Theorem mpllsslem 19435

Description: If 𝐴 is an ideal of subsets (a nonempty collection closed under subset and binary union) of the set 𝐷 of finite bags (the primary applications being 𝐴 = Fin and 𝐴 = 𝒫 𝐵 for some 𝐵), then the set of all power series whose coefficient functions are supported on an element of 𝐴 is a linear subspace of the set of all power series. (Contributed by Mario Carneiro, 12-Jan-2015.) (Revised by AV, 16-Jul-2019.)

Hypotheses

Ref	Expression
mplsubglem.s	⊢ 𝑆 = (𝐼 mPwSer 𝑅)
mplsubglem.b	⊢ 𝐵 = (Base‘𝑆)
mplsubglem.z	⊢ 0 = (0g‘𝑅)
mplsubglem.d	⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}
mplsubglem.i	⊢ (𝜑 → 𝐼 ∈ 𝑊)
mplsubglem.0	⊢ (𝜑 → ∅ ∈ 𝐴)
mplsubglem.a	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥 ∪ 𝑦) ∈ 𝐴)
mplsubglem.y	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ⊆ 𝑥)) → 𝑦 ∈ 𝐴)
mplsubglem.u	⊢ (𝜑 → 𝑈 = {𝑔 ∈ 𝐵 ∣ (𝑔 supp 0 ) ∈ 𝐴})
mpllsslem.r	⊢ (𝜑 → 𝑅 ∈ Ring)

Assertion

Ref	Expression
mpllsslem	⊢ (𝜑 → 𝑈 ∈ (LSubSp‘𝑆))

Distinct variable groups:   𝑓,𝑔,𝑥,𝑦, 0   𝐴,𝑓,𝑔,𝑥,𝑦   𝐵,𝑓,𝑔   𝐷,𝑔   𝑓,𝐼   𝜑,𝑥,𝑦   𝑆,𝑓,𝑔,𝑦

Allowed substitution hints:   𝜑(𝑓,𝑔)   𝐵(𝑥,𝑦)   𝐷(𝑥,𝑦,𝑓)   𝑅(𝑥,𝑦,𝑓,𝑔)   𝑆(𝑥)   𝑈(𝑥,𝑦,𝑓,𝑔)   𝐼(𝑥,𝑦,𝑔)   𝑊(𝑥,𝑦,𝑓,𝑔)

