Definition df-rnghm

Description: Define the set of non-unital ring homomorphisms from r to s . (Contributed by AV, 20-Feb-2020)

	Ref	Expression
Assertion	df-rnghm	⊢ RngHom = ( 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 ) ) ) } )

Detailed syntax breakdown

Step	Hyp	Ref	Expression
0		crnghm	⊢ RngHom
1		vr	⊢ 𝑟
2		crng	⊢ Rng
3		vs	⊢ 𝑠
4		cbs	⊢ Base
5	1	cv	⊢ 𝑟
6	5 4	cfv	⊢ ( Base ‘ 𝑟 )
7		vv	⊢ 𝑣
8	3	cv	⊢ 𝑠
9	8 4	cfv	⊢ ( Base ‘ 𝑠 )
10		vw	⊢ 𝑤
11		vf	⊢ 𝑓
12	10	cv	⊢ 𝑤
13		cmap	⊢ ↑m
14	7	cv	⊢ 𝑣
15	12 14 13	co	⊢ ( 𝑤 ↑m 𝑣 )
16		vx	⊢ 𝑥
17		vy	⊢ 𝑦
18	11	cv	⊢ 𝑓
19	16	cv	⊢ 𝑥
20		cplusg	⊢ +g
21	5 20	cfv	⊢ ( +g ‘ 𝑟 )
22	17	cv	⊢ 𝑦
23	19 22 21	co	⊢ ( 𝑥 ( +g ‘ 𝑟 ) 𝑦 )
24	23 18	cfv	⊢ ( 𝑓 ‘ ( 𝑥 ( +g ‘ 𝑟 ) 𝑦 ) )
25	19 18	cfv	⊢ ( 𝑓 ‘ 𝑥 )
26	8 20	cfv	⊢ ( +g ‘ 𝑠 )
27	22 18	cfv	⊢ ( 𝑓 ‘ 𝑦 )
28	25 27 26	co	⊢ ( ( 𝑓 ‘ 𝑥 ) ( +g ‘ 𝑠 ) ( 𝑓 ‘ 𝑦 ) )
29	24 28	wceq	⊢ ( 𝑓 ‘ ( 𝑥 ( +g ‘ 𝑟 ) 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ( +g ‘ 𝑠 ) ( 𝑓 ‘ 𝑦 ) )
30		cmulr	⊢ .r
31	5 30	cfv	⊢ ( .r ‘ 𝑟 )
32	19 22 31	co	⊢ ( 𝑥 ( .r ‘ 𝑟 ) 𝑦 )
33	32 18	cfv	⊢ ( 𝑓 ‘ ( 𝑥 ( .r ‘ 𝑟 ) 𝑦 ) )
34	8 30	cfv	⊢ ( .r ‘ 𝑠 )
35	25 27 34	co	⊢ ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ 𝑠 ) ( 𝑓 ‘ 𝑦 ) )
36	33 35	wceq	⊢ ( 𝑓 ‘ ( 𝑥 ( .r ‘ 𝑟 ) 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ 𝑠 ) ( 𝑓 ‘ 𝑦 ) )
37	29 36	wa	⊢ ( ( 𝑓 ‘ ( 𝑥 ( +g ‘ 𝑟 ) 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ( +g ‘ 𝑠 ) ( 𝑓 ‘ 𝑦 ) ) ∧ ( 𝑓 ‘ ( 𝑥 ( .r ‘ 𝑟 ) 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ 𝑠 ) ( 𝑓 ‘ 𝑦 ) ) )
38	37 17 14	wral	⊢ ∀ 𝑦 ∈ 𝑣 ( ( 𝑓 ‘ ( 𝑥 ( +g ‘ 𝑟 ) 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ( +g ‘ 𝑠 ) ( 𝑓 ‘ 𝑦 ) ) ∧ ( 𝑓 ‘ ( 𝑥 ( .r ‘ 𝑟 ) 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ 𝑠 ) ( 𝑓 ‘ 𝑦 ) ) )
39	38 16 14	wral	⊢ ∀ 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 ) ) )
40	39 11 15	crab	⊢ { 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 ) ) ) }
41	10 9 40	csb	⊢ ⦋ ( 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 ) ) ) }
42	7 6 41	csb	⊢ ⦋ ( 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 ) ) ) }
43	1 3 2 2 42	cmpo	⊢ ( 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 ) ) ) } )
44	0 43	wceq	⊢ RngHom = ( 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 ) ) ) } )