Theorem resspsrmul 22292

Description: A restricted power series algebra has the same multiplication operation. (Contributed by Mario Carneiro, 3-Jul-2015.)

Hypotheses

Ref	Expression
resspsr.s	⊢ 𝑆 = (𝐼 mPwSer 𝑅)
resspsr.h	⊢ 𝐻 = (𝑅 ↾s 𝑇)
resspsr.u	⊢ 𝑈 = (𝐼 mPwSer 𝐻)
resspsr.b	⊢ 𝐵 = (Base‘𝑈)
resspsr.p	⊢ 𝑃 = (𝑆 ↾s 𝐵)
resspsr.2	⊢ (𝜑 → 𝑇 ∈ (SubRing‘𝑅))

Assertion

Ref	Expression
resspsrmul	⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋(.r‘𝑈)𝑌) = (𝑋(.r‘𝑃)𝑌))

Proof of Theorem resspsrmul

Dummy variables 𝑥 𝑘 𝑓 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2760	. . . . . . . 8 ⊢ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}
2	1	psrbaglefi 22243	. . . . . . 7 ⊢ (𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} → {𝑦 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∣ 𝑦 ∘r ≤ 𝑘} ∈ Fin)
3	2	adantl 487	. . . . . 6 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → {𝑦 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∣ 𝑦 ∘r ≤ 𝑘} ∈ Fin)
4		resspsr.2	. . . . . . . . 9 ⊢ (𝜑 → 𝑇 ∈ (SubRing‘𝑅))
5		subrgsubg 20838	. . . . . . . . 9 ⊢ (𝑇 ∈ (SubRing‘𝑅) → 𝑇 ∈ (SubGrp‘𝑅))
6	4, 5	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝑅))
7		subgsubm 19368	. . . . . . . 8 ⊢ (𝑇 ∈ (SubGrp‘𝑅) → 𝑇 ∈ (SubMnd‘𝑅))
8	6, 7	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝑇 ∈ (SubMnd‘𝑅))
9	8	ad2antrr 739	. . . . . 6 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → 𝑇 ∈ (SubMnd‘𝑅))
10	4	ad3antrrr 743	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑥 ∈ {𝑦 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∣ 𝑦 ∘r ≤ 𝑘}) → 𝑇 ∈ (SubRing‘𝑅))
11		resspsr.u	. . . . . . . . . . . 12 ⊢ 𝑈 = (𝐼 mPwSer 𝐻)
12		eqid 2760	. . . . . . . . . . . 12 ⊢ (Base‘𝐻) = (Base‘𝐻)
13		resspsr.b	. . . . . . . . . . . 12 ⊢ 𝐵 = (Base‘𝑈)
14		simprl 783	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 𝑋 ∈ 𝐵)
15	11, 12, 1, 13, 14	psrelbas 22252	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 𝑋:{𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝐻))
16	15	adantr 486	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → 𝑋:{𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝐻))
17		elrabi 3640	. . . . . . . . . 10 ⊢ (𝑥 ∈ {𝑦 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∣ 𝑦 ∘r ≤ 𝑘} → 𝑥 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin})
18		ffvelcdm 7086	. . . . . . . . . 10 ⊢ ((𝑋:{𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝐻) ∧ 𝑥 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (𝑋‘𝑥) ∈ (Base‘𝐻))
19	16, 17, 18	syl2an 608	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑥 ∈ {𝑦 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∣ 𝑦 ∘r ≤ 𝑘}) → (𝑋‘𝑥) ∈ (Base‘𝐻))
20		resspsr.h	. . . . . . . . . . 11 ⊢ 𝐻 = (𝑅 ↾s 𝑇)
21	20	subrgbas 20842	. . . . . . . . . 10 ⊢ (𝑇 ∈ (SubRing‘𝑅) → 𝑇 = (Base‘𝐻))
22	10, 21	syl 18	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑥 ∈ {𝑦 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∣ 𝑦 ∘r ≤ 𝑘}) → 𝑇 = (Base‘𝐻))
23	19, 22	eleqtrrd 2863	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑥 ∈ {𝑦 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∣ 𝑦 ∘r ≤ 𝑘}) → (𝑋‘𝑥) ∈ 𝑇)
24		simprr 785	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 𝑌 ∈ 𝐵)
25	11, 12, 1, 13, 24	psrelbas 22252	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 𝑌:{𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝐻))
26	25	ad2antrr 739	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑥 ∈ {𝑦 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∣ 𝑦 ∘r ≤ 𝑘}) → 𝑌:{𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝐻))
27		ssrab2 4004	. . . . . . . . . . 11 ⊢ {𝑦 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∣ 𝑦 ∘r ≤ 𝑘} ⊆ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}
28		simplr 781	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑥 ∈ {𝑦 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∣ 𝑦 ∘r ≤ 𝑘}) → 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin})
