Metamath Proof Explorer


Theorem estrcco

Description: Composition in the category of extensible structures. (Contributed by AV, 7-Mar-2020)

Ref Expression
Hypotheses estrcbas.c 𝐶 = ( ExtStrCat ‘ 𝑈 )
estrcbas.u ( 𝜑𝑈𝑉 )
estrcco.o · = ( comp ‘ 𝐶 )
estrcco.x ( 𝜑𝑋𝑈 )
estrcco.y ( 𝜑𝑌𝑈 )
estrcco.z ( 𝜑𝑍𝑈 )
estrcco.a 𝐴 = ( Base ‘ 𝑋 )
estrcco.b 𝐵 = ( Base ‘ 𝑌 )
estrcco.d 𝐷 = ( Base ‘ 𝑍 )
estrcco.f ( 𝜑𝐹 : 𝐴𝐵 )
estrcco.g ( 𝜑𝐺 : 𝐵𝐷 )
Assertion estrcco ( 𝜑 → ( 𝐺 ( ⟨ 𝑋 , 𝑌· 𝑍 ) 𝐹 ) = ( 𝐺𝐹 ) )

Proof

Step Hyp Ref Expression
1 estrcbas.c 𝐶 = ( ExtStrCat ‘ 𝑈 )
2 estrcbas.u ( 𝜑𝑈𝑉 )
3 estrcco.o · = ( comp ‘ 𝐶 )
4 estrcco.x ( 𝜑𝑋𝑈 )
5 estrcco.y ( 𝜑𝑌𝑈 )
6 estrcco.z ( 𝜑𝑍𝑈 )
7 estrcco.a 𝐴 = ( Base ‘ 𝑋 )
8 estrcco.b 𝐵 = ( Base ‘ 𝑌 )
9 estrcco.d 𝐷 = ( Base ‘ 𝑍 )
10 estrcco.f ( 𝜑𝐹 : 𝐴𝐵 )
11 estrcco.g ( 𝜑𝐺 : 𝐵𝐷 )
12 1 2 3 estrccofval ( 𝜑· = ( 𝑣 ∈ ( 𝑈 × 𝑈 ) , 𝑧𝑈 ↦ ( 𝑔 ∈ ( ( Base ‘ 𝑧 ) ↑m ( Base ‘ ( 2nd𝑣 ) ) ) , 𝑓 ∈ ( ( Base ‘ ( 2nd𝑣 ) ) ↑m ( Base ‘ ( 1st𝑣 ) ) ) ↦ ( 𝑔𝑓 ) ) ) )
13 fveq2 ( 𝑧 = 𝑍 → ( Base ‘ 𝑧 ) = ( Base ‘ 𝑍 ) )
14 13 adantl ( ( 𝑣 = ⟨ 𝑋 , 𝑌 ⟩ ∧ 𝑧 = 𝑍 ) → ( Base ‘ 𝑧 ) = ( Base ‘ 𝑍 ) )
15 14 adantl ( ( 𝜑 ∧ ( 𝑣 = ⟨ 𝑋 , 𝑌 ⟩ ∧ 𝑧 = 𝑍 ) ) → ( Base ‘ 𝑧 ) = ( Base ‘ 𝑍 ) )
16 simprl ( ( 𝜑 ∧ ( 𝑣 = ⟨ 𝑋 , 𝑌 ⟩ ∧ 𝑧 = 𝑍 ) ) → 𝑣 = ⟨ 𝑋 , 𝑌 ⟩ )
17 16 fveq2d ( ( 𝜑 ∧ ( 𝑣 = ⟨ 𝑋 , 𝑌 ⟩ ∧ 𝑧 = 𝑍 ) ) → ( 2nd𝑣 ) = ( 2nd ‘ ⟨ 𝑋 , 𝑌 ⟩ ) )
18 op2ndg ( ( 𝑋𝑈𝑌𝑈 ) → ( 2nd ‘ ⟨ 𝑋 , 𝑌 ⟩ ) = 𝑌 )
19 4 5 18 syl2anc ( 𝜑 → ( 2nd ‘ ⟨ 𝑋 , 𝑌 ⟩ ) = 𝑌 )
20 19 adantr ( ( 𝜑 ∧ ( 𝑣 = ⟨ 𝑋 , 𝑌 ⟩ ∧ 𝑧 = 𝑍 ) ) → ( 2nd ‘ ⟨ 𝑋 , 𝑌 ⟩ ) = 𝑌 )
