Theorem psraddcl

Description: Closure of the power series addition operation. (Contributed by Mario Carneiro, 28-Dec-2014) Generalize to magmas. (Revised by SN, 12-Apr-2025)

	Ref	Expression
Hypotheses	psraddcl.s	⊢ 𝑆 = ( 𝐼 mPwSer 𝑅 )
	psraddcl.b	⊢ 𝐵 = ( Base ‘ 𝑆 )
	psraddcl.p	⊢ + = ( +g ‘ 𝑆 )
	psraddcl.r	⊢ ( 𝜑 → 𝑅 ∈ Mgm )
	psraddcl.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
	psraddcl.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
Assertion	psraddcl	⊢ ( 𝜑 → ( 𝑋 + 𝑌 ) ∈ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		psraddcl.s	⊢ 𝑆 = ( 𝐼 mPwSer 𝑅 )
2		psraddcl.b	⊢ 𝐵 = ( Base ‘ 𝑆 )
3		psraddcl.p	⊢ + = ( +g ‘ 𝑆 )
4		psraddcl.r	⊢ ( 𝜑 → 𝑅 ∈ Mgm )
5		psraddcl.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
6		psraddcl.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
7		eqid	⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 )
8		eqid	⊢ ( +g ‘ 𝑅 ) = ( +g ‘ 𝑅 )
9	7 8	mgmcl	⊢ ( ( 𝑅 ∈ Mgm ∧ 𝑥 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ ( Base ‘ 𝑅 ) ) → ( 𝑥 ( +g ‘ 𝑅 ) 𝑦 ) ∈ ( Base ‘ 𝑅 ) )
10	9	3expb	⊢ ( ( 𝑅 ∈ Mgm ∧ ( 𝑥 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ ( Base ‘ 𝑅 ) ) ) → ( 𝑥 ( +g ‘ 𝑅 ) 𝑦 ) ∈ ( Base ‘ 𝑅 ) )
11	4 10	sylan	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ ( Base ‘ 𝑅 ) ) ) → ( 𝑥 ( +g ‘ 𝑅 ) 𝑦 ) ∈ ( Base ‘ 𝑅 ) )
12		eqid	⊢ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } = { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin }
13	1 7 12 2 5	psrelbas	⊢ ( 𝜑 → 𝑋 : { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ⟶ ( Base ‘ 𝑅 ) )
14	1 7 12 2 6	psrelbas	⊢ ( 𝜑 → 𝑌 : { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ⟶ ( Base ‘ 𝑅 ) )
15		ovex	⊢ ( ℕ0 ↑m 𝐼 ) ∈ V
16	15	rabex	⊢ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ∈ V
17	16	a1i	⊢ ( 𝜑 → { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ∈ V )
18		inidm	⊢ ( { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ∩ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ) = { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin }
19	11 13 14 17 17 18	off	⊢ ( 𝜑 → ( 𝑋 ∘f ( +g ‘ 𝑅 ) 𝑌 ) : { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ⟶ ( Base ‘ 𝑅 ) )
20		fvex	⊢ ( Base ‘ 𝑅 ) ∈ V
21	20 16	elmap	⊢ ( ( 𝑋 ∘f ( +g ‘ 𝑅 ) 𝑌 ) ∈ ( ( Base ‘ 𝑅 ) ↑m { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ) ↔ ( 𝑋 ∘f ( +g ‘ 𝑅 ) 𝑌 ) : { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ⟶ ( Base ‘ 𝑅 ) )
22	19 21	sylibr	⊢ ( 𝜑 → ( 𝑋 ∘f ( +g ‘ 𝑅 ) 𝑌 ) ∈ ( ( Base ‘ 𝑅 ) ↑m { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ) )
23	1 2 8 3 5 6	psradd	⊢ ( 𝜑 → ( 𝑋 + 𝑌 ) = ( 𝑋 ∘f ( +g ‘ 𝑅 ) 𝑌 ) )
24		reldmpsr	⊢ Rel dom mPwSer
25	24 1 2	elbasov	⊢ ( 𝑋 ∈ 𝐵 → ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) )
26	5 25	syl	⊢ ( 𝜑 → ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) )
27	26	simpld	⊢ ( 𝜑 → 𝐼 ∈ V )
28	1 7 12 2 27	psrbas	⊢ ( 𝜑 → 𝐵 = ( ( Base ‘ 𝑅 ) ↑m { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ) )
29	22 23 28	3eltr4d	⊢ ( 𝜑 → ( 𝑋 + 𝑌 ) ∈ 𝐵 )