Theorem psrlmod 21941

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 2741	. 2 ⊢ (𝜑 → (Base‘𝑆) = (Base‘𝑆))
2		eqidd 2741	. 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 21929	. 2 ⊢ (𝜑 → 𝑅 = (Scalar‘𝑆))
7		eqidd 2741	. 2 ⊢ (𝜑 → ( ·𝑠 ‘𝑆) = ( ·𝑠 ‘𝑆))
8		eqidd 2741	. 2 ⊢ (𝜑 → (Base‘𝑅) = (Base‘𝑅))
9		eqidd 2741	. 2 ⊢ (𝜑 → (+g‘𝑅) = (+g‘𝑅))
10		eqidd 2741	. 2 ⊢ (𝜑 → (.r‘𝑅) = (.r‘𝑅))
11		eqidd 2741	. 2 ⊢ (𝜑 → (1r‘𝑅) = (1r‘𝑅))
12		ringgrp 20217	. . . 4 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp)
13	5, 12	syl 17	. . 3 ⊢ (𝜑 → 𝑅 ∈ Grp)
14	3, 4, 13	psrgrp 21938	. 2 ⊢ (𝜑 → 𝑆 ∈ Grp)
15		eqid 2740	. . 3 ⊢ ( ·𝑠 ‘𝑆) = ( ·𝑠 ‘𝑆)
16		eqid 2740	. . 3 ⊢ (Base‘𝑅) = (Base‘𝑅)
17		eqid 2740	. . 3 ⊢ (Base‘𝑆) = (Base‘𝑆)
18	5	3ad2ant1 1139	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑅 ∈ Ring)
19		simp2 1143	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑥 ∈ (Base‘𝑅))
20		simp3 1144	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑦 ∈ (Base‘𝑆))
21	3, 15, 16, 17, 18, 19, 20	psrvscacl 21933	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑥( ·𝑠 ‘𝑆)𝑦) ∈ (Base‘𝑆))
22		ovex 7396	. . . . . . 7 ⊢ (ℕ0 ↑m 𝐼) ∈ V
23	22	rabex 5274	. . . . . 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 1201	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑥 ∈ (Base‘𝑅))
26		fconst6g 6723	. . . . . 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 17	. . . . 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 2740	. . . . . 6 ⊢ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}
29		simpr2 1202	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑦 ∈ (Base‘𝑆))
30	3, 16, 28, 17, 29	psrelbas 21917	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑦:{𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑅))
31		simpr3 1203	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑧 ∈ (Base‘𝑆))
32	3, 16, 28, 17, 31	psrelbas 21917	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑧:{𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑅))
33	5	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑅 ∈ Ring)
34		eqid 2740	. . . . . . 7 ⊢ (+g‘𝑅) = (+g‘𝑅)
35		eqid 2740	. . . . . . 7 ⊢ (.r‘𝑅) = (.r‘𝑅)
36	16, 34, 35	ringdi 20240	. . . . . 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 586	. . . . 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 7669	. . . 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 2740	. . . . . 6 ⊢ (+g‘𝑆) = (+g‘𝑆)
40	3, 17, 34, 39, 29, 31	psradd 21920	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑦(+g‘𝑆)𝑧) = (𝑦 ∘f (+g‘𝑅)𝑧))
41	40	oveq2d 7379	. . . 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 21931	. . . . 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 21931	. . . . 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 7381	. . . 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 2785	. . 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 18935	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Mgm)
47	46	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑅 ∈ Mgm)
48	3, 17, 39, 47, 29, 31	psraddcl 21921	. . . 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 21931	. . 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 1191	. . . 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 21933	. . . 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 21920	. . 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 2785	. 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 1201	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑥 ∈ (Base‘𝑅))
55		simpr3 1203	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑧 ∈ (Base‘𝑆))
56	3, 15, 16, 17, 35, 28, 54, 55	psrvsca 21931	. . . . 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 1202	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑦 ∈ (Base‘𝑅))
58	3, 15, 16, 17, 35, 28, 57, 55	psrvsca 21931	. . . . 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 7381	. . . 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 21917	. . . . 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 17	. . . . 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 6723	. . . . . 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 17	. . . . 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 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑅 ∈ Ring)
66	16, 34, 35	ringdir 20241	. . . . . 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 586	. . . . 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 7670	. . . 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 7657	. . . . 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 7378	. . . 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 2782	. . 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 20257	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(+g‘𝑅)𝑦) ∈ (Base‘𝑅))
73	65, 54, 57, 72	syl3anc 1379	. . . 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 21931	. . 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 21933	. . . 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 21933	. . . 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 21920	. . 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 2785	. 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 7379	. . . 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 20232	. . . . . 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 586	. . . . 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 7667	. . . 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 7657	. . . . 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 7378	. . . 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 2782	. . 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 20229	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(.r‘𝑅)𝑦) ∈ (Base‘𝑅))
87	65, 54, 57, 86	syl3anc 1379	. . . 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 21931	. . 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 21931	. . 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 2785	. 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 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆)) → 𝑅 ∈ Ring)
92		eqid 2740	. . . . . 6 ⊢ (1r‘𝑅) = (1r‘𝑅)
93	16, 92	ringidcl 20244	. . . . 5 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ (Base‘𝑅))
94	91, 93	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆)) → (1r‘𝑅) ∈ (Base‘𝑅))
95		simpr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆)) → 𝑥 ∈ (Base‘𝑆))
96	3, 15, 16, 17, 35, 28, 94, 95	psrvsca 21931	. . 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 21917	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆)) → 𝑥:{𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑅))
99	16, 35, 92	ringlidm 20248	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑟 ∈ (Base‘𝑅)) → ((1r‘𝑅)(.r‘𝑅)𝑟) = 𝑟)
100	91, 99	sylan 586	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑟 ∈ (Base‘𝑅)) → ((1r‘𝑅)(.r‘𝑅)𝑟) = 𝑟)
101	97, 98, 94, 100	caofid0l 7660	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆)) → (({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {(1r‘𝑅)}) ∘f (.r‘𝑅)𝑥) = 𝑥)
102	96, 101	eqtrd 2775	. 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 20863	1 ⊢ (𝜑 → 𝑆 ∈ LMod)

