Theorem rnghmval

Description: The set of the non-unital ring homomorphisms between two non-unital rings. (Contributed by AV, 22-Feb-2020)

	Ref	Expression
Hypotheses	isrnghm.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
	isrnghm.t	⊢ · = ( .r ‘ 𝑅 )
	isrnghm.m	⊢ ∗ = ( .r ‘ 𝑆 )
	rnghmval.c	⊢ 𝐶 = ( Base ‘ 𝑆 )
	rnghmval.p	⊢ + = ( +g ‘ 𝑅 )
	rnghmval.a	⊢ ✚ = ( +g ‘ 𝑆 )
Assertion	rnghmval	⊢ ( ( 𝑅 ∈ Rng ∧ 𝑆 ∈ Rng ) → ( 𝑅 RngHomo 𝑆 ) = { 𝑓 ∈ ( 𝐶 ↑m 𝐵 ) ∣ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( ( 𝑓 ‘ ( 𝑥 + 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ✚ ( 𝑓 ‘ 𝑦 ) ) ∧ ( 𝑓 ‘ ( 𝑥 · 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ∗ ( 𝑓 ‘ 𝑦 ) ) ) } )

Proof

Step	Hyp	Ref	Expression
1		isrnghm.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
2		isrnghm.t	⊢ · = ( .r ‘ 𝑅 )
3		isrnghm.m	⊢ ∗ = ( .r ‘ 𝑆 )
4		rnghmval.c	⊢ 𝐶 = ( Base ‘ 𝑆 )
5		rnghmval.p	⊢ + = ( +g ‘ 𝑅 )
6		rnghmval.a	⊢ ✚ = ( +g ‘ 𝑆 )
7		df-rnghomo	⊢ RngHomo = ( 𝑟 ∈ Rng , 𝑠 ∈ Rng ↦ ⦋ ( Base ‘ 𝑟 ) / 𝑣 ⦌ ⦋ ( Base ‘ 𝑠 ) / 𝑤 ⦌ { 𝑓 ∈ ( 𝑤 ↑m 𝑣 ) ∣ ∀ 𝑥 ∈ 𝑣 ∀ 𝑦 ∈ 𝑣 ( ( 𝑓 ‘ ( 𝑥 ( +g ‘ 𝑟 ) 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ( +g ‘ 𝑠 ) ( 𝑓 ‘ 𝑦 ) ) ∧ ( 𝑓 ‘ ( 𝑥 ( .r ‘ 𝑟 ) 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ 𝑠 ) ( 𝑓 ‘ 𝑦 ) ) ) } )
8	7	a1i	⊢ ( ( 𝑅 ∈ Rng ∧ 𝑆 ∈ Rng ) → RngHomo = ( 𝑟 ∈ Rng , 𝑠 ∈ Rng ↦ ⦋ ( Base ‘ 𝑟 ) / 𝑣 ⦌ ⦋ ( Base ‘ 𝑠 ) / 𝑤 ⦌ { 𝑓 ∈ ( 𝑤 ↑m 𝑣 ) ∣ ∀ 𝑥 ∈ 𝑣 ∀ 𝑦 ∈ 𝑣 ( ( 𝑓 ‘ ( 𝑥 ( +g ‘ 𝑟 ) 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ( +g ‘ 𝑠 ) ( 𝑓 ‘ 𝑦 ) ) ∧ ( 𝑓 ‘ ( 𝑥 ( .r ‘ 𝑟 ) 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ 𝑠 ) ( 𝑓 ‘ 𝑦 ) ) ) } ) )
9		fveq2	⊢ ( 𝑟 = 𝑅 → ( Base ‘ 𝑟 ) = ( Base ‘ 𝑅 ) )
10	9 1	eqtr4di	⊢ ( 𝑟 = 𝑅 → ( Base ‘ 𝑟 ) = 𝐵 )