29		simpr 490	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑥 ∈ {𝑦 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∣ 𝑦 ∘r ≤ 𝑘}) → 𝑥 ∈ {𝑦 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∣ 𝑦 ∘r ≤ 𝑘})
30		eqid 2760	. . . . . . . . . . . . 13 ⊢ {𝑦 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∣ 𝑦 ∘r ≤ 𝑘} = {𝑦 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∣ 𝑦 ∘r ≤ 𝑘}
31	1, 30	psrbagconcl 22244	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑥 ∈ {𝑦 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∣ 𝑦 ∘r ≤ 𝑘}) → (𝑘 ∘f − 𝑥) ∈ {𝑦 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∣ 𝑦 ∘r ≤ 𝑘})
32	28, 29, 31	syl2anc 596	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑥 ∈ {𝑦 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∣ 𝑦 ∘r ≤ 𝑘}) → (𝑘 ∘f − 𝑥) ∈ {𝑦 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∣ 𝑦 ∘r ≤ 𝑘})
33	27, 32	sselid 3907	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑥 ∈ {𝑦 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∣ 𝑦 ∘r ≤ 𝑘}) → (𝑘 ∘f − 𝑥) ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin})
34	26, 33	ffvelcdmd 7090	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑥 ∈ {𝑦 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∣ 𝑦 ∘r ≤ 𝑘}) → (𝑌‘(𝑘 ∘f − 𝑥)) ∈ (Base‘𝐻))
35	34, 22	eleqtrrd 2863	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑥 ∈ {𝑦 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∣ 𝑦 ∘r ≤ 𝑘}) → (𝑌‘(𝑘 ∘f − 𝑥)) ∈ 𝑇)
36		eqid 2760	. . . . . . . . 9 ⊢ (.r‘𝑅) = (.r‘𝑅)
37	36	subrgmcl 20845	. . . . . . . 8 ⊢ ((𝑇 ∈ (SubRing‘𝑅) ∧ (𝑋‘𝑥) ∈ 𝑇 ∧ (𝑌‘(𝑘 ∘f − 𝑥)) ∈ 𝑇) → ((𝑋‘𝑥)(.r‘𝑅)(𝑌‘(𝑘 ∘f − 𝑥))) ∈ 𝑇)
38	10, 23, 35, 37	syl3anc 1398	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑥 ∈ {𝑦 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∣ 𝑦 ∘r ≤ 𝑘}) → ((𝑋‘𝑥)(.r‘𝑅)(𝑌‘(𝑘 ∘f − 𝑥))) ∈ 𝑇)
39	38	fmpttd 7120	. . . . . 6 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (𝑥 ∈ {𝑦 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∣ 𝑦 ∘r ≤ 𝑘} ↦ ((𝑋‘𝑥)(.r‘𝑅)(𝑌‘(𝑘 ∘f − 𝑥)))):{𝑦 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∣ 𝑦 ∘r ≤ 𝑘}⟶𝑇)
40	3, 9, 39, 20	gsumsubm 19040	. . . . 5 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (𝑅 Σg (𝑥 ∈ {𝑦 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∣ 𝑦 ∘r ≤ 𝑘} ↦ ((𝑋‘𝑥)(.r‘𝑅)(𝑌‘(𝑘 ∘f − 𝑥))))) = (𝐻 Σg (𝑥 ∈ {𝑦 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∣ 𝑦 ∘r ≤ 𝑘} ↦ ((𝑋‘𝑥)(.r‘𝑅)(𝑌‘(𝑘 ∘f − 𝑥))))))
41	20, 36	ressmulr 17487	. . . . . . . . . 10 ⊢ (𝑇 ∈ (SubRing‘𝑅) → (.r‘𝑅) = (.r‘𝐻))
42	4, 41	syl 18	. . . . . . . . 9 ⊢ (𝜑 → (.r‘𝑅) = (.r‘𝐻))
43	42	ad3antrrr 743	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑥 ∈ {𝑦 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∣ 𝑦 ∘r ≤ 𝑘}) → (.r‘𝑅) = (.r‘𝐻))
44	43	oveqd 7441	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑥 ∈ {𝑦 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∣ 𝑦 ∘r ≤ 𝑘}) → ((𝑋‘𝑥)(.r‘𝑅)(𝑌‘(𝑘 ∘f − 𝑥))) = ((𝑋‘𝑥)(.r‘𝐻)(𝑌‘(𝑘 ∘f − 𝑥))))
45	44	mpteq2dva 5197	. . . . . 6 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (𝑥 ∈ {𝑦 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∣ 𝑦 ∘r ≤ 𝑘} ↦ ((𝑋‘𝑥)(.r‘𝑅)(𝑌‘(𝑘 ∘f − 𝑥)))) = (𝑥 ∈ {𝑦 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∣ 𝑦 ∘r ≤ 𝑘} ↦ ((𝑋‘𝑥)(.r‘𝐻)(𝑌‘(𝑘 ∘f − 𝑥)))))
46	45	oveq2d 7440	. . . . 5 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (𝐻 Σg (𝑥 ∈ {𝑦 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∣ 𝑦 ∘r ≤ 𝑘} ↦ ((𝑋‘𝑥)(.r‘𝑅)(𝑌‘(𝑘 ∘f − 𝑥))))) = (𝐻 Σg (𝑥 ∈ {𝑦 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∣ 𝑦 ∘r ≤ 𝑘} ↦ ((𝑋‘𝑥)(.r‘𝐻)(𝑌‘(𝑘 ∘f − 𝑥))))))
47	40, 46	eqtrd 2795	. . . 4 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (𝑅 Σg (𝑥 ∈ {𝑦 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∣ 𝑦 ∘r ≤ 𝑘} ↦ ((𝑋‘𝑥)(.r‘𝑅)(𝑌‘(𝑘 ∘f − 𝑥))))) = (𝐻 Σg (𝑥 ∈ {𝑦 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∣ 𝑦 ∘r ≤ 𝑘} ↦ ((𝑋‘𝑥)(.r‘𝐻)(𝑌‘(𝑘 ∘f − 𝑥))))))
