Step |
Hyp |
Ref |
Expression |
1 |
|
xpcval.t |
⊢ 𝑇 = ( 𝐶 ×c 𝐷 ) |
2 |
|
xpcval.x |
⊢ 𝑋 = ( Base ‘ 𝐶 ) |
3 |
|
xpcval.y |
⊢ 𝑌 = ( Base ‘ 𝐷 ) |
4 |
|
xpcval.h |
⊢ 𝐻 = ( Hom ‘ 𝐶 ) |
5 |
|
xpcval.j |
⊢ 𝐽 = ( Hom ‘ 𝐷 ) |
6 |
|
xpcval.o1 |
⊢ · = ( comp ‘ 𝐶 ) |
7 |
|
xpcval.o2 |
⊢ ∙ = ( comp ‘ 𝐷 ) |
8 |
|
xpcval.c |
⊢ ( 𝜑 → 𝐶 ∈ 𝑉 ) |
9 |
|
xpcval.d |
⊢ ( 𝜑 → 𝐷 ∈ 𝑊 ) |
10 |
|
xpcval.b |
⊢ ( 𝜑 → 𝐵 = ( 𝑋 × 𝑌 ) ) |
11 |
|
xpcval.k |
⊢ ( 𝜑 → 𝐾 = ( 𝑢 ∈ 𝐵 , 𝑣 ∈ 𝐵 ↦ ( ( ( 1st ‘ 𝑢 ) 𝐻 ( 1st ‘ 𝑣 ) ) × ( ( 2nd ‘ 𝑢 ) 𝐽 ( 2nd ‘ 𝑣 ) ) ) ) ) |
12 |
|
xpcval.o |
⊢ ( 𝜑 → 𝑂 = ( 𝑥 ∈ ( 𝐵 × 𝐵 ) , 𝑦 ∈ 𝐵 ↦ ( 𝑔 ∈ ( ( 2nd ‘ 𝑥 ) 𝐾 𝑦 ) , 𝑓 ∈ ( 𝐾 ‘ 𝑥 ) ↦ 〈 ( ( 1st ‘ 𝑔 ) ( 〈 ( 1st ‘ ( 1st ‘ 𝑥 ) ) , ( 1st ‘ ( 2nd ‘ 𝑥 ) ) 〉 · ( 1st ‘ 𝑦 ) ) ( 1st ‘ 𝑓 ) ) , ( ( 2nd ‘ 𝑔 ) ( 〈 ( 2nd ‘ ( 1st ‘ 𝑥 ) ) , ( 2nd ‘ ( 2nd ‘ 𝑥 ) ) 〉 ∙ ( 2nd ‘ 𝑦 ) ) ( 2nd ‘ 𝑓 ) ) 〉 ) ) ) |
13 |
|
df-xpc |
⊢ ×c = ( 𝑟 ∈ V , 𝑠 ∈ V ↦ ⦋ ( ( Base ‘ 𝑟 ) × ( Base ‘ 𝑠 ) ) / 𝑏 ⦌ ⦋ ( 𝑢 ∈ 𝑏 , 𝑣 ∈ 𝑏 ↦ ( ( ( 1st ‘ 𝑢 ) ( Hom ‘ 𝑟 ) ( 1st ‘ 𝑣 ) ) × ( ( 2nd ‘ 𝑢 ) ( Hom ‘ 𝑠 ) ( 2nd ‘ 𝑣 ) ) ) ) / ℎ ⦌ { 〈 ( Base ‘ ndx ) , 𝑏 〉 , 〈 ( Hom ‘ ndx ) , ℎ 〉 , 〈 ( comp ‘ ndx ) , ( 𝑥 ∈ ( 𝑏 × 𝑏 ) , 𝑦 ∈ 𝑏 ↦ ( 𝑔 ∈ ( ( 2nd ‘ 𝑥 ) ℎ 𝑦 ) , 𝑓 ∈ ( ℎ ‘ 𝑥 ) ↦ 〈 ( ( 1st ‘ 𝑔 ) ( 〈 ( 1st ‘ ( 1st ‘ 𝑥 ) ) , ( 1st ‘ ( 2nd ‘ 𝑥 ) ) 〉 ( comp ‘ 𝑟 ) ( 1st ‘ 𝑦 ) ) ( 1st ‘ 𝑓 ) ) , ( ( 2nd ‘ 𝑔 ) ( 〈 ( 2nd ‘ ( 1st ‘ 𝑥 ) ) , ( 2nd ‘ ( 2nd ‘ 𝑥 ) ) 〉 ( comp ‘ 𝑠 ) ( 2nd ‘ 𝑦 ) ) ( 2nd ‘ 𝑓 ) ) 〉 ) ) 〉 } ) |
14 |
13
|
a1i |
⊢ ( 𝜑 → ×c = ( 𝑟 ∈ V , 𝑠 ∈ V ↦ ⦋ ( ( Base ‘ 𝑟 ) × ( Base ‘ 𝑠 ) ) / 𝑏 ⦌ ⦋ ( 𝑢 ∈ 𝑏 , 𝑣 ∈ 𝑏 ↦ ( ( ( 1st ‘ 𝑢 ) ( Hom ‘ 𝑟 ) ( 1st ‘ 𝑣 ) ) × ( ( 2nd ‘ 𝑢 ) ( Hom ‘ 𝑠 ) ( 2nd ‘ 𝑣 ) ) ) ) / ℎ ⦌ { 〈 ( Base ‘ ndx ) , 𝑏 〉 , 〈 ( Hom ‘ ndx ) , ℎ 〉 , 〈 ( comp ‘ ndx ) , ( 𝑥 ∈ ( 𝑏 × 𝑏 ) , 