11	10	csbeq1d	⊢ ( 𝑟 = 𝑅 → ⦋ ( Base ‘ 𝑟 ) / 𝑣 ⦌ ⦋ ( Base ‘ 𝑠 ) / 𝑤 ⦌ { 𝑓 ∈ ( 𝑤 ↑m 𝑣 ) ∣ ∀ 𝑥 ∈ 𝑣 ∀ 𝑦 ∈ 𝑣 ( ( 𝑓 ‘ ( 𝑥 ( +g ‘ 𝑟 ) 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ( +g ‘ 𝑠 ) ( 𝑓 ‘ 𝑦 ) ) ∧ ( 𝑓 ‘ ( 𝑥 ( .r ‘ 𝑟 ) 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ 𝑠 ) ( 𝑓 ‘ 𝑦 ) ) ) } = ⦋ 𝐵 / 𝑣 ⦌ ⦋ ( Base ‘ 𝑠 ) / 𝑤 ⦌ { 𝑓 ∈ ( 𝑤 ↑m 𝑣 ) ∣ ∀ 𝑥 ∈ 𝑣 ∀ 𝑦 ∈ 𝑣 ( ( 𝑓 ‘ ( 𝑥 ( +g ‘ 𝑟 ) 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ( +g ‘ 𝑠 ) ( 𝑓 ‘ 𝑦 ) ) ∧ ( 𝑓 ‘ ( 𝑥 ( .r ‘ 𝑟 ) 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ 𝑠 ) ( 𝑓 ‘ 𝑦 ) ) ) } )
12		fveq2	⊢ ( 𝑠 = 𝑆 → ( Base ‘ 𝑠 ) = ( Base ‘ 𝑆 ) )
13	12 4	eqtr4di	⊢ ( 𝑠 = 𝑆 → ( Base ‘ 𝑠 ) = 𝐶 )
14	13	csbeq1d	⊢ ( 𝑠 = 𝑆 → ⦋ ( Base ‘ 𝑠 ) / 𝑤 ⦌ { 𝑓 ∈ ( 𝑤 ↑m 𝑣 ) ∣ ∀ 𝑥 ∈ 𝑣 ∀ 𝑦 ∈ 𝑣 ( ( 𝑓 ‘ ( 𝑥 ( +g ‘ 𝑟 ) 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ( +g ‘ 𝑠 ) ( 𝑓 ‘ 𝑦 ) ) ∧ ( 𝑓 ‘ ( 𝑥 ( .r ‘ 𝑟 ) 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ 𝑠 ) ( 𝑓 ‘ 𝑦 ) ) ) } = ⦋ 𝐶 / 𝑤 ⦌ { 𝑓 ∈ ( 𝑤 ↑m 𝑣 ) ∣ ∀ 𝑥 ∈ 𝑣 ∀ 𝑦 ∈ 𝑣 ( ( 𝑓 ‘ ( 𝑥 ( +g ‘ 𝑟 ) 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ( +g ‘ 𝑠 ) ( 𝑓 ‘ 𝑦 ) ) ∧ ( 𝑓 ‘ ( 𝑥 ( .r ‘ 𝑟 ) 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ 𝑠 ) ( 𝑓 ‘ 𝑦 ) ) ) } )
15	14	csbeq2dv	⊢ ( 𝑠 = 𝑆 → ⦋ 𝐵 / 𝑣 ⦌ ⦋ ( Base ‘ 𝑠 ) / 𝑤 ⦌ { 𝑓 ∈ ( 𝑤 ↑m 𝑣 ) ∣ ∀ 𝑥 ∈ 𝑣 ∀ 𝑦 ∈ 𝑣 ( ( 𝑓 ‘ ( 𝑥 ( +g ‘ 𝑟 ) 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ( +g ‘ 𝑠 ) ( 𝑓 ‘ 𝑦 ) ) ∧ ( 𝑓 ‘ ( 𝑥 ( .r ‘ 𝑟 ) 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ 𝑠 ) ( 𝑓 ‘ 𝑦 ) ) ) } = ⦋ 𝐵 / 𝑣 ⦌ ⦋ 𝐶 / 𝑤 ⦌ { 𝑓 ∈ ( 𝑤 ↑m 𝑣 ) ∣ ∀ 𝑥 ∈ 𝑣 ∀ 𝑦 ∈ 𝑣 ( ( 𝑓 ‘ ( 𝑥 ( +g ‘ 𝑟 ) 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ( +g ‘ 𝑠 ) ( 𝑓 ‘ 𝑦 ) ) ∧ ( 𝑓 ‘ ( 𝑥 ( .r ‘ 𝑟 ) 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ 𝑠 ) ( 𝑓 ‘ 𝑦 ) ) ) } )
