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑓 = 〈 𝐹 , 𝐺 〉 ∧ 𝑔 = 〈 𝐾 , 𝐿 〉 ) ) ∧ 𝑟 = 𝐹 ) ∧ 𝑠 = 𝐾 ) → ( ( 𝑥 ( 2nd ‘ 𝑓 ) 𝑦 ) ‘ ℎ ) = ( ( 𝑥 𝐺 𝑦 ) ‘ ℎ ) ) |
48 |
38 39 47
|
oveq123d |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑓 = 〈 𝐹 , 𝐺 〉 ∧ 𝑔 = 〈 𝐾 , 𝐿 〉 ) ) ∧ 𝑟 = 𝐹 ) ∧ 𝑠 = 𝐾 ) → ( ( 𝑎 ‘ 𝑦 ) ( 〈 ( 𝑟 ‘ 𝑥 ) , ( 𝑟 ‘ 𝑦 ) 〉 · ( 𝑠 ‘ 𝑦 ) ) ( ( 𝑥 ( 2nd ‘ 𝑓 ) 𝑦 ) ‘ ℎ ) ) = ( ( 𝑎 ‘ 𝑦 ) ( 〈 ( 𝐹 ‘ 𝑥 ) , ( 𝐹 ‘ 𝑦 ) 〉 · ( 𝐾 ‘ 𝑦 ) ) ( ( 𝑥 𝐺 𝑦 ) ‘ ℎ ) ) ) |
49 |
30 32
|
opeq12d |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑓 = 〈 𝐹 , 𝐺 〉 ∧ 𝑔 = 〈 𝐾 , 𝐿 〉 ) ) ∧ 𝑟 = 𝐹 ) ∧ 𝑠 = 𝐾 ) → 〈 ( 𝑟 ‘ 𝑥 ) , ( 𝑠 ‘ 𝑥 ) 〉 = 〈 ( 𝐹 ‘ 𝑥 ) , ( 𝐾 ‘ 𝑥 ) 〉 ) |
50 |
49 37
|
oveq12d |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑓 = 〈 𝐹 , 𝐺 〉 ∧ 𝑔 = 〈 𝐾 , 𝐿 〉 ) ) ∧ 𝑟 = 𝐹 ) ∧ 𝑠 = 𝐾 ) → ( 〈 ( 𝑟 ‘ 𝑥 ) , ( 𝑠 ‘ 𝑥 ) 〉 · ( 𝑠 ‘ 𝑦 ) ) = ( 〈 ( 𝐹 ‘ 𝑥 ) , ( 𝐾 ‘ 𝑥 ) 〉 · ( 𝐾 ‘ 𝑦 ) ) ) |
51 |
21
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑓 = 〈 𝐹 , 𝐺 〉 ∧ 𝑔 = 〈 𝐾 , 𝐿 〉 ) ) ∧ 𝑟 = 𝐹 ) ∧ 𝑠 = 𝐾 ) → 𝑔 = 〈 𝐾 , 𝐿 〉 ) |
52 |
51
|
fveq2d |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑓 = 〈 𝐹 , 𝐺 〉 ∧ 𝑔 = 〈 𝐾 , 𝐿 〉 ) ) ∧ 𝑟 = 𝐹 ) ∧ 𝑠 = 𝐾 ) → ( 2nd ‘ 𝑔 ) = ( 2nd ‘ 〈 𝐾 , 𝐿 〉 ) ) |
53 |
|
op2ndg |
⊢ ( ( 𝐾 ∈ V ∧ 𝐿 ∈ V ) → ( 2nd ‘ 〈 𝐾 , 𝐿 〉 ) = 𝐿 ) |
54 |
24 53
|
syl |
⊢ ( 𝜑 → ( 2nd ‘ 〈 𝐾 , 𝐿 〉 ) = 𝐿 ) |
