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Theorem mpllsslem 22020

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 = (0_g‘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 21967	. 2 ⊢ (𝜑 → 𝑅 = (Scalar‘𝑆))
5		eqidd 2738	. 2 ⊢ (𝜑 → (Base‘𝑅) = (Base‘𝑅))
6		mplsubglem.b	. . 3 ⊢ 𝐵 = (Base‘𝑆)
7	6	a1i 11	. 2 ⊢ (𝜑 → 𝐵 = (Base‘𝑆))
8		eqidd 2738	. 2 ⊢ (𝜑 → (+_g‘𝑆) = (+_g‘𝑆))
9		eqidd 2738	. 2 ⊢ (𝜑 → ( ·_𝑠 ‘𝑆) = ( ·_𝑠 ‘𝑆))
10		eqidd 2738	. 2 ⊢ (𝜑 → (LSubSp‘𝑆) = (LSubSp‘𝑆))
11		mplsubglem.z	. . . 4 ⊢ 0 = (0_g‘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 20235	. . . . 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 22019	. . 3 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝑆))
20	6	subgss 19145	. . 3 ⊢ (𝑈 ∈ (SubGrp‘𝑆) → 𝑈 ⊆ 𝐵)
21	19, 20	syl 17	. 2 ⊢ (𝜑 → 𝑈 ⊆ 𝐵)
22		eqid 2737	. . . 4 ⊢ (0_g‘𝑆) = (0_g‘𝑆)
23	22	subg0cl 19152	. . 3 ⊢ (𝑈 ∈ (SubGrp‘𝑆) → (0_g‘𝑆) ∈ 𝑈)
24		ne0i 4341	. . 3 ⊢ ((0_g‘𝑆) ∈ 𝑈 → 𝑈 ≠ ∅)
25	19, 23, 24	3syl 18	. 2 ⊢ (𝜑 → 𝑈 ≠ ∅)
26	19	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈 ∧ 𝑤 ∈ 𝑈)) → 𝑈 ∈ (SubGrp‘𝑆))
27		eqid 2737	. . . . . 6 ⊢ ( ·_𝑠 ‘𝑆) = ( ·_𝑠 ‘𝑆)
28		eqid 2737	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
29	3	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) → 𝑅 ∈ Ring)
30		simprl 771	. . . . . 6 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) → 𝑢 ∈ (Base‘𝑅))
31		simprr 773	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) → 𝑣 ∈ 𝑈)
32	16	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) → 𝑈 = {𝑔 ∈ 𝐵 ∣ (𝑔 supp 0 ) ∈ 𝐴})
33	32	eleq2d 2827	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) → (𝑣 ∈ 𝑈 ↔ 𝑣 ∈ {𝑔 ∈ 𝐵 ∣ (𝑔 supp 0 ) ∈ 𝐴}))
34		oveq1 7438	. . . . . . . . . . 11 ⊢ (𝑔 = 𝑣 → (𝑔 supp 0 ) = (𝑣 supp 0 ))
35	34	eleq1d 2826	. . . . . . . . . 10 ⊢ (𝑔 = 𝑣 → ((𝑔 supp 0 ) ∈ 𝐴 ↔ (𝑣 supp 0 ) ∈ 𝐴))
36	35	elrab 3692	. . . . . . . . 9 ⊢ (𝑣 ∈ {𝑔 ∈ 𝐵 ∣ (𝑔 supp 0 ) ∈ 𝐴} ↔ (𝑣 ∈ 𝐵 ∧ (𝑣 supp 0 ) ∈ 𝐴))
37	33, 36	bitrdi 287	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) → (𝑣 ∈ 𝑈 ↔ (𝑣 ∈ 𝐵 ∧ (𝑣 supp 0 ) ∈ 𝐴)))
38	31, 37	mpbid 232	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) → (𝑣 ∈ 𝐵 ∧ (𝑣 supp 0 ) ∈ 𝐴))
39	38	simpld 494	. . . . . 6 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) → 𝑣 ∈ 𝐵)
40	1, 27, 28, 6, 29, 30, 39	psrvscacl 21971	. . . . 5 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) → (𝑢( ·_𝑠 ‘𝑆)𝑣) ∈ 𝐵)
41		ovex 7464	. . . . . . 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		sseq2 4010	. . . . . . . . 9 ⊢ (𝑥 = (𝑣 supp 0 ) → (𝑦 ⊆ 𝑥 ↔ 𝑦 ⊆ (𝑣 supp 0 )))
44	43	imbi1d 341	. . . . . . . 8 ⊢ (𝑥 = (𝑣 supp 0 ) → ((𝑦 ⊆ 𝑥 → 𝑦 ∈ 𝐴) ↔ (𝑦 ⊆ (𝑣 supp 0 ) → 𝑦 ∈ 𝐴)))
45	44	albidv 1920	. . . . . . 7 ⊢ (𝑥 = (𝑣 supp 0 ) → (∀𝑦(𝑦 ⊆ 𝑥 → 𝑦 ∈ 𝐴) ↔ ∀𝑦(𝑦 ⊆ (𝑣 supp 0 ) → 𝑦 ∈ 𝐴)))
