Metamath Proof Explorer

Theorem xpcval

Description: Value of the binary product of categories. (Contributed by Mario Carneiro, 10-Jan-2017)

Ref Expression
Hypotheses xpcval.t 𝑇 = ( 𝐶 ×c 𝐷 )
xpcval.x 𝑋 = ( Base ‘ 𝐶 )
xpcval.y 𝑌 = ( Base ‘ 𝐷 )
xpcval.h 𝐻 = ( Hom ‘ 𝐶 )
xpcval.j 𝐽 = ( Hom ‘ 𝐷 )
xpcval.o1 · = ( comp ‘ 𝐶 )
xpcval.o2 = ( comp ‘ 𝐷 )
xpcval.c ( 𝜑𝐶𝑉 )
xpcval.d ( 𝜑𝐷𝑊 )
xpcval.b ( 𝜑𝐵 = ( 𝑋 × 𝑌 ) )
xpcval.k ( 𝜑𝐾 = ( 𝑢𝐵 , 𝑣𝐵 ↦ ( ( ( 1st𝑢 ) 𝐻 ( 1st𝑣 ) ) × ( ( 2nd𝑢 ) 𝐽 ( 2nd𝑣 ) ) ) ) )
xpcval.o ( 𝜑𝑂 = ( 𝑥 ∈ ( 𝐵 × 𝐵 ) , 𝑦𝐵 ↦ ( 𝑔 ∈ ( ( 2nd𝑥 ) 𝐾 𝑦 ) , 𝑓 ∈ ( 𝐾𝑥 ) ↦ ⟨ ( ( 1st𝑔 ) ( ⟨ ( 1st ‘ ( 1st𝑥 ) ) , ( 1st ‘ ( 2nd𝑥 ) ) ⟩ · ( 1st𝑦 ) ) ( 1st𝑓 ) ) , ( ( 2nd𝑔 ) ( ⟨ ( 2nd ‘ ( 1st𝑥 ) ) , ( 2nd ‘ ( 2nd𝑥 ) ) ⟩ ( 2nd𝑦 ) ) ( 2nd𝑓 ) ) ⟩ ) ) )
Assertion xpcval ( 𝜑𝑇 = { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( Hom ‘ ndx ) , 𝐾 ⟩ , ⟨ ( comp ‘ ndx ) , 𝑂 ⟩ } )

Proof

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 ( 𝜑𝐶𝑉 )
9 xpcval.d ( 𝜑𝐷𝑊 )
10 xpcval.b ( 𝜑𝐵 = ( 𝑋 × 𝑌 ) )
11 xpcval.k ( 𝜑𝐾 = ( 𝑢𝐵 , 𝑣𝐵 ↦ ( ( ( 1st𝑢 ) 𝐻 ( 1st𝑣 ) ) × ( ( 2nd𝑢 ) 𝐽 ( 2nd𝑣 ) ) ) ) )
12 xpcval.o ( 𝜑𝑂 = ( 𝑥 ∈ ( 𝐵 × 𝐵 ) , 𝑦𝐵 ↦ ( 𝑔 ∈ ( ( 2nd𝑥 ) 𝐾 𝑦 ) , 𝑓 ∈ ( 𝐾𝑥 ) ↦ ⟨ ( ( 1st𝑔 ) ( ⟨ ( 1st ‘ ( 1st𝑥 ) ) , ( 1st ‘ ( 2nd𝑥 ) ) ⟩ · ( 1st𝑦 ) ) ( 1st𝑓 ) ) , ( ( 2nd𝑔 ) ( ⟨ ( 2nd ‘ ( 1st𝑥 ) ) , ( 2nd ‘ ( 2nd𝑥 ) ) ⟩ ( 2nd𝑦 ) ) ( 2nd𝑓 ) ) ⟩ ) ) )
