Theorem resspsrmul 21386

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 2736	. . . . . . . 8 ⊢ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}
2	1	psrbaglefi 21334	. . . . . . 7 ⊢ (𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} → {𝑦 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∣ 𝑦 ∘r ≤ 𝑘} ∈ Fin)
3	2	adantl 482	. . . . . 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 20228	. . . . . . . . 9 ⊢ (𝑇 ∈ (SubRing‘𝑅) → 𝑇 ∈ (SubGrp‘𝑅))
6	4, 5	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝑅))
7		subgsubm 18950	. . . . . . . 8 ⊢ (𝑇 ∈ (SubGrp‘𝑅) → 𝑇 ∈ (SubMnd‘𝑅))
8	6, 7	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑇 ∈ (SubMnd‘𝑅))
9	8	ad2antrr 724	. . . . . 6 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → 𝑇 ∈ (SubMnd‘𝑅))
10	4	ad3antrrr 728	. . . . . . . 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 2736	. . . . . . . . . . . 12 ⊢ (Base‘𝐻) = (Base‘𝐻)
13		resspsr.b	. . . . . . . . . . . 12 ⊢ 𝐵 = (Base‘𝑈)
14		simprl 769	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 𝑋 ∈ 𝐵)
15	11, 12, 1, 13, 14	psrelbas 21347	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 𝑋:{𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝐻))
16	15	adantr 481	. . . . . . . . . 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 3639	. . . . . . . . . 10 ⊢ (𝑥 ∈ {𝑦 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∣ 𝑦 ∘r ≤ 𝑘} → 𝑥 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin})
18		ffvelcdm 7032	. . . . . . . . . 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 596	. . . . . . . . 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 20231	. . . . . . . . . 10 ⊢ (𝑇 ∈ (SubRing‘𝑅) → 𝑇 = (Base‘𝐻))
22	10, 21	syl 17	. . . . . . . . 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 2841	. . . . . . . 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 771	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 𝑌 ∈ 𝐵)
25	11, 12, 1, 13, 24	psrelbas 21347	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 𝑌:{𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝐻))
26	25	ad2antrr 724	. . . . . . . . . 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 4037	. . . . . . . . . . 11 ⊢ {𝑦 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∣ 𝑦 ∘r ≤ 𝑘} ⊆ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}
28		simplr 767	. . . . . . . . . . . 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 485	. . . . . . . . . . . 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 2736	. . . . . . . . . . . . 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 21336	. . . . . . . . . . . 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 584	. . . . . . . . . . 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 3942	. . . . . . . . . 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 7036	. . . . . . . . 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 2841	. . . . . . . 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 2736	. . . . . . . . 9 ⊢ (.r‘𝑅) = (.r‘𝑅)
37	36	subrgmcl 20234	. . . . . . . 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 1371	. . . . . . 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 7063	. . . . . 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 18645	. . . . 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 17188	. . . . . . . . . 10 ⊢ (𝑇 ∈ (SubRing‘𝑅) → (.r‘𝑅) = (.r‘𝐻))
42	4, 41	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (.r‘𝑅) = (.r‘𝐻))
43	42	ad3antrrr 728	. . . . . . . 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 7374	. . . . . . 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 5205	. . . . . 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 7373	. . . . 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 2776	. . . 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 5205	. . 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 2736	. . . 4 ⊢ (Base‘𝑆) = (Base‘𝑆)
