Theorem psrlmod 21876

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 2731	. 2 ⊢ (𝜑 → (Base‘𝑆) = (Base‘𝑆))
2		eqidd 2731	. 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 21863	. 2 ⊢ (𝜑 → 𝑅 = (Scalar‘𝑆))
7		eqidd 2731	. 2 ⊢ (𝜑 → ( ·𝑠 ‘𝑆) = ( ·𝑠 ‘𝑆))
8		eqidd 2731	. 2 ⊢ (𝜑 → (Base‘𝑅) = (Base‘𝑅))
9		eqidd 2731	. 2 ⊢ (𝜑 → (+g‘𝑅) = (+g‘𝑅))
10		eqidd 2731	. 2 ⊢ (𝜑 → (.r‘𝑅) = (.r‘𝑅))
11		eqidd 2731	. 2 ⊢ (𝜑 → (1r‘𝑅) = (1r‘𝑅))
12		ringgrp 20154	. . . 4 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp)
13	5, 12	syl 17	. . 3 ⊢ (𝜑 → 𝑅 ∈ Grp)
14	3, 4, 13	psrgrp 21872	. 2 ⊢ (𝜑 → 𝑆 ∈ Grp)
15		eqid 2730	. . 3 ⊢ ( ·𝑠 ‘𝑆) = ( ·𝑠 ‘𝑆)
16		eqid 2730	. . 3 ⊢ (Base‘𝑅) = (Base‘𝑅)
17		eqid 2730	. . 3 ⊢ (Base‘𝑆) = (Base‘𝑆)
18	5	3ad2ant1 1133	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑅 ∈ Ring)
19		simp2 1137	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑥 ∈ (Base‘𝑅))
20		simp3 1138	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑦 ∈ (Base‘𝑆))
21	3, 15, 16, 17, 18, 19, 20	psrvscacl 21867	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑥( ·𝑠 ‘𝑆)𝑦) ∈ (Base‘𝑆))
22		ovex 7423	. . . . . . 7 ⊢ (ℕ0 ↑m 𝐼) ∈ V
23	22	rabex 5297	. . . . . 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 1195	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑥 ∈ (Base‘𝑅))
26		fconst6g 6752	. . . . . 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 2730	. . . . . 6 ⊢ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}
29		simpr2 1196	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑦 ∈ (Base‘𝑆))
30	3, 16, 28, 17, 29	psrelbas 21850	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑦:{𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑅))
31		simpr3 1197	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑧 ∈ (Base‘𝑆))
32	3, 16, 28, 17, 31	psrelbas 21850	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑧:{𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑅))
33	5	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑅 ∈ Ring)
34		eqid 2730	. . . . . . 7 ⊢ (+g‘𝑅) = (+g‘𝑅)
35		eqid 2730	. . . . . . 7 ⊢ (.r‘𝑅) = (.r‘𝑅)
36	16, 34, 35	ringdi 20177	. . . . . 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 580	. . . . 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 7698	. . . 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 2730	. . . . . 6 ⊢ (+g‘𝑆) = (+g‘𝑆)
40	3, 17, 34, 39, 29, 31	psradd 21853	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑦(+g‘𝑆)𝑧) = (𝑦 ∘f (+g‘𝑅)𝑧))
41	40	oveq2d 7406	. . . 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 21865	. . . . 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 21865	. . . . 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 7408	. . . 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 2775	. . 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 18900	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Mgm)
47	46	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑅 ∈ Mgm)
48	3, 17, 39, 47, 29, 31	psraddcl 21854	. . . 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 21865	. . 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 1185	. . . 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 21867	. . . 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 21853	. . 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 2775	. 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 1195	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑥 ∈ (Base‘𝑅))
55		simpr3 1197	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑧 ∈ (Base‘𝑆))
56	3, 15, 16, 17, 35, 28, 54, 55	psrvsca 21865	. . . . 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 1196	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑦 ∈ (Base‘𝑅))
58	3, 15, 16, 17, 35, 28, 57, 55	psrvsca 21865	. . . . 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 7408	. . . 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 21850	. . . . 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 6752	. . . . . 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 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑅 ∈ Ring)
66	16, 34, 35	ringdir 20178	. . . . . 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 580	. . . . 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 7699	. . . 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 7686	. . . . 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 7405	. . . 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 2772	. . 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 20194	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(+g‘𝑅)𝑦) ∈ (Base‘𝑅))
73	65, 54, 57, 72	syl3anc 1373	. . . 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 21865	. . 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 21867	. . . 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 21867	. . . 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 21853	. . 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 2775	. 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 7406	. . . 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 20169	. . . . . 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 580	. . . . 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 7696	. . . 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 7686	. . . . 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 7405	. . . 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 2772	. . 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 20166	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(.r‘𝑅)𝑦) ∈ (Base‘𝑅))
87	65, 54, 57, 86	syl3anc 1373	. . . 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 21865	. . 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 21865	. . 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 2775	. 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 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆)) → 𝑅 ∈ Ring)
92		eqid 2730	. . . . . 6 ⊢ (1r‘𝑅) = (1r‘𝑅)
93	16, 92	ringidcl 20181	. . . . 5 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ (Base‘𝑅))
94	91, 93	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆)) → (1r‘𝑅) ∈ (Base‘𝑅))
95		simpr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆)) → 𝑥 ∈ (Base‘𝑆))
96	3, 15, 16, 17, 35, 28, 94, 95	psrvsca 21865	. . 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 21850	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆)) → 𝑥:{𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑅))
99	16, 35, 92	ringlidm 20185	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑟 ∈ (Base‘𝑅)) → ((1r‘𝑅)(.r‘𝑅)𝑟) = 𝑟)
100	91, 99	sylan 580	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑟 ∈ (Base‘𝑅)) → ((1r‘𝑅)(.r‘𝑅)𝑟) = 𝑟)
101	97, 98, 94, 100	caofid0l 7689	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆)) → (({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {(1r‘𝑅)}) ∘f (.r‘𝑅)𝑥) = 𝑥)
102	96, 101	eqtrd 2765	. 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 20779	1 ⊢ (𝜑 → 𝑆 ∈ LMod)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  {crab 3408  Vcvv 3450  {csn 4592   × cxp 5639  ^◡ccnv 5640   “ cima 5644  ⟶wf 6510  ‘cfv 6514  (class class class)co 7390   ∘_f cof 7654   ↑_m cmap 8802  Fincfn 8921  ℕcn 12193  ℕ₀cn0 12449  Basecbs 17186  +_gcplusg 17227  ._rcmulr 17228   ·_𝑠 cvsca 17231  Mgmcmgm 18572  Grpcgrp 18872  1_rcur 20097  Ringcrg 20149  LModclmod 20773   mPwSer cmps 21820

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-tp 4597  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-of 7656  df-om 7846  df-1st 7971  df-2nd 7972  df-supp 8143  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-er 8674  df-map 8804  df-ixp 8874  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-fsupp 9320  df-sup 9400  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-nn 12194  df-2 12256  df-3 12257  df-4 12258  df-5 12259  df-6 12260  df-7 12261  df-8 12262  df-9 12263  df-n0 12450  df-z 12537  df-dec 12657  df-uz 12801  df-fz 13476  df-struct 17124  df-sets 17141  df-slot 17159  df-ndx 17171  df-base 17187  df-plusg 17240  df-mulr 17241  df-sca 17243  df-vsca 17244  df-ip 17245  df-tset 17246  df-ple 17247  df-ds 17249  df-hom 17251  df-cco 17252  df-0g 17411  df-prds 17417  df-pws 17419  df-mgm 18574  df-sgrp 18653  df-mnd 18669  df-grp 18875  df-minusg 18876  df-mgp 20057  df-ur 20098  df-ring 20151  df-lmod 20775  df-psr 21825

This theorem is referenced by:  psrassa  21889  psrasclcl  21896  mpllmod  21934  mplbas2  21956  psdascl  22062  opsrlmod  22137