psrlmod - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > psrlmod		Structured version Visualization version GIF version

Theorem psrlmod 21512

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 2733	. 2 ⊢ (𝜑 → (Base‘𝑆) = (Base‘𝑆))
2		eqidd 2733	. 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 21499	. 2 ⊢ (𝜑 → 𝑅 = (Scalar‘𝑆))
7		eqidd 2733	. 2 ⊢ (𝜑 → ( ·_𝑠 ‘𝑆) = ( ·_𝑠 ‘𝑆))
8		eqidd 2733	. 2 ⊢ (𝜑 → (Base‘𝑅) = (Base‘𝑅))
9		eqidd 2733	. 2 ⊢ (𝜑 → (+_g‘𝑅) = (+_g‘𝑅))
10		eqidd 2733	. 2 ⊢ (𝜑 → (._r‘𝑅) = (._r‘𝑅))
11		eqidd 2733	. 2 ⊢ (𝜑 → (1_r‘𝑅) = (1_r‘𝑅))
12		ringgrp 20054	. . . 4 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp)
13	5, 12	syl 17	. . 3 ⊢ (𝜑 → 𝑅 ∈ Grp)
14	3, 4, 13	psrgrp 21508	. 2 ⊢ (𝜑 → 𝑆 ∈ Grp)
15		eqid 2732	. . 3 ⊢ ( ·_𝑠 ‘𝑆) = ( ·_𝑠 ‘𝑆)
16		eqid 2732	. . 3 ⊢ (Base‘𝑅) = (Base‘𝑅)
17		eqid 2732	. . 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 21503	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑥( ·_𝑠 ‘𝑆)𝑦) ∈ (Base‘𝑆))
22		ovex 7438	. . . . . . 7 ⊢ (ℕ₀ ↑_m 𝐼) ∈ V
23	22	rabex 5331	. . . . . 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 1194	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑥 ∈ (Base‘𝑅))
26		fconst6g 6777	. . . . . 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 2732	. . . . . 6 ⊢ {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}
29		simpr2 1195	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑦 ∈ (Base‘𝑆))
30	3, 16, 28, 17, 29	psrelbas 21489	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑦:{𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑅))
31		simpr3 1196	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑧 ∈ (Base‘𝑆))
32	3, 16, 28, 17, 31	psrelbas 21489	. . . . 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 2732	. . . . . . 7 ⊢ (+_g‘𝑅) = (+_g‘𝑅)
35		eqid 2732	. . . . . . 7 ⊢ (._r‘𝑅) = (._r‘𝑅)
36	16, 34, 35	ringdi 20074	. . . . . 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 7705	. . . 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 2732	. . . . . 6 ⊢ (+_g‘𝑆) = (+_g‘𝑆)
40	3, 17, 34, 39, 29, 31	psradd 21492	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑦(+_g‘𝑆)𝑧) = (𝑦 ∘_f (+_g‘𝑅)𝑧))
41	40	oveq2d 7421	. . . 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 21501	. . . . 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 21501	. . . . 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 7423	. . . 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 2782	. . 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 481	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑅 ∈ Grp)
47	3, 17, 39, 46, 29, 31	psraddcl 21493	. . . 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 21501	. . 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 1184	. . . 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 21503	. . . 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 21492	. . 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 2782	. 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 1194	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑥 ∈ (Base‘𝑅))
54		simpr3 1196	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑧 ∈ (Base‘𝑆))
55	3, 15, 16, 17, 35, 28, 53, 54	psrvsca 21501	. . . . 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 1195	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑦 ∈ (Base‘𝑅))
57	3, 15, 16, 17, 35, 28, 56, 54	psrvsca 21501	. . . . 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 7423	. . . 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 21489	. . . . 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 6777	. . . . . 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 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑅 ∈ Ring)
65	16, 34, 35	ringdir 20075	. . . . . 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 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)))
67	59, 60, 61, 63, 66	caofdir 7706	. . . 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 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)}))
