Metamath Proof Explorer


Theorem rngccofvalALTV

Description: Composition in the category of non-unital rings. (New usage is discouraged.) (Contributed by AV, 27-Feb-2020)

Ref Expression
Hypotheses rngcbasALTV.c 𝐶 = ( RngCatALTV ‘ 𝑈 )
rngcbasALTV.b 𝐵 = ( Base ‘ 𝐶 )
rngcbasALTV.u ( 𝜑𝑈𝑉 )
rngccofvalALTV.o · = ( comp ‘ 𝐶 )
Assertion rngccofvalALTV ( 𝜑· = ( 𝑣 ∈ ( 𝐵 × 𝐵 ) , 𝑧𝐵 ↦ ( 𝑔 ∈ ( ( 2nd𝑣 ) RngHomo 𝑧 ) , 𝑓 ∈ ( ( 1st𝑣 ) RngHomo ( 2nd𝑣 ) ) ↦ ( 𝑔𝑓 ) ) ) )

Proof

Step Hyp Ref Expression
1 rngcbasALTV.c 𝐶 = ( RngCatALTV ‘ 𝑈 )
2 rngcbasALTV.b 𝐵 = ( Base ‘ 𝐶 )
3 rngcbasALTV.u ( 𝜑𝑈𝑉 )
4 rngccofvalALTV.o · = ( comp ‘ 𝐶 )
5 1 2 3 rngcbasALTV ( 𝜑𝐵 = ( 𝑈 ∩ Rng ) )
6 eqid ( Hom ‘ 𝐶 ) = ( Hom ‘ 𝐶 )
7 1 2 3 6 rngchomfvalALTV ( 𝜑 → ( Hom ‘ 𝐶 ) = ( 𝑥𝐵 , 𝑦𝐵 ↦ ( 𝑥 RngHomo 𝑦 ) ) )
8 eqidd ( 𝜑 → ( 𝑣 ∈ ( 𝐵 × 𝐵 ) , 𝑧𝐵 ↦ ( 𝑔 ∈ ( ( 2nd𝑣 ) RngHomo 𝑧 ) , 𝑓 ∈ ( ( 1st𝑣 ) RngHomo ( 2nd𝑣 ) ) ↦ ( 𝑔𝑓 ) ) ) = ( 𝑣 ∈ ( 𝐵 × 𝐵 ) , 𝑧𝐵 ↦ ( 𝑔 ∈ ( ( 2nd𝑣 ) RngHomo 𝑧 ) , 𝑓 ∈ ( ( 1st𝑣 ) RngHomo ( 2nd𝑣 ) ) ↦ ( 𝑔𝑓 ) ) ) )
9 1 3 5 7 8 rngcvalALTV ( 𝜑𝐶 = { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( Hom ‘ ndx ) , ( Hom ‘ 𝐶 ) ⟩ , ⟨ ( comp ‘ ndx ) , ( 𝑣 ∈ ( 𝐵 × 𝐵 ) , 𝑧𝐵 ↦ ( 𝑔 ∈ ( ( 2nd𝑣 ) RngHomo 𝑧 ) , 𝑓 ∈ ( ( 1st𝑣 ) RngHomo ( 2nd𝑣 ) ) ↦ ( 𝑔𝑓 ) ) ) ⟩ } )
10 9 fveq2d ( 𝜑 → ( comp ‘ 𝐶 ) = ( comp ‘ { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( Hom ‘ ndx ) , ( Hom ‘ 𝐶 ) ⟩ , ⟨ ( comp ‘ ndx ) , ( 𝑣 ∈ ( 𝐵 × 𝐵 ) , 𝑧𝐵 ↦ ( 𝑔 ∈ ( ( 2nd𝑣 ) RngHomo 𝑧 ) , 𝑓 ∈ ( ( 1st𝑣 ) RngHomo ( 2nd𝑣 ) ) ↦ ( 𝑔𝑓 ) ) ) ⟩ } ) )
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 ) , 𝑧𝐵 ↦ ( 𝑔 ∈ ( ( 2nd𝑣 ) RngHomo 𝑧 ) , 𝑓 ∈ ( ( 1st𝑣 ) RngHomo ( 2nd𝑣 ) ) ↦ ( 𝑔𝑓 ) ) ) ∈ V