46	15	expr 456	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑦 ⊆ 𝑥 → 𝑦 ∈ 𝐴))
47	46	alrimiv 1927	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑦(𝑦 ⊆ 𝑥 → 𝑦 ∈ 𝐴))
48	47	ralrimiva 3146	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦(𝑦 ⊆ 𝑥 → 𝑦 ∈ 𝐴))
49	48	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) → ∀𝑥 ∈ 𝐴 ∀𝑦(𝑦 ⊆ 𝑥 → 𝑦 ∈ 𝐴))
50	38	simprd 495	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) → (𝑣 supp 0 ) ∈ 𝐴)
51	45, 49, 50	rspcdva 3623	. . . . . 6 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) → ∀𝑦(𝑦 ⊆ (𝑣 supp 0 ) → 𝑦 ∈ 𝐴))
52	1, 28, 12, 6, 40	psrelbas 21954	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) → (𝑢( ·_𝑠 ‘𝑆)𝑣):𝐷⟶(Base‘𝑅))
53		eqid 2737	. . . . . . . . 9 ⊢ (._r‘𝑅) = (._r‘𝑅)
54	30	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) ∧ 𝑘 ∈ (𝐷 ∖ (𝑣 supp 0 ))) → 𝑢 ∈ (Base‘𝑅))
55	39	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) ∧ 𝑘 ∈ (𝐷 ∖ (𝑣 supp 0 ))) → 𝑣 ∈ 𝐵)
56		eldifi 4131	. . . . . . . . . 10 ⊢ (𝑘 ∈ (𝐷 ∖ (𝑣 supp 0 )) → 𝑘 ∈ 𝐷)
57	56	adantl 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) ∧ 𝑘 ∈ (𝐷 ∖ (𝑣 supp 0 ))) → 𝑘 ∈ 𝐷)
58	1, 27, 28, 6, 53, 12, 54, 55, 57	psrvscaval 21970	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) ∧ 𝑘 ∈ (𝐷 ∖ (𝑣 supp 0 ))) → ((𝑢( ·_𝑠 ‘𝑆)𝑣)‘𝑘) = (𝑢(._r‘𝑅)(𝑣‘𝑘)))
59	1, 28, 12, 6, 39	psrelbas 21954	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) → 𝑣:𝐷⟶(Base‘𝑅))
60		ssidd 4007	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) → (𝑣 supp 0 ) ⊆ (𝑣 supp 0 ))
61		ovex 7464	. . . . . . . . . . . 12 ⊢ (ℕ₀ ↑_m 𝐼) ∈ V
62	12, 61	rabex2 5341	. . . . . . . . . . 11 ⊢ 𝐷 ∈ V
63	62	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) → 𝐷 ∈ V)
64	11	fvexi 6920	. . . . . . . . . . 11 ⊢ 0 ∈ V
65	64	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) → 0 ∈ V)
66	59, 60, 63, 65	suppssr 8220	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) ∧ 𝑘 ∈ (𝐷 ∖ (𝑣 supp 0 ))) → (𝑣‘𝑘) = 0 )
67	66	oveq2d 7447	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) ∧ 𝑘 ∈ (𝐷 ∖ (𝑣 supp 0 ))) → (𝑢(._r‘𝑅)(𝑣‘𝑘)) = (𝑢(._r‘𝑅) 0 ))
68	28, 53, 11	ringrz 20291	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝑢 ∈ (Base‘𝑅)) → (𝑢(._r‘𝑅) 0 ) = 0 )
69	3, 30, 68	syl2an2r 685	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) → (𝑢(._r‘𝑅) 0 ) = 0 )
70	69	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) ∧ 𝑘 ∈ (𝐷 ∖ (𝑣 supp 0 ))) → (𝑢(._r‘𝑅) 0 ) = 0 )
71	58, 67, 70	3eqtrd 2781	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) ∧ 𝑘 ∈ (𝐷 ∖ (𝑣 supp 0 ))) → ((𝑢( ·_𝑠 ‘𝑆)𝑣)‘𝑘) = 0 )
72	52, 71	suppss 8219	. . . . . 6 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) → ((𝑢( ·_𝑠 ‘𝑆)𝑣) supp 0 ) ⊆ (𝑣 supp 0 ))
73		sseq1 4009	. . . . . . . 8 ⊢ (𝑦 = ((𝑢( ·_𝑠 ‘𝑆)𝑣) supp 0 ) → (𝑦 ⊆ (𝑣 supp 0 ) ↔ ((𝑢( ·_𝑠 ‘𝑆)𝑣) supp 0 ) ⊆ (𝑣 supp 0 )))