48	47	mpteq2dva 5197	. . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ (𝑅 Σg (𝑥 ∈ {𝑦 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∣ 𝑦 ∘r ≤ 𝑘} ↦ ((𝑋‘𝑥)(.r‘𝑅)(𝑌‘(𝑘 ∘f − 𝑥)))))) = (𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ (𝐻 Σg (𝑥 ∈ {𝑦 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∣ 𝑦 ∘r ≤ 𝑘} ↦ ((𝑋‘𝑥)(.r‘𝐻)(𝑌‘(𝑘 ∘f − 𝑥)))))))
49		resspsr.s	. . . 4 ⊢ 𝑆 = (𝐼 mPwSer 𝑅)
50		eqid 2760	. . . 4 ⊢ (Base‘𝑆) = (Base‘𝑆)
51		eqid 2760	. . . 4 ⊢ (.r‘𝑆) = (.r‘𝑆)
52		fvex 6903	. . . . . . . 8 ⊢ (Base‘𝑅) ∈ V
53	4, 21	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝑇 = (Base‘𝐻))
54		eqid 2760	. . . . . . . . . . 11 ⊢ (Base‘𝑅) = (Base‘𝑅)
55	54	subrgss 20833	. . . . . . . . . 10 ⊢ (𝑇 ∈ (SubRing‘𝑅) → 𝑇 ⊆ (Base‘𝑅))
56	4, 55	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝑇 ⊆ (Base‘𝑅))
57	53, 56	eqsstrrd 3944	. . . . . . . 8 ⊢ (𝜑 → (Base‘𝐻) ⊆ (Base‘𝑅))
58		mapss 8925	. . . . . . . 8 ⊢ (((Base‘𝑅) ∈ V ∧ (Base‘𝐻) ⊆ (Base‘𝑅)) → ((Base‘𝐻) ↑m {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ⊆ ((Base‘𝑅) ↑m {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}))
59	52, 57, 58	sylancr 599	. . . . . . 7 ⊢ (𝜑 → ((Base‘𝐻) ↑m {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ⊆ ((Base‘𝑅) ↑m {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}))
60	59	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ((Base‘𝐻) ↑m {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ⊆ ((Base‘𝑅) ↑m {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}))
61		reldmpsr 22231	. . . . . . . . . 10 ⊢ Rel dom mPwSer
62	61, 11, 13	elbasov 17403	. . . . . . . . 9 ⊢ (𝑋 ∈ 𝐵 → (𝐼 ∈ V ∧ 𝐻 ∈ V))
63	62	ad2antrl 741	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝐼 ∈ V ∧ 𝐻 ∈ V))
64	63	simpld 500	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 𝐼 ∈ V)
65	11, 12, 1, 13, 64	psrbas 22251	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 𝐵 = ((Base‘𝐻) ↑m {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}))
66	49, 54, 1, 50, 64	psrbas 22251	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (Base‘𝑆) = ((Base‘𝑅) ↑m {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}))
67	60, 65, 66	3sstr4d 3964	. . . . 5 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 𝐵 ⊆ (Base‘𝑆))
68	67, 14	sseldd 3910	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 𝑋 ∈ (Base‘𝑆))
69	67, 24	sseldd 3910	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 𝑌 ∈ (Base‘𝑆))
70	49, 50, 36, 51, 1, 68, 69	psrmulfval 22260	. . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋(.r‘𝑆)𝑌) = (𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ (𝑅 Σg (𝑥 ∈ {𝑦 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∣ 𝑦 ∘r ≤ 𝑘} ↦ ((𝑋‘𝑥)(.r‘𝑅)(𝑌‘(𝑘 ∘f − 𝑥)))))))
71		eqid 2760	. . . 4 ⊢ (.r‘𝐻) = (.r‘𝐻)
72		eqid 2760	. . . 4 ⊢ (.r‘𝑈) = (.r‘𝑈)
73	11, 13, 71, 72, 1, 14, 24	psrmulfval 22260	. . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋(.r‘𝑈)𝑌) = (𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ (𝐻 Σg (𝑥 ∈ {𝑦 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∣ 𝑦 ∘r ≤ 𝑘} ↦ ((𝑋‘𝑥)(.r‘𝐻)(𝑌‘(𝑘 ∘f − 𝑥)))))))
74	48, 70, 73	3eqtr4rd 2806	. 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋(.r‘𝑈)𝑌) = (𝑋(.r‘𝑆)𝑌))
75	13	fvexi 6904	. . . 4 ⊢ 𝐵 ∈ V
76		resspsr.p	. . . . 5 ⊢ 𝑃 = (𝑆 ↾s 𝐵)
77	76, 51	ressmulr 17487	. . . 4 ⊢ (𝐵 ∈ V → (.r‘𝑆) = (.r‘𝑃))
78	75, 77	mp1i 14	. . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (.r‘𝑆) = (.r‘𝑃))
79	78	oveqd 7441	. 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋(.r‘𝑆)𝑌) = (𝑋(.r‘𝑃)𝑌))
80	74, 79	eqtrd 2795	1 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋(.r‘𝑈)𝑌) = (𝑋(.r‘𝑃)𝑌))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  {crab 3412  Vcvv 3450   ⊆ wss 3896   class class class wbr 5102   ↦ cmpt 5185  ^◡ccnv 5650   “ cima 5654  ⟶wf 6539  ‘cfv 6543  (class class class)co 7424   ∘_f cof 7695   ∘_r cofr 7696   ↑_m cmap 8855  Fincfn 8981   ≤ cle 11351   − cmin 11548  ℕcn 12342  ℕ₀cn0 12613  Basecbs 17396   ↾_s cress 17417  ._rcmulr 17438   Σ_g cgsu 17620  SubMndcsubmnd 18986  SubGrpcsubg 19339  SubRingcsubrg 20830   mPwSer cmps 22221