13 df-xpc ×c = ( 𝑟 ∈ V , 𝑠 ∈ V ↦ ( ( Base ‘ 𝑟 ) × ( Base ‘ 𝑠 ) ) / 𝑏 ( 𝑢𝑏 , 𝑣𝑏 ↦ ( ( ( 1st𝑢 ) ( Hom ‘ 𝑟 ) ( 1st𝑣 ) ) × ( ( 2nd𝑢 ) ( Hom ‘ 𝑠 ) ( 2nd𝑣 ) ) ) ) / { ⟨ ( Base ‘ ndx ) , 𝑏 ⟩ , ⟨ ( Hom ‘ ndx ) , ⟩ , ⟨ ( comp ‘ ndx ) , ( 𝑥 ∈ ( 𝑏 × 𝑏 ) , 𝑦𝑏 ↦ ( 𝑔 ∈ ( ( 2nd𝑥 ) 𝑦 ) , 𝑓 ∈ ( 𝑥 ) ↦ ⟨ ( ( 1st𝑔 ) ( ⟨ ( 1st ‘ ( 1st𝑥 ) ) , ( 1st ‘ ( 2nd𝑥 ) ) ⟩ ( comp ‘ 𝑟 ) ( 1st𝑦 ) ) ( 1st𝑓 ) ) , ( ( 2nd𝑔 ) ( ⟨ ( 2nd ‘ ( 1st𝑥 ) ) , ( 2nd ‘ ( 2nd𝑥 ) ) ⟩ ( comp ‘ 𝑠 ) ( 2nd𝑦 ) ) ( 2nd𝑓 ) ) ⟩ ) ) ⟩ } )
14 13 a1i ( 𝜑 → ×c = ( 𝑟 ∈ V , 𝑠 ∈ V ↦ ( ( Base ‘ 𝑟 ) × ( Base ‘ 𝑠 ) ) / 𝑏 ( 𝑢𝑏 , 𝑣𝑏 ↦ ( ( ( 1st𝑢 ) ( Hom ‘ 𝑟 ) ( 1st𝑣 ) ) × ( ( 2nd𝑢 ) ( Hom ‘ 𝑠 ) ( 2nd𝑣 ) ) ) ) / { ⟨ ( Base ‘ ndx ) , 𝑏 ⟩ , ⟨ ( Hom ‘ ndx ) , ⟩ , ⟨ ( comp ‘ ndx ) , ( 𝑥 ∈ ( 𝑏 × 𝑏 ) , 𝑦𝑏 ↦ ( 𝑔 ∈ ( ( 2nd𝑥 ) 𝑦 ) , 𝑓 ∈ ( 𝑥 ) ↦ ⟨ ( ( 1st𝑔 ) ( ⟨ ( 1st ‘ ( 1st𝑥 ) ) , ( 1st ‘ ( 2nd𝑥 ) ) ⟩ ( comp ‘ 𝑟 ) ( 1st𝑦 ) ) ( 1st𝑓 ) ) , ( ( 2nd𝑔 ) ( ⟨ ( 2nd ‘ ( 1st𝑥 ) ) , ( 2nd ‘ ( 2nd𝑥 ) ) ⟩ ( comp ‘ 𝑠 ) ( 2nd𝑦 ) ) ( 2nd𝑓 ) ) ⟩ ) ) ⟩ } ) )
15 fvex ( Base ‘ 𝑟 ) ∈ V
16 fvex ( Base ‘ 𝑠 ) ∈ V
17 15 16 xpex ( ( Base ‘ 𝑟 ) × ( Base ‘ 𝑠 ) ) ∈ V
18 17 a1i ( ( 𝜑 ∧ ( 𝑟 = 𝐶𝑠 = 𝐷 ) ) → ( ( Base ‘ 𝑟 ) × ( Base ‘ 𝑠 ) ) ∈ V )
19 simprl ( ( 𝜑 ∧ ( 𝑟 = 𝐶𝑠 = 𝐷 ) ) → 𝑟 = 𝐶 )
20 19 fveq2d ( ( 𝜑 ∧ ( 𝑟 = 𝐶𝑠 = 𝐷 ) ) → ( Base ‘ 𝑟 ) = ( Base ‘ 𝐶 ) )
21 20 2 eqtr4di ( ( 𝜑 ∧ ( 𝑟 = 𝐶𝑠 = 𝐷 ) ) → ( Base ‘ 𝑟 ) = 𝑋 )
22 simprr ( ( 𝜑 ∧ ( 𝑟 = 𝐶𝑠 = 𝐷 ) ) → 𝑠 = 𝐷 )
23 22 fveq2d ( ( 𝜑 ∧ ( 𝑟 = 𝐶𝑠 = 𝐷 ) ) → ( Base ‘ 𝑠 ) = ( Base ‘ 𝐷 ) )
