Metamath Proof Explorer


Theorem rngchomrnghmresALTV

Description: The value of the functionalized Hom-set operation in the category of non-unital rings (in a universe) as restriction of the non-unital ring homomorphisms. (Contributed by AV, 2-Mar-2020) (New usage is discouraged.)

Ref Expression
Hypotheses rngchomrnghmresALTV.c 𝐶 = ( RngCatALTV ‘ 𝑈 )
rngchomrnghmresALTV.b 𝐵 = ( Rng ∩ 𝑈 )
rngchomrnghmresALTV.u ( 𝜑𝑈𝑉 )
rngchomrnghmresALTV.f 𝐹 = ( Homf𝐶 )
Assertion rngchomrnghmresALTV ( 𝜑𝐹 = ( RngHomo ↾ ( 𝐵 × 𝐵 ) ) )

Proof

Step Hyp Ref Expression
1 rngchomrnghmresALTV.c 𝐶 = ( RngCatALTV ‘ 𝑈 )
2 rngchomrnghmresALTV.b 𝐵 = ( Rng ∩ 𝑈 )
3 rngchomrnghmresALTV.u ( 𝜑𝑈𝑉 )
4 rngchomrnghmresALTV.f 𝐹 = ( Homf𝐶 )
5 eqid ( Base ‘ 𝐶 ) = ( Base ‘ 𝐶 )
6 1 5 3 rngcbasALTV ( 𝜑 → ( Base ‘ 𝐶 ) = ( 𝑈 ∩ Rng ) )
7 inss2 ( 𝑈 ∩ Rng ) ⊆ Rng
8 6 7 eqsstrdi ( 𝜑 → ( Base ‘ 𝐶 ) ⊆ Rng )
9 resmpo ( ( ( Base ‘ 𝐶 ) ⊆ Rng ∧ ( Base ‘ 𝐶 ) ⊆ Rng ) → ( ( 𝑥 ∈ Rng , 𝑦 ∈ Rng ↦ ( 𝑥 RngHomo 𝑦 ) ) ↾ ( ( Base ‘ 𝐶 ) × ( Base ‘ 𝐶 ) ) ) = ( 𝑥 ∈ ( Base ‘ 𝐶 ) , 𝑦 ∈ ( Base ‘ 𝐶 ) ↦ ( 𝑥 RngHomo 𝑦 ) ) )
10 8 8 9 syl2anc ( 𝜑 → ( ( 𝑥 ∈ Rng , 𝑦 ∈ Rng ↦ ( 𝑥 RngHomo 𝑦 ) ) ↾ ( ( Base ‘ 𝐶 ) × ( Base ‘ 𝐶 ) ) ) = ( 𝑥 ∈ ( Base ‘ 𝐶 ) , 𝑦 ∈ ( Base ‘ 𝐶 ) ↦ ( 𝑥 RngHomo 𝑦 ) ) )
11 df-rnghomo RngHomo = ( 𝑟 ∈ Rng , 𝑠 ∈ Rng ↦ ( Base ‘ 𝑟 ) / 𝑣 ( Base ‘ 𝑠 ) / 𝑤 { 𝑓 ∈ ( 𝑤m 𝑣 ) ∣ ∀ 𝑥𝑣𝑦𝑣 ( ( 𝑓 ‘ ( 𝑥 ( +g𝑟 ) 𝑦 ) ) = ( ( 𝑓𝑥 ) ( +g𝑠 ) ( 𝑓𝑦 ) ) ∧ ( 𝑓 ‘ ( 𝑥 ( .r𝑟 ) 𝑦 ) ) = ( ( 𝑓𝑥 ) ( .r𝑠 ) ( 𝑓𝑦 ) ) ) } )