16	11 15	sylan9eq	⊢ ( ( 𝑟 = 𝑅 ∧ 𝑠 = 𝑆 ) → ⦋ ( Base ‘ 𝑟 ) / 𝑣 ⦌ ⦋ ( Base ‘ 𝑠 ) / 𝑤 ⦌ { 𝑓 ∈ ( 𝑤 ↑m 𝑣 ) ∣ ∀ 𝑥 ∈ 𝑣 ∀ 𝑦 ∈ 𝑣 ( ( 𝑓 ‘ ( 𝑥 ( +g ‘ 𝑟 ) 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ( +g ‘ 𝑠 ) ( 𝑓 ‘ 𝑦 ) ) ∧ ( 𝑓 ‘ ( 𝑥 ( .r ‘ 𝑟 ) 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ 𝑠 ) ( 𝑓 ‘ 𝑦 ) ) ) } = ⦋ 𝐵 / 𝑣 ⦌ ⦋ 𝐶 / 𝑤 ⦌ { 𝑓 ∈ ( 𝑤 ↑m 𝑣 ) ∣ ∀ 𝑥 ∈ 𝑣 ∀ 𝑦 ∈ 𝑣 ( ( 𝑓 ‘ ( 𝑥 ( +g ‘ 𝑟 ) 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ( +g ‘ 𝑠 ) ( 𝑓 ‘ 𝑦 ) ) ∧ ( 𝑓 ‘ ( 𝑥 ( .r ‘ 𝑟 ) 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ 𝑠 ) ( 𝑓 ‘ 𝑦 ) ) ) } )
17	16	adantl	⊢ ( ( ( 𝑅 ∈ Rng ∧ 𝑆 ∈ Rng ) ∧ ( 𝑟 = 𝑅 ∧ 𝑠 = 𝑆 ) ) → ⦋ ( Base ‘ 𝑟 ) / 𝑣 ⦌ ⦋ ( Base ‘ 𝑠 ) / 𝑤 ⦌ { 𝑓 ∈ ( 𝑤 ↑m 𝑣 ) ∣ ∀ 𝑥 ∈ 𝑣 ∀ 𝑦 ∈ 𝑣 ( ( 𝑓 ‘ ( 𝑥 ( +g ‘ 𝑟 ) 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ( +g ‘ 𝑠 ) ( 𝑓 ‘ 𝑦 ) ) ∧ ( 𝑓 ‘ ( 𝑥 ( .r ‘ 𝑟 ) 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ 𝑠 ) ( 𝑓 ‘ 𝑦 ) ) ) } = ⦋ 𝐵 / 𝑣 ⦌ ⦋ 𝐶 / 𝑤 ⦌ { 𝑓 ∈ ( 𝑤 ↑m 𝑣 ) ∣ ∀ 𝑥 ∈ 𝑣 ∀ 𝑦 ∈ 𝑣 ( ( 𝑓 ‘ ( 𝑥 ( +g ‘ 𝑟 ) 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ( +g ‘ 𝑠 ) ( 𝑓 ‘ 𝑦 ) ) ∧ ( 𝑓 ‘ ( 𝑥 ( .r ‘ 𝑟 ) 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ 𝑠 ) ( 𝑓 ‘ 𝑦 ) ) ) } )
18	1	fvexi	⊢ 𝐵 ∈ V
19	4	fvexi	⊢ 𝐶 ∈ V
20		oveq12	⊢ ( ( 𝑤 = 𝐶 ∧ 𝑣 = 𝐵 ) → ( 𝑤 ↑m 𝑣 ) = ( 𝐶 ↑m 𝐵 ) )
21	20	ancoms	⊢ ( ( 𝑣 = 𝐵 ∧ 𝑤 = 𝐶 ) → ( 𝑤 ↑m 𝑣 ) = ( 𝐶 ↑m 𝐵 ) )
22		raleq	⊢ ( 𝑣 = 𝐵 → ( ∀ 𝑦 ∈ 𝑣 ( ( 𝑓 ‘ ( 𝑥 ( +g ‘ 𝑟 ) 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ( +g ‘ 𝑠 ) ( 𝑓 ‘ 𝑦 ) ) ∧ ( 𝑓 ‘ ( 𝑥 ( .r ‘ 𝑟 ) 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ 𝑠 ) ( 𝑓 ‘ 𝑦 ) ) ) ↔ ∀ 𝑦 ∈ 𝐵 ( ( 𝑓 ‘ ( 𝑥 ( +g ‘ 𝑟 ) 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ( +g ‘ 𝑠 ) ( 𝑓 ‘ 𝑦 ) ) ∧ ( 𝑓 ‘ ( 𝑥 ( .r ‘ 𝑟 ) 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ 𝑠 ) ( 𝑓 ‘ 𝑦 ) ) ) ) )
23	22	raleqbi1dv	⊢ ( 𝑣 = 𝐵 → ( ∀ 𝑥 ∈ 𝑣 ∀ 𝑦 ∈ 𝑣 ( ( 𝑓 ‘ ( 𝑥 ( +g ‘ 𝑟 ) 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ( +g ‘ 𝑠 ) ( 𝑓 ‘ 𝑦 ) ) ∧ ( 𝑓 ‘ ( 𝑥 ( .r ‘ 𝑟 ) 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ 𝑠 ) ( 𝑓 ‘ 𝑦 ) ) ) ↔ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( ( 𝑓 ‘ ( 𝑥 ( +g ‘ 𝑟 ) 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ( +g ‘ 𝑠 ) ( 𝑓 ‘ 𝑦 ) ) ∧ ( 𝑓 ‘ ( 𝑥 ( .r ‘ 𝑟 ) 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ 𝑠 ) ( 𝑓 ‘ 𝑦 ) ) ) ) )