24 23 3 eqtr4di ( ( 𝜑 ∧ ( 𝑟 = 𝐶𝑠 = 𝐷 ) ) → ( Base ‘ 𝑠 ) = 𝑌 )
25 21 24 xpeq12d ( ( 𝜑 ∧ ( 𝑟 = 𝐶𝑠 = 𝐷 ) ) → ( ( Base ‘ 𝑟 ) × ( Base ‘ 𝑠 ) ) = ( 𝑋 × 𝑌 ) )
26 10 adantr ( ( 𝜑 ∧ ( 𝑟 = 𝐶𝑠 = 𝐷 ) ) → 𝐵 = ( 𝑋 × 𝑌 ) )
27 25 26 eqtr4d ( ( 𝜑 ∧ ( 𝑟 = 𝐶𝑠 = 𝐷 ) ) → ( ( Base ‘ 𝑟 ) × ( Base ‘ 𝑠 ) ) = 𝐵 )
28 vex 𝑏 ∈ V
29 28 28 mpoex ( 𝑢𝑏 , 𝑣𝑏 ↦ ( ( ( 1st𝑢 ) ( Hom ‘ 𝑟 ) ( 1st𝑣 ) ) × ( ( 2nd𝑢 ) ( Hom ‘ 𝑠 ) ( 2nd𝑣 ) ) ) ) ∈ V
30 29 a1i ( ( ( 𝜑 ∧ ( 𝑟 = 𝐶𝑠 = 𝐷 ) ) ∧ 𝑏 = 𝐵 ) → ( 𝑢𝑏 , 𝑣𝑏 ↦ ( ( ( 1st𝑢 ) ( Hom ‘ 𝑟 ) ( 1st𝑣 ) ) × ( ( 2nd𝑢 ) ( Hom ‘ 𝑠 ) ( 2nd𝑣 ) ) ) ) ∈ V )
31 simpr ( ( ( 𝜑 ∧ ( 𝑟 = 𝐶𝑠 = 𝐷 ) ) ∧ 𝑏 = 𝐵 ) → 𝑏 = 𝐵 )
32 simplrl ( ( ( 𝜑 ∧ ( 𝑟 = 𝐶𝑠 = 𝐷 ) ) ∧ 𝑏 = 𝐵 ) → 𝑟 = 𝐶 )
33 32 fveq2d ( ( ( 𝜑 ∧ ( 𝑟 = 𝐶𝑠 = 𝐷 ) ) ∧ 𝑏 = 𝐵 ) → ( Hom ‘ 𝑟 ) = ( Hom ‘ 𝐶 ) )
34 33 4 eqtr4di ( ( ( 𝜑 ∧ ( 𝑟 = 𝐶𝑠 = 𝐷 ) ) ∧ 𝑏 = 𝐵 ) → ( Hom ‘ 𝑟 ) = 𝐻 )
35 34 oveqd ( ( ( 𝜑 ∧ ( 𝑟 = 𝐶𝑠 = 𝐷 ) ) ∧ 𝑏 = 𝐵 ) → ( ( 1st𝑢 ) ( Hom ‘ 𝑟 ) ( 1st𝑣 ) ) = ( ( 1st𝑢 ) 𝐻 ( 1st𝑣 ) ) )
36 simplrr ( ( ( 𝜑 ∧ ( 𝑟 = 𝐶𝑠 = 𝐷 ) ) ∧ 𝑏 = 𝐵 ) → 𝑠 = 𝐷 )
37 36 fveq2d ( ( ( 𝜑 ∧ ( 𝑟 = 𝐶𝑠 = 𝐷 ) ) ∧ 𝑏 = 𝐵 ) → ( Hom ‘ 𝑠 ) = ( Hom ‘ 𝐷 ) )
38 37 5 eqtr4di ( ( ( 𝜑 ∧ ( 𝑟 = 𝐶𝑠 = 𝐷 ) ) ∧ 𝑏 = 𝐵 ) → ( Hom ‘ 𝑠 ) = 𝐽 )
39 38 oveqd ( ( ( 𝜑 ∧ ( 𝑟 = 𝐶𝑠 = 𝐷 ) ) ∧ 𝑏 = 𝐵 ) → ( ( 2nd𝑢 ) ( Hom ‘ 𝑠 ) ( 2nd𝑣 ) ) = ( ( 2nd𝑢 ) 𝐽 ( 2nd𝑣 ) ) )
40 35 39 xpeq12d ( ( ( 𝜑 ∧ ( 𝑟 = 𝐶𝑠 = 𝐷 ) ) ∧ 𝑏 = 𝐵 ) → ( ( ( 1st𝑢 ) ( Hom ‘ 𝑟 ) ( 1st𝑣 ) ) × ( ( 2nd𝑢 ) ( Hom ‘ 𝑠 ) ( 2nd𝑣 ) ) ) = ( ( ( 1st𝑢 ) 𝐻 ( 1st𝑣 ) ) × ( ( 2nd𝑢 ) 𝐽 ( 2nd𝑣 ) ) ) )