𝑦 ∈ 𝑏 ↦ ( 𝑔 ∈ ( ( 2nd ‘ 𝑥 ) ℎ 𝑦 ) , 𝑓 ∈ ( ℎ ‘ 𝑥 ) ↦ 〈 ( ( 1st ‘ 𝑔 ) ( 〈 ( 1st ‘ ( 1st ‘ 𝑥 ) ) , ( 1st ‘ ( 2nd ‘ 𝑥 ) ) 〉 ( comp ‘ 𝑟 ) ( 1st ‘ 𝑦 ) ) ( 1st ‘ 𝑓 ) ) , ( ( 2nd ‘ 𝑔 ) ( 〈 ( 2nd ‘ ( 1st ‘ 𝑥 ) ) , ( 2nd ‘ ( 2nd ‘ 𝑥 ) ) 〉 ( comp ‘ 𝑠 ) ( 2nd ‘ 𝑦 ) ) ( 2nd ‘ 𝑓 ) ) 〉 ) ) 〉 } ) ) |
15 |
|
fvex |
⊢ ( Base ‘ 𝑟 ) ∈ V |
16 |
|
fvex |
⊢ ( Base ‘ 𝑠 ) ∈ V |
17 |
15 16
|
xpex |
⊢ ( ( Base ‘ 𝑟 ) × ( Base ‘ 𝑠 ) ) ∈ V |
18 |
17
|
a1i |
⊢ ( ( 𝜑 ∧ ( 𝑟 = 𝐶 ∧ 𝑠 = 𝐷 ) ) → ( ( Base ‘ 𝑟 ) × ( Base ‘ 𝑠 ) ) ∈ V ) |
19 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( 𝑟 = 𝐶 ∧ 𝑠 = 𝐷 ) ) → 𝑟 = 𝐶 ) |
20 |
19
|
fveq2d |
⊢ ( ( 𝜑 ∧ ( 𝑟 = 𝐶 ∧ 𝑠 = 𝐷 ) ) → ( Base ‘ 𝑟 ) = ( Base ‘ 𝐶 ) ) |
21 |
20 2
|
eqtr4di |
⊢ ( ( 𝜑 ∧ ( 𝑟 = 𝐶 ∧ 𝑠 = 𝐷 ) ) → ( Base ‘ 𝑟 ) = 𝑋 ) |
22 |
|
simprr |
⊢ ( ( 𝜑 ∧ ( 𝑟 = 𝐶 ∧ 𝑠 = 𝐷 ) ) → 𝑠 = 𝐷 ) |
23 |
22
|
fveq2d |
⊢ ( ( 𝜑 ∧ ( 𝑟 = 𝐶 ∧ 𝑠 = 𝐷 ) ) → ( Base ‘ 𝑠 ) = ( Base ‘ 𝐷 ) ) |
24 |
23 3
|
eqtr4di |
⊢ ( ( 𝜑 ∧ ( 𝑟 = 𝐶 ∧ 𝑠 = 𝐷 ) ) → ( Base ‘ 𝑠 ) = 𝑌 ) |
25 |
21 24
|
xpeq12d |
⊢ ( ( 𝜑 ∧ ( 𝑟 = 𝐶 ∧ 𝑠 = 𝐷 ) ) → ( ( Base ‘ 𝑟 ) × ( Base ‘ 𝑠 ) ) = ( 𝑋 × 𝑌 ) ) |
26 |
10
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑟 = 𝐶 ∧ 𝑠 = 𝐷 ) ) → 𝐵 = ( 𝑋 × 𝑌 ) ) |
27 |
25 26
|
eqtr4d |
⊢ ( ( 𝜑 ∧ ( 𝑟 = 𝐶 ∧ 𝑠 = 𝐷 ) ) → ( ( Base ‘ 𝑟 ) × ( Base ‘ 𝑠 ) ) = 𝐵 ) |
28 |
|
vex |
⊢ 𝑏 ∈ V |
29 |
28 28
|
mpoex |
⊢ ( 𝑢 ∈ 𝑏 , 𝑣 ∈ 𝑏 ↦ ( ( ( 1st ‘ 𝑢 ) ( Hom ‘ 𝑟 ) ( 1st ‘ 𝑣 ) ) × ( ( 2nd ‘ 𝑢 ) ( Hom ‘ 𝑠 ) ( 2nd ‘ 𝑣 ) ) ) ) ∈ V |
30 |
29
|
a1i |
⊢ ( ( ( 𝜑 ∧ ( 𝑟 = 𝐶 ∧ 𝑠 = 𝐷 ) ) ∧ 𝑏 = 𝐵 ) → ( 𝑢 ∈ 𝑏 , 𝑣 ∈ 𝑏 ↦ ( ( ( 1st ‘ 𝑢 ) ( Hom ‘ 𝑟 ) ( 1st ‘ 𝑣 ) ) × ( ( 2nd ‘ 𝑢 ) ( Hom ‘ 𝑠 ) ( 2nd ‘ 𝑣 ) ) ) ) ∈ V ) |
31 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ ( 𝑟 = 𝐶 ∧ 𝑠 = 𝐷 ) ) ∧ 𝑏 = 𝐵 ) → 𝑏 = 𝐵 ) |