55 |
54
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑓 = 〈 𝐹 , 𝐺 〉 ∧ 𝑔 = 〈 𝐾 , 𝐿 〉 ) ) ∧ 𝑟 = 𝐹 ) ∧ 𝑠 = 𝐾 ) → ( 2nd ‘ 〈 𝐾 , 𝐿 〉 ) = 𝐿 ) |
56 |
52 55
|
eqtrd |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑓 = 〈 𝐹 , 𝐺 〉 ∧ 𝑔 = 〈 𝐾 , 𝐿 〉 ) ) ∧ 𝑟 = 𝐹 ) ∧ 𝑠 = 𝐾 ) → ( 2nd ‘ 𝑔 ) = 𝐿 ) |
57 |
56
|
oveqd |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑓 = 〈 𝐹 , 𝐺 〉 ∧ 𝑔 = 〈 𝐾 , 𝐿 〉 ) ) ∧ 𝑟 = 𝐹 ) ∧ 𝑠 = 𝐾 ) → ( 𝑥 ( 2nd ‘ 𝑔 ) 𝑦 ) = ( 𝑥 𝐿 𝑦 ) ) |
58 |
57
|
fveq1d |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑓 = 〈 𝐹 , 𝐺 〉 ∧ 𝑔 = 〈 𝐾 , 𝐿 〉 ) ) ∧ 𝑟 = 𝐹 ) ∧ 𝑠 = 𝐾 ) → ( ( 𝑥 ( 2nd ‘ 𝑔 ) 𝑦 ) ‘ ℎ ) = ( ( 𝑥 𝐿 𝑦 ) ‘ ℎ ) ) |
59 |
|
eqidd |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑓 = 〈 𝐹 , 𝐺 〉 ∧ 𝑔 = 〈 𝐾 , 𝐿 〉 ) ) ∧ 𝑟 = 𝐹 ) ∧ 𝑠 = 𝐾 ) → ( 𝑎 ‘ 𝑥 ) = ( 𝑎 ‘ 𝑥 ) ) |
60 |
50 58 59
|
oveq123d |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑓 = 〈 𝐹 , 𝐺 〉 ∧ 𝑔 = 〈 𝐾 , 𝐿 〉 ) ) ∧ 𝑟 = 𝐹 ) ∧ 𝑠 = 𝐾 ) → ( ( ( 𝑥 ( 2nd ‘ 𝑔 ) 𝑦 ) ‘ ℎ ) ( 〈 ( 𝑟 ‘ 𝑥 ) , ( 𝑠 ‘ 𝑥 ) 〉 · ( 𝑠 ‘ 𝑦 ) ) ( 𝑎 ‘ 𝑥 ) ) = ( ( ( 𝑥 𝐿 𝑦 ) ‘ ℎ ) ( 〈 ( 𝐹 ‘ 𝑥 ) , ( 𝐾 ‘ 𝑥 ) 〉 · ( 𝐾 ‘ 𝑦 ) ) ( 𝑎 ‘ 𝑥 ) ) ) |
61 |
48 60
|
