Theorem psrlmod 22069

Description: The ring of power series is a left module. (Contributed by Mario Carneiro, 29-Dec-2014.)

Hypotheses

Ref	Expression
psrring.s	⊢ 𝑆 = (𝐼 mPwSer 𝑅)
psrring.i	⊢ (𝜑 → 𝐼 ∈ 𝑉)
psrring.r	⊢ (𝜑 → 𝑅 ∈ Ring)

Assertion

Ref	Expression
psrlmod	⊢ (𝜑 → 𝑆 ∈ LMod)

Proof of Theorem psrlmod

Dummy variables 𝑥 𝑓 𝑦 𝑧 𝑟 𝑠 𝑡 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqidd 2766	. 2 ⊢ (𝜑 → (Base‘𝑆) = (Base‘𝑆))
2		eqidd 2766	. 2 ⊢ (𝜑 → (+g‘𝑆) = (+g‘𝑆))
3		psrring.s	. . 3 ⊢ 𝑆 = (𝐼 mPwSer 𝑅)
4		psrring.i	. . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
5		psrring.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ Ring)
6	3, 4, 5	psrsca 22057	. 2 ⊢ (𝜑 → 𝑅 = (Scalar‘𝑆))
7		eqidd 2766	. 2 ⊢ (𝜑 → ( ·𝑠 ‘𝑆) = ( ·𝑠 ‘𝑆))
8		eqidd 2766	. 2 ⊢ (𝜑 → (Base‘𝑅) = (Base‘𝑅))
9		eqidd 2766	. 2 ⊢ (𝜑 → (+g‘𝑅) = (+g‘𝑅))
10		eqidd 2766	. 2 ⊢ (𝜑 → (.r‘𝑅) = (.r‘𝑅))
11		eqidd 2766	. 2 ⊢ (𝜑 → (1r‘𝑅) = (1r‘𝑅))
12		ringgrp 20311	. . . 4 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp)
13	5, 12	syl 18	. . 3 ⊢ (𝜑 → 𝑅 ∈ Grp)
14	3, 4, 13	psrgrp 22066	. 2 ⊢ (𝜑 → 𝑆 ∈ Grp)
15		eqid 2765	. . 3 ⊢ ( ·𝑠 ‘𝑆) = ( ·𝑠 ‘𝑆)
16		eqid 2765	. . 3 ⊢ (Base‘𝑅) = (Base‘𝑅)
17		eqid 2765	. . 3 ⊢ (Base‘𝑆) = (Base‘𝑆)
18	5	3ad2ant1 1149	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑅 ∈ Ring)
19		simp2 1153	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑥 ∈ (Base‘𝑅))
20		simp3 1154	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑦 ∈ (Base‘𝑆))
21	3, 15, 16, 17, 18, 19, 20	psrvscacl 22061	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑥( ·𝑠 ‘𝑆)𝑦) ∈ (Base‘𝑆))
22		ovex 7433	. . . . . . 7 ⊢ (ℕ0 ↑m 𝐼) ∈ V
23	22	rabex 5300	. . . . . 6 ⊢ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∈ V
24	23	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∈ V)
25		simpr1 1211	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑥 ∈ (Base‘𝑅))
26		fconst6g 6757	. . . . . 6 ⊢ (𝑥 ∈ (Base‘𝑅) → ({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑥}):{𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑅))
27	25, 26	syl 18	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → ({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑥}):{𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑅))
28		eqid 2765	. . . . . 6 ⊢ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}
29		simpr2 1212	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑦 ∈ (Base‘𝑆))
30	3, 16, 28, 17, 29	psrelbas 22045	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑦:{𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑅))
31		simpr3 1213	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑧 ∈ (Base‘𝑆))
32	3, 16, 28, 17, 31	psrelbas 22045	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑧:{𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑅))
33	5	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑅 ∈ Ring)
34		eqid 2765	. . . . . . 7 ⊢ (+g‘𝑅) = (+g‘𝑅)
35		eqid 2765	. . . . . . 7 ⊢ (.r‘𝑅) = (.r‘𝑅)
36	16, 34, 35	ringdi 20334	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ (𝑟 ∈ (Base‘𝑅) ∧ 𝑠 ∈ (Base‘𝑅) ∧ 𝑡 ∈ (Base‘𝑅))) → (𝑟(.r‘𝑅)(𝑠(+g‘𝑅)𝑡)) = ((𝑟(.r‘𝑅)𝑠)(+g‘𝑅)(𝑟(.r‘𝑅)𝑡)))
37	33, 36	sylan 591	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) ∧ (𝑟 ∈ (Base‘𝑅) ∧ 𝑠 ∈ (Base‘𝑅) ∧ 𝑡 ∈ (Base‘𝑅))) → (𝑟(.r‘𝑅)(𝑠(+g‘𝑅)𝑡)) = ((𝑟(.r‘𝑅)𝑠)(+g‘𝑅)(𝑟(.r‘𝑅)𝑡)))
38	24, 27, 30, 32, 37	caofdi 7706	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → (({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑥}) ∘f (.r‘𝑅)(𝑦 ∘f (+g‘𝑅)𝑧)) = ((({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑥}) ∘f (.r‘𝑅)𝑦) ∘f (+g‘𝑅)(({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑥}) ∘f (.r‘𝑅)𝑧)))