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  {crab 3392  Vcvv 3432  {csn 4562   × cxp 5623  ^◡ccnv 5624   “ cima 5628  ⟶wf 6488  ‘cfv 6492  (class class class)co 7363   ∘_f cof 7625   ↑_m cmap 8770  Fincfn 8890  ℕcn 12172  ℕ₀cn0 12435  Basecbs 17177  +_gcplusg 17218  ._rcmulr 17219   ·_𝑠 cvsca 17222  Mgmcmgm 18604  Grpcgrp 18907  1_rcur 20160  Ringcrg 20212  LModclmod 20857   mPwSer cmps 21886

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685  ax-cnex 11092  ax-resscn 11093  ax-1cn 11094  ax-icn 11095  ax-addcl 11096  ax-addrcl 11097  ax-mulcl 11098  ax-mulrcl 11099  ax-mulcom 11100  ax-addass 11101  ax-mulass 11102  ax-distr 11103  ax-i2m1 11104  ax-1ne0 11105  ax-1rid 11106  ax-rnegex 11107  ax-rrecex 11108  ax-cnre 11109  ax-pre-lttri 11110  ax-pre-lttrn 11111  ax-pre-ltadd 11112  ax-pre-mulgt0 11113

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-nel 3040  df-ral 3055  df-rex 3065  df-rmo 3345  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-tp 4567  df-op 4569  df-uni 4846  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7320  df-ov 7366  df-oprab 7367  df-mpo 7368  df-of 7627  df-om 7814  df-1st 7938  df-2nd 7939  df-supp 8108  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-rdg 8346  df-1o 8402  df-er 8640  df-map 8772  df-ixp 8843  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-fsupp 9272  df-sup 9352  df-pnf 11179  df-mnf 11180  df-xr 11181  df-ltxr 11182  df-le 11183  df-sub 11377  df-neg 11378  df-nn 12173  df-2 12242  df-3 12243  df-4 12244  df-5 12245  df-6 12246  df-7 12247  df-8 12248  df-9 12249  df-n0 12436  df-z 12523  df-dec 12643  df-uz 12787  df-fz 13460  df-struct 17115  df-sets 17132  df-slot 17150  df-ndx 17162  df-base 17178  df-plusg 17231  df-mulr 17232  df-sca 17234  df-vsca 17235  df-ip 17236  df-tset 17237  df-ple 17238  df-ds 17240  df-hom 17242  df-cco 17243  df-0g 17402  df-prds 17408  df-pws 17410  df-mgm 18606  df-sgrp 18685  df-mnd 18701  df-grp 18910  df-minusg 18911  df-mgp 20120  df-ur 20161  df-ring 20214  df-lmod 20859  df-psr 21891

This theorem is referenced by:  psrassa  21954  psrasclcl  21961  mpllmod  21999  mplbas2  22025  psdascl  22163  opsrlmod  22237