Proof of Theorem mpllsslem

Dummy variables 𝑘 𝑢 𝑣 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mplsubglem.s	. . 3 ⊢ 𝑆 = (𝐼 mPwSer 𝑅)
2		mplsubglem.i	. . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑊)
3		mpllsslem.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ Ring)
4	1, 2, 3	psrsca 19389	. 2 ⊢ (𝜑 → 𝑅 = (Scalar‘𝑆))
5		eqidd 2623	. 2 ⊢ (𝜑 → (Base‘𝑅) = (Base‘𝑅))
6		mplsubglem.b	. . 3 ⊢ 𝐵 = (Base‘𝑆)
7	6	a1i 11	. 2 ⊢ (𝜑 → 𝐵 = (Base‘𝑆))
8		eqidd 2623	. 2 ⊢ (𝜑 → (+g‘𝑆) = (+g‘𝑆))
9		eqidd 2623	. 2 ⊢ (𝜑 → ( ·𝑠 ‘𝑆) = ( ·𝑠 ‘𝑆))
10		eqidd 2623	. 2 ⊢ (𝜑 → (LSubSp‘𝑆) = (LSubSp‘𝑆))
11		mplsubglem.z	. . . 4 ⊢ 0 = (0g‘𝑅)
12		mplsubglem.d	. . . 4 ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}
13		mplsubglem.0	. . . 4 ⊢ (𝜑 → ∅ ∈ 𝐴)
14		mplsubglem.a	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥 ∪ 𝑦) ∈ 𝐴)
15		mplsubglem.y	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ⊆ 𝑥)) → 𝑦 ∈ 𝐴)
16		mplsubglem.u	. . . 4 ⊢ (𝜑 → 𝑈 = {𝑔 ∈ 𝐵 ∣ (𝑔 supp 0 ) ∈ 𝐴})
17		ringgrp 18552	. . . . 5 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp)
18	3, 17	syl 17	. . . 4 ⊢ (𝜑 → 𝑅 ∈ Grp)
19	1, 6, 11, 12, 2, 13, 14, 15, 16, 18	mplsubglem 19434	. . 3 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝑆))
20	6	subgss 17595	. . 3 ⊢ (𝑈 ∈ (SubGrp‘𝑆) → 𝑈 ⊆ 𝐵)
21	19, 20	syl 17	. 2 ⊢ (𝜑 → 𝑈 ⊆ 𝐵)
22		eqid 2622	. . . 4 ⊢ (0g‘𝑆) = (0g‘𝑆)
23	22	subg0cl 17602	. . 3 ⊢ (𝑈 ∈ (SubGrp‘𝑆) → (0g‘𝑆) ∈ 𝑈)
24		ne0i 3921	. . 3 ⊢ ((0g‘𝑆) ∈ 𝑈 → 𝑈 ≠ ∅)
25	19, 23, 24	3syl 18	. 2 ⊢ (𝜑 → 𝑈 ≠ ∅)
26	19	adantr 481	. . 3 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈 ∧ 𝑤 ∈ 𝑈)) → 𝑈 ∈ (SubGrp‘𝑆))
27		eqid 2622	. . . . . 6 ⊢ ( ·𝑠 ‘𝑆) = ( ·𝑠 ‘𝑆)
28		eqid 2622	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
29	3	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) → 𝑅 ∈ Ring)
30		simprl 794	. . . . . 6 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) → 𝑢 ∈ (Base‘𝑅))
31		simprr 796	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) → 𝑣 ∈ 𝑈)
32	16	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) → 𝑈 = {𝑔 ∈ 𝐵 ∣ (𝑔 supp 0 ) ∈ 𝐴})
33	32	eleq2d 2687	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) → (𝑣 ∈ 𝑈 ↔ 𝑣 ∈ {𝑔 ∈ 𝐵 ∣ (𝑔 supp 0 ) ∈ 𝐴}))
34		oveq1 6657	. . . . . . . . . . 11 ⊢ (𝑔 = 𝑣 → (𝑔 supp 0 ) = (𝑣 supp 0 ))