39		eqid 2765	. . . . . 6 ⊢ (+g‘𝑆) = (+g‘𝑆)
40	3, 17, 34, 39, 29, 31	psradd 22048	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑦(+g‘𝑆)𝑧) = (𝑦 ∘f (+g‘𝑅)𝑧))
41	40	oveq2d 7416	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → (({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑥}) ∘f (.r‘𝑅)(𝑦(+g‘𝑆)𝑧)) = (({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑥}) ∘f (.r‘𝑅)(𝑦 ∘f (+g‘𝑅)𝑧)))
42	3, 15, 16, 17, 35, 28, 25, 29	psrvsca 22059	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑥( ·𝑠 ‘𝑆)𝑦) = (({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑥}) ∘f (.r‘𝑅)𝑦))
43	3, 15, 16, 17, 35, 28, 25, 31	psrvsca 22059	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑥( ·𝑠 ‘𝑆)𝑧) = (({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑥}) ∘f (.r‘𝑅)𝑧))
44	42, 43	oveq12d 7418	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → ((𝑥( ·𝑠 ‘𝑆)𝑦) ∘f (+g‘𝑅)(𝑥( ·𝑠 ‘𝑆)𝑧)) = ((({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑥}) ∘f (.r‘𝑅)𝑦) ∘f (+g‘𝑅)(({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑥}) ∘f (.r‘𝑅)𝑧)))
45	38, 41, 44	3eqtr4d 2810	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → (({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑥}) ∘f (.r‘𝑅)(𝑦(+g‘𝑆)𝑧)) = ((𝑥( ·𝑠 ‘𝑆)𝑦) ∘f (+g‘𝑅)(𝑥( ·𝑠 ‘𝑆)𝑧)))
46	13	grpmgmd 19018	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Mgm)
47	46	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑅 ∈ Mgm)
48	3, 17, 39, 47, 29, 31	psraddcl 22049	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑦(+g‘𝑆)𝑧) ∈ (Base‘𝑆))
49	3, 15, 16, 17, 35, 28, 25, 48	psrvsca 22059	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑥( ·𝑠 ‘𝑆)(𝑦(+g‘𝑆)𝑧)) = (({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑥}) ∘f (.r‘𝑅)(𝑦(+g‘𝑆)𝑧)))
50	21	3adant3r3 1201	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑥( ·𝑠 ‘𝑆)𝑦) ∈ (Base‘𝑆))
51	3, 15, 16, 17, 33, 25, 31	psrvscacl 22061	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑥( ·𝑠 ‘𝑆)𝑧) ∈ (Base‘𝑆))
52	3, 17, 34, 39, 50, 51	psradd 22048	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → ((𝑥( ·𝑠 ‘𝑆)𝑦)(+g‘𝑆)(𝑥( ·𝑠 ‘𝑆)𝑧)) = ((𝑥( ·𝑠 ‘𝑆)𝑦) ∘f (+g‘𝑅)(𝑥( ·𝑠 ‘𝑆)𝑧)))
53	45, 49, 52	3eqtr4d 2810	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑥( ·𝑠 ‘𝑆)(𝑦(+g‘𝑆)𝑧)) = ((𝑥( ·𝑠 ‘𝑆)𝑦)(+g‘𝑆)(𝑥( ·𝑠 ‘𝑆)𝑧)))
54		simpr1 1211	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑥 ∈ (Base‘𝑅))
55		simpr3 1213	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑧 ∈ (Base‘𝑆))
56	3, 15, 16, 17, 35, 28, 54, 55	psrvsca 22059	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑥( ·𝑠 ‘𝑆)𝑧) = (({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑥}) ∘f (.r‘𝑅)𝑧))
57		simpr2 1212	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑦 ∈ (Base‘𝑅))
58	3, 15, 16, 17, 35, 28, 57, 55	psrvsca 22059	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑦( ·𝑠 ‘𝑆)𝑧) = (({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑦}) ∘f (.r‘𝑅)𝑧))
59	56, 58	oveq12d 7418	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → ((𝑥( ·𝑠 ‘𝑆)𝑧) ∘f (+g‘𝑅)(𝑦( ·𝑠 ‘𝑆)𝑧)) = ((({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑥}) ∘f (.r‘𝑅)𝑧) ∘f (+g‘𝑅)(({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑦}) ∘f (.r‘𝑅)𝑧)))
60	23	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∈ V)
61	3, 16, 28, 17, 55	psrelbas 22045	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑧:{𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑅))
62	54, 26	syl 18	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → ({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑥}):{𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑅))