41 31 31 40 mpoeq123dv ( ( ( 𝜑 ∧ ( 𝑟 = 𝐶𝑠 = 𝐷 ) ) ∧ 𝑏 = 𝐵 ) → ( 𝑢𝑏 , 𝑣𝑏 ↦ ( ( ( 1st𝑢 ) ( Hom ‘ 𝑟 ) ( 1st𝑣 ) ) × ( ( 2nd𝑢 ) ( Hom ‘ 𝑠 ) ( 2nd𝑣 ) ) ) ) = ( 𝑢𝐵 , 𝑣𝐵 ↦ ( ( ( 1st𝑢 ) 𝐻 ( 1st𝑣 ) ) × ( ( 2nd𝑢 ) 𝐽 ( 2nd𝑣 ) ) ) ) )
42 11 ad2antrr ( ( ( 𝜑 ∧ ( 𝑟 = 𝐶𝑠 = 𝐷 ) ) ∧ 𝑏 = 𝐵 ) → 𝐾 = ( 𝑢𝐵 , 𝑣𝐵 ↦ ( ( ( 1st𝑢 ) 𝐻 ( 1st𝑣 ) ) × ( ( 2nd𝑢 ) 𝐽 ( 2nd𝑣 ) ) ) ) )
43 41 42 eqtr4d ( ( ( 𝜑 ∧ ( 𝑟 = 𝐶𝑠 = 𝐷 ) ) ∧ 𝑏 = 𝐵 ) → ( 𝑢𝑏 , 𝑣𝑏 ↦ ( ( ( 1st𝑢 ) ( Hom ‘ 𝑟 ) ( 1st𝑣 ) ) × ( ( 2nd𝑢 ) ( Hom ‘ 𝑠 ) ( 2nd𝑣 ) ) ) ) = 𝐾 )
44 simplr ( ( ( ( 𝜑 ∧ ( 𝑟 = 𝐶𝑠 = 𝐷 ) ) ∧ 𝑏 = 𝐵 ) ∧ = 𝐾 ) → 𝑏 = 𝐵 )
45 44 opeq2d ( ( ( ( 𝜑 ∧ ( 𝑟 = 𝐶𝑠 = 𝐷 ) ) ∧ 𝑏 = 𝐵 ) ∧ = 𝐾 ) → ⟨ ( Base ‘ ndx ) , 𝑏 ⟩ = ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ )
46 simpr ( ( ( ( 𝜑 ∧ ( 𝑟 = 𝐶𝑠 = 𝐷 ) ) ∧ 𝑏 = 𝐵 ) ∧ = 𝐾 ) → = 𝐾 )
47 46 opeq2d ( ( ( ( 𝜑 ∧ ( 𝑟 = 𝐶𝑠 = 𝐷 ) ) ∧ 𝑏 = 𝐵 ) ∧ = 𝐾 ) → ⟨ ( Hom ‘ ndx ) , ⟩ = ⟨ ( Hom ‘ ndx ) , 𝐾 ⟩ )
48 44 44 xpeq12d ( ( ( ( 𝜑 ∧ ( 𝑟 = 𝐶𝑠 = 𝐷 ) ) ∧ 𝑏 = 𝐵 ) ∧ = 𝐾 ) → ( 𝑏 × 𝑏 ) = ( 𝐵 × 𝐵 ) )
49 46 oveqd ( ( ( ( 𝜑 ∧ ( 𝑟 = 𝐶𝑠 = 𝐷 ) ) ∧ 𝑏 = 𝐵 ) ∧ = 𝐾 ) → ( ( 2nd𝑥 ) 𝑦 ) = ( ( 2nd𝑥 ) 𝐾 𝑦 ) )
50 46 fveq1d ( ( ( ( 𝜑 ∧ ( 𝑟 = 𝐶𝑠 = 𝐷 ) ) ∧ 𝑏 = 𝐵 ) ∧ = 𝐾 ) → ( 𝑥 ) = ( 𝐾𝑥 ) )
51 32 adantr ( ( ( ( 𝜑 ∧ ( 𝑟 = 𝐶𝑠 = 𝐷 ) ) ∧ 𝑏 = 𝐵 ) ∧ = 𝐾 ) → 𝑟 = 𝐶 )
52 51 fveq2d ( ( ( ( 𝜑 ∧ ( 𝑟 = 𝐶𝑠 = 𝐷 ) ) ∧ 𝑏 = 𝐵 ) ∧ = 𝐾 ) → ( comp ‘ 𝑟 ) = ( comp ‘ 𝐶 ) )