12 ovex ( 𝑤m 𝑣 ) ∈ V
13 12 rabex { 𝑓 ∈ ( 𝑤m 𝑣 ) ∣ ∀ 𝑥𝑣𝑦𝑣 ( ( 𝑓 ‘ ( 𝑥 ( +g𝑟 ) 𝑦 ) ) = ( ( 𝑓𝑥 ) ( +g𝑠 ) ( 𝑓𝑦 ) ) ∧ ( 𝑓 ‘ ( 𝑥 ( .r𝑟 ) 𝑦 ) ) = ( ( 𝑓𝑥 ) ( .r𝑠 ) ( 𝑓𝑦 ) ) ) } ∈ V
14 13 csbex ( Base ‘ 𝑠 ) / 𝑤 { 𝑓 ∈ ( 𝑤m 𝑣 ) ∣ ∀ 𝑥𝑣𝑦𝑣 ( ( 𝑓 ‘ ( 𝑥 ( +g𝑟 ) 𝑦 ) ) = ( ( 𝑓𝑥 ) ( +g𝑠 ) ( 𝑓𝑦 ) ) ∧ ( 𝑓 ‘ ( 𝑥 ( .r𝑟 ) 𝑦 ) ) = ( ( 𝑓𝑥 ) ( .r𝑠 ) ( 𝑓𝑦 ) ) ) } ∈ V
15 14 csbex ( Base ‘ 𝑟 ) / 𝑣 ( Base ‘ 𝑠 ) / 𝑤 { 𝑓 ∈ ( 𝑤m 𝑣 ) ∣ ∀ 𝑥𝑣𝑦𝑣 ( ( 𝑓 ‘ ( 𝑥 ( +g𝑟 ) 𝑦 ) ) = ( ( 𝑓𝑥 ) ( +g𝑠 ) ( 𝑓𝑦 ) ) ∧ ( 𝑓 ‘ ( 𝑥 ( .r𝑟 ) 𝑦 ) ) = ( ( 𝑓𝑥 ) ( .r𝑠 ) ( 𝑓𝑦 ) ) ) } ∈ V
16 11 15 fnmpoi RngHomo Fn ( Rng × Rng )
17 16 a1i ( 𝜑 → RngHomo Fn ( Rng × Rng ) )
18 fnov ( RngHomo Fn ( Rng × Rng ) ↔ RngHomo = ( 𝑥 ∈ Rng , 𝑦 ∈ Rng ↦ ( 𝑥 RngHomo 𝑦 ) ) )
19 17 18 sylib ( 𝜑 → RngHomo = ( 𝑥 ∈ Rng , 𝑦 ∈ Rng ↦ ( 𝑥 RngHomo 𝑦 ) ) )
20 incom ( 𝑈 ∩ Rng ) = ( Rng ∩ 𝑈 )
21 20 a1i ( 𝜑 → ( 𝑈 ∩ Rng ) = ( Rng ∩ 𝑈 ) )
22 2 a1i ( 𝜑𝐵 = ( Rng ∩ 𝑈 ) )
23 21 6 22 3eqtr4rd ( 𝜑𝐵 = ( Base ‘ 𝐶 ) )
24 23 sqxpeqd ( 𝜑 → ( 𝐵 × 𝐵 ) = ( ( Base ‘ 𝐶 ) × ( Base ‘ 𝐶 ) ) )
25 19 24 reseq12d ( 𝜑 → ( RngHomo ↾ ( 𝐵 × 𝐵 ) ) = ( ( 𝑥 ∈ Rng , 𝑦 ∈ Rng ↦ ( 𝑥 RngHomo 𝑦 ) ) ↾ ( ( Base ‘ 𝐶 ) × ( Base ‘ 𝐶 ) ) ) )
26 1 5 3 4 rngchomffvalALTV ( 𝜑𝐹 = ( 𝑥 ∈ ( Base ‘ 𝐶 ) , 𝑦 ∈ ( Base ‘ 𝐶 ) ↦ ( 𝑥 RngHomo 𝑦 ) ) )
27 10 25 26 3eqtr4rd ( 𝜑𝐹 = ( RngHomo ↾ ( 𝐵 × 𝐵 ) ) )