eqeq12d |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑓 = 〈 𝐹 , 𝐺 〉 ∧ 𝑔 = 〈 𝐾 , 𝐿 〉 ) ) ∧ 𝑟 = 𝐹 ) ∧ 𝑠 = 𝐾 ) → ( ( ( 𝑎 ‘ 𝑦 ) ( 〈 ( 𝑟 ‘ 𝑥 ) , ( 𝑟 ‘ 𝑦 ) 〉 · ( 𝑠 ‘ 𝑦 ) ) ( ( 𝑥 ( 2nd ‘ 𝑓 ) 𝑦 ) ‘ ℎ ) ) = ( ( ( 𝑥 ( 2nd ‘ 𝑔 ) 𝑦 ) ‘ ℎ ) ( 〈 ( 𝑟 ‘ 𝑥 ) , ( 𝑠 ‘ 𝑥 ) 〉 · ( 𝑠 ‘ 𝑦 ) ) ( 𝑎 ‘ 𝑥 ) ) ↔ ( ( 𝑎 ‘ 𝑦 ) ( 〈 ( 𝐹 ‘ 𝑥 ) , ( 𝐹 ‘ 𝑦 ) 〉 · ( 𝐾 ‘ 𝑦 ) ) ( ( 𝑥 𝐺 𝑦 ) ‘ ℎ ) ) = ( ( ( 𝑥 𝐿 𝑦 ) ‘ ℎ ) ( 〈 ( 𝐹 ‘ 𝑥 ) , ( 𝐾 ‘ 𝑥 ) 〉 · ( 𝐾 ‘ 𝑦 ) ) ( 𝑎 ‘ 𝑥 ) ) ) ) |
62 |
61
|
ralbidv |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑓 = 〈 𝐹 , 𝐺 〉 ∧ 𝑔 = 〈 𝐾 , 𝐿 〉 ) ) ∧ 𝑟 = 𝐹 ) ∧ 𝑠 = 𝐾 ) → ( ∀ ℎ ∈ ( 𝑥 𝐻 𝑦 ) ( ( 𝑎 ‘ 𝑦 ) ( 〈 ( 𝑟 ‘ 𝑥 ) , ( 𝑟 ‘ 𝑦 ) 〉 · ( 𝑠 ‘ 𝑦 ) ) ( ( 𝑥 ( 2nd ‘ 𝑓 ) 𝑦 ) ‘ ℎ ) ) = ( ( ( 𝑥 ( 2nd ‘ 𝑔 ) 𝑦 ) ‘ ℎ ) ( 〈 ( 𝑟 ‘ 𝑥 ) , ( 𝑠 ‘ 𝑥 ) 〉 · ( 𝑠 ‘ 𝑦 ) ) ( 𝑎 ‘ 𝑥 ) ) ↔ ∀ ℎ ∈ ( 𝑥 𝐻 𝑦 ) ( ( 𝑎 ‘ 𝑦 ) ( 〈 ( 𝐹 ‘ 𝑥 ) , ( 𝐹 ‘ 𝑦 ) 〉 · ( 𝐾 ‘ 𝑦 ) ) ( ( 𝑥 𝐺 𝑦 ) ‘ ℎ ) ) = ( ( ( 𝑥 𝐿 𝑦 ) ‘ ℎ ) ( 〈 ( 𝐹 ‘ 𝑥 ) , ( 𝐾 ‘ 𝑥 ) 〉 · ( 𝐾 ‘ 𝑦 ) ) ( 𝑎 ‘ 𝑥 ) ) ) ) |
63 |
62
|
2ralbidv |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑓 = 〈 𝐹 , 𝐺 〉 ∧ 𝑔 = 〈 𝐾 , 𝐿 〉 ) ) ∧ 𝑟 = 𝐹 ) ∧ 𝑠 = 𝐾 ) → ( ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ∀ ℎ ∈ ( 𝑥 𝐻 𝑦 ) ( ( 𝑎 ‘ 𝑦 ) ( 〈 ( 𝑟 ‘ 𝑥 ) , ( 𝑟 ‘ 𝑦 ) 〉 · ( 𝑠 ‘ 𝑦 ) ) ( ( 𝑥 ( 2nd ‘ 𝑓 ) 𝑦 ) ‘ ℎ ) ) = ( ( ( 𝑥 ( 2nd ‘ 𝑔 ) 𝑦 ) ‘ ℎ ) ( 〈 ( 𝑟 ‘ 𝑥 ) , ( 𝑠 ‘ 𝑥 ) 〉 · ( 𝑠 ‘ 𝑦 ) ) ( 𝑎 ‘ 𝑥 ) ) ↔ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ∀ ℎ ∈ ( 𝑥 𝐻 𝑦 ) ( ( 𝑎 ‘ 𝑦 ) ( 〈 ( 𝐹 ‘ 𝑥 ) , ( 𝐹 ‘ 𝑦 ) 〉 · ( 𝐾 ‘ 𝑦 ) ) ( ( 𝑥 𝐺 𝑦 ) ‘ ℎ ) ) = ( ( ( 𝑥 𝐿 𝑦 ) ‘ ℎ ) ( 〈 ( 𝐹 ‘ 𝑥 ) , ( 𝐾 ‘ 𝑥 ) 〉 · ( 𝐾 ‘ 𝑦 ) ) ( 𝑎 ‘ 𝑥 ) ) ) ) |