53 52 6 eqtr4di ( ( ( ( 𝜑 ∧ ( 𝑟 = 𝐶𝑠 = 𝐷 ) ) ∧ 𝑏 = 𝐵 ) ∧ = 𝐾 ) → ( comp ‘ 𝑟 ) = · )
54 53 oveqd ( ( ( ( 𝜑 ∧ ( 𝑟 = 𝐶𝑠 = 𝐷 ) ) ∧ 𝑏 = 𝐵 ) ∧ = 𝐾 ) → ( ⟨ ( 1st ‘ ( 1st𝑥 ) ) , ( 1st ‘ ( 2nd𝑥 ) ) ⟩ ( comp ‘ 𝑟 ) ( 1st𝑦 ) ) = ( ⟨ ( 1st ‘ ( 1st𝑥 ) ) , ( 1st ‘ ( 2nd𝑥 ) ) ⟩ · ( 1st𝑦 ) ) )
55 54 oveqd ( ( ( ( 𝜑 ∧ ( 𝑟 = 𝐶𝑠 = 𝐷 ) ) ∧ 𝑏 = 𝐵 ) ∧ = 𝐾 ) → ( ( 1st𝑔 ) ( ⟨ ( 1st ‘ ( 1st𝑥 ) ) , ( 1st ‘ ( 2nd𝑥 ) ) ⟩ ( comp ‘ 𝑟 ) ( 1st𝑦 ) ) ( 1st𝑓 ) ) = ( ( 1st𝑔 ) ( ⟨ ( 1st ‘ ( 1st𝑥 ) ) , ( 1st ‘ ( 2nd𝑥 ) ) ⟩ · ( 1st𝑦 ) ) ( 1st𝑓 ) ) )
56 36 adantr ( ( ( ( 𝜑 ∧ ( 𝑟 = 𝐶𝑠 = 𝐷 ) ) ∧ 𝑏 = 𝐵 ) ∧ = 𝐾 ) → 𝑠 = 𝐷 )
57 56 fveq2d ( ( ( ( 𝜑 ∧ ( 𝑟 = 𝐶𝑠 = 𝐷 ) ) ∧ 𝑏 = 𝐵 ) ∧ = 𝐾 ) → ( comp ‘ 𝑠 ) = ( comp ‘ 𝐷 ) )
58 57 7 eqtr4di ( ( ( ( 𝜑 ∧ ( 𝑟 = 𝐶𝑠 = 𝐷 ) ) ∧ 𝑏 = 𝐵 ) ∧ = 𝐾 ) → ( comp ‘ 𝑠 ) = )
59 58 oveqd ( ( ( ( 𝜑 ∧ ( 𝑟 = 𝐶𝑠 = 𝐷 ) ) ∧ 𝑏 = 𝐵 ) ∧ = 𝐾 ) → ( ⟨ ( 2nd ‘ ( 1st𝑥 ) ) , ( 2nd ‘ ( 2nd𝑥 ) ) ⟩ ( comp ‘ 𝑠 ) ( 2nd𝑦 ) ) = ( ⟨ ( 2nd ‘ ( 1st𝑥 ) ) , ( 2nd ‘ ( 2nd𝑥 ) ) ⟩ ( 2nd𝑦 ) ) )
60 59 oveqd ( ( ( ( 𝜑 ∧ ( 𝑟 = 𝐶𝑠 = 𝐷 ) ) ∧ 𝑏 = 𝐵 ) ∧ = 𝐾 ) → ( ( 2nd𝑔 ) ( ⟨ ( 2nd ‘ ( 1st𝑥 ) ) , ( 2nd ‘ ( 2nd𝑥 ) ) ⟩ ( comp ‘ 𝑠 ) ( 2nd𝑦 ) ) ( 2nd𝑓 ) ) = ( ( 2nd𝑔 ) ( ⟨ ( 2nd ‘ ( 1st𝑥 ) ) , ( 2nd ‘ ( 2nd𝑥 ) ) ⟩ ( 2nd𝑦 ) ) ( 2nd𝑓 ) ) )
