Theorem psrlmod 20639

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 2799	. 2 ⊢ (𝜑 → (Base‘𝑆) = (Base‘𝑆))
2		eqidd 2799	. 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 20627	. 2 ⊢ (𝜑 → 𝑅 = (Scalar‘𝑆))
7		eqidd 2799	. 2 ⊢ (𝜑 → ( ·𝑠 ‘𝑆) = ( ·𝑠 ‘𝑆))
8		eqidd 2799	. 2 ⊢ (𝜑 → (Base‘𝑅) = (Base‘𝑅))
9		eqidd 2799	. 2 ⊢ (𝜑 → (+g‘𝑅) = (+g‘𝑅))
10		eqidd 2799	. 2 ⊢ (𝜑 → (.r‘𝑅) = (.r‘𝑅))
11		eqidd 2799	. 2 ⊢ (𝜑 → (1r‘𝑅) = (1r‘𝑅))
12		ringgrp 19295	. . . 4 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp)
13	5, 12	syl 17	. . 3 ⊢ (𝜑 → 𝑅 ∈ Grp)
14	3, 4, 13	psrgrp 20636	. 2 ⊢ (𝜑 → 𝑆 ∈ Grp)
15		eqid 2798	. . 3 ⊢ ( ·𝑠 ‘𝑆) = ( ·𝑠 ‘𝑆)
16		eqid 2798	. . 3 ⊢ (Base‘𝑅) = (Base‘𝑅)
17		eqid 2798	. . 3 ⊢ (Base‘𝑆) = (Base‘𝑆)
18	5	3ad2ant1 1130	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑅 ∈ Ring)
19		simp2 1134	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑥 ∈ (Base‘𝑅))
20		simp3 1135	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑦 ∈ (Base‘𝑆))
21	3, 15, 16, 17, 18, 19, 20	psrvscacl 20631	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑥( ·𝑠 ‘𝑆)𝑦) ∈ (Base‘𝑆))
22		ovex 7168	. . . . . . 7 ⊢ (ℕ0 ↑m 𝐼) ∈ V
23	22	rabex 5199	. . . . . 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 1191	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑥 ∈ (Base‘𝑅))
26		fconst6g 6542	. . . . . 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 2798	. . . . . 6 ⊢ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}
29		simpr2 1192	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑦 ∈ (Base‘𝑆))
30	3, 16, 28, 17, 29	psrelbas 20617	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑦:{𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑅))
31		simpr3 1193	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑧 ∈ (Base‘𝑆))
32	3, 16, 28, 17, 31	psrelbas 20617	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑧:{𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑅))
33	5	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑅 ∈ Ring)
34		eqid 2798	. . . . . . 7 ⊢ (+g‘𝑅) = (+g‘𝑅)
35		eqid 2798	. . . . . . 7 ⊢ (.r‘𝑅) = (.r‘𝑅)
36	16, 34, 35	ringdi 19312	. . . . . 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 583	. . . . 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 7425	. . . 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 2798	. . . . . 6 ⊢ (+g‘𝑆) = (+g‘𝑆)
40	3, 17, 34, 39, 29, 31	psradd 20620	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑦(+g‘𝑆)𝑧) = (𝑦 ∘f (+g‘𝑅)𝑧))
41	40	oveq2d 7151	. . . 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 20629	. . . . 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 20629	. . . . 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 7153	. . . 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 2843	. . 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	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑅 ∈ Grp)
47	3, 17, 39, 46, 29, 31	psraddcl 20621	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑦(+g‘𝑆)𝑧) ∈ (Base‘𝑆))
48	3, 15, 16, 17, 35, 28, 25, 47	psrvsca 20629	. . 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)))
49	21	3adant3r3 1181	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑥( ·𝑠 ‘𝑆)𝑦) ∈ (Base‘𝑆))
50	3, 15, 16, 17, 33, 25, 31	psrvscacl 20631	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑥( ·𝑠 ‘𝑆)𝑧) ∈ (Base‘𝑆))
51	3, 17, 34, 39, 49, 50	psradd 20620	. . 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)))
52	45, 48, 51	3eqtr4d 2843	. 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)))
53		simpr1 1191	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑥 ∈ (Base‘𝑅))
54		simpr3 1193	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑧 ∈ (Base‘𝑆))
55	3, 15, 16, 17, 35, 28, 53, 54	psrvsca 20629	. . . . 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))
56		simpr2 1192	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑦 ∈ (Base‘𝑅))
57	3, 15, 16, 17, 35, 28, 56, 54	psrvsca 20629	. . . . 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))
58	55, 57	oveq12d 7153	. . . 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)))
59	23	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∈ V)
60	3, 16, 28, 17, 54	psrelbas 20617	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑧:{𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑅))
61	53, 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))
62		fconst6g 6542	. . . . . 6 ⊢ (𝑦 ∈ (Base‘𝑅) → ({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑦}):{𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑅))
63	56, 62	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))
64	5	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑅 ∈ Ring)
65	16, 34, 35	ringdir 19313	. . . . . 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)))
66	64, 65	sylan 583	. . . . 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)))
67	59, 60, 61, 63, 66	caofdir 7426	. . . 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)))
68	59, 53, 56	ofc12 7414	. . . . 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)}))
69	68	oveq1d 7150	. . . 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))
70	58, 67, 69	3eqtr2rd 2840	. . 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)))
71	16, 34	ringacl 19324	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(+g‘𝑅)𝑦) ∈ (Base‘𝑅))
72	64, 53, 56, 71	syl3anc 1368	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑥(+g‘𝑅)𝑦) ∈ (Base‘𝑅))
73	3, 15, 16, 17, 35, 28, 72, 54	psrvsca 20629	. . 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))
74	3, 15, 16, 17, 64, 53, 54	psrvscacl 20631	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑥( ·𝑠 ‘𝑆)𝑧) ∈ (Base‘𝑆))
75	3, 15, 16, 17, 64, 56, 54	psrvscacl 20631	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑦( ·𝑠 ‘𝑆)𝑧) ∈ (Base‘𝑆))
76	3, 17, 34, 39, 74, 75	psradd 20620	. . 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)))
77	70, 73, 76	3eqtr4d 2843	. 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)))
78	57	oveq2d 7151	. . . 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)))
79	16, 35	ringass 19310	. . . . . 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)))
80	64, 79	sylan 583	. . . . 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)))
81	59, 61, 63, 60, 80	caofass 7423	. . . 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)))
82	59, 53, 56	ofc12 7414	. . . . 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)}))
83	82	oveq1d 7150	. . . 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))
84	78, 81, 83	3eqtr2rd 2840	. . 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)))
85	16, 35	ringcl 19307	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(.r‘𝑅)𝑦) ∈ (Base‘𝑅))
86	64, 53, 56, 85	syl3anc 1368	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑥(.r‘𝑅)𝑦) ∈ (Base‘𝑅))
87	3, 15, 16, 17, 35, 28, 86, 54	psrvsca 20629	. . 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))
88	3, 15, 16, 17, 35, 28, 53, 75	psrvsca 20629	. . 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)))
89	84, 87, 88	3eqtr4d 2843	. 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)))
90	5	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆)) → 𝑅 ∈ Ring)
91		eqid 2798	. . . . . 6 ⊢ (1r‘𝑅) = (1r‘𝑅)
92	16, 91	ringidcl 19314	. . . . 5 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ (Base‘𝑅))
93	90, 92	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆)) → (1r‘𝑅) ∈ (Base‘𝑅))
94		simpr 488	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆)) → 𝑥 ∈ (Base‘𝑆))
95	3, 15, 16, 17, 35, 28, 93, 94	psrvsca 20629	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆)) → ((1r‘𝑅)( ·𝑠 ‘𝑆)𝑥) = (({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {(1r‘𝑅)}) ∘f (.r‘𝑅)𝑥))
96	23	a1i 11	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆)) → {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∈ V)
97	3, 16, 28, 17, 94	psrelbas 20617	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆)) → 𝑥:{𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑅))
98	16, 35, 91	ringlidm 19317	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑟 ∈ (Base‘𝑅)) → ((1r‘𝑅)(.r‘𝑅)𝑟) = 𝑟)
99	90, 98	sylan 583	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑟 ∈ (Base‘𝑅)) → ((1r‘𝑅)(.r‘𝑅)𝑟) = 𝑟)
100	96, 97, 93, 99	caofid0l 7417	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆)) → (({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} × {(1r‘𝑅)}) ∘f (.r‘𝑅)𝑥) = 𝑥)
101	95, 100	eqtrd 2833	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆)) → ((1r‘𝑅)( ·𝑠 ‘𝑆)𝑥) = 𝑥)
102	1, 2, 6, 7, 8, 9, 10, 11, 5, 14, 21, 52, 77, 89, 101	islmodd 19633	1 ⊢ (𝜑 → 𝑆 ∈ LMod)