32 |
|
simplrl |
⊢ ( ( ( 𝜑 ∧ ( 𝑟 = 𝐶 ∧ 𝑠 = 𝐷 ) ) ∧ 𝑏 = 𝐵 ) → 𝑟 = 𝐶 ) |
33 |
32
|
fveq2d |
⊢ ( ( ( 𝜑 ∧ ( 𝑟 = 𝐶 ∧ 𝑠 = 𝐷 ) ) ∧ 𝑏 = 𝐵 ) → ( Hom ‘ 𝑟 ) = ( Hom ‘ 𝐶 ) ) |
34 |
33 4
|
eqtr4di |
⊢ ( ( ( 𝜑 ∧ ( 𝑟 = 𝐶 ∧ 𝑠 = 𝐷 ) ) ∧ 𝑏 = 𝐵 ) → ( Hom ‘ 𝑟 ) = 𝐻 ) |
35 |
34
|
oveqd |
⊢ ( ( ( 𝜑 ∧ ( 𝑟 = 𝐶 ∧ 𝑠 = 𝐷 ) ) ∧ 𝑏 = 𝐵 ) → ( ( 1st ‘ 𝑢 ) ( Hom ‘ 𝑟 ) ( 1st ‘ 𝑣 ) ) = ( ( 1st ‘ 𝑢 ) 𝐻 ( 1st ‘ 𝑣 ) ) ) |
36 |
|
simplrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑟 = 𝐶 ∧ 𝑠 = 𝐷 ) ) ∧ 𝑏 = 𝐵 ) → 𝑠 = 𝐷 ) |
37 |
36
|
fveq2d |
⊢ ( ( ( 𝜑 ∧ ( 𝑟 = 𝐶 ∧ 𝑠 = 𝐷 ) ) ∧ 𝑏 = 𝐵 ) → ( Hom ‘ 𝑠 ) = ( Hom ‘ 𝐷 ) ) |
38 |
37 5
|
eqtr4di |
⊢ ( ( ( 𝜑 ∧ ( 𝑟 = 𝐶 ∧ 𝑠 = 𝐷 ) ) ∧ 𝑏 = 𝐵 ) → ( Hom ‘ 𝑠 ) = 𝐽 ) |
39 |
38
|
oveqd |
⊢ ( ( ( 𝜑 ∧ ( 𝑟 = 𝐶 ∧ 𝑠 = 𝐷 ) ) ∧ 𝑏 = 𝐵 ) → ( ( 2nd ‘ 𝑢 ) ( Hom ‘ 𝑠 ) ( 2nd ‘ 𝑣 ) ) = ( ( 2nd ‘ 𝑢 ) 𝐽 ( 2nd ‘ 𝑣 ) ) ) |
40 |
35 39
|
xpeq12d |
⊢ ( ( ( 𝜑 ∧ ( 𝑟 = 𝐶 ∧ 𝑠 = 𝐷 ) ) ∧ 𝑏 = 𝐵 ) → ( ( ( 1st ‘ 𝑢 ) ( Hom ‘ 𝑟 ) ( 1st ‘ 𝑣 ) ) × ( ( 2nd ‘ 𝑢 ) ( Hom ‘ 𝑠 ) ( 2nd ‘ 𝑣 ) ) ) = ( ( ( 1st ‘ 𝑢 ) 𝐻 ( 1st ‘ 𝑣 ) ) × ( ( 2nd ‘ 𝑢 ) 𝐽 ( 2nd ‘ 𝑣 ) ) ) ) |
41 |
31 31 40
|
mpoeq123dv |
⊢ ( ( ( 𝜑 ∧ ( 𝑟 = 𝐶 ∧ 𝑠 = 𝐷 ) ) ∧ 𝑏 = 𝐵 ) → ( 𝑢 ∈ 𝑏 , 𝑣 ∈ 𝑏 ↦ ( ( ( 1st ‘ 𝑢 ) ( Hom ‘ 𝑟 ) ( 1st ‘ 𝑣 ) ) × ( ( 2nd ‘ 𝑢 ) ( Hom ‘ 𝑠 ) ( 2nd ‘ 𝑣 ) ) ) ) = ( 𝑢 ∈ 𝐵 , 𝑣 ∈ 𝐵 ↦ ( ( ( 1st ‘ 𝑢 ) 𝐻 ( 1st ‘ 𝑣 ) ) × ( ( 2nd ‘ 𝑢 ) 𝐽 ( 2nd ‘ 𝑣 ) ) ) ) ) |
42 |
11
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑟 = 𝐶 ∧ 𝑠 = 𝐷 ) ) ∧ 𝑏 = 𝐵 ) → 𝐾 = ( 𝑢 ∈ 𝐵 , 𝑣 ∈ 𝐵 ↦ ( ( ( 1st ‘ 𝑢 ) 𝐻 ( 1st ‘ 𝑣 ) ) × ( ( 2nd ‘ 𝑢 ) 𝐽 ( 2nd ‘ 𝑣 ) ) ) ) ) |
43 |
41 42
|
eqtr4d |
⊢ ( ( ( 𝜑 ∧ ( 𝑟 = 𝐶 ∧ 𝑠 = 𝐷 ) ) ∧ 𝑏 = 𝐵 ) → ( 𝑢 ∈ 𝑏 , 𝑣 ∈ 𝑏 ↦ ( ( ( 1st ‘ 𝑢 ) ( Hom ‘ 𝑟 ) ( 1st ‘ 𝑣 ) ) × ( ( 2nd ‘ 𝑢 ) ( Hom ‘ 𝑠 ) ( 2nd ‘ 𝑣 ) ) ) ) = 𝐾 ) |
44 |
|