24	23	adantr	⊢ ( ( 𝑣 = 𝐵 ∧ 𝑤 = 𝐶 ) → ( ∀ 𝑥 ∈ 𝑣 ∀ 𝑦 ∈ 𝑣 ( ( 𝑓 ‘ ( 𝑥 ( +g ‘ 𝑟 ) 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ( +g ‘ 𝑠 ) ( 𝑓 ‘ 𝑦 ) ) ∧ ( 𝑓 ‘ ( 𝑥 ( .r ‘ 𝑟 ) 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ 𝑠 ) ( 𝑓 ‘ 𝑦 ) ) ) ↔ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( ( 𝑓 ‘ ( 𝑥 ( +g ‘ 𝑟 ) 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ( +g ‘ 𝑠 ) ( 𝑓 ‘ 𝑦 ) ) ∧ ( 𝑓 ‘ ( 𝑥 ( .r ‘ 𝑟 ) 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ 𝑠 ) ( 𝑓 ‘ 𝑦 ) ) ) ) )
25	21 24	rabeqbidv	⊢ ( ( 𝑣 = 𝐵 ∧ 𝑤 = 𝐶 ) → { 𝑓 ∈ ( 𝑤 ↑m 𝑣 ) ∣ ∀ 𝑥 ∈ 𝑣 ∀ 𝑦 ∈ 𝑣 ( ( 𝑓 ‘ ( 𝑥 ( +g ‘ 𝑟 ) 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ( +g ‘ 𝑠 ) ( 𝑓 ‘ 𝑦 ) ) ∧ ( 𝑓 ‘ ( 𝑥 ( .r ‘ 𝑟 ) 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ 𝑠 ) ( 𝑓 ‘ 𝑦 ) ) ) } = { 𝑓 ∈ ( 𝐶 ↑m 𝐵 ) ∣ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( ( 𝑓 ‘ ( 𝑥 ( +g ‘ 𝑟 ) 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ( +g ‘ 𝑠 ) ( 𝑓 ‘ 𝑦 ) ) ∧ ( 𝑓 ‘ ( 𝑥 ( .r ‘ 𝑟 ) 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ 𝑠 ) ( 𝑓 ‘ 𝑦 ) ) ) } )
26	18 19 25	csbie2	⊢ ⦋ 𝐵 / 𝑣 ⦌ ⦋ 𝐶 / 𝑤 ⦌ { 𝑓 ∈ ( 𝑤 ↑m 𝑣 ) ∣ ∀ 𝑥 ∈ 𝑣 ∀ 𝑦 ∈ 𝑣 ( ( 𝑓 ‘ ( 𝑥 ( +g ‘ 𝑟 ) 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ( +g ‘ 𝑠 ) ( 𝑓 ‘ 𝑦 ) ) ∧ ( 𝑓 ‘ ( 𝑥 ( .r ‘ 𝑟 ) 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ 𝑠 ) ( 𝑓 ‘ 𝑦 ) ) ) } = { 𝑓 ∈ ( 𝐶 ↑m 𝐵 ) ∣ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( ( 𝑓 ‘ ( 𝑥 ( +g ‘ 𝑟 ) 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ( +g ‘ 𝑠 ) ( 𝑓 ‘ 𝑦 ) ) ∧ ( 𝑓 ‘ ( 𝑥 ( .r ‘ 𝑟 ) 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ 𝑠 ) ( 𝑓 ‘ 𝑦 ) ) ) }
27		fveq2	⊢ ( 𝑟 = 𝑅 → ( +g ‘ 𝑟 ) = ( +g ‘ 𝑅 ) )
28	27 5	eqtr4di	⊢ ( 𝑟 = 𝑅 → ( +g ‘ 𝑟 ) = + )