64 |
34 63
|
rabeqbidv |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑓 = 〈 𝐹 , 𝐺 〉 ∧ 𝑔 = 〈 𝐾 , 𝐿 〉 ) ) ∧ 𝑟 = 𝐹 ) ∧ 𝑠 = 𝐾 ) → { 𝑎 ∈ X 𝑥 ∈ 𝐵 ( ( 𝑟 ‘ 𝑥 ) 𝐽 ( 𝑠 ‘ 𝑥 ) ) ∣ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ∀ ℎ ∈ ( 𝑥 𝐻 𝑦 ) ( ( 𝑎 ‘ 𝑦 ) ( 〈 ( 𝑟 ‘ 𝑥 ) , ( 𝑟 ‘ 𝑦 ) 〉 · ( 𝑠 ‘ 𝑦 ) ) ( ( 𝑥 ( 2nd ‘ 𝑓 ) 𝑦 ) ‘ ℎ ) ) = ( ( ( 𝑥 ( 2nd ‘ 𝑔 ) 𝑦 ) ‘ ℎ ) ( 〈 ( 𝑟 ‘ 𝑥 ) , ( 𝑠 ‘ 𝑥 ) 〉 · ( 𝑠 ‘ 𝑦 ) ) ( 𝑎 ‘ 𝑥 ) ) } = { 𝑎 ∈ X 𝑥 ∈ 𝐵 ( ( 𝐹 ‘ 𝑥 ) 𝐽 ( 𝐾 ‘ 𝑥 ) ) ∣ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ∀ ℎ ∈ ( 𝑥 𝐻 𝑦 ) ( ( 𝑎 ‘ 𝑦 ) ( 〈 ( 𝐹 ‘ 𝑥 ) , ( 𝐹 ‘ 𝑦 ) 〉 · ( 𝐾 ‘ 𝑦 ) ) ( ( 𝑥 𝐺 𝑦 ) ‘ ℎ ) ) = ( ( ( 𝑥 𝐿 𝑦 ) ‘ ℎ ) ( 〈 ( 𝐹 ‘ 𝑥 ) , ( 𝐾 ‘ 𝑥 ) 〉 · ( 𝐾 ‘ 𝑦 ) ) ( 𝑎 ‘ 𝑥 ) ) } ) |
65 |
20 28 64
|
csbied2 |
⊢ ( ( ( 𝜑 ∧ ( 𝑓 = 〈 𝐹 , 𝐺 〉 ∧ 𝑔 = 〈 𝐾 , 𝐿 〉 ) ) ∧ 𝑟 = 𝐹 ) → ⦋ ( 1st ‘ 𝑔 ) / 𝑠 ⦌ { 𝑎 ∈ X 𝑥 ∈ 𝐵 ( ( 𝑟 ‘ 𝑥 ) 𝐽 ( 𝑠 ‘ 𝑥 ) ) ∣ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ∀ ℎ ∈ ( 𝑥 𝐻 𝑦 ) ( ( 𝑎 ‘ 𝑦 ) ( 〈 ( 𝑟 ‘ 𝑥 ) , ( 𝑟 ‘ 𝑦 ) 〉 · ( 𝑠 ‘ 𝑦 ) ) ( ( 𝑥 ( 2nd ‘ 𝑓 ) 𝑦 ) ‘ ℎ ) ) = ( ( ( 𝑥 ( 2nd ‘ 𝑔 ) 𝑦 ) ‘ ℎ ) ( 〈 ( 𝑟 ‘ 𝑥 ) , ( 𝑠 ‘ 𝑥 ) 〉 · ( 𝑠 ‘ 𝑦 ) ) ( 𝑎 ‘ 𝑥 ) ) } = { 𝑎 ∈ X 𝑥 ∈ 𝐵 ( ( 𝐹 ‘ 𝑥 ) 𝐽 ( 𝐾 ‘ 𝑥 ) ) ∣ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ∀ ℎ ∈ ( 𝑥 𝐻 𝑦 ) ( ( 𝑎 ‘ 𝑦 ) ( 〈 ( 𝐹 ‘ 𝑥 ) , ( 𝐹 ‘ 𝑦 ) 〉 · ( 𝐾 ‘ 𝑦 ) ) ( ( 𝑥 𝐺 𝑦 ) ‘ ℎ ) ) = ( ( ( 𝑥 𝐿 𝑦 ) ‘ ℎ ) ( 〈 ( 𝐹 ‘ 𝑥 ) , ( 𝐾 ‘ 𝑥 ) 〉 · ( 𝐾 ‘ 𝑦 ) ) ( 𝑎 ‘ 𝑥 ) ) } ) |