simplr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑟 = 𝐶 ∧ 𝑠 = 𝐷 ) ) ∧ 𝑏 = 𝐵 ) ∧ ℎ = 𝐾 ) → 𝑏 = 𝐵 ) |
45 |
44
|
opeq2d |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑟 = 𝐶 ∧ 𝑠 = 𝐷 ) ) ∧ 𝑏 = 𝐵 ) ∧ ℎ = 𝐾 ) → 〈 ( Base ‘ ndx ) , 𝑏 〉 = 〈 ( Base ‘ ndx ) , 𝐵 〉 ) |
46 |
|
simpr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑟 = 𝐶 ∧ 𝑠 = 𝐷 ) ) ∧ 𝑏 = 𝐵 ) ∧ ℎ = 𝐾 ) → ℎ = 𝐾 ) |
47 |
46
|
opeq2d |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑟 = 𝐶 ∧ 𝑠 = 𝐷 ) ) ∧ 𝑏 = 𝐵 ) ∧ ℎ = 𝐾 ) → 〈 ( Hom ‘ ndx ) , ℎ 〉 = 〈 ( Hom ‘ ndx ) , 𝐾 〉 ) |
48 |
44 44
|
xpeq12d |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑟 = 𝐶 ∧ 𝑠 = 𝐷 ) ) ∧ 𝑏 = 𝐵 ) ∧ ℎ = 𝐾 ) → ( 𝑏 × 𝑏 ) = ( 𝐵 × 𝐵 ) ) |
49 |
46
|
oveqd |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑟 = 𝐶 ∧ 𝑠 = 𝐷 ) ) ∧ 𝑏 = 𝐵 ) ∧ ℎ = 𝐾 ) → ( ( 2nd ‘ 𝑥 ) ℎ 𝑦 ) = ( ( 2nd ‘ 𝑥 ) 𝐾 𝑦 ) ) |
50 |
46
|
fveq1d |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑟 = 𝐶 ∧ 𝑠 = 𝐷 ) ) ∧ 𝑏 = 𝐵 ) ∧ ℎ = 𝐾 ) → ( ℎ ‘ 𝑥 ) = ( 𝐾 ‘ 𝑥 ) ) |
51 |
32
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑟 = 𝐶 ∧ 𝑠 = 𝐷 ) ) ∧ 𝑏 = 𝐵 ) ∧ ℎ = 𝐾 ) → 𝑟 = 𝐶 ) |
52 |
51
|
fveq2d |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑟 = 𝐶 ∧ 𝑠 = 𝐷 ) ) ∧ 𝑏 = 𝐵 ) ∧ ℎ = 𝐾 ) → ( comp ‘ 𝑟 ) = ( comp ‘ 𝐶 ) ) |
53 |
52 6
|
eqtr4di |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑟 = 𝐶 ∧ 𝑠 = 𝐷 ) ) ∧ 𝑏 = 𝐵 ) ∧ ℎ = 𝐾 ) → ( comp ‘ 𝑟 ) = · ) |
54 |
53
|
oveqd |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑟 = 𝐶 ∧ 𝑠 = 𝐷 ) ) ∧ 𝑏 = 𝐵 ) ∧ ℎ = 𝐾 ) → ( 〈 ( 1st ‘ ( 1st ‘ 𝑥 ) ) , ( 1st ‘ ( 2nd ‘ 𝑥 ) ) 〉 ( comp ‘ 𝑟 ) ( 1st ‘ 𝑦 ) ) = ( 〈 ( 1st ‘ ( 1st ‘ 𝑥 ) ) , ( 1st ‘ ( 2nd ‘ 𝑥 ) ) 〉 · ( 1st ‘ 𝑦 ) ) ) |
55 |
54
|