74		eleq1 2829	. . . . . . . 8 ⊢ (𝑦 = ((𝑢( ·_𝑠 ‘𝑆)𝑣) supp 0 ) → (𝑦 ∈ 𝐴 ↔ ((𝑢( ·_𝑠 ‘𝑆)𝑣) supp 0 ) ∈ 𝐴))
75	73, 74	imbi12d 344	. . . . . . 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)))
76	75	spcgv 3596	. . . . . 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)))
77	42, 51, 72, 76	syl3c 66	. . . . 5 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) → ((𝑢( ·_𝑠 ‘𝑆)𝑣) supp 0 ) ∈ 𝐴)
78	32	eleq2d 2827	. . . . . 6 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) → ((𝑢( ·_𝑠 ‘𝑆)𝑣) ∈ 𝑈 ↔ (𝑢( ·_𝑠 ‘𝑆)𝑣) ∈ {𝑔 ∈ 𝐵 ∣ (𝑔 supp 0 ) ∈ 𝐴}))
79		oveq1 7438	. . . . . . . 8 ⊢ (𝑔 = (𝑢( ·_𝑠 ‘𝑆)𝑣) → (𝑔 supp 0 ) = ((𝑢( ·_𝑠 ‘𝑆)𝑣) supp 0 ))
80	79	eleq1d 2826	. . . . . . 7 ⊢ (𝑔 = (𝑢( ·_𝑠 ‘𝑆)𝑣) → ((𝑔 supp 0 ) ∈ 𝐴 ↔ ((𝑢( ·_𝑠 ‘𝑆)𝑣) supp 0 ) ∈ 𝐴))
81	80	elrab 3692	. . . . . 6 ⊢ ((𝑢( ·_𝑠 ‘𝑆)𝑣) ∈ {𝑔 ∈ 𝐵 ∣ (𝑔 supp 0 ) ∈ 𝐴} ↔ ((𝑢( ·_𝑠 ‘𝑆)𝑣) ∈ 𝐵 ∧ ((𝑢( ·_𝑠 ‘𝑆)𝑣) supp 0 ) ∈ 𝐴))
82	78, 81	bitrdi 287	. . . . 5 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) → ((𝑢( ·_𝑠 ‘𝑆)𝑣) ∈ 𝑈 ↔ ((𝑢( ·_𝑠 ‘𝑆)𝑣) ∈ 𝐵 ∧ ((𝑢( ·_𝑠 ‘𝑆)𝑣) supp 0 ) ∈ 𝐴)))
83	40, 77, 82	mpbir2and 713	. . . 4 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) → (𝑢( ·_𝑠 ‘𝑆)𝑣) ∈ 𝑈)
84	83	3adantr3 1172	. . 3 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈 ∧ 𝑤 ∈ 𝑈)) → (𝑢( ·_𝑠 ‘𝑆)𝑣) ∈ 𝑈)
85		simpr3 1197	. . 3 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈 ∧ 𝑤 ∈ 𝑈)) → 𝑤 ∈ 𝑈)
86		eqid 2737	. . . 4 ⊢ (+_g‘𝑆) = (+_g‘𝑆)
87	86	subgcl 19154	. . 3 ⊢ ((𝑈 ∈ (SubGrp‘𝑆) ∧ (𝑢( ·_𝑠 ‘𝑆)𝑣) ∈ 𝑈 ∧ 𝑤 ∈ 𝑈) → ((𝑢( ·_𝑠 ‘𝑆)𝑣)(+_g‘𝑆)𝑤) ∈ 𝑈)
88	26, 84, 85, 87	syl3anc 1373	. 2 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈 ∧ 𝑤 ∈ 𝑈)) → ((𝑢( ·_𝑠 ‘𝑆)𝑣)(+_g‘𝑆)𝑤) ∈ 𝑈)
89	4, 5, 7, 8, 9, 10, 21, 25, 88	islssd 20933	1 ⊢ (𝜑 → 𝑈 ∈ (LSubSp‘𝑆))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 ∀wal 1538 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∀wral 3061 {crab 3436 Vcvv 3480 ∖ cdif 3948 ∪ cun 3949 ⊆ wss 3951 ∅c0 4333 ^◡ccnv 5684 “ cima 5688 ‘cfv 6561 (class class class)co 7431 supp csupp 8185 ↑_m cmap 8866 Fincfn 8985 ℕcn 12266 ℕ₀cn0 12526 Basecbs 17247 +_gcplusg 17297 ._rcmulr 17298 ·_𝑠 cvsca 17301 0_gc0g 17484 Grpcgrp 18951 SubGrpcsubg 19138 Ringcrg 20230 LSubSpclss 20929 mPwSer cmps 21924

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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-map 8868 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-sup 9482 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-fz 13548 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-hom 17321 df-cco 17322 df-0g 17486 df-prds 17492 df-pws 17494 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-minusg 18955 df-subg 19141 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-lss 20930 df-psr 21929

This theorem is referenced by: mpllss 22023

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