66 |
10 19 65
|
csbied2 |
⊢ ( ( 𝜑 ∧ ( 𝑓 = 〈 𝐹 , 𝐺 〉 ∧ 𝑔 = 〈 𝐾 , 𝐿 〉 ) ) → ⦋ ( 1st ‘ 𝑓 ) / 𝑟 ⦌ ⦋ ( 1st ‘ 𝑔 ) / 𝑠 ⦌ { 𝑎 ∈ X 𝑥 ∈ 𝐵 ( ( 𝑟 ‘ 𝑥 ) 𝐽 ( 𝑠 ‘ 𝑥 ) ) ∣ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ∀ ℎ ∈ ( 𝑥 𝐻 𝑦 ) ( ( 𝑎 ‘ 𝑦 ) ( 〈 ( 𝑟 ‘ 𝑥 ) , ( 𝑟 ‘ 𝑦 ) 〉 · ( 𝑠 ‘ 𝑦 ) ) ( ( 𝑥 ( 2nd ‘ 𝑓 ) 𝑦 ) ‘ ℎ ) ) = ( ( ( 𝑥 ( 2nd ‘ 𝑔 ) 𝑦 ) ‘ ℎ ) ( 〈 ( 𝑟 ‘ 𝑥 ) , ( 𝑠 ‘ 𝑥 ) 〉 · ( 𝑠 ‘ 𝑦 ) ) ( 𝑎 ‘ 𝑥 ) ) } = { 𝑎 ∈ X 𝑥 ∈ 𝐵 ( ( 𝐹 ‘ 𝑥 ) 𝐽 ( 𝐾 ‘ 𝑥 ) ) ∣ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ∀ ℎ ∈ ( 𝑥 𝐻 𝑦 ) ( ( 𝑎 ‘ 𝑦 ) ( 〈 ( 𝐹 ‘ 𝑥 ) , ( 𝐹 ‘ 𝑦 ) 〉 · ( 𝐾 ‘ 𝑦 ) ) ( ( 𝑥 𝐺 𝑦 ) ‘ ℎ ) ) = ( ( ( 𝑥 𝐿 𝑦 ) ‘ ℎ ) ( 〈 ( 𝐹 ‘ 𝑥 ) , ( 𝐾 ‘ 𝑥 ) 〉 · ( 𝐾 ‘ 𝑦 ) ) ( 𝑎 ‘ 𝑥 ) ) } ) |
67 |
|
df-br |
⊢ ( 𝐹 ( 𝐶 Func 𝐷 ) 𝐺 ↔ 〈 𝐹 , 𝐺 〉 ∈ ( 𝐶 Func 𝐷 ) ) |
68 |
6 67
|
sylib |
⊢ ( 𝜑 → 〈 𝐹 , 𝐺 〉 ∈ ( 𝐶 Func 𝐷 ) ) |
69 |
|
df-br |
⊢ ( 𝐾 ( 𝐶 Func 𝐷 ) 𝐿 ↔ 〈 𝐾 , 𝐿 〉 ∈ ( 𝐶 Func 𝐷 ) ) |
70 |
7 69
|
sylib |
⊢ ( 𝜑 → 〈 𝐾 , 𝐿 〉 ∈ ( 𝐶 Func 𝐷 ) ) |
71 |
|
ovex |
⊢ ( ( 𝐹 ‘ 𝑥 ) 𝐽 ( 𝐾 ‘ 𝑥 ) ) ∈ V |
72 |
71
|
rgenw |
⊢ ∀ 𝑥 ∈ 𝐵 ( ( 𝐹 ‘ 𝑥 ) 𝐽 ( 𝐾 ‘ 𝑥 ) ) ∈ V |
73 |
|
ixpexg |
⊢ ( ∀ 𝑥 ∈ 𝐵 ( ( 𝐹 ‘ 𝑥 ) 𝐽 ( 𝐾 ‘ 𝑥 ) ) ∈ V → X 𝑥 ∈ 𝐵 ( ( 𝐹 ‘ 𝑥 ) 𝐽 ( 𝐾 ‘ 𝑥 ) ) ∈ V ) |
74 |
72 73
|
ax-mp |