15 catstr { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( Hom ‘ ndx ) , ( Hom ‘ 𝐶 ) ⟩ , ⟨ ( comp ‘ ndx ) , ( 𝑣 ∈ ( 𝐵 × 𝐵 ) , 𝑧𝐵 ↦ ( 𝑔 ∈ ( ( 2nd𝑣 ) RngHomo 𝑧 ) , 𝑓 ∈ ( ( 1st𝑣 ) RngHomo ( 2nd𝑣 ) ) ↦ ( 𝑔𝑓 ) ) ) ⟩ } Struct ⟨ 1 , 1 5 ⟩
16 ccoid comp = Slot ( comp ‘ ndx )
17 snsstp3 { ⟨ ( comp ‘ ndx ) , ( 𝑣 ∈ ( 𝐵 × 𝐵 ) , 𝑧𝐵 ↦ ( 𝑔 ∈ ( ( 2nd𝑣 ) RngHomo 𝑧 ) , 𝑓 ∈ ( ( 1st𝑣 ) RngHomo ( 2nd𝑣 ) ) ↦ ( 𝑔𝑓 ) ) ) ⟩ } ⊆ { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( Hom ‘ ndx ) , ( Hom ‘ 𝐶 ) ⟩ , ⟨ ( comp ‘ ndx ) , ( 𝑣 ∈ ( 𝐵 × 𝐵 ) , 𝑧𝐵 ↦ ( 𝑔 ∈ ( ( 2nd𝑣 ) RngHomo 𝑧 ) , 𝑓 ∈ ( ( 1st𝑣 ) RngHomo ( 2nd𝑣 ) ) ↦ ( 𝑔𝑓 ) ) ) ⟩ }
18 15 16 17 strfv ( ( 𝑣 ∈ ( 𝐵 × 𝐵 ) , 𝑧𝐵 ↦ ( 𝑔 ∈ ( ( 2nd𝑣 ) RngHomo 𝑧 ) , 𝑓 ∈ ( ( 1st𝑣 ) RngHomo ( 2nd𝑣 ) ) ↦ ( 𝑔𝑓 ) ) ) ∈ V → ( 𝑣 ∈ ( 𝐵 × 𝐵 ) , 𝑧𝐵 ↦ ( 𝑔 ∈ ( ( 2nd𝑣 ) RngHomo 𝑧 ) , 𝑓 ∈ ( ( 1st𝑣 ) RngHomo ( 2nd𝑣 ) ) ↦ ( 𝑔𝑓 ) ) ) = ( comp ‘ { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( Hom ‘ ndx ) , ( Hom ‘ 𝐶 ) ⟩ , ⟨ ( comp ‘ ndx ) , ( 𝑣 ∈ ( 𝐵 × 𝐵 ) , 𝑧𝐵 ↦ ( 𝑔 ∈ ( ( 2nd𝑣 ) RngHomo 𝑧 ) , 𝑓 ∈ ( ( 1st𝑣 ) RngHomo ( 2nd𝑣 ) ) ↦ ( 𝑔𝑓 ) ) ) ⟩ } ) )
19 14 18 ax-mp ( 𝑣 ∈ ( 𝐵 × 𝐵 ) , 𝑧𝐵 ↦ ( 𝑔 ∈ ( ( 2nd𝑣 ) RngHomo 𝑧 ) , 𝑓 ∈ ( ( 1st𝑣 ) RngHomo ( 2nd𝑣 ) ) ↦ ( 𝑔𝑓 ) ) ) = ( comp ‘ { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( Hom ‘ ndx ) , ( Hom ‘ 𝐶 ) ⟩ , ⟨ ( comp ‘ ndx ) , ( 𝑣 ∈ ( 𝐵 × 𝐵 ) , 𝑧𝐵 ↦ ( 𝑔 ∈ ( ( 2nd𝑣 ) RngHomo 𝑧 ) , 𝑓 ∈ ( ( 1st𝑣 ) RngHomo ( 2nd𝑣 ) ) ↦ ( 𝑔𝑓 ) ) ) ⟩ } )
20 10 4 19 3eqtr4g ( 𝜑· = ( 𝑣 ∈ ( 𝐵 × 𝐵 ) , 𝑧𝐵 ↦ ( 𝑔 ∈ ( ( 2nd𝑣 ) RngHomo 𝑧 ) , 𝑓 ∈ ( ( 1st𝑣 ) RngHomo ( 2nd𝑣 ) ) ↦ ( 𝑔𝑓 ) ) ) )