21 17 20 eqtrd ( ( 𝜑 ∧ ( 𝑣 = ⟨ 𝑋 , 𝑌 ⟩ ∧ 𝑧 = 𝑍 ) ) → ( 2nd𝑣 ) = 𝑌 )
22 21 fveq2d ( ( 𝜑 ∧ ( 𝑣 = ⟨ 𝑋 , 𝑌 ⟩ ∧ 𝑧 = 𝑍 ) ) → ( Base ‘ ( 2nd𝑣 ) ) = ( Base ‘ 𝑌 ) )
23 15 22 oveq12d ( ( 𝜑 ∧ ( 𝑣 = ⟨ 𝑋 , 𝑌 ⟩ ∧ 𝑧 = 𝑍 ) ) → ( ( Base ‘ 𝑧 ) ↑m ( Base ‘ ( 2nd𝑣 ) ) ) = ( ( Base ‘ 𝑍 ) ↑m ( Base ‘ 𝑌 ) ) )
24 16 fveq2d ( ( 𝜑 ∧ ( 𝑣 = ⟨ 𝑋 , 𝑌 ⟩ ∧ 𝑧 = 𝑍 ) ) → ( 1st𝑣 ) = ( 1st ‘ ⟨ 𝑋 , 𝑌 ⟩ ) )
25 24 fveq2d ( ( 𝜑 ∧ ( 𝑣 = ⟨ 𝑋 , 𝑌 ⟩ ∧ 𝑧 = 𝑍 ) ) → ( Base ‘ ( 1st𝑣 ) ) = ( Base ‘ ( 1st ‘ ⟨ 𝑋 , 𝑌 ⟩ ) ) )
26 op1stg ( ( 𝑋𝑈𝑌𝑈 ) → ( 1st ‘ ⟨ 𝑋 , 𝑌 ⟩ ) = 𝑋 )
27 4 5 26 syl2anc ( 𝜑 → ( 1st ‘ ⟨ 𝑋 , 𝑌 ⟩ ) = 𝑋 )
28 27 fveq2d ( 𝜑 → ( Base ‘ ( 1st ‘ ⟨ 𝑋 , 𝑌 ⟩ ) ) = ( Base ‘ 𝑋 ) )
29 28 adantr ( ( 𝜑 ∧ ( 𝑣 = ⟨ 𝑋 , 𝑌 ⟩ ∧ 𝑧 = 𝑍 ) ) → ( Base ‘ ( 1st ‘ ⟨ 𝑋 , 𝑌 ⟩ ) ) = ( Base ‘ 𝑋 ) )
30 25 29 eqtrd ( ( 𝜑 ∧ ( 𝑣 = ⟨ 𝑋 , 𝑌 ⟩ ∧ 𝑧 = 𝑍 ) ) → ( Base ‘ ( 1st𝑣 ) ) = ( Base ‘ 𝑋 ) )
31 22 30 oveq12d ( ( 𝜑 ∧ ( 𝑣 = ⟨ 𝑋 , 𝑌 ⟩ ∧ 𝑧 = 𝑍 ) ) → ( ( Base ‘ ( 2nd𝑣 ) ) ↑m ( Base ‘ ( 1st𝑣 ) ) ) = ( ( Base ‘ 𝑌 ) ↑m ( Base ‘ 𝑋 ) ) )
32 eqidd ( ( 𝜑 ∧ ( 𝑣 = ⟨ 𝑋 , 𝑌 ⟩ ∧ 𝑧 = 𝑍 ) ) → ( 𝑔𝑓 ) = ( 𝑔𝑓 ) )
33 23 31 32 mpoeq123dv ( ( 𝜑 ∧ ( 𝑣 = ⟨ 𝑋 , 𝑌 ⟩ ∧ 𝑧 = 𝑍 ) ) → ( 𝑔 ∈ ( ( Base ‘ 𝑧 ) ↑m ( Base ‘ ( 2nd𝑣 ) ) ) , 𝑓 ∈ ( ( Base ‘ ( 2nd𝑣 ) ) ↑m ( Base ‘ ( 1st𝑣 ) ) ) ↦ ( 𝑔𝑓 ) ) = ( 𝑔 ∈ ( ( Base ‘ 𝑍 ) ↑m ( Base ‘ 𝑌 ) ) , 𝑓 ∈ ( ( Base ‘ 𝑌 ) ↑m ( Base ‘ 𝑋 ) ) ↦ ( 𝑔𝑓 ) ) )
34 4 5 opelxpd ( 𝜑 → ⟨ 𝑋 , 𝑌 ⟩ ∈ ( 𝑈 × 𝑈 ) )
35 ovex ( ( Base ‘ 𝑍 ) ↑m ( Base ‘ 𝑌 ) ) ∈ V
36 ovex ( ( Base ‘ 𝑌 ) ↑m ( Base ‘ 𝑋 ) ) ∈ V
37 35 36 mpoex ( 𝑔 ∈ ( ( Base ‘ 𝑍 ) ↑m ( Base ‘ 𝑌 ) ) , 𝑓 ∈ ( ( Base ‘ 𝑌 ) ↑m ( Base ‘ 𝑋 ) ) ↦ ( 𝑔𝑓 ) ) ∈ V
