Theorem oppccofval

Description: Composition in the opposite category. (Contributed by Mario Carneiro, 2-Jan-2017)

	Ref	Expression
Hypotheses	oppcco.b	⊢ 𝐵 = ( Base ‘ 𝐶 )
	oppcco.c	⊢ · = ( comp ‘ 𝐶 )
	oppcco.o	⊢ 𝑂 = ( oppCat ‘ 𝐶 )
	oppcco.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
	oppcco.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
	oppcco.z	⊢ ( 𝜑 → 𝑍 ∈ 𝐵 )
Assertion	oppccofval	⊢ ( 𝜑 → ( ⟨ 𝑋 , 𝑌 ⟩ ( comp ‘ 𝑂 ) 𝑍 ) = tpos ( ⟨ 𝑍 , 𝑌 ⟩ · 𝑋 ) )

Proof

Step	Hyp	Ref	Expression
1		oppcco.b	⊢ 𝐵 = ( Base ‘ 𝐶 )
2		oppcco.c	⊢ · = ( comp ‘ 𝐶 )
3		oppcco.o	⊢ 𝑂 = ( oppCat ‘ 𝐶 )
4		oppcco.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
5		oppcco.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
6		oppcco.z	⊢ ( 𝜑 → 𝑍 ∈ 𝐵 )
7		elfvex	⊢ ( 𝑋 ∈ ( Base ‘ 𝐶 ) → 𝐶 ∈ V )
8	7 1	eleq2s	⊢ ( 𝑋 ∈ 𝐵 → 𝐶 ∈ V )
9		eqid	⊢ ( Hom ‘ 𝐶 ) = ( Hom ‘ 𝐶 )
10	1 9 2 3	oppcval	⊢ ( 𝐶 ∈ V → 𝑂 = ( ( 𝐶 sSet ⟨ ( Hom ‘ ndx ) , tpos ( Hom ‘ 𝐶 ) ⟩ ) sSet ⟨ ( comp ‘ ndx ) , ( 𝑢 ∈ ( 𝐵 × 𝐵 ) , 𝑧 ∈ 𝐵 ↦ tpos ( ⟨ 𝑧 , ( 2nd ‘ 𝑢 ) ⟩ · ( 1st ‘ 𝑢 ) ) ) ⟩ ) )
11	4 8 10	3syl	⊢ ( 𝜑 → 𝑂 = ( ( 𝐶 sSet ⟨ ( Hom ‘ ndx ) , tpos ( Hom ‘ 𝐶 ) ⟩ ) sSet ⟨ ( comp ‘ ndx ) , ( 𝑢 ∈ ( 𝐵 × 𝐵 ) , 𝑧 ∈ 𝐵 ↦ tpos ( ⟨ 𝑧 , ( 2nd ‘ 𝑢 ) ⟩ · ( 1st ‘ 𝑢 ) ) ) ⟩ ) )