⊢ X 𝑥 ∈ 𝐵 ( ( 𝐹 ‘ 𝑥 ) 𝐽 ( 𝐾 ‘ 𝑥 ) ) ∈ V |
75 |
74
|
rabex |
⊢ { 𝑎 ∈ X 𝑥 ∈ 𝐵 ( ( 𝐹 ‘ 𝑥 ) 𝐽 ( 𝐾 ‘ 𝑥 ) ) ∣ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ∀ ℎ ∈ ( 𝑥 𝐻 𝑦 ) ( ( 𝑎 ‘ 𝑦 ) ( 〈 ( 𝐹 ‘ 𝑥 ) , ( 𝐹 ‘ 𝑦 ) 〉 · ( 𝐾 ‘ 𝑦 ) ) ( ( 𝑥 𝐺 𝑦 ) ‘ ℎ ) ) = ( ( ( 𝑥 𝐿 𝑦 ) ‘ ℎ ) ( 〈 ( 𝐹 ‘ 𝑥 ) , ( 𝐾 ‘ 𝑥 ) 〉 · ( 𝐾 ‘ 𝑦 ) ) ( 𝑎 ‘ 𝑥 ) ) } ∈ V |
76 |
75
|
a1i |
⊢ ( 𝜑 → { 𝑎 ∈ X 𝑥 ∈ 𝐵 ( ( 𝐹 ‘ 𝑥 ) 𝐽 ( 𝐾 ‘ 𝑥 ) ) ∣ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ∀ ℎ ∈ ( 𝑥 𝐻 𝑦 ) ( ( 𝑎 ‘ 𝑦 ) ( 〈 ( 𝐹 ‘ 𝑥 ) , ( 𝐹 ‘ 𝑦 ) 〉 · ( 𝐾 ‘ 𝑦 ) ) ( ( 𝑥 𝐺 𝑦 ) ‘ ℎ ) ) = ( ( ( 𝑥 𝐿 𝑦 ) ‘ ℎ ) ( 〈 ( 𝐹 ‘ 𝑥 ) , ( 𝐾 ‘ 𝑥 ) 〉 · ( 𝐾 ‘ 𝑦 ) ) ( 𝑎 ‘ 𝑥 ) ) } ∈ V ) |
77 |
9 66 68 70 76
|
ovmpod |
⊢ ( 𝜑 → ( 〈 𝐹 , 𝐺 〉 𝑁 〈 𝐾 , 𝐿 〉 ) = { 𝑎 ∈ X 𝑥 ∈ 𝐵 ( ( 𝐹 ‘ 𝑥 ) 𝐽 ( 𝐾 ‘ 𝑥 ) ) ∣ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ∀ ℎ ∈ ( 𝑥 𝐻 𝑦 ) ( ( 𝑎 ‘ 𝑦 ) ( 〈 ( 𝐹 ‘ 𝑥 ) , ( 𝐹 ‘ 𝑦 ) 〉 · ( 𝐾 ‘ 𝑦 ) ) ( ( 𝑥 𝐺 𝑦 ) ‘ ℎ ) ) = ( ( ( 𝑥 𝐿 𝑦 ) ‘ ℎ ) ( 〈 ( 𝐹 ‘ 𝑥 ) , ( 𝐾 ‘ 𝑥 ) 〉 · ( 𝐾 ‘ 𝑦 ) ) ( 𝑎 ‘ 𝑥 ) ) } ) |
78 |
77
|
eleq2d |
⊢ ( 𝜑 → ( 𝐴 ∈ ( 〈 𝐹 , 𝐺 〉 𝑁 〈 𝐾 , 𝐿 〉 ) ↔ 𝐴 ∈ { 𝑎 ∈ X 𝑥 ∈ 𝐵 ( ( 𝐹 ‘ 𝑥 ) 𝐽 ( 𝐾 ‘ 𝑥 ) ) ∣ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ∀ ℎ ∈ ( 𝑥 𝐻 𝑦 ) ( ( 𝑎 ‘ 𝑦 ) ( 〈 ( 𝐹 ‘ 𝑥 ) , ( 𝐹 ‘ 𝑦 ) 〉 · ( 𝐾 ‘ 𝑦 ) ) ( ( 𝑥 𝐺 𝑦 ) ‘ ℎ ) ) = ( ( ( 𝑥 𝐿 𝑦 ) ‘ ℎ ) ( 〈 ( 𝐹 ‘ 𝑥 ) , ( 𝐾 ‘ 𝑥 ) 〉 · ( 𝐾 ‘ 𝑦 ) ) ( 𝑎 ‘ 𝑥 ) ) } ) ) |