63		fconst6g 6757	. . . . . 6 ⊢ (𝑦 ∈ (Base‘𝑅) → ({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑦}):{𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑅))
64	57, 63	syl 18	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → ({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑦}):{𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑅))
65	5	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑅 ∈ Ring)
66	16, 34, 35	ringdir 20335	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ (𝑟 ∈ (Base‘𝑅) ∧ 𝑠 ∈ (Base‘𝑅) ∧ 𝑡 ∈ (Base‘𝑅))) → ((𝑟(+g‘𝑅)𝑠)(.r‘𝑅)𝑡) = ((𝑟(.r‘𝑅)𝑡)(+g‘𝑅)(𝑠(.r‘𝑅)𝑡)))
67	65, 66	sylan 591	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) ∧ (𝑟 ∈ (Base‘𝑅) ∧ 𝑠 ∈ (Base‘𝑅) ∧ 𝑡 ∈ (Base‘𝑅))) → ((𝑟(+g‘𝑅)𝑠)(.r‘𝑅)𝑡) = ((𝑟(.r‘𝑅)𝑡)(+g‘𝑅)(𝑠(.r‘𝑅)𝑡)))
68	60, 61, 62, 64, 67	caofdir 7707	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → ((({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑥}) ∘f (+g‘𝑅)({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑦})) ∘f (.r‘𝑅)𝑧) = ((({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑥}) ∘f (.r‘𝑅)𝑧) ∘f (+g‘𝑅)(({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑦}) ∘f (.r‘𝑅)𝑧)))
69	60, 54, 57	ofc12 7694	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → (({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑥}) ∘f (+g‘𝑅)({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑦})) = ({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {(𝑥(+g‘𝑅)𝑦)}))
70	69	oveq1d 7415	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → ((({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑥}) ∘f (+g‘𝑅)({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑦})) ∘f (.r‘𝑅)𝑧) = (({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {(𝑥(+g‘𝑅)𝑦)}) ∘f (.r‘𝑅)𝑧))
71	59, 68, 70	3eqtr2rd 2807	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → (({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {(𝑥(+g‘𝑅)𝑦)}) ∘f (.r‘𝑅)𝑧) = ((𝑥( ·𝑠 ‘𝑆)𝑧) ∘f (+g‘𝑅)(𝑦( ·𝑠 ‘𝑆)𝑧)))
72	16, 34	ringacl 20352	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(+g‘𝑅)𝑦) ∈ (Base‘𝑅))
73	65, 54, 57, 72	syl3anc 1394	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑥(+g‘𝑅)𝑦) ∈ (Base‘𝑅))
74	3, 15, 16, 17, 35, 28, 73, 55	psrvsca 22059	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → ((𝑥(+g‘𝑅)𝑦)( ·𝑠 ‘𝑆)𝑧) = (({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {(𝑥(+g‘𝑅)𝑦)}) ∘f (.r‘𝑅)𝑧))
75	3, 15, 16, 17, 65, 54, 55	psrvscacl 22061	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑥( ·𝑠 ‘𝑆)𝑧) ∈ (Base‘𝑆))
76	3, 15, 16, 17, 65, 57, 55	psrvscacl 22061	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑦( ·𝑠 ‘𝑆)𝑧) ∈ (Base‘𝑆))
77	3, 17, 34, 39, 75, 76	psradd 22048	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → ((𝑥( ·𝑠 ‘𝑆)𝑧)(+g‘𝑆)(𝑦( ·𝑠 ‘𝑆)𝑧)) = ((𝑥( ·𝑠 ‘𝑆)𝑧) ∘f (+g‘𝑅)(𝑦( ·𝑠 ‘𝑆)𝑧)))
78	71, 74, 77	3eqtr4d 2810	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → ((𝑥(+g‘𝑅)𝑦)( ·𝑠 ‘𝑆)𝑧) = ((𝑥( ·𝑠 ‘𝑆)𝑧)(+g‘𝑆)(𝑦( ·𝑠 ‘𝑆)𝑧)))
79	58	oveq2d 7416	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → (({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑥}) ∘f (.r‘𝑅)(𝑦( ·𝑠 ‘𝑆)𝑧)) = (({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑥}) ∘f (.r‘𝑅)(({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑦}) ∘f (.r‘𝑅)𝑧)))
80	16, 35	ringass 20326	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ (𝑟 ∈ (Base‘𝑅) ∧ 𝑠 ∈ (Base‘𝑅) ∧ 𝑡 ∈ (Base‘𝑅))) → ((𝑟(.r‘𝑅)𝑠)(.r‘𝑅)𝑡) = (𝑟(.r‘𝑅)(𝑠(.r‘𝑅)𝑡)))