29	28	oveqdr	⊢ ( ( 𝑟 = 𝑅 ∧ 𝑠 = 𝑆 ) → ( 𝑥 ( +g ‘ 𝑟 ) 𝑦 ) = ( 𝑥 + 𝑦 ) )
30	29	fveq2d	⊢ ( ( 𝑟 = 𝑅 ∧ 𝑠 = 𝑆 ) → ( 𝑓 ‘ ( 𝑥 ( +g ‘ 𝑟 ) 𝑦 ) ) = ( 𝑓 ‘ ( 𝑥 + 𝑦 ) ) )
31		fveq2	⊢ ( 𝑠 = 𝑆 → ( +g ‘ 𝑠 ) = ( +g ‘ 𝑆 ) )
32	31 6	eqtr4di	⊢ ( 𝑠 = 𝑆 → ( +g ‘ 𝑠 ) = ✚ )
33	32	adantl	⊢ ( ( 𝑟 = 𝑅 ∧ 𝑠 = 𝑆 ) → ( +g ‘ 𝑠 ) = ✚ )
34	33	oveqd	⊢ ( ( 𝑟 = 𝑅 ∧ 𝑠 = 𝑆 ) → ( ( 𝑓 ‘ 𝑥 ) ( +g ‘ 𝑠 ) ( 𝑓 ‘ 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ✚ ( 𝑓 ‘ 𝑦 ) ) )
35	30 34	eqeq12d	⊢ ( ( 𝑟 = 𝑅 ∧ 𝑠 = 𝑆 ) → ( ( 𝑓 ‘ ( 𝑥 ( +g ‘ 𝑟 ) 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ( +g ‘ 𝑠 ) ( 𝑓 ‘ 𝑦 ) ) ↔ ( 𝑓 ‘ ( 𝑥 + 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ✚ ( 𝑓 ‘ 𝑦 ) ) ) )
36		fveq2	⊢ ( 𝑟 = 𝑅 → ( .r ‘ 𝑟 ) = ( .r ‘ 𝑅 ) )
37	36 2	eqtr4di	⊢ ( 𝑟 = 𝑅 → ( .r ‘ 𝑟 ) = · )
38	37	oveqdr	⊢ ( ( 𝑟 = 𝑅 ∧ 𝑠 = 𝑆 ) → ( 𝑥 ( .r ‘ 𝑟 ) 𝑦 ) = ( 𝑥 · 𝑦 ) )
39	38	fveq2d	⊢ ( ( 𝑟 = 𝑅 ∧ 𝑠 = 𝑆 ) → ( 𝑓 ‘ ( 𝑥 ( .r ‘ 𝑟 ) 𝑦 ) ) = ( 𝑓 ‘ ( 𝑥 · 𝑦 ) ) )
40		fveq2	⊢ ( 𝑠 = 𝑆 → ( .r ‘ 𝑠 ) = ( .r ‘ 𝑆 ) )
41	40 3	eqtr4di	⊢ ( 𝑠 = 𝑆 → ( .r ‘ 𝑠 ) = ∗ )
42	41	adantl	⊢ ( ( 𝑟 = 𝑅 ∧ 𝑠 = 𝑆 ) → ( .r ‘ 𝑠 ) = ∗ )
43	42	oveqd	⊢ ( ( 𝑟 = 𝑅 ∧ 𝑠 = 𝑆 ) → ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ 𝑠 ) ( 𝑓 ‘ 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ∗ ( 𝑓 ‘ 𝑦 ) ) )
44	39 43	eqeq12d	⊢ ( ( 𝑟 = 𝑅 ∧ 𝑠 = 𝑆 ) → ( ( 𝑓 ‘ ( 𝑥 ( .r ‘ 𝑟 ) 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ 𝑠 ) ( 𝑓 ‘ 𝑦 ) ) ↔ ( 𝑓 ‘ ( 𝑥 · 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ∗ ( 𝑓 ‘ 𝑦 ) ) ) )
45	35 44	anbi12d	⊢ ( ( 𝑟 = 𝑅 ∧ 𝑠 = 𝑆 ) → ( ( ( 𝑓 ‘ ( 𝑥 ( +g ‘ 𝑟 ) 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ( +g ‘ 𝑠 ) ( 𝑓 ‘ 𝑦 ) ) ∧ ( 𝑓 ‘ ( 𝑥 ( .r ‘ 𝑟 ) 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ 𝑠 ) ( 𝑓 ‘ 𝑦 ) ) ) ↔ ( ( 𝑓 ‘ ( 𝑥 + 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ✚ ( 𝑓 ‘ 𝑦 ) ) ∧ ( 𝑓 ‘ ( 𝑥 · 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ∗ ( 𝑓 ‘ 𝑦 ) ) ) ) )