79 |
|
fveq1 |
⊢ ( 𝑎 = 𝐴 → ( 𝑎 ‘ 𝑦 ) = ( 𝐴 ‘ 𝑦 ) ) |
80 |
79
|
oveq1d |
⊢ ( 𝑎 = 𝐴 → ( ( 𝑎 ‘ 𝑦 ) ( 〈 ( 𝐹 ‘ 𝑥 ) , ( 𝐹 ‘ 𝑦 ) 〉 · ( 𝐾 ‘ 𝑦 ) ) ( ( 𝑥 𝐺 𝑦 ) ‘ ℎ ) ) = ( ( 𝐴 ‘ 𝑦 ) ( 〈 ( 𝐹 ‘ 𝑥 ) , ( 𝐹 ‘ 𝑦 ) 〉 · ( 𝐾 ‘ 𝑦 ) ) ( ( 𝑥 𝐺 𝑦 ) ‘ ℎ ) ) ) |
81 |
|
fveq1 |
⊢ ( 𝑎 = 𝐴 → ( 𝑎 ‘ 𝑥 ) = ( 𝐴 ‘ 𝑥 ) ) |
82 |
81
|
oveq2d |
⊢ ( 𝑎 = 𝐴 → ( ( ( 𝑥 𝐿 𝑦 ) ‘ ℎ ) ( 〈 ( 𝐹 ‘ 𝑥 ) , ( 𝐾 ‘ 𝑥 ) 〉 · ( 𝐾 ‘ 𝑦 ) ) ( 𝑎 ‘ 𝑥 ) ) = ( ( ( 𝑥 𝐿 𝑦 ) ‘ ℎ ) ( 〈 ( 𝐹 ‘ 𝑥 ) , ( 𝐾 ‘ 𝑥 ) 〉 · ( 𝐾 ‘ 𝑦 ) ) ( 𝐴 ‘ 𝑥 ) ) ) |
83 |
80 82
|
eqeq12d |
⊢ ( 𝑎 = 𝐴 → ( ( ( 𝑎 ‘ 𝑦 ) ( 〈 ( 𝐹 ‘ 𝑥 ) , ( 𝐹 ‘ 𝑦 ) 〉 · ( 𝐾 ‘ 𝑦 ) ) ( ( 𝑥 𝐺 𝑦 ) ‘ ℎ ) ) = ( ( ( 𝑥 𝐿 𝑦 ) ‘ ℎ ) ( 〈 ( 𝐹 ‘ 𝑥 ) , ( 𝐾 ‘ 𝑥 ) 〉 · ( 𝐾 ‘ 𝑦 ) ) ( 𝑎 ‘ 𝑥 ) ) ↔ ( ( 𝐴 ‘ 𝑦 ) ( 〈 ( 𝐹 ‘ 𝑥 ) , ( 𝐹 ‘ 𝑦 ) 〉 · ( 𝐾 ‘ 𝑦 ) ) ( ( 𝑥 𝐺 𝑦 ) ‘ ℎ ) ) = ( ( ( 𝑥 𝐿 𝑦 ) ‘ ℎ ) ( 〈 ( 𝐹 ‘ 𝑥 ) , ( 𝐾 ‘ 𝑥 ) 〉 · ( 𝐾 ‘ 𝑦 ) ) ( 𝐴 ‘ 𝑥 ) ) ) ) |
84 |
83
|
ralbidv |
⊢ ( 𝑎 = 𝐴 → ( ∀ ℎ ∈ ( 𝑥 𝐻 𝑦 ) ( ( 𝑎 ‘ 𝑦 ) ( 〈 ( 𝐹 ‘ 𝑥 ) , ( 𝐹 ‘ 𝑦 ) 〉 · ( 𝐾 ‘ 𝑦 ) ) ( ( 𝑥 𝐺 𝑦 ) ‘ ℎ ) ) = ( ( ( 𝑥 𝐿 𝑦 ) ‘ ℎ ) ( 〈 ( 𝐹 ‘ 𝑥 ) , ( 𝐾 ‘ 𝑥 ) 〉 · ( 𝐾 ‘ 𝑦 ) ) ( 𝑎 ‘ 𝑥 ) ) ↔ ∀ ℎ ∈ ( 𝑥 𝐻 𝑦 ) ( ( 𝐴 ‘ 𝑦 ) ( 〈 ( 𝐹 ‘ 𝑥 ) , ( 𝐹 ‘ 𝑦 ) 〉 · ( 𝐾 ‘ 𝑦 ) ) ( ( 𝑥 𝐺 𝑦 ) ‘ ℎ ) ) = ( ( ( 𝑥 𝐿 𝑦 ) ‘ ℎ ) ( 〈 ( 𝐹 ‘ 𝑥 ) , ( 𝐾 ‘ 𝑥 ) 〉 · ( 𝐾 ‘ 𝑦 ) ) ( 𝐴 ‘ 𝑥 ) ) ) ) |
85 |
84
|
2ralbidv |