oveqd |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑟 = 𝐶 ∧ 𝑠 = 𝐷 ) ) ∧ 𝑏 = 𝐵 ) ∧ ℎ = 𝐾 ) → ( ( 1st ‘ 𝑔 ) ( 〈 ( 1st ‘ ( 1st ‘ 𝑥 ) ) , ( 1st ‘ ( 2nd ‘ 𝑥 ) ) 〉 ( comp ‘ 𝑟 ) ( 1st ‘ 𝑦 ) ) ( 1st ‘ 𝑓 ) ) = ( ( 1st ‘ 𝑔 ) ( 〈 ( 1st ‘ ( 1st ‘ 𝑥 ) ) , ( 1st ‘ ( 2nd ‘ 𝑥 ) ) 〉 · ( 1st ‘ 𝑦 ) ) ( 1st ‘ 𝑓 ) ) ) |
56 |
36
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑟 = 𝐶 ∧ 𝑠 = 𝐷 ) ) ∧ 𝑏 = 𝐵 ) ∧ ℎ = 𝐾 ) → 𝑠 = 𝐷 ) |
57 |
56
|
fveq2d |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑟 = 𝐶 ∧ 𝑠 = 𝐷 ) ) ∧ 𝑏 = 𝐵 ) ∧ ℎ = 𝐾 ) → ( comp ‘ 𝑠 ) = ( comp ‘ 𝐷 ) ) |
58 |
57 7
|
eqtr4di |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑟 = 𝐶 ∧ 𝑠 = 𝐷 ) ) ∧ 𝑏 = 𝐵 ) ∧ ℎ = 𝐾 ) → ( comp ‘ 𝑠 ) = ∙ ) |
59 |
58
|
oveqd |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑟 = 𝐶 ∧ 𝑠 = 𝐷 ) ) ∧ 𝑏 = 𝐵 ) ∧ ℎ = 𝐾 ) → ( 〈 ( 2nd ‘ ( 1st ‘ 𝑥 ) ) , ( 2nd ‘ ( 2nd ‘ 𝑥 ) ) 〉 ( comp ‘ 𝑠 ) ( 2nd ‘ 𝑦 ) ) = ( 〈 ( 2nd ‘ ( 1st ‘ 𝑥 ) ) , ( 2nd ‘ ( 2nd ‘ 𝑥 ) ) 〉 ∙ ( 2nd ‘ 𝑦 ) ) ) |
60 |
59
|
oveqd |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑟 = 𝐶 ∧ 𝑠 = 𝐷 ) ) ∧ 𝑏 = 𝐵 ) ∧ ℎ = 𝐾 ) → ( ( 2nd ‘ 𝑔 ) ( 〈 ( 2nd ‘ ( 1st ‘ 𝑥 ) ) , ( 2nd ‘ ( 2nd ‘ 𝑥 ) ) 〉 ( comp ‘ 𝑠 ) ( 2nd ‘ 𝑦 ) ) ( 2nd ‘ 𝑓 ) ) = ( ( 2nd ‘ 𝑔 ) ( 〈 ( 2nd ‘ ( 1st ‘ 𝑥 ) ) , ( 2nd ‘ ( 2nd ‘ 𝑥 ) ) 〉 ∙ ( 2nd ‘ 𝑦 ) ) ( 2nd ‘ 𝑓 ) ) ) |
61 |
55 60
|
opeq12d |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑟 = 𝐶 ∧ 𝑠 = 𝐷 ) ) ∧ 𝑏 = 𝐵 ) ∧ ℎ = 𝐾 ) → 〈 ( ( 1st ‘ 𝑔 ) ( 〈 ( 1st ‘ ( 1st ‘ 𝑥 ) ) , ( 1st ‘ ( 2nd ‘ 𝑥 ) ) 〉 ( comp ‘ 𝑟 ) ( 1st ‘ 𝑦 ) ) ( 1st ‘ 𝑓 ) ) , ( ( 2nd ‘ 𝑔 ) ( 〈 ( 2nd ‘ ( 1st ‘ 𝑥 ) ) , ( 2nd ‘ ( 2nd ‘ 𝑥 ) ) 〉 ( comp ‘ 𝑠 ) ( 2nd ‘ 𝑦 ) ) ( 2nd ‘ 𝑓 ) ) 〉 = 〈 ( ( 1st ‘ 𝑔 ) ( 〈 ( 1st ‘ ( 1st ‘ 𝑥 ) ) , ( 1st ‘ ( 2nd ‘ 𝑥 ) ) 〉 · ( 1st ‘ 𝑦 ) ) ( 1st ‘ 𝑓 ) ) , ( ( 2nd ‘ 𝑔 ) ( 〈 ( 2nd ‘ ( 1st ‘ 𝑥 ) ) , ( 2nd ‘ ( 2nd ‘ 𝑥 ) ) 〉 ∙ ( 2nd ‘ 𝑦 ) ) ( 2nd ‘ 𝑓 ) ) 〉 ) |