35	34	eleq1d 2686	. . . . . . . . . 10 ⊢ (𝑔 = 𝑣 → ((𝑔 supp 0 ) ∈ 𝐴 ↔ (𝑣 supp 0 ) ∈ 𝐴))
36	35	elrab 3363	. . . . . . . . 9 ⊢ (𝑣 ∈ {𝑔 ∈ 𝐵 ∣ (𝑔 supp 0 ) ∈ 𝐴} ↔ (𝑣 ∈ 𝐵 ∧ (𝑣 supp 0 ) ∈ 𝐴))
37	33, 36	syl6bb 276	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) → (𝑣 ∈ 𝑈 ↔ (𝑣 ∈ 𝐵 ∧ (𝑣 supp 0 ) ∈ 𝐴)))
38	31, 37	mpbid 222	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) → (𝑣 ∈ 𝐵 ∧ (𝑣 supp 0 ) ∈ 𝐴))
39	38	simpld 475	. . . . . 6 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) → 𝑣 ∈ 𝐵)
40	1, 27, 28, 6, 29, 30, 39	psrvscacl 19393	. . . . 5 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) → (𝑢( ·𝑠 ‘𝑆)𝑣) ∈ 𝐵)
41		ovex 6678	. . . . . . 7 ⊢ ((𝑢( ·𝑠 ‘𝑆)𝑣) supp 0 ) ∈ V
42	41	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) → ((𝑢( ·𝑠 ‘𝑆)𝑣) supp 0 ) ∈ V)
43	38	simprd 479	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) → (𝑣 supp 0 ) ∈ 𝐴)
44	15	expr 643	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑦 ⊆ 𝑥 → 𝑦 ∈ 𝐴))
45	44	alrimiv 1855	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑦(𝑦 ⊆ 𝑥 → 𝑦 ∈ 𝐴))
46	45	ralrimiva 2966	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦(𝑦 ⊆ 𝑥 → 𝑦 ∈ 𝐴))
47	46	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) → ∀𝑥 ∈ 𝐴 ∀𝑦(𝑦 ⊆ 𝑥 → 𝑦 ∈ 𝐴))
48		sseq2 3627	. . . . . . . . . 10 ⊢ (𝑥 = (𝑣 supp 0 ) → (𝑦 ⊆ 𝑥 ↔ 𝑦 ⊆ (𝑣 supp 0 )))
49	48	imbi1d 331	. . . . . . . . 9 ⊢ (𝑥 = (𝑣 supp 0 ) → ((𝑦 ⊆ 𝑥 → 𝑦 ∈ 𝐴) ↔ (𝑦 ⊆ (𝑣 supp 0 ) → 𝑦 ∈ 𝐴)))
50	49	albidv 1849	. . . . . . . 8 ⊢ (𝑥 = (𝑣 supp 0 ) → (∀𝑦(𝑦 ⊆ 𝑥 → 𝑦 ∈ 𝐴) ↔ ∀𝑦(𝑦 ⊆ (𝑣 supp 0 ) → 𝑦 ∈ 𝐴)))
51	50	rspcv 3305	. . . . . . 7 ⊢ ((𝑣 supp 0 ) ∈ 𝐴 → (∀𝑥 ∈ 𝐴 ∀𝑦(𝑦 ⊆ 𝑥 → 𝑦 ∈ 𝐴) → ∀𝑦(𝑦 ⊆ (𝑣 supp 0 ) → 𝑦 ∈ 𝐴)))
52	43, 47, 51	sylc 65	. . . . . 6 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) → ∀𝑦(𝑦 ⊆ (𝑣 supp 0 ) → 𝑦 ∈ 𝐴))
53	1, 28, 12, 6, 40	psrelbas 19379	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) → (𝑢( ·𝑠 ‘𝑆)𝑣):𝐷⟶(Base‘𝑅))
54		eqid 2622	. . . . . . . . 9 ⊢ (.r‘𝑅) = (.r‘𝑅)
55	30	adantr 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) ∧ 𝑘 ∈ (𝐷 ∖ (𝑣 supp 0 ))) → 𝑢 ∈ (Base‘𝑅))
56	39	adantr 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) ∧ 𝑘 ∈ (𝐷 ∖ (𝑣 supp 0 ))) → 𝑣 ∈ 𝐵)
57		eldifi 3732	. . . . . . . . . 10 ⊢ (𝑘 ∈ (𝐷 ∖ (𝑣 supp 0 )) → 𝑘 ∈ 𝐷)