12	11	fveq2d	⊢ ( 𝜑 → ( comp ‘ 𝑂 ) = ( comp ‘ ( ( 𝐶 sSet ⟨ ( Hom ‘ ndx ) , tpos ( Hom ‘ 𝐶 ) ⟩ ) sSet ⟨ ( comp ‘ ndx ) , ( 𝑢 ∈ ( 𝐵 × 𝐵 ) , 𝑧 ∈ 𝐵 ↦ tpos ( ⟨ 𝑧 , ( 2nd ‘ 𝑢 ) ⟩ · ( 1st ‘ 𝑢 ) ) ) ⟩ ) ) )
13		ovex	⊢ ( 𝐶 sSet ⟨ ( Hom ‘ ndx ) , tpos ( Hom ‘ 𝐶 ) ⟩ ) ∈ V
14	1	fvexi	⊢ 𝐵 ∈ V
15	14 14	xpex	⊢ ( 𝐵 × 𝐵 ) ∈ V
16	15 14	mpoex	⊢ ( 𝑢 ∈ ( 𝐵 × 𝐵 ) , 𝑧 ∈ 𝐵 ↦ tpos ( ⟨ 𝑧 , ( 2nd ‘ 𝑢 ) ⟩ · ( 1st ‘ 𝑢 ) ) ) ∈ V
17		ccoid	⊢ comp = Slot ( comp ‘ ndx )
18	17	setsid	⊢ ( ( ( 𝐶 sSet ⟨ ( Hom ‘ ndx ) , tpos ( Hom ‘ 𝐶 ) ⟩ ) ∈ V ∧ ( 𝑢 ∈ ( 𝐵 × 𝐵 ) , 𝑧 ∈ 𝐵 ↦ tpos ( ⟨ 𝑧 , ( 2nd ‘ 𝑢 ) ⟩ · ( 1st ‘ 𝑢 ) ) ) ∈ V ) → ( 𝑢 ∈ ( 𝐵 × 𝐵 ) , 𝑧 ∈ 𝐵 ↦ tpos ( ⟨ 𝑧 , ( 2nd ‘ 𝑢 ) ⟩ · ( 1st ‘ 𝑢 ) ) ) = ( comp ‘ ( ( 𝐶 sSet ⟨ ( Hom ‘ ndx ) , tpos ( Hom ‘ 𝐶 ) ⟩ ) sSet ⟨ ( comp ‘ ndx ) , ( 𝑢 ∈ ( 𝐵 × 𝐵 ) , 𝑧 ∈ 𝐵 ↦ tpos ( ⟨ 𝑧 , ( 2nd ‘ 𝑢 ) ⟩ · ( 1st ‘ 𝑢 ) ) ) ⟩ ) ) )
19	13 16 18	mp2an	⊢ ( 𝑢 ∈ ( 𝐵 × 𝐵 ) , 𝑧 ∈ 𝐵 ↦ tpos ( ⟨ 𝑧 , ( 2nd ‘ 𝑢 ) ⟩ · ( 1st ‘ 𝑢 ) ) ) = ( comp ‘ ( ( 𝐶 sSet ⟨ ( Hom ‘ ndx ) , tpos ( Hom ‘ 𝐶 ) ⟩ ) sSet ⟨ ( comp ‘ ndx ) , ( 𝑢 ∈ ( 𝐵 × 𝐵 ) , 𝑧 ∈ 𝐵 ↦ tpos ( ⟨ 𝑧 , ( 2nd ‘ 𝑢 ) ⟩ · ( 1st ‘ 𝑢 ) ) ) ⟩ ) )
20	12 19	eqtr4di	⊢ ( 𝜑 → ( comp ‘ 𝑂 ) = ( 𝑢 ∈ ( 𝐵 × 𝐵 ) , 𝑧 ∈ 𝐵 ↦ tpos ( ⟨ 𝑧 , ( 2nd ‘ 𝑢 ) ⟩ · ( 1st ‘ 𝑢 ) ) ) )
21		simprr	⊢ ( ( 𝜑 ∧ ( 𝑢 = ⟨ 𝑋 , 𝑌 ⟩ ∧ 𝑧 = 𝑍 ) ) → 𝑧 = 𝑍 )
22		simprl	⊢ ( ( 𝜑 ∧ ( 𝑢 = ⟨ 𝑋 , 𝑌 ⟩ ∧ 𝑧 = 𝑍 ) ) → 𝑢 = ⟨ 𝑋 , 𝑌 ⟩ )