51		eqid 2736	. . . 4 ⊢ (.r‘𝑆) = (.r‘𝑆)
52		fvex 6855	. . . . . . . 8 ⊢ (Base‘𝑅) ∈ V
53	4, 21	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑇 = (Base‘𝐻))
54		eqid 2736	. . . . . . . . . . 11 ⊢ (Base‘𝑅) = (Base‘𝑅)
55	54	subrgss 20223	. . . . . . . . . 10 ⊢ (𝑇 ∈ (SubRing‘𝑅) → 𝑇 ⊆ (Base‘𝑅))
56	4, 55	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑇 ⊆ (Base‘𝑅))
57	53, 56	eqsstrrd 3983	. . . . . . . 8 ⊢ (𝜑 → (Base‘𝐻) ⊆ (Base‘𝑅))
58		mapss 8827	. . . . . . . 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 587	. . . . . . 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 481	. . . . . 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 21316	. . . . . . . . . 10 ⊢ Rel dom mPwSer
62	61, 11, 13	elbasov 17090	. . . . . . . . 9 ⊢ (𝑋 ∈ 𝐵 → (𝐼 ∈ V ∧ 𝐻 ∈ V))
63	62	ad2antrl 726	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝐼 ∈ V ∧ 𝐻 ∈ V))
64	63	simpld 495	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 𝐼 ∈ V)
65	11, 12, 1, 13, 64	psrbas 21346	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 𝐵 = ((Base‘𝐻) ↑m {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}))
66	49, 54, 1, 50, 64	psrbas 21346	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (Base‘𝑆) = ((Base‘𝑅) ↑m {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}))
67	60, 65, 66	3sstr4d 3991	. . . . 5 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 𝐵 ⊆ (Base‘𝑆))
68	67, 14	sseldd 3945	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 𝑋 ∈ (Base‘𝑆))
69	67, 24	sseldd 3945	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 𝑌 ∈ (Base‘𝑆))
70	49, 50, 36, 51, 1, 68, 69	psrmulfval 21353	. . 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 2736	. . . 4 ⊢ (.r‘𝐻) = (.r‘𝐻)
72		eqid 2736	. . . 4 ⊢ (.r‘𝑈) = (.r‘𝑈)
73	11, 13, 71, 72, 1, 14, 24	psrmulfval 21353	. . 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 2787	. 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋(.r‘𝑈)𝑌) = (𝑋(.r‘𝑆)𝑌))
75	13	fvexi 6856	. . . 4 ⊢ 𝐵 ∈ V
76		resspsr.p	. . . . 5 ⊢ 𝑃 = (𝑆 ↾s 𝐵)
77	76, 51	ressmulr 17188	. . . 4 ⊢ (𝐵 ∈ V → (.r‘𝑆) = (.r‘𝑃))
78	75, 77	mp1i 13	. . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (.r‘𝑆) = (.r‘𝑃))
79	78	oveqd 7374	. 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋(.r‘𝑆)𝑌) = (𝑋(.r‘𝑃)𝑌))
80	74, 79	eqtrd 2776	1 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋(.r‘𝑈)𝑌) = (𝑋(.r‘𝑃)𝑌))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  {crab 3407  Vcvv 3445   ⊆ wss 3910   class class class wbr 5105   ↦ cmpt 5188  ^◡ccnv 5632   “ cima 5636  ⟶wf 6492  ‘cfv 6496  (class class class)co 7357   ∘_f cof 7615   ∘_r cofr 7616   ↑_m cmap 8765  Fincfn 8883   ≤ cle 11190   − cmin 11385  ℕcn 12153  ℕ₀cn0 12413  Basecbs 17083   ↾_s cress 17112  ._rcmulr 17134   Σ_g cgsu 17322  SubMndcsubmnd 18600  SubGrpcsubg 18922  SubRingcsubrg 20218   mPwSer cmps 21306

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-ofr 7618  df-om 7803  df-1st 7921  df-2nd 7922  df-supp 8093  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-pm 8768  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9306  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-z 12500  df-uz 12764  df-fz 13425  df-seq 13907  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-mulr 17147  df-sca 17149  df-vsca 17150  df-tset 17152  df-0g 17323  df-gsum 17324  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-submnd 18602  df-grp 18751  df-minusg 18752  df-subg 18925  df-mgp 19897  df-ring 19966  df-subrg 20220  df-psr 21311

This theorem is referenced by:  subrgpsr  21388  ressmplmul  21431