69	68	oveq1d 7420	. . . 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 2779	. . 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 20088	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(+_g‘𝑅)𝑦) ∈ (Base‘𝑅))
72	64, 53, 56, 71	syl3anc 1371	. . . 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 21501	. . 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 21503	. . . 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 21503	. . . 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 21492	. . 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 2782	. 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 7421	. . . 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 20069	. . . . . 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 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)))
81	59, 61, 63, 60, 80	caofass 7703	. . . 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 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)}))
83	82	oveq1d 7420	. . . 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 2779	. . 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 20066	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(._r‘𝑅)𝑦) ∈ (Base‘𝑅))
86	64, 53, 56, 85	syl3anc 1371	. . . 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 21501	. . 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 21501	. . 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 2782	. 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 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆)) → 𝑅 ∈ Ring)
91		eqid 2732	. . . . . 6 ⊢ (1_r‘𝑅) = (1_r‘𝑅)
92	16, 91	ringidcl 20076	. . . . 5 ⊢ (𝑅 ∈ Ring → (1_r‘𝑅) ∈ (Base‘𝑅))
93	90, 92	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆)) → (1_r‘𝑅) ∈ (Base‘𝑅))
94		simpr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆)) → 𝑥 ∈ (Base‘𝑆))
95	3, 15, 16, 17, 35, 28, 93, 94	psrvsca 21501	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆)) → ((1_r‘𝑅)( ·_𝑠 ‘𝑆)𝑥) = (({𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin} × {(1_r‘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 21489	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆)) → 𝑥:{𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑅))
98	16, 35, 91	ringlidm 20079	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑟 ∈ (Base‘𝑅)) → ((1_r‘𝑅)(._r‘𝑅)𝑟) = 𝑟)
99	90, 98	sylan 580	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑟 ∈ (Base‘𝑅)) → ((1_r‘𝑅)(._r‘𝑅)𝑟) = 𝑟)
100	96, 97, 93, 99	caofid0l 7697	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆)) → (({𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin} × {(1_r‘𝑅)}) ∘_f (._r‘𝑅)𝑥) = 𝑥)
101	95, 100	eqtrd 2772	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆)) → ((1_r‘𝑅)( ·_𝑠 ‘𝑆)𝑥) = 𝑥)
102	1, 2, 6, 7, 8, 9, 10, 11, 5, 14, 21, 52, 77, 89, 101	islmodd 20469	1 ⊢ (𝜑 → 𝑆 ∈ LMod)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 {crab 3432 Vcvv 3474 {csn 4627 × cxp 5673 ^◡ccnv 5674 “ cima 5678 ⟶wf 6536 ‘cfv 6540 (class class class)co 7405 ∘_f cof 7664 ↑_m cmap 8816 Fincfn 8935 ℕcn 12208 ℕ₀cn0 12468 Basecbs 17140 +_gcplusg 17193 ._rcmulr 17194 ·_𝑠 cvsca 17197 Grpcgrp 18815 1_rcur 19998 Ringcrg 20049 LModclmod 20463 mPwSer cmps 21448

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 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183

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 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7666 df-om 7852 df-1st 7971 df-2nd 7972 df-supp 8143 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8699 df-map 8818 df-ixp 8888 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fsupp 9358 df-sup 9433 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-fz 13481 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-plusg 17206 df-mulr 17207 df-sca 17209 df-vsca 17210 df-ip 17211 df-tset 17212 df-ple 17213 df-ds 17215 df-hom 17217 df-cco 17218 df-0g 17383 df-prds 17389 df-pws 17391 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-grp 18818 df-minusg 18819 df-mgp 19982 df-ur 19999 df-ring 20051 df-lmod 20465 df-psr 21453

This theorem is referenced by: psrassa 21525 mpllmod 21568 mplbas2 21588 opsrlmod 21759

W3C validator