81	65, 80	sylan 591	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) ∧ (𝑟 ∈ (Base‘𝑅) ∧ 𝑠 ∈ (Base‘𝑅) ∧ 𝑡 ∈ (Base‘𝑅))) → ((𝑟(.r‘𝑅)𝑠)(.r‘𝑅)𝑡) = (𝑟(.r‘𝑅)(𝑠(.r‘𝑅)𝑡)))
82	60, 62, 64, 61, 81	caofass 7704	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → ((({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑥}) ∘f (.r‘𝑅)({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑦})) ∘f (.r‘𝑅)𝑧) = (({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑥}) ∘f (.r‘𝑅)(({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑦}) ∘f (.r‘𝑅)𝑧)))
83	60, 54, 57	ofc12 7694	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → (({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑥}) ∘f (.r‘𝑅)({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑦})) = ({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {(𝑥(.r‘𝑅)𝑦)}))
84	83	oveq1d 7415	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → ((({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑥}) ∘f (.r‘𝑅)({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑦})) ∘f (.r‘𝑅)𝑧) = (({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {(𝑥(.r‘𝑅)𝑦)}) ∘f (.r‘𝑅)𝑧))
85	79, 82, 84	3eqtr2rd 2807	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → (({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {(𝑥(.r‘𝑅)𝑦)}) ∘f (.r‘𝑅)𝑧) = (({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑥}) ∘f (.r‘𝑅)(𝑦( ·𝑠 ‘𝑆)𝑧)))
86	16, 35	ringcl 20323	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(.r‘𝑅)𝑦) ∈ (Base‘𝑅))
87	65, 54, 57, 86	syl3anc 1394	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑥(.r‘𝑅)𝑦) ∈ (Base‘𝑅))
88	3, 15, 16, 17, 35, 28, 87, 55	psrvsca 22059	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → ((𝑥(.r‘𝑅)𝑦)( ·𝑠 ‘𝑆)𝑧) = (({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {(𝑥(.r‘𝑅)𝑦)}) ∘f (.r‘𝑅)𝑧))
89	3, 15, 16, 17, 35, 28, 54, 76	psrvsca 22059	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑥( ·𝑠 ‘𝑆)(𝑦( ·𝑠 ‘𝑆)𝑧)) = (({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑥}) ∘f (.r‘𝑅)(𝑦( ·𝑠 ‘𝑆)𝑧)))
90	85, 88, 89	3eqtr4d 2810	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → ((𝑥(.r‘𝑅)𝑦)( ·𝑠 ‘𝑆)𝑧) = (𝑥( ·𝑠 ‘𝑆)(𝑦( ·𝑠 ‘𝑆)𝑧)))
91	5	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆)) → 𝑅 ∈ Ring)
92		eqid 2765	. . . . . 6 ⊢ (1r‘𝑅) = (1r‘𝑅)
93	16, 92	ringidcl 20339	. . . . 5 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ (Base‘𝑅))
94	91, 93	syl 18	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆)) → (1r‘𝑅) ∈ (Base‘𝑅))
95		simpr 489	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆)) → 𝑥 ∈ (Base‘𝑆))
96	3, 15, 16, 17, 35, 28, 94, 95	psrvsca 22059	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆)) → ((1r‘𝑅)( ·𝑠 ‘𝑆)𝑥) = (({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {(1r‘𝑅)}) ∘f (.r‘𝑅)𝑥))
97	23	a1i 11	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆)) → {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∈ V)
98	3, 16, 28, 17, 95	psrelbas 22045	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆)) → 𝑥:{𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑅))
99	16, 35, 92	ringlidm 20343	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑟 ∈ (Base‘𝑅)) → ((1r‘𝑅)(.r‘𝑅)𝑟) = 𝑟)
100	91, 99	sylan 591	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑟 ∈ (Base‘𝑅)) → ((1r‘𝑅)(.r‘𝑅)𝑟) = 𝑟)
101	97, 98, 94, 100	caofid0l 7697	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆)) → (({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {(1r‘𝑅)}) ∘f (.r‘𝑅)𝑥) = 𝑥)
102	96, 101	eqtrd 2800	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆)) → ((1r‘𝑅)( ·𝑠 ‘𝑆)𝑥) = 𝑥)
103	1, 2, 6, 7, 8, 9, 10, 11, 5, 14, 21, 53, 78, 90, 102	islmodd 20956	1 ⊢ (𝜑 → 𝑆 ∈ LMod)