61 55 60 opeq12d ( ( ( ( 𝜑 ∧ ( 𝑟 = 𝐶𝑠 = 𝐷 ) ) ∧ 𝑏 = 𝐵 ) ∧ = 𝐾 ) → ⟨ ( ( 1st𝑔 ) ( ⟨ ( 1st ‘ ( 1st𝑥 ) ) , ( 1st ‘ ( 2nd𝑥 ) ) ⟩ ( comp ‘ 𝑟 ) ( 1st𝑦 ) ) ( 1st𝑓 ) ) , ( ( 2nd𝑔 ) ( ⟨ ( 2nd ‘ ( 1st𝑥 ) ) , ( 2nd ‘ ( 2nd𝑥 ) ) ⟩ ( comp ‘ 𝑠 ) ( 2nd𝑦 ) ) ( 2nd𝑓 ) ) ⟩ = ⟨ ( ( 1st𝑔 ) ( ⟨ ( 1st ‘ ( 1st𝑥 ) ) , ( 1st ‘ ( 2nd𝑥 ) ) ⟩ · ( 1st𝑦 ) ) ( 1st𝑓 ) ) , ( ( 2nd𝑔 ) ( ⟨ ( 2nd ‘ ( 1st𝑥 ) ) , ( 2nd ‘ ( 2nd𝑥 ) ) ⟩ ( 2nd𝑦 ) ) ( 2nd𝑓 ) ) ⟩ )
62 49 50 61 mpoeq123dv ( ( ( ( 𝜑 ∧ ( 𝑟 = 𝐶𝑠 = 𝐷 ) ) ∧ 𝑏 = 𝐵 ) ∧ = 𝐾 ) → ( 𝑔 ∈ ( ( 2nd𝑥 ) 𝑦 ) , 𝑓 ∈ ( 𝑥 ) ↦ ⟨ ( ( 1st𝑔 ) ( ⟨ ( 1st ‘ ( 1st𝑥 ) ) , ( 1st ‘ ( 2nd𝑥 ) ) ⟩ ( comp ‘ 𝑟 ) ( 1st𝑦 ) ) ( 1st𝑓 ) ) , ( ( 2nd𝑔 ) ( ⟨ ( 2nd ‘ ( 1st𝑥 ) ) , ( 2nd ‘ ( 2nd𝑥 ) ) ⟩ ( comp ‘ 𝑠 ) ( 2nd𝑦 ) ) ( 2nd𝑓 ) ) ⟩ ) = ( 𝑔 ∈ ( ( 2nd𝑥 ) 𝐾 𝑦 ) , 𝑓 ∈ ( 𝐾𝑥 ) ↦ ⟨ ( ( 1st𝑔 ) ( ⟨ ( 1st ‘ ( 1st𝑥 ) ) , ( 1st ‘ ( 2nd𝑥 ) ) ⟩ · ( 1st𝑦 ) ) ( 1st𝑓 ) ) , ( ( 2nd𝑔 ) ( ⟨ ( 2nd ‘ ( 1st𝑥 ) ) , ( 2nd ‘ ( 2nd𝑥 ) ) ⟩ ( 2nd𝑦 ) ) ( 2nd𝑓 ) ) ⟩ ) )
63 48 44 62 mpoeq123dv ( ( ( ( 𝜑 ∧ ( 𝑟 = 𝐶𝑠 = 𝐷 ) ) ∧ 𝑏 = 𝐵 ) ∧ = 𝐾 ) → ( 𝑥 ∈ ( 𝑏 × 𝑏 ) , 𝑦𝑏 ↦ ( 𝑔 ∈ ( ( 2nd𝑥 ) 𝑦 ) , 𝑓 ∈ ( 𝑥 ) ↦ ⟨ ( ( 1st𝑔 ) ( ⟨ ( 1st ‘ ( 1st𝑥 ) ) , ( 1st ‘ ( 2nd𝑥 ) ) ⟩ ( comp ‘ 𝑟 ) ( 1st𝑦 ) ) ( 1st𝑓 ) ) , ( ( 2nd𝑔 ) ( ⟨ ( 2nd ‘ ( 1st𝑥 ) ) , ( 2nd ‘ ( 2nd𝑥 ) ) ⟩ ( comp ‘ 𝑠 ) ( 2nd𝑦 ) ) ( 2nd𝑓 ) ) ⟩ ) ) = ( 𝑥 ∈ ( 𝐵 × 𝐵 ) , 𝑦𝐵 ↦ ( 𝑔 ∈ ( ( 2nd𝑥 ) 𝐾 𝑦 ) , 𝑓 ∈ ( 𝐾𝑥 ) ↦ ⟨ ( ( 1st𝑔 ) ( ⟨ ( 1st ‘ ( 1st𝑥 ) ) , ( 1st ‘ ( 2nd𝑥 ) ) ⟩ · ( 1st𝑦 ) ) ( 1st𝑓 ) ) , ( ( 2nd𝑔 ) ( ⟨ ( 2nd ‘ ( 1st𝑥 ) ) , ( 2nd ‘ ( 2nd𝑥 ) ) ⟩ ( 2nd𝑦 ) ) ( 2nd𝑓 ) ) ⟩ ) ) )