62 |
49 50 61
|
mpoeq123dv |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑟 = 𝐶 ∧ 𝑠 = 𝐷 ) ) ∧ 𝑏 = 𝐵 ) ∧ ℎ = 𝐾 ) → ( 𝑔 ∈ ( ( 2nd ‘ 𝑥 ) ℎ 𝑦 ) , 𝑓 ∈ ( ℎ ‘ 𝑥 ) ↦ 〈 ( ( 1st ‘ 𝑔 ) ( 〈 ( 1st ‘ ( 1st ‘ 𝑥 ) ) , ( 1st ‘ ( 2nd ‘ 𝑥 ) ) 〉 ( comp ‘ 𝑟 ) ( 1st ‘ 𝑦 ) ) ( 1st ‘ 𝑓 ) ) , ( ( 2nd ‘ 𝑔 ) ( 〈 ( 2nd ‘ ( 1st ‘ 𝑥 ) ) , ( 2nd ‘ ( 2nd ‘ 𝑥 ) ) 〉 ( comp ‘ 𝑠 ) ( 2nd ‘ 𝑦 ) ) ( 2nd ‘ 𝑓 ) ) 〉 ) = ( 𝑔 ∈ ( ( 2nd ‘ 𝑥 ) 𝐾 𝑦 ) , 𝑓 ∈ ( 𝐾 ‘ 𝑥 ) ↦ 〈 ( ( 1st ‘ 𝑔 ) ( 〈 ( 1st ‘ ( 1st ‘ 𝑥 ) ) , ( 1st ‘ ( 2nd ‘ 𝑥 ) ) 〉 · ( 1st ‘ 𝑦 ) ) ( 1st ‘ 𝑓 ) ) , ( ( 2nd ‘ 𝑔 ) ( 〈 ( 2nd ‘ ( 1st ‘ 𝑥 ) ) , ( 2nd ‘ ( 2nd ‘ 𝑥 ) ) 〉 ∙ ( 2nd ‘ 𝑦 ) ) ( 2nd ‘ 𝑓 ) ) 〉 ) ) |
63 |
48 44 62
|
mpoeq123dv |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑟 = 𝐶 ∧ 𝑠 = 𝐷 ) ) ∧ 𝑏 = 𝐵 ) ∧ ℎ = 𝐾 ) → ( 𝑥 ∈ ( 𝑏 × 𝑏 ) , 𝑦 ∈ 𝑏 ↦ ( 𝑔 ∈ ( ( 2nd ‘ 𝑥 ) ℎ 𝑦 ) , 𝑓 ∈ ( ℎ ‘ 𝑥 ) ↦ 〈 ( ( 1st ‘ 𝑔 ) ( 〈 ( 1st ‘ ( 1st ‘ 𝑥 ) ) , ( 1st ‘ ( 2nd ‘ 𝑥 ) ) 〉 ( comp ‘ 𝑟 ) ( 1st ‘ 𝑦 ) ) ( 1st ‘ 𝑓 ) ) , ( ( 2nd ‘ 𝑔 ) ( 〈 ( 2nd ‘ ( 1st ‘ 𝑥 ) ) , ( 2nd ‘ ( 2nd ‘ 𝑥 ) ) 〉 ( comp ‘ 𝑠 ) ( 2nd ‘ 𝑦 ) ) ( 2nd ‘ 𝑓 ) ) 〉 ) ) = ( 𝑥 ∈ ( 𝐵 × 𝐵 ) , 𝑦 ∈ 𝐵 ↦ ( 𝑔 ∈ ( ( 2nd ‘ 𝑥 ) 𝐾 𝑦 ) , 𝑓 ∈ ( 𝐾 ‘ 𝑥 ) ↦ 〈 ( ( 1st ‘ 𝑔 ) ( 〈 ( 1st ‘ ( 1st ‘ 𝑥 ) ) , ( 1st ‘ ( 2nd ‘ 𝑥 ) ) 〉 · ( 1st ‘ 𝑦 ) ) ( 1st ‘ 𝑓 ) ) , ( ( 2nd ‘ 𝑔 ) ( 〈 ( 2nd ‘ ( 1st ‘ 𝑥 ) ) , ( 2nd ‘ ( 2nd ‘ 𝑥 ) ) 〉 ∙ ( 2nd ‘ 𝑦 ) ) ( 2nd ‘ 𝑓 ) ) 〉 ) ) ) |
64 |
12
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑟 = 𝐶 ∧ 𝑠 = 𝐷 ) ) ∧ 𝑏 = 𝐵 ) ∧ ℎ = 𝐾 ) → 𝑂 = ( 𝑥 ∈ ( 𝐵 × 𝐵 ) , 𝑦 ∈ 𝐵 ↦ ( 𝑔 ∈ ( ( 2nd ‘ 𝑥 ) 𝐾 𝑦 ) , 𝑓 ∈ ( 𝐾 ‘ 𝑥 ) ↦ 〈 ( ( 1st ‘ 𝑔 ) ( 〈 ( 1st ‘ ( 1st ‘ 𝑥 ) ) , ( 1st ‘ ( 2nd ‘ 𝑥 ) ) 〉 · ( 1st ‘ 𝑦 ) ) ( 1st ‘ 𝑓 ) ) , ( ( 2nd ‘ 𝑔 ) ( 〈 ( 2nd ‘ ( 1st ‘ 𝑥 ) ) , ( 2nd ‘ ( 2nd ‘ 𝑥 ) ) 〉 ∙ ( 2nd ‘ 𝑦 ) ) ( 2nd ‘ 𝑓 ) ) 〉 ) ) ) |