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  {crab 3110  Vcvv 3441  {csn 4525   × cxp 5517  ^◡ccnv 5518   “ cima 5522  ⟶wf 6320  ‘cfv 6324  (class class class)co 7135   ∘_f cof 7387   ↑_m cmap 8389  Fincfn 8492  ℕcn 11625  ℕ₀cn0 11885  Basecbs 16475  +_gcplusg 16557  ._rcmulr 16558   ·_𝑠 cvsca 16561  Grpcgrp 18095  1_rcur 19244  Ringcrg 19290  LModclmod 19627   mPwSer cmps 20589

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-of 7389  df-om 7561  df-1st 7671  df-2nd 7672  df-supp 7814  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-oadd 8089  df-er 8272  df-map 8391  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-fsupp 8818  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11626  df-2 11688  df-3 11689  df-4 11690  df-5 11691  df-6 11692  df-7 11693  df-8 11694  df-9 11695  df-n0 11886  df-z 11970  df-uz 12232  df-fz 12886  df-struct 16477  df-ndx 16478  df-slot 16479  df-base 16481  df-sets 16482  df-plusg 16570  df-mulr 16571  df-sca 16573  df-vsca 16574  df-tset 16576  df-0g 16707  df-mgm 17844  df-sgrp 17893  df-mnd 17904  df-grp 18098  df-minusg 18099  df-mgp 19233  df-ur 19245  df-ring 19292  df-lmod 19629  df-psr 20594

This theorem is referenced by:  psrassa  20652  mpllmod  20690  mplbas2  20710  opsrlmod  20875