23	22	fveq2d	⊢ ( ( 𝜑 ∧ ( 𝑢 = ⟨ 𝑋 , 𝑌 ⟩ ∧ 𝑧 = 𝑍 ) ) → ( 2nd ‘ 𝑢 ) = ( 2nd ‘ ⟨ 𝑋 , 𝑌 ⟩ ) )
24	5	adantr	⊢ ( ( 𝜑 ∧ ( 𝑢 = ⟨ 𝑋 , 𝑌 ⟩ ∧ 𝑧 = 𝑍 ) ) → 𝑌 ∈ 𝐵 )
25		op2ndg	⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 2nd ‘ ⟨ 𝑋 , 𝑌 ⟩ ) = 𝑌 )
26	4 24 25	syl2an2r	⊢ ( ( 𝜑 ∧ ( 𝑢 = ⟨ 𝑋 , 𝑌 ⟩ ∧ 𝑧 = 𝑍 ) ) → ( 2nd ‘ ⟨ 𝑋 , 𝑌 ⟩ ) = 𝑌 )
27	23 26	eqtrd	⊢ ( ( 𝜑 ∧ ( 𝑢 = ⟨ 𝑋 , 𝑌 ⟩ ∧ 𝑧 = 𝑍 ) ) → ( 2nd ‘ 𝑢 ) = 𝑌 )
28	21 27	opeq12d	⊢ ( ( 𝜑 ∧ ( 𝑢 = ⟨ 𝑋 , 𝑌 ⟩ ∧ 𝑧 = 𝑍 ) ) → ⟨ 𝑧 , ( 2nd ‘ 𝑢 ) ⟩ = ⟨ 𝑍 , 𝑌 ⟩ )
29	22	fveq2d	⊢ ( ( 𝜑 ∧ ( 𝑢 = ⟨ 𝑋 , 𝑌 ⟩ ∧ 𝑧 = 𝑍 ) ) → ( 1st ‘ 𝑢 ) = ( 1st ‘ ⟨ 𝑋 , 𝑌 ⟩ ) )
30		op1stg	⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 1st ‘ ⟨ 𝑋 , 𝑌 ⟩ ) = 𝑋 )
31	4 24 30	syl2an2r	⊢ ( ( 𝜑 ∧ ( 𝑢 = ⟨ 𝑋 , 𝑌 ⟩ ∧ 𝑧 = 𝑍 ) ) → ( 1st ‘ ⟨ 𝑋 , 𝑌 ⟩ ) = 𝑋 )
32	29 31	eqtrd	⊢ ( ( 𝜑 ∧ ( 𝑢 = ⟨ 𝑋 , 𝑌 ⟩ ∧ 𝑧 = 𝑍 ) ) → ( 1st ‘ 𝑢 ) = 𝑋 )
33	28 32	oveq12d	⊢ ( ( 𝜑 ∧ ( 𝑢 = ⟨ 𝑋 , 𝑌 ⟩ ∧ 𝑧 = 𝑍 ) ) → ( ⟨ 𝑧 , ( 2nd ‘ 𝑢 ) ⟩ · ( 1st ‘ 𝑢 ) ) = ( ⟨ 𝑍 , 𝑌 ⟩ · 𝑋 ) )
34	33	tposeqd	⊢ ( ( 𝜑 ∧ ( 𝑢 = ⟨ 𝑋 , 𝑌 ⟩ ∧ 𝑧 = 𝑍 ) ) → tpos ( ⟨ 𝑧 , ( 2nd ‘ 𝑢 ) ⟩ · ( 1st ‘ 𝑢 ) ) = tpos ( ⟨ 𝑍 , 𝑌 ⟩ · 𝑋 ) )
35	4 5	opelxpd	⊢ ( 𝜑 → ⟨ 𝑋 , 𝑌 ⟩ ∈ ( 𝐵 × 𝐵 ) )
36		ovex	⊢ ( ⟨ 𝑍 , 𝑌 ⟩ · 𝑋 ) ∈ V
37	36	tposex	⊢ tpos ( ⟨ 𝑍 , 𝑌 ⟩ · 𝑋 ) ∈ V
38	37	a1i	⊢ ( 𝜑 → tpos ( ⟨ 𝑍 , 𝑌 ⟩ · 𝑋 ) ∈ V )
39	20 34 35 6 38	ovmpod	⊢ ( 𝜑 → ( ⟨ 𝑋 , 𝑌 ⟩ ( comp ‘ 𝑂 ) 𝑍 ) = tpos ( ⟨ 𝑍 , 𝑌 ⟩ · 𝑋 ) )