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1101   = wceq 1563   ∈ wcel 2145  {crab 3417  Vcvv 3457  {csn 4585   × cxp 5650  ^◡ccnv 5651   “ cima 5655  ⟶wf 6521  ‘cfv 6525  (class class class)co 7400   ∘_f cof 7662   ↑_m cmap 8812  Fincfn 8931  ℕcn 12224  ℕ₀cn0 12495  Basecbs 17259  +_gcplusg 17300  ._rcmulr 17301   ·_𝑠 cvsca 17304  Mgmcmgm 18686  Grpcgrp 18990  1_rcur 20254  Ringcrg 20306  LModclmod 20950   mPwSer cmps 22014

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722  ax-cnex 11144  ax-resscn 11145  ax-1cn 11146  ax-icn 11147  ax-addcl 11148  ax-addrcl 11149  ax-mulcl 11150  ax-mulrcl 11151  ax-mulcom 11152  ax-addass 11153  ax-mulass 11154  ax-distr 11155  ax-i2m1 11156  ax-1ne0 11157  ax-1rid 11158  ax-rnegex 11159  ax-rrecex 11160  ax-cnre 11161  ax-pre-lttri 11162  ax-pre-lttrn 11163  ax-pre-ltadd 11164  ax-pre-mulgt0 11165

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-uni 4869  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5547  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-we 5607  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-pred 6292  df-ord 6353  df-on 6354  df-lim 6355  df-suc 6356  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-riota 7357  df-ov 7403  df-oprab 7404  df-mpo 7405  df-of 7664  df-om 7851  df-1st 7974  df-2nd 7975  df-supp 8145  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-rdg 8385  df-1o 8441  df-er 8682  df-map 8814  df-ixp 8884  df-en 8932  df-dom 8933  df-sdom 8934  df-fin 8935  df-fsupp 9310  df-sup 9390  df-pnf 11233  df-mnf 11234  df-xr 11235  df-ltxr 11236  df-le 11237  df-sub 11431  df-neg 11432  df-nn 12225  df-2 12294  df-3 12295  df-4 12296  df-5 12297  df-6 12298  df-7 12299  df-8 12300  df-9 12301  df-n0 12496  df-z 12583  df-dec 12703  df-uz 12854  df-fz 13527  df-struct 17197  df-sets 17214  df-slot 17232  df-ndx 17244  df-base 17260  df-plusg 17313  df-mulr 17314  df-sca 17316  df-vsca 17317  df-ip 17318  df-tset 17319  df-ple 17320  df-ds 17322  df-hom 17324  df-cco 17325  df-0g 17484  df-prds 17490  df-pws 17492  df-mgm 18688  df-sgrp 18767  df-mnd 18783  df-grp 18993  df-minusg 18994  df-mgp 20208  df-ur 20255  df-ring 20308  df-lmod 20952  df-psr 22019

This theorem is referenced by:  psrassa  22082  psrasclcl  22089  mpllmod  22127  mplbas2  22153  psdascl  22291  opsrlmod  22365