65 |
63 64
|
eqtr4d |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑟 = 𝐶 ∧ 𝑠 = 𝐷 ) ) ∧ 𝑏 = 𝐵 ) ∧ ℎ = 𝐾 ) → ( 𝑥 ∈ ( 𝑏 × 𝑏 ) , 𝑦 ∈ 𝑏 ↦ ( 𝑔 ∈ ( ( 2nd ‘ 𝑥 ) ℎ 𝑦 ) , 𝑓 ∈ ( ℎ ‘ 𝑥 ) ↦ 〈 ( ( 1st ‘ 𝑔 ) ( 〈 ( 1st ‘ ( 1st ‘ 𝑥 ) ) , ( 1st ‘ ( 2nd ‘ 𝑥 ) ) 〉 ( comp ‘ 𝑟 ) ( 1st ‘ 𝑦 ) ) ( 1st ‘ 𝑓 ) ) , ( ( 2nd ‘ 𝑔 ) ( 〈 ( 2nd ‘ ( 1st ‘ 𝑥 ) ) , ( 2nd ‘ ( 2nd ‘ 𝑥 ) ) 〉 ( comp ‘ 𝑠 ) ( 2nd ‘ 𝑦 ) ) ( 2nd ‘ 𝑓 ) ) 〉 ) ) = 𝑂 ) |
66 |
65
|
opeq2d |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑟 = 𝐶 ∧ 𝑠 = 𝐷 ) ) ∧ 𝑏 = 𝐵 ) ∧ ℎ = 𝐾 ) → 〈 ( comp ‘ ndx ) , ( 𝑥 ∈ ( 𝑏 × 𝑏 ) , 𝑦 ∈ 𝑏 ↦ ( 𝑔 ∈ ( ( 2nd ‘ 𝑥 ) ℎ 𝑦 ) , 𝑓 ∈ ( ℎ ‘ 𝑥 ) ↦ 〈 ( ( 1st ‘ 𝑔 ) ( 〈 ( 1st ‘ ( 1st ‘ 𝑥 ) ) , ( 1st ‘ ( 2nd ‘ 𝑥 ) ) 〉 ( comp ‘ 𝑟 ) ( 1st ‘ 𝑦 ) ) ( 1st ‘ 𝑓 ) ) , ( ( 2nd ‘ 𝑔 ) ( 〈 ( 2nd ‘ ( 1st ‘ 𝑥 ) ) , ( 2nd ‘ ( 2nd ‘ 𝑥 ) ) 〉 ( comp ‘ 𝑠 ) ( 2nd ‘ 𝑦 ) ) ( 2nd ‘ 𝑓 ) ) 〉 ) ) 〉 = 〈 ( comp ‘ ndx ) , 𝑂 〉 ) |
67 |
45 47 66
|
tpeq123d |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑟 = 𝐶 ∧ 𝑠 = 𝐷 ) ) ∧ 𝑏 = 𝐵 ) ∧ ℎ = 𝐾 ) → { 〈 ( Base ‘ ndx ) , 𝑏 〉 , 〈 ( Hom ‘ ndx ) , ℎ 〉 , 〈 ( comp ‘ ndx ) , ( 𝑥 ∈ ( 𝑏 × 𝑏 ) , 𝑦 ∈ 𝑏 ↦ ( 𝑔 ∈ ( ( 2nd ‘ 𝑥 ) ℎ 𝑦 ) , 𝑓 ∈ ( ℎ ‘ 𝑥 ) ↦ 〈 ( ( 1st ‘ 𝑔 ) ( 〈 ( 1st ‘ ( 1st ‘ 𝑥 ) ) , ( 1st ‘ ( 2nd ‘ 𝑥 ) ) 〉 ( comp ‘ 𝑟 ) ( 1st ‘ 𝑦 ) ) ( 1st ‘ 𝑓 ) ) , ( ( 2nd ‘ 𝑔 ) ( 〈 ( 2nd ‘ ( 1st ‘ 𝑥 ) ) , ( 2nd ‘ ( 2nd ‘ 𝑥 ) ) 〉 ( comp ‘ 𝑠 ) ( 2nd ‘ 𝑦 ) ) ( 2nd ‘ 𝑓 ) ) 〉 ) ) 〉 } = { 〈 ( Base ‘ ndx ) , 𝐵 〉 , 〈 ( Hom ‘ ndx ) , 𝐾 〉 , 〈 ( comp ‘ ndx ) , 𝑂 〉 } ) |
68 |
30 43 67
|
csbied2 |