38 37 a1i ( 𝜑 → ( 𝑔 ∈ ( ( Base ‘ 𝑍 ) ↑m ( Base ‘ 𝑌 ) ) , 𝑓 ∈ ( ( Base ‘ 𝑌 ) ↑m ( Base ‘ 𝑋 ) ) ↦ ( 𝑔𝑓 ) ) ∈ V )
39 12 33 34 6 38 ovmpod ( 𝜑 → ( ⟨ 𝑋 , 𝑌· 𝑍 ) = ( 𝑔 ∈ ( ( Base ‘ 𝑍 ) ↑m ( Base ‘ 𝑌 ) ) , 𝑓 ∈ ( ( Base ‘ 𝑌 ) ↑m ( Base ‘ 𝑋 ) ) ↦ ( 𝑔𝑓 ) ) )
40 simpl ( ( 𝑔 = 𝐺𝑓 = 𝐹 ) → 𝑔 = 𝐺 )
41 simpr ( ( 𝑔 = 𝐺𝑓 = 𝐹 ) → 𝑓 = 𝐹 )
42 40 41 coeq12d ( ( 𝑔 = 𝐺𝑓 = 𝐹 ) → ( 𝑔𝑓 ) = ( 𝐺𝐹 ) )
43 42 adantl ( ( 𝜑 ∧ ( 𝑔 = 𝐺𝑓 = 𝐹 ) ) → ( 𝑔𝑓 ) = ( 𝐺𝐹 ) )
44 8 a1i ( 𝜑𝐵 = ( Base ‘ 𝑌 ) )
45 44 eqcomd ( 𝜑 → ( Base ‘ 𝑌 ) = 𝐵 )
46 9 a1i ( 𝜑𝐷 = ( Base ‘ 𝑍 ) )
47 46 eqcomd ( 𝜑 → ( Base ‘ 𝑍 ) = 𝐷 )
48 45 47 feq23d ( 𝜑 → ( 𝐺 : ( Base ‘ 𝑌 ) ⟶ ( Base ‘ 𝑍 ) ↔ 𝐺 : 𝐵𝐷 ) )
49 11 48 mpbird ( 𝜑𝐺 : ( Base ‘ 𝑌 ) ⟶ ( Base ‘ 𝑍 ) )
50 fvexd ( 𝜑 → ( Base ‘ 𝑍 ) ∈ V )
51 fvexd ( 𝜑 → ( Base ‘ 𝑌 ) ∈ V )
52 50 51 elmapd ( 𝜑 → ( 𝐺 ∈ ( ( Base ‘ 𝑍 ) ↑m ( Base ‘ 𝑌 ) ) ↔ 𝐺 : ( Base ‘ 𝑌 ) ⟶ ( Base ‘ 𝑍 ) ) )
53 49 52 mpbird ( 𝜑𝐺 ∈ ( ( Base ‘ 𝑍 ) ↑m ( Base ‘ 𝑌 ) ) )
54 7 a1i ( 𝜑𝐴 = ( Base ‘ 𝑋 ) )
55 54 eqcomd ( 𝜑 → ( Base ‘ 𝑋 ) = 𝐴 )
56 55 45 feq23d ( 𝜑 → ( 𝐹 : ( Base ‘ 𝑋 ) ⟶ ( Base ‘ 𝑌 ) ↔ 𝐹 : 𝐴𝐵 ) )
57 10 56 mpbird ( 𝜑𝐹 : ( Base ‘ 𝑋 ) ⟶ ( Base ‘ 𝑌 ) )
58 fvexd ( 𝜑 → ( Base ‘ 𝑋 ) ∈ V )
59 51 58 elmapd ( 𝜑 → ( 𝐹 ∈ ( ( Base ‘ 𝑌 ) ↑m ( Base ‘ 𝑋 ) ) ↔ 𝐹 : ( Base ‘ 𝑋 ) ⟶ ( Base ‘ 𝑌 ) ) )
60 57 59 mpbird ( 𝜑𝐹 ∈ ( ( Base ‘ 𝑌 ) ↑m ( Base ‘ 𝑋 ) ) )
61 coexg ( ( 𝐺 ∈ ( ( Base ‘ 𝑍 ) ↑m ( Base ‘ 𝑌 ) ) ∧ 𝐹 ∈ ( ( Base ‘ 𝑌 ) ↑m ( Base ‘ 𝑋 ) ) ) → ( 𝐺𝐹 ) ∈ V )
62 53 60 61 syl2anc ( 𝜑 → ( 𝐺𝐹 ) ∈ V )
63 39 43 53 60 62 ovmpod ( 𝜑 → ( 𝐺 ( ⟨ 𝑋 , 𝑌· 𝑍 ) 𝐹 ) = ( 𝐺𝐹 ) )