Theorem ringccofvalALTV

Description: Composition in the category of rings. (Contributed by AV, 14-Feb-2020) (New usage is discouraged.)

	Ref	Expression
Hypotheses	ringcbasALTV.c	⊢ 𝐶 = ( RingCatALTV ‘ 𝑈 )
	ringcbasALTV.b	⊢ 𝐵 = ( Base ‘ 𝐶 )
	ringcbasALTV.u	⊢ ( 𝜑 → 𝑈 ∈ 𝑉 )
	ringccoALTV.o	⊢ · = ( comp ‘ 𝐶 )
Assertion	ringccofvalALTV	⊢ ( 𝜑 → · = ( 𝑣 ∈ ( 𝐵 × 𝐵 ) , 𝑧 ∈ 𝐵 ↦ ( 𝑔 ∈ ( ( 2nd ‘ 𝑣 ) RingHom 𝑧 ) , 𝑓 ∈ ( ( 1st ‘ 𝑣 ) RingHom ( 2nd ‘ 𝑣 ) ) ↦ ( 𝑔 ∘ 𝑓 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		ringcbasALTV.c	⊢ 𝐶 = ( RingCatALTV ‘ 𝑈 )
2		ringcbasALTV.b	⊢ 𝐵 = ( Base ‘ 𝐶 )
3		ringcbasALTV.u	⊢ ( 𝜑 → 𝑈 ∈ 𝑉 )
4		ringccoALTV.o	⊢ · = ( comp ‘ 𝐶 )
5	1 2 3	ringcbasALTV	⊢ ( 𝜑 → 𝐵 = ( 𝑈 ∩ Ring ) )
6		eqid	⊢ ( Hom ‘ 𝐶 ) = ( Hom ‘ 𝐶 )
7	1 2 3 6	ringchomfvalALTV	⊢ ( 𝜑 → ( Hom ‘ 𝐶 ) = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 𝑥 RingHom 𝑦 ) ) )
8		eqidd	⊢ ( 𝜑 → ( 𝑣 ∈ ( 𝐵 × 𝐵 ) , 𝑧 ∈ 𝐵 ↦ ( 𝑔 ∈ ( ( 2nd ‘ 𝑣 ) RingHom 𝑧 ) , 𝑓 ∈ ( ( 1st ‘ 𝑣 ) RingHom ( 2nd ‘ 𝑣 ) ) ↦ ( 𝑔 ∘ 𝑓 ) ) ) = ( 𝑣 ∈ ( 𝐵 × 𝐵 ) , 𝑧 ∈ 𝐵 ↦ ( 𝑔 ∈ ( ( 2nd ‘ 𝑣 ) RingHom 𝑧 ) , 𝑓 ∈ ( ( 1st ‘ 𝑣 ) RingHom ( 2nd ‘ 𝑣 ) ) ↦ ( 𝑔 ∘ 𝑓 ) ) ) )
9	1 3 5 7 8	ringcvalALTV	⊢ ( 𝜑 → 𝐶 = { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( Hom ‘ ndx ) , ( Hom ‘ 𝐶 ) ⟩ , ⟨ ( comp ‘ ndx ) , ( 𝑣 ∈ ( 𝐵 × 𝐵 ) , 𝑧 ∈ 𝐵 ↦ ( 𝑔 ∈ ( ( 2nd ‘ 𝑣 ) RingHom 𝑧 ) , 𝑓 ∈ ( ( 1st ‘ 𝑣 ) RingHom ( 2nd ‘ 𝑣 ) ) ↦ ( 𝑔 ∘ 𝑓 ) ) ) ⟩ } )
10	9	fveq2d	⊢ ( 𝜑 → ( comp ‘ 𝐶 ) = ( comp ‘ { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( Hom ‘ ndx ) , ( Hom ‘ 𝐶 ) ⟩ , ⟨ ( comp ‘ ndx ) , ( 𝑣 ∈ ( 𝐵 × 𝐵 ) , 𝑧 ∈ 𝐵 ↦ ( 𝑔 ∈ ( ( 2nd ‘ 𝑣 ) RingHom 𝑧 ) , 𝑓 ∈ ( ( 1st ‘ 𝑣 ) RingHom ( 2nd ‘ 𝑣 ) ) ↦ ( 𝑔 ∘ 𝑓 ) ) ) ⟩ } ) )
11	2	fvexi	⊢ 𝐵 ∈ V
12		sqxpexg	⊢ ( 𝐵 ∈ V → ( 𝐵 × 𝐵 ) ∈ V )
13	11 12	ax-mp	⊢ ( 𝐵 × 𝐵 ) ∈ V