46	45	2ralbidv	⊢ ( ( 𝑟 = 𝑅 ∧ 𝑠 = 𝑆 ) → ( ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( ( 𝑓 ‘ ( 𝑥 ( +g ‘ 𝑟 ) 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ( +g ‘ 𝑠 ) ( 𝑓 ‘ 𝑦 ) ) ∧ ( 𝑓 ‘ ( 𝑥 ( .r ‘ 𝑟 ) 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ 𝑠 ) ( 𝑓 ‘ 𝑦 ) ) ) ↔ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( ( 𝑓 ‘ ( 𝑥 + 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ✚ ( 𝑓 ‘ 𝑦 ) ) ∧ ( 𝑓 ‘ ( 𝑥 · 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ∗ ( 𝑓 ‘ 𝑦 ) ) ) ) )
47	46	rabbidv	⊢ ( ( 𝑟 = 𝑅 ∧ 𝑠 = 𝑆 ) → { 𝑓 ∈ ( 𝐶 ↑m 𝐵 ) ∣ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( ( 𝑓 ‘ ( 𝑥 ( +g ‘ 𝑟 ) 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ( +g ‘ 𝑠 ) ( 𝑓 ‘ 𝑦 ) ) ∧ ( 𝑓 ‘ ( 𝑥 ( .r ‘ 𝑟 ) 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ 𝑠 ) ( 𝑓 ‘ 𝑦 ) ) ) } = { 𝑓 ∈ ( 𝐶 ↑m 𝐵 ) ∣ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( ( 𝑓 ‘ ( 𝑥 + 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ✚ ( 𝑓 ‘ 𝑦 ) ) ∧ ( 𝑓 ‘ ( 𝑥 · 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ∗ ( 𝑓 ‘ 𝑦 ) ) ) } )
48	26 47	syl5eq	⊢ ( ( 𝑟 = 𝑅 ∧ 𝑠 = 𝑆 ) → ⦋ 𝐵 / 𝑣 ⦌ ⦋ 𝐶 / 𝑤 ⦌ { 𝑓 ∈ ( 𝑤 ↑m 𝑣 ) ∣ ∀ 𝑥 ∈ 𝑣 ∀ 𝑦 ∈ 𝑣 ( ( 𝑓 ‘ ( 𝑥 ( +g ‘ 𝑟 ) 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ( +g ‘ 𝑠 ) ( 𝑓 ‘ 𝑦 ) ) ∧ ( 𝑓 ‘ ( 𝑥 ( .r ‘ 𝑟 ) 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ 𝑠 ) ( 𝑓 ‘ 𝑦 ) ) ) } = { 𝑓 ∈ ( 𝐶 ↑m 𝐵 ) ∣ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( ( 𝑓 ‘ ( 𝑥 + 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ✚ ( 𝑓 ‘ 𝑦 ) ) ∧ ( 𝑓 ‘ ( 𝑥 · 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ∗ ( 𝑓 ‘ 𝑦 ) ) ) } )