58	57	adantl 482	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) ∧ 𝑘 ∈ (𝐷 ∖ (𝑣 supp 0 ))) → 𝑘 ∈ 𝐷)
59	1, 27, 28, 6, 54, 12, 55, 56, 58	psrvscaval 19392	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) ∧ 𝑘 ∈ (𝐷 ∖ (𝑣 supp 0 ))) → ((𝑢( ·𝑠 ‘𝑆)𝑣)‘𝑘) = (𝑢(.r‘𝑅)(𝑣‘𝑘)))
60	1, 28, 12, 6, 39	psrelbas 19379	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) → 𝑣:𝐷⟶(Base‘𝑅))
61		ssid 3624	. . . . . . . . . . 11 ⊢ (𝑣 supp 0 ) ⊆ (𝑣 supp 0 )
62	61	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) → (𝑣 supp 0 ) ⊆ (𝑣 supp 0 ))
63		ovex 6678	. . . . . . . . . . . 12 ⊢ (ℕ0 ↑𝑚 𝐼) ∈ V
64	12, 63	rabex2 4815	. . . . . . . . . . 11 ⊢ 𝐷 ∈ V
65	64	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) → 𝐷 ∈ V)
66		fvex 6201	. . . . . . . . . . . 12 ⊢ (0g‘𝑅) ∈ V
67	11, 66	eqeltri 2697	. . . . . . . . . . 11 ⊢ 0 ∈ V
68	67	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) → 0 ∈ V)
69	60, 62, 65, 68	suppssr 7326	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) ∧ 𝑘 ∈ (𝐷 ∖ (𝑣 supp 0 ))) → (𝑣‘𝑘) = 0 )
70	69	oveq2d 6666	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) ∧ 𝑘 ∈ (𝐷 ∖ (𝑣 supp 0 ))) → (𝑢(.r‘𝑅)(𝑣‘𝑘)) = (𝑢(.r‘𝑅) 0 ))
71	28, 54, 11	ringrz 18588	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝑢 ∈ (Base‘𝑅)) → (𝑢(.r‘𝑅) 0 ) = 0 )
72	3, 30, 71	syl2an2r 876	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) → (𝑢(.r‘𝑅) 0 ) = 0 )
73	72	adantr 481	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) ∧ 𝑘 ∈ (𝐷 ∖ (𝑣 supp 0 ))) → (𝑢(.r‘𝑅) 0 ) = 0 )
74	59, 70, 73	3eqtrd 2660	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) ∧ 𝑘 ∈ (𝐷 ∖ (𝑣 supp 0 ))) → ((𝑢( ·𝑠 ‘𝑆)𝑣)‘𝑘) = 0 )
75	53, 74	suppss 7325	. . . . . 6 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) → ((𝑢( ·𝑠 ‘𝑆)𝑣) supp 0 ) ⊆ (𝑣 supp 0 ))
76		sseq1 3626	. . . . . . . 8 ⊢ (𝑦 = ((𝑢( ·𝑠 ‘𝑆)𝑣) supp 0 ) → (𝑦 ⊆ (𝑣 supp 0 ) ↔ ((𝑢( ·𝑠 ‘𝑆)𝑣) supp 0 ) ⊆ (𝑣 supp 0 )))
77		eleq1 2689	. . . . . . . 8 ⊢ (𝑦 = ((𝑢( ·𝑠 ‘𝑆)𝑣) supp 0 ) → (𝑦 ∈ 𝐴 ↔ ((𝑢( ·𝑠 ‘𝑆)𝑣) supp 0 ) ∈ 𝐴))
78	76, 77	imbi12d 334	. . . . . . 7 ⊢ (𝑦 = ((𝑢( ·𝑠 ‘𝑆)𝑣) supp 0 ) → ((𝑦 ⊆ (𝑣 supp 0 ) → 𝑦 ∈ 𝐴) ↔ (((𝑢( ·𝑠 ‘𝑆)𝑣) supp 0 ) ⊆ (𝑣 supp 0 ) → ((𝑢( ·𝑠 ‘𝑆)𝑣) supp 0 ) ∈ 𝐴)))
79	78	spcgv 3293	. . . . . 6 ⊢ (((𝑢( ·𝑠 ‘𝑆)𝑣) supp 0 ) ∈ V → (∀𝑦(𝑦 ⊆ (𝑣 supp 0 ) → 𝑦 ∈ 𝐴) → (((𝑢( ·𝑠 ‘𝑆)𝑣) supp 0 ) ⊆ (𝑣 supp 0 ) → ((𝑢( ·𝑠 ‘𝑆)𝑣) supp 0 ) ∈ 𝐴)))