14	13 11	mpoex	⊢ ( 𝑣 ∈ ( 𝐵 × 𝐵 ) , 𝑧 ∈ 𝐵 ↦ ( 𝑔 ∈ ( ( 2nd ‘ 𝑣 ) RingHom 𝑧 ) , 𝑓 ∈ ( ( 1st ‘ 𝑣 ) RingHom ( 2nd ‘ 𝑣 ) ) ↦ ( 𝑔 ∘ 𝑓 ) ) ) ∈ V
15		catstr	⊢ { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( Hom ‘ ndx ) , ( Hom ‘ 𝐶 ) ⟩ , ⟨ ( comp ‘ ndx ) , ( 𝑣 ∈ ( 𝐵 × 𝐵 ) , 𝑧 ∈ 𝐵 ↦ ( 𝑔 ∈ ( ( 2nd ‘ 𝑣 ) RingHom 𝑧 ) , 𝑓 ∈ ( ( 1st ‘ 𝑣 ) RingHom ( 2nd ‘ 𝑣 ) ) ↦ ( 𝑔 ∘ 𝑓 ) ) ) ⟩ } Struct ⟨ 1 , ; 1 5 ⟩
16		ccoid	⊢ comp = Slot ( comp ‘ ndx )
17		snsstp3	⊢ { ⟨ ( comp ‘ ndx ) , ( 𝑣 ∈ ( 𝐵 × 𝐵 ) , 𝑧 ∈ 𝐵 ↦ ( 𝑔 ∈ ( ( 2nd ‘ 𝑣 ) RingHom 𝑧 ) , 𝑓 ∈ ( ( 1st ‘ 𝑣 ) RingHom ( 2nd ‘ 𝑣 ) ) ↦ ( 𝑔 ∘ 𝑓 ) ) ) ⟩ } ⊆ { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( Hom ‘ ndx ) , ( Hom ‘ 𝐶 ) ⟩ , ⟨ ( comp ‘ ndx ) , ( 𝑣 ∈ ( 𝐵 × 𝐵 ) , 𝑧 ∈ 𝐵 ↦ ( 𝑔 ∈ ( ( 2nd ‘ 𝑣 ) RingHom 𝑧 ) , 𝑓 ∈ ( ( 1st ‘ 𝑣 ) RingHom ( 2nd ‘ 𝑣 ) ) ↦ ( 𝑔 ∘ 𝑓 ) ) ) ⟩ }
18	15 16 17	strfv	⊢ ( ( 𝑣 ∈ ( 𝐵 × 𝐵 ) , 𝑧 ∈ 𝐵 ↦ ( 𝑔 ∈ ( ( 2nd ‘ 𝑣 ) RingHom 𝑧 ) , 𝑓 ∈ ( ( 1st ‘ 𝑣 ) RingHom ( 2nd ‘ 𝑣 ) ) ↦ ( 𝑔 ∘ 𝑓 ) ) ) ∈ V → ( 𝑣 ∈ ( 𝐵 × 𝐵 ) , 𝑧 ∈ 𝐵 ↦ ( 𝑔 ∈ ( ( 2nd ‘ 𝑣 ) RingHom 𝑧 ) , 𝑓 ∈ ( ( 1st ‘ 𝑣 ) RingHom ( 2nd ‘ 𝑣 ) ) ↦ ( 𝑔 ∘ 𝑓 ) ) ) = ( comp ‘ { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( Hom ‘ ndx ) , ( Hom ‘ 𝐶 ) ⟩ , ⟨ ( comp ‘ ndx ) , ( 𝑣 ∈ ( 𝐵 × 𝐵 ) , 𝑧 ∈ 𝐵 ↦ ( 𝑔 ∈ ( ( 2nd ‘ 𝑣 ) RingHom 𝑧 ) , 𝑓 ∈ ( ( 1st ‘ 𝑣 ) RingHom ( 2nd ‘ 𝑣 ) ) ↦ ( 𝑔 ∘ 𝑓 ) ) ) ⟩ } ) )
19	14 18	ax-mp	⊢ ( 𝑣 ∈ ( 𝐵 × 𝐵 ) , 𝑧 ∈ 𝐵 ↦ ( 𝑔 ∈ ( ( 2nd ‘ 𝑣 ) RingHom 𝑧 ) , 𝑓 ∈ ( ( 1st ‘ 𝑣 ) RingHom ( 2nd ‘ 𝑣 ) ) ↦ ( 𝑔 ∘ 𝑓 ) ) ) = ( comp ‘ { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( Hom ‘ ndx ) , ( Hom ‘ 𝐶 ) ⟩ , ⟨ ( comp ‘ ndx ) , ( 𝑣 ∈ ( 𝐵 × 𝐵 ) , 𝑧 ∈ 𝐵 ↦ ( 𝑔 ∈ ( ( 2nd ‘ 𝑣 ) RingHom 𝑧 ) , 𝑓 ∈ ( ( 1st ‘ 𝑣 ) RingHom ( 2nd ‘ 𝑣 ) ) ↦ ( 𝑔 ∘ 𝑓 ) ) ) ⟩ } )
20	10 4 19	3eqtr4g	⊢ ( 𝜑 → · = ( 𝑣 ∈ ( 𝐵 × 𝐵 ) , 𝑧 ∈ 𝐵 ↦ ( 𝑔 ∈ ( ( 2nd ‘ 𝑣 ) RingHom 𝑧 ) , 𝑓 ∈ ( ( 1st ‘ 𝑣 ) RingHom ( 2nd ‘ 𝑣 ) ) ↦ ( 𝑔 ∘ 𝑓 ) ) ) )