49	48	adantl	⊢ ( ( ( 𝑅 ∈ Rng ∧ 𝑆 ∈ Rng ) ∧ ( 𝑟 = 𝑅 ∧ 𝑠 = 𝑆 ) ) → ⦋ 𝐵 / 𝑣 ⦌ ⦋ 𝐶 / 𝑤 ⦌ { 𝑓 ∈ ( 𝑤 ↑m 𝑣 ) ∣ ∀ 𝑥 ∈ 𝑣 ∀ 𝑦 ∈ 𝑣 ( ( 𝑓 ‘ ( 𝑥 ( +g ‘ 𝑟 ) 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ( +g ‘ 𝑠 ) ( 𝑓 ‘ 𝑦 ) ) ∧ ( 𝑓 ‘ ( 𝑥 ( .r ‘ 𝑟 ) 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ 𝑠 ) ( 𝑓 ‘ 𝑦 ) ) ) } = { 𝑓 ∈ ( 𝐶 ↑m 𝐵 ) ∣ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( ( 𝑓 ‘ ( 𝑥 + 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ✚ ( 𝑓 ‘ 𝑦 ) ) ∧ ( 𝑓 ‘ ( 𝑥 · 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ∗ ( 𝑓 ‘ 𝑦 ) ) ) } )
50	17 49	eqtrd	⊢ ( ( ( 𝑅 ∈ Rng ∧ 𝑆 ∈ Rng ) ∧ ( 𝑟 = 𝑅 ∧ 𝑠 = 𝑆 ) ) → ⦋ ( Base ‘ 𝑟 ) / 𝑣 ⦌ ⦋ ( Base ‘ 𝑠 ) / 𝑤 ⦌ { 𝑓 ∈ ( 𝑤 ↑m 𝑣 ) ∣ ∀ 𝑥 ∈ 𝑣 ∀ 𝑦 ∈ 𝑣 ( ( 𝑓 ‘ ( 𝑥 ( +g ‘ 𝑟 ) 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ( +g ‘ 𝑠 ) ( 𝑓 ‘ 𝑦 ) ) ∧ ( 𝑓 ‘ ( 𝑥 ( .r ‘ 𝑟 ) 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ 𝑠 ) ( 𝑓 ‘ 𝑦 ) ) ) } = { 𝑓 ∈ ( 𝐶 ↑m 𝐵 ) ∣ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( ( 𝑓 ‘ ( 𝑥 + 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ✚ ( 𝑓 ‘ 𝑦 ) ) ∧ ( 𝑓 ‘ ( 𝑥 · 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ∗ ( 𝑓 ‘ 𝑦 ) ) ) } )
51		simpl	⊢ ( ( 𝑅 ∈ Rng ∧ 𝑆 ∈ Rng ) → 𝑅 ∈ Rng )
52		simpr	⊢ ( ( 𝑅 ∈ Rng ∧ 𝑆 ∈ Rng ) → 𝑆 ∈ Rng )
53		ovex	⊢ ( 𝐶 ↑m 𝐵 ) ∈ V
54	53	rabex	⊢ { 𝑓 ∈ ( 𝐶 ↑m 𝐵 ) ∣ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( ( 𝑓 ‘ ( 𝑥 + 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ✚ ( 𝑓 ‘ 𝑦 ) ) ∧ ( 𝑓 ‘ ( 𝑥 · 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ∗ ( 𝑓 ‘ 𝑦 ) ) ) } ∈ V
55	54	a1i	⊢ ( ( 𝑅 ∈ Rng ∧ 𝑆 ∈ Rng ) → { 𝑓 ∈ ( 𝐶 ↑m 𝐵 ) ∣ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( ( 𝑓 ‘ ( 𝑥 + 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ✚ ( 𝑓 ‘ 𝑦 ) ) ∧ ( 𝑓 ‘ ( 𝑥 · 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ∗ ( 𝑓 ‘ 𝑦 ) ) ) } ∈ V )
56	8 50 51 52 55	ovmpod	⊢ ( ( 𝑅 ∈ Rng ∧ 𝑆 ∈ Rng ) → ( 𝑅 RngHomo 𝑆 ) = { 𝑓 ∈ ( 𝐶 ↑m 𝐵 ) ∣ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( ( 𝑓 ‘ ( 𝑥 + 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ✚ ( 𝑓 ‘ 𝑦 ) ) ∧ ( 𝑓 ‘ ( 𝑥 · 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ∗ ( 𝑓 ‘ 𝑦 ) ) ) } )