64 12 ad3antrrr ( ( ( ( 𝜑 ∧ ( 𝑟 = 𝐶𝑠 = 𝐷 ) ) ∧ 𝑏 = 𝐵 ) ∧ = 𝐾 ) → 𝑂 = ( 𝑥 ∈ ( 𝐵 × 𝐵 ) , 𝑦𝐵 ↦ ( 𝑔 ∈ ( ( 2nd𝑥 ) 𝐾 𝑦 ) , 𝑓 ∈ ( 𝐾𝑥 ) ↦ ⟨ ( ( 1st𝑔 ) ( ⟨ ( 1st ‘ ( 1st𝑥 ) ) , ( 1st ‘ ( 2nd𝑥 ) ) ⟩ · ( 1st𝑦 ) ) ( 1st𝑓 ) ) , ( ( 2nd𝑔 ) ( ⟨ ( 2nd ‘ ( 1st𝑥 ) ) , ( 2nd ‘ ( 2nd𝑥 ) ) ⟩ ( 2nd𝑦 ) ) ( 2nd𝑓 ) ) ⟩ ) ) )
65 63 64 eqtr4d ( ( ( ( 𝜑 ∧ ( 𝑟 = 𝐶𝑠 = 𝐷 ) ) ∧ 𝑏 = 𝐵 ) ∧ = 𝐾 ) → ( 𝑥 ∈ ( 𝑏 × 𝑏 ) , 𝑦𝑏 ↦ ( 𝑔 ∈ ( ( 2nd𝑥 ) 𝑦 ) , 𝑓 ∈ ( 𝑥 ) ↦ ⟨ ( ( 1st𝑔 ) ( ⟨ ( 1st ‘ ( 1st𝑥 ) ) , ( 1st ‘ ( 2nd𝑥 ) ) ⟩ ( comp ‘ 𝑟 ) ( 1st𝑦 ) ) ( 1st𝑓 ) ) , ( ( 2nd𝑔 ) ( ⟨ ( 2nd ‘ ( 1st𝑥 ) ) , ( 2nd ‘ ( 2nd𝑥 ) ) ⟩ ( comp ‘ 𝑠 ) ( 2nd𝑦 ) ) ( 2nd𝑓 ) ) ⟩ ) ) = 𝑂 )
66 65 opeq2d ( ( ( ( 𝜑 ∧ ( 𝑟 = 𝐶𝑠 = 𝐷 ) ) ∧ 𝑏 = 𝐵 ) ∧ = 𝐾 ) → ⟨ ( comp ‘ ndx ) , ( 𝑥 ∈ ( 𝑏 × 𝑏 ) , 𝑦𝑏 ↦ ( 𝑔 ∈ ( ( 2nd𝑥 ) 𝑦 ) , 𝑓 ∈ ( 𝑥 ) ↦ ⟨ ( ( 1st𝑔 ) ( ⟨ ( 1st ‘ ( 1st𝑥 ) ) , ( 1st ‘ ( 2nd𝑥 ) ) ⟩ ( comp ‘ 𝑟 ) ( 1st𝑦 ) ) ( 1st𝑓 ) ) , ( ( 2nd𝑔 ) ( ⟨ ( 2nd ‘ ( 1st𝑥 ) ) , ( 2nd ‘ ( 2nd𝑥 ) ) ⟩ ( comp ‘ 𝑠 ) ( 2nd𝑦 ) ) ( 2nd𝑓 ) ) ⟩ ) ) ⟩ = ⟨ ( comp ‘ ndx ) , 𝑂 ⟩ )
67 45 47 66 tpeq123d ( ( ( ( 𝜑 ∧ ( 𝑟 = 𝐶𝑠 = 𝐷 ) ) ∧ 𝑏 = 𝐵 ) ∧ = 𝐾 ) → { ⟨ ( Base ‘ ndx ) , 𝑏 ⟩ , ⟨ ( Hom ‘ ndx ) , ⟩ , ⟨ ( comp ‘ ndx ) , ( 𝑥 ∈ ( 𝑏 × 𝑏 ) , 𝑦𝑏 ↦ ( 𝑔 ∈ ( ( 2nd𝑥 ) 𝑦 ) , 𝑓 ∈ ( 𝑥 ) ↦ ⟨ ( ( 1st𝑔 ) ( ⟨ ( 1st ‘ ( 1st𝑥 ) ) , ( 1st ‘ ( 2nd𝑥 ) ) ⟩ ( comp ‘ 𝑟 ) ( 1st𝑦 ) ) ( 1st𝑓 ) ) , ( ( 2nd𝑔 ) ( ⟨ ( 2nd ‘ ( 1st𝑥 ) ) , ( 2nd ‘ ( 2nd𝑥 ) ) ⟩ ( comp ‘ 𝑠 ) ( 2nd𝑦 ) ) ( 2nd𝑓 ) ) ⟩ ) ) ⟩ } = { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( Hom ‘ ndx ) , 𝐾 ⟩ , ⟨ ( comp ‘ ndx ) , 𝑂 ⟩ } )