⊢ ( 𝑎 = 𝐴 → ( ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ∀ ℎ ∈ ( 𝑥 𝐻 𝑦 ) ( ( 𝑎 ‘ 𝑦 ) ( 〈 ( 𝐹 ‘ 𝑥 ) , ( 𝐹 ‘ 𝑦 ) 〉 · ( 𝐾 ‘ 𝑦 ) ) ( ( 𝑥 𝐺 𝑦 ) ‘ ℎ ) ) = ( ( ( 𝑥 𝐿 𝑦 ) ‘ ℎ ) ( 〈 ( 𝐹 ‘ 𝑥 ) , ( 𝐾 ‘ 𝑥 ) 〉 · ( 𝐾 ‘ 𝑦 ) ) ( 𝑎 ‘ 𝑥 ) ) ↔ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ∀ ℎ ∈ ( 𝑥 𝐻 𝑦 ) ( ( 𝐴 ‘ 𝑦 ) ( 〈 ( 𝐹 ‘ 𝑥 ) , ( 𝐹 ‘ 𝑦 ) 〉 · ( 𝐾 ‘ 𝑦 ) ) ( ( 𝑥 𝐺 𝑦 ) ‘ ℎ ) ) = ( ( ( 𝑥 𝐿 𝑦 ) ‘ ℎ ) ( 〈 ( 𝐹 ‘ 𝑥 ) , ( 𝐾 ‘ 𝑥 ) 〉 · ( 𝐾 ‘ 𝑦 ) ) ( 𝐴 ‘ 𝑥 ) ) ) ) |
86 |
85
|
elrab |
⊢ ( 𝐴 ∈ { 𝑎 ∈ X 𝑥 ∈ 𝐵 ( ( 𝐹 ‘ 𝑥 ) 𝐽 ( 𝐾 ‘ 𝑥 ) ) ∣ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ∀ ℎ ∈ ( 𝑥 𝐻 𝑦 ) ( ( 𝑎 ‘ 𝑦 ) ( 〈 ( 𝐹 ‘ 𝑥 ) , ( 𝐹 ‘ 𝑦 ) 〉 · ( 𝐾 ‘ 𝑦 ) ) ( ( 𝑥 𝐺 𝑦 ) ‘ ℎ ) ) = ( ( ( 𝑥 𝐿 𝑦 ) ‘ ℎ ) ( 〈 ( 𝐹 ‘ 𝑥 ) , ( 𝐾 ‘ 𝑥 ) 〉 · ( 𝐾 ‘ 𝑦 ) ) ( 𝑎 ‘ 𝑥 ) ) } ↔ ( 𝐴 ∈ X 𝑥 ∈ 𝐵 ( ( 𝐹 ‘ 𝑥 ) 𝐽 ( 𝐾 ‘ 𝑥 ) ) ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ∀ ℎ ∈ ( 𝑥 𝐻 𝑦 ) ( ( 𝐴 ‘ 𝑦 ) ( 〈 ( 𝐹 ‘ 𝑥 ) , ( 𝐹 ‘ 𝑦 ) 〉 · ( 𝐾 ‘ 𝑦 ) ) ( ( 𝑥 𝐺 𝑦 ) ‘ ℎ ) ) = ( ( ( 𝑥 𝐿 𝑦 ) ‘ ℎ ) ( 〈 ( 𝐹 ‘ 𝑥 ) , ( 𝐾 ‘ 𝑥 ) 〉 · ( 𝐾 ‘ 𝑦 ) ) ( 𝐴 ‘ 𝑥 ) ) ) ) |
87 |
78 86
|
bitrdi |
⊢ ( 𝜑 → ( 𝐴 ∈ ( 〈 𝐹 , 𝐺 〉 𝑁 〈 𝐾 , 𝐿 〉 ) ↔ ( 𝐴 ∈ X 𝑥 ∈ 𝐵 ( ( 𝐹 ‘ 𝑥 ) 𝐽 ( 𝐾 ‘ 𝑥 ) ) ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ∀ ℎ ∈ ( 𝑥 𝐻 𝑦 ) ( ( 𝐴 ‘ 𝑦 ) ( 〈 ( 𝐹 ‘ 𝑥 ) , ( 𝐹 ‘ 𝑦 ) 〉 · ( 𝐾 ‘ 𝑦 ) ) ( ( 𝑥 𝐺 𝑦 ) ‘ ℎ ) ) = ( ( ( 𝑥 𝐿 𝑦 ) ‘ ℎ ) ( 〈 ( 𝐹 ‘ 𝑥 ) , ( 𝐾 ‘ 𝑥 ) 〉 · ( 𝐾 ‘ 𝑦 ) ) ( 𝐴 ‘ 𝑥 ) ) ) ) ) |