80	42, 52, 75, 79	syl3c 66	. . . . 5 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) → ((𝑢( ·𝑠 ‘𝑆)𝑣) supp 0 ) ∈ 𝐴)
81	32	eleq2d 2687	. . . . . 6 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) → ((𝑢( ·𝑠 ‘𝑆)𝑣) ∈ 𝑈 ↔ (𝑢( ·𝑠 ‘𝑆)𝑣) ∈ {𝑔 ∈ 𝐵 ∣ (𝑔 supp 0 ) ∈ 𝐴}))
82		oveq1 6657	. . . . . . . 8 ⊢ (𝑔 = (𝑢( ·𝑠 ‘𝑆)𝑣) → (𝑔 supp 0 ) = ((𝑢( ·𝑠 ‘𝑆)𝑣) supp 0 ))
83	82	eleq1d 2686	. . . . . . 7 ⊢ (𝑔 = (𝑢( ·𝑠 ‘𝑆)𝑣) → ((𝑔 supp 0 ) ∈ 𝐴 ↔ ((𝑢( ·𝑠 ‘𝑆)𝑣) supp 0 ) ∈ 𝐴))
84	83	elrab 3363	. . . . . 6 ⊢ ((𝑢( ·𝑠 ‘𝑆)𝑣) ∈ {𝑔 ∈ 𝐵 ∣ (𝑔 supp 0 ) ∈ 𝐴} ↔ ((𝑢( ·𝑠 ‘𝑆)𝑣) ∈ 𝐵 ∧ ((𝑢( ·𝑠 ‘𝑆)𝑣) supp 0 ) ∈ 𝐴))
85	81, 84	syl6bb 276	. . . . 5 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) → ((𝑢( ·𝑠 ‘𝑆)𝑣) ∈ 𝑈 ↔ ((𝑢( ·𝑠 ‘𝑆)𝑣) ∈ 𝐵 ∧ ((𝑢( ·𝑠 ‘𝑆)𝑣) supp 0 ) ∈ 𝐴)))
86	40, 80, 85	mpbir2and 957	. . . 4 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) → (𝑢( ·𝑠 ‘𝑆)𝑣) ∈ 𝑈)
87	86	3adantr3 1222	. . 3 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈 ∧ 𝑤 ∈ 𝑈)) → (𝑢( ·𝑠 ‘𝑆)𝑣) ∈ 𝑈)
88		simpr3 1069	. . 3 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈 ∧ 𝑤 ∈ 𝑈)) → 𝑤 ∈ 𝑈)
89		eqid 2622	. . . 4 ⊢ (+g‘𝑆) = (+g‘𝑆)
90	89	subgcl 17604	. . 3 ⊢ ((𝑈 ∈ (SubGrp‘𝑆) ∧ (𝑢( ·𝑠 ‘𝑆)𝑣) ∈ 𝑈 ∧ 𝑤 ∈ 𝑈) → ((𝑢( ·𝑠 ‘𝑆)𝑣)(+g‘𝑆)𝑤) ∈ 𝑈)
91	26, 87, 88, 90	syl3anc 1326	. 2 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈 ∧ 𝑤 ∈ 𝑈)) → ((𝑢( ·𝑠 ‘𝑆)𝑣)(+g‘𝑆)𝑤) ∈ 𝑈)
92	4, 5, 7, 8, 9, 10, 21, 25, 91	islssd 18936	1 ⊢ (𝜑 → 𝑈 ∈ (LSubSp‘𝑆))

Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1037  ∀wal 1481   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912  {crab 2916  Vcvv 3200   ∖ cdif 3571   ∪ cun 3572   ⊆ wss 3574  ∅c0 3915  ^◡ccnv 5113   “ cima 5117  ‘cfv 5888  (class class class)co 6650   supp csupp 7295   ↑_𝑚 cmap 7857  Fincfn 7955  ℕcn 11020  ℕ₀cn0 11292  Basecbs 15857  +_gcplusg 15941  ._rcmulr 15942   ·_𝑠 cvsca 15945  0_gc0g 16100  Grpcgrp 17422  SubGrpcsubg 17588  Ringcrg 18547  LSubSpclss 18932   mPwSer cmps 19351

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-sca 15957  df-vsca 15958  df-tset 15960  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-subg 17591  df-mgp 18490  df-ring 18549  df-lss 18933  df-psr 19356

This theorem is referenced by:  mpllss  19438