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391  ax-un 7755  ax-cnex 11263  ax-resscn 11264  ax-1cn 11265  ax-icn 11266  ax-addcl 11267  ax-addrcl 11268  ax-mulcl 11269  ax-mulrcl 11270  ax-mulcom 11271  ax-addass 11272  ax-mulass 11273  ax-distr 11274  ax-i2m1 11275  ax-1ne0 11276  ax-1rid 11277  ax-rnegex 11278  ax-rrecex 11279  ax-cnre 11280  ax-pre-lttri 11281  ax-pre-lttrn 11282  ax-pre-ltadd 11283  ax-pre-mulgt0 11284

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6307  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7381  df-ov 7427  df-oprab 7428  df-mpo 7429  df-of 7697  df-ofr 7698  df-om 7882  df-1st 8005  df-2nd 8006  df-supp 8185  df-frecs 8307  df-wrecs 8338  df-recs 8387  df-rdg 8426  df-1o 8484  df-er 8725  df-map 8857  df-pm 8858  df-ixp 8934  df-en 8982  df-dom 8983  df-sdom 8984  df-fin 8985  df-fsupp 9362  df-pnf 11352  df-mnf 11353  df-xr 11354  df-ltxr 11355  df-le 11356  df-sub 11550  df-neg 11551  df-nn 12343  df-2 12412  df-3 12413  df-4 12414  df-5 12415  df-6 12416  df-7 12417  df-8 12418  df-9 12419  df-n0 12614  df-z 12701  df-uz 12973  df-fz 13647  df-seq 14152  df-struct 17334  df-sets 17351  df-slot 17369  df-ndx 17381  df-base 17397  df-ress 17418  df-plusg 17450  df-mulr 17451  df-sca 17453  df-vsca 17454  df-tset 17456  df-0g 17621  df-gsum 17622  df-mgm 18825  df-sgrp 18917  df-mnd 18933  df-submnd 18988  df-grp 19156  df-minusg 19157  df-subg 19342  df-cmn 20005  df-abl 20006  df-mgp 20370  df-rng 20384  df-ur 20417  df-ring 20470  df-subrng 20807  df-subrg 20831  df-psr 22226

This theorem is used by:  subrgpsr  22294  ressmplmul  22347