68 30 43 67 csbied2 ( ( ( 𝜑 ∧ ( 𝑟 = 𝐶𝑠 = 𝐷 ) ) ∧ 𝑏 = 𝐵 ) → ( 𝑢𝑏 , 𝑣𝑏 ↦ ( ( ( 1st𝑢 ) ( Hom ‘ 𝑟 ) ( 1st𝑣 ) ) × ( ( 2nd𝑢 ) ( Hom ‘ 𝑠 ) ( 2nd𝑣 ) ) ) ) / { ⟨ ( Base ‘ ndx ) , 𝑏 ⟩ , ⟨ ( Hom ‘ ndx ) , ⟩ , ⟨ ( comp ‘ ndx ) , ( 𝑥 ∈ ( 𝑏 × 𝑏 ) , 𝑦𝑏 ↦ ( 𝑔 ∈ ( ( 2nd𝑥 ) 𝑦 ) , 𝑓 ∈ ( 𝑥 ) ↦ ⟨ ( ( 1st𝑔 ) ( ⟨ ( 1st ‘ ( 1st𝑥 ) ) , ( 1st ‘ ( 2nd𝑥 ) ) ⟩ ( comp ‘ 𝑟 ) ( 1st𝑦 ) ) ( 1st𝑓 ) ) , ( ( 2nd𝑔 ) ( ⟨ ( 2nd ‘ ( 1st𝑥 ) ) , ( 2nd ‘ ( 2nd𝑥 ) ) ⟩ ( comp ‘ 𝑠 ) ( 2nd𝑦 ) ) ( 2nd𝑓 ) ) ⟩ ) ) ⟩ } = { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( Hom ‘ ndx ) , 𝐾 ⟩ , ⟨ ( comp ‘ ndx ) , 𝑂 ⟩ } )
69 18 27 68 csbied2 ( ( 𝜑 ∧ ( 𝑟 = 𝐶𝑠 = 𝐷 ) ) → ( ( Base ‘ 𝑟 ) × ( Base ‘ 𝑠 ) ) / 𝑏 ( 𝑢𝑏 , 𝑣𝑏 ↦ ( ( ( 1st𝑢 ) ( Hom ‘ 𝑟 ) ( 1st𝑣 ) ) × ( ( 2nd𝑢 ) ( Hom ‘ 𝑠 ) ( 2nd𝑣 ) ) ) ) / { ⟨ ( Base ‘ ndx ) , 𝑏 ⟩ , ⟨ ( Hom ‘ ndx ) , ⟩ , ⟨ ( comp ‘ ndx ) , ( 𝑥 ∈ ( 𝑏 × 𝑏 ) , 𝑦𝑏 ↦ ( 𝑔 ∈ ( ( 2nd𝑥 ) 𝑦 ) , 𝑓 ∈ ( 𝑥 ) ↦ ⟨ ( ( 1st𝑔 ) ( ⟨ ( 1st ‘ ( 1st𝑥 ) ) , ( 1st ‘ ( 2nd𝑥 ) ) ⟩ ( comp ‘ 𝑟 ) ( 1st𝑦 ) ) ( 1st𝑓 ) ) , ( ( 2nd𝑔 ) ( ⟨ ( 2nd ‘ ( 1st𝑥 ) ) , ( 2nd ‘ ( 2nd𝑥 ) ) ⟩ ( comp ‘ 𝑠 ) ( 2nd𝑦 ) ) ( 2nd𝑓 ) ) ⟩ ) ) ⟩ } = { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( Hom ‘ ndx ) , 𝐾 ⟩ , ⟨ ( comp ‘ ndx ) , 𝑂 ⟩ } )
70 8 elexd ( 𝜑𝐶 ∈ V )
71 9 elexd ( 𝜑𝐷 ∈ 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 ( 𝜑𝑇 = { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( Hom ‘ ndx ) , 𝐾 ⟩ , ⟨ ( comp ‘ ndx ) , 𝑂 ⟩ } )