⊢ ( ( ( 𝜑 ∧ ( 𝑟 = 𝐶 ∧ 𝑠 = 𝐷 ) ) ∧ 𝑏 = 𝐵 ) → ⦋ ( 𝑢 ∈ 𝑏 , 𝑣 ∈ 𝑏 ↦ ( ( ( 1st ‘ 𝑢 ) ( Hom ‘ 𝑟 ) ( 1st ‘ 𝑣 ) ) × ( ( 2nd ‘ 𝑢 ) ( Hom ‘ 𝑠 ) ( 2nd ‘ 𝑣 ) ) ) ) / ℎ ⦌ { 〈 ( Base ‘ ndx ) , 𝑏 〉 , 〈 ( Hom ‘ ndx ) , ℎ 〉 , 〈 ( comp ‘ ndx ) , ( 𝑥 ∈ ( 𝑏 × 𝑏 ) , 𝑦 ∈ 𝑏 ↦ ( 𝑔 ∈ ( ( 2nd ‘ 𝑥 ) ℎ 𝑦 ) , 𝑓 ∈ ( ℎ ‘ 𝑥 ) ↦ 〈 ( ( 1st ‘ 𝑔 ) ( 〈 ( 1st ‘ ( 1st ‘ 𝑥 ) ) , ( 1st ‘ ( 2nd ‘ 𝑥 ) ) 〉 ( comp ‘ 𝑟 ) ( 1st ‘ 𝑦 ) ) ( 1st ‘ 𝑓 ) ) , ( ( 2nd ‘ 𝑔 ) ( 〈 ( 2nd ‘ ( 1st ‘ 𝑥 ) ) , ( 2nd ‘ ( 2nd ‘ 𝑥 ) ) 〉 ( comp ‘ 𝑠 ) ( 2nd ‘ 𝑦 ) ) ( 2nd ‘ 𝑓 ) ) 〉 ) ) 〉 } = { 〈 ( Base ‘ ndx ) , 𝐵 〉 , 〈 ( Hom ‘ ndx ) , 𝐾 〉 , 〈 ( comp ‘ ndx ) , 𝑂 〉 } ) |
69 |
18 27 68
|
csbied2 |
⊢ ( ( 𝜑 ∧ ( 𝑟 = 𝐶 ∧ 𝑠 = 𝐷 ) ) → ⦋ ( ( Base ‘ 𝑟 ) × ( Base ‘ 𝑠 ) ) / 𝑏 ⦌ ⦋ ( 𝑢 ∈ 𝑏 , 𝑣 ∈ 𝑏 ↦ ( ( ( 1st ‘ 𝑢 ) ( Hom ‘ 𝑟 ) ( 1st ‘ 𝑣 ) ) × ( ( 2nd ‘ 𝑢 ) ( Hom ‘ 𝑠 ) ( 2nd ‘ 𝑣 ) ) ) ) / ℎ ⦌ { 〈 ( Base ‘ ndx ) , 𝑏 〉 , 〈 ( Hom ‘ ndx ) , ℎ 〉 , 〈 ( comp ‘ ndx ) , ( 𝑥 ∈ ( 𝑏 × 𝑏 ) , 𝑦 ∈ 𝑏 ↦ ( 𝑔 ∈ ( ( 2nd ‘ 𝑥 ) ℎ 𝑦 ) , 𝑓 ∈ ( ℎ ‘ 𝑥 ) ↦ 〈 ( ( 1st ‘ 𝑔 ) ( 〈 ( 1st ‘ ( 1st ‘ 𝑥 ) ) , ( 1st ‘ ( 2nd ‘ 𝑥 ) ) 〉 ( comp ‘ 𝑟 ) ( 1st ‘ 𝑦 ) ) ( 1st ‘ 𝑓 ) ) , ( ( 2nd ‘ 𝑔 ) ( 〈 ( 2nd ‘ ( 1st ‘ 𝑥 ) ) , ( 2nd ‘ ( 2nd ‘ 𝑥 ) ) 〉 ( comp ‘ 𝑠 ) ( 2nd ‘ 𝑦 ) ) ( 2nd ‘ 𝑓 ) ) 〉 ) ) 〉 } = { 〈 ( Base ‘ ndx ) , 𝐵 〉 , 〈 ( Hom ‘ ndx ) , 𝐾 〉 , 〈 ( comp ‘ ndx ) , 𝑂 〉 } ) |
70 |
8
|
elexd |
⊢ ( 𝜑 → 𝐶 ∈ V ) |
71 |
9
|
elexd |
⊢ ( 𝜑 → 𝐷 ∈ V ) |
72 |
|
tpex |
⊢ { 〈 ( Base ‘ ndx ) , 𝐵 〉 , 〈 ( Hom ‘ ndx ) , 𝐾 〉 , 〈 ( comp ‘ ndx ) , 𝑂 〉 } ∈ V |
73 |
72
|
a1i |
⊢ ( 𝜑 → { 〈 ( Base ‘ ndx ) , 𝐵 〉 , 〈 ( Hom ‘ ndx ) , 𝐾 〉 , 〈 ( comp ‘ ndx ) , 𝑂 〉 } ∈ V ) |
74 |
14 69 70 71 73
|
ovmpod |
⊢ ( 𝜑 → ( 𝐶 ×c 𝐷 ) = { 〈 ( Base ‘ ndx ) , 𝐵 〉 , 〈 ( Hom ‘ ndx ) , 𝐾 〉 , 〈 ( comp ‘ ndx ) , 𝑂 〉 } ) |
75 |
1 74
|
syl5eq |
⊢ ( 𝜑 → 𝑇 = { 〈 ( Base ‘ ndx ) , 𝐵 〉 , 〈 ( Hom ‘ ndx ) , 𝐾 〉 , 〈 ( comp ‘ ndx ) , 𝑂 〉 } ) |