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Theorem mpllsslem 21406

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 21357	. 2 ⊢ (𝜑 → 𝑅 = (Scalar‘𝑆))
5		eqidd 2737	. 2 ⊢ (𝜑 → (Base‘𝑅) = (Base‘𝑅))
6		mplsubglem.b	. . 3 ⊢ 𝐵 = (Base‘𝑆)
7	6	a1i 11	. 2 ⊢ (𝜑 → 𝐵 = (Base‘𝑆))
8		eqidd 2737	. 2 ⊢ (𝜑 → (+_g‘𝑆) = (+_g‘𝑆))
9		eqidd 2737	. 2 ⊢ (𝜑 → ( ·_𝑠 ‘𝑆) = ( ·_𝑠 ‘𝑆))
10		eqidd 2737	. 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 19969	. . . . 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 21405	. . 3 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝑆))
20	6	subgss 18929	. . 3 ⊢ (𝑈 ∈ (SubGrp‘𝑆) → 𝑈 ⊆ 𝐵)
21	19, 20	syl 17	. 2 ⊢ (𝜑 → 𝑈 ⊆ 𝐵)
22		eqid 2736	. . . 4 ⊢ (0_g‘𝑆) = (0_g‘𝑆)
23	22	subg0cl 18936	. . 3 ⊢ (𝑈 ∈ (SubGrp‘𝑆) → (0_g‘𝑆) ∈ 𝑈)
24		ne0i 4294	. . 3 ⊢ ((0_g‘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 2736	. . . . . 6 ⊢ ( ·_𝑠 ‘𝑆) = ( ·_𝑠 ‘𝑆)
28		eqid 2736	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
29	3	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) → 𝑅 ∈ Ring)
30		simprl 769	. . . . . 6 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) → 𝑢 ∈ (Base‘𝑅))
31		simprr 771	. . . . . . . 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 2823	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) → (𝑣 ∈ 𝑈 ↔ 𝑣 ∈ {𝑔 ∈ 𝐵 ∣ (𝑔 supp 0 ) ∈ 𝐴}))
34		oveq1 7364	. . . . . . . . . . 11 ⊢ (𝑔 = 𝑣 → (𝑔 supp 0 ) = (𝑣 supp 0 ))
35	34	eleq1d 2822	. . . . . . . . . 10 ⊢ (𝑔 = 𝑣 → ((𝑔 supp 0 ) ∈ 𝐴 ↔ (𝑣 supp 0 ) ∈ 𝐴))
36	35	elrab 3645	. . . . . . . . 9 ⊢ (𝑣 ∈ {𝑔 ∈ 𝐵 ∣ (𝑔 supp 0 ) ∈ 𝐴} ↔ (𝑣 ∈ 𝐵 ∧ (𝑣 supp 0 ) ∈ 𝐴))
37	33, 36	bitrdi 286	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) → (𝑣 ∈ 𝑈 ↔ (𝑣 ∈ 𝐵 ∧ (𝑣 supp 0 ) ∈ 𝐴)))
38	31, 37	mpbid 231	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) → (𝑣 ∈ 𝐵 ∧ (𝑣 supp 0 ) ∈ 𝐴))
39	38	simpld 495	. . . . . 6 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) → 𝑣 ∈ 𝐵)
40	1, 27, 28, 6, 29, 30, 39	psrvscacl 21361	. . . . 5 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) → (𝑢( ·_𝑠 ‘𝑆)𝑣) ∈ 𝐵)
41		ovex 7390	. . . . . . 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 3970	. . . . . . . . 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 1923	. . . . . . 7 ⊢ (𝑥 = (𝑣 supp 0 ) → (∀𝑦(𝑦 ⊆ 𝑥 → 𝑦 ∈ 𝐴) ↔ ∀𝑦(𝑦 ⊆ (𝑣 supp 0 ) → 𝑦 ∈ 𝐴)))
46	15	expr 457	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑦 ⊆ 𝑥 → 𝑦 ∈ 𝐴))
47	46	alrimiv 1930	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑦(𝑦 ⊆ 𝑥 → 𝑦 ∈ 𝐴))
48	47	ralrimiva 3143	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦(𝑦 ⊆ 𝑥 → 𝑦 ∈ 𝐴))
49	48	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) → ∀𝑥 ∈ 𝐴 ∀𝑦(𝑦 ⊆ 𝑥 → 𝑦 ∈ 𝐴))
50	38	simprd 496	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) → (𝑣 supp 0 ) ∈ 𝐴)
51	45, 49, 50	rspcdva 3582	. . . . . 6 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) → ∀𝑦(𝑦 ⊆ (𝑣 supp 0 ) → 𝑦 ∈ 𝐴))
52	1, 28, 12, 6, 40	psrelbas 21347	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) → (𝑢( ·_𝑠 ‘𝑆)𝑣):𝐷⟶(Base‘𝑅))
53		eqid 2736	. . . . . . . . 9 ⊢ (._r‘𝑅) = (._r‘𝑅)
54	30	adantr 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) ∧ 𝑘 ∈ (𝐷 ∖ (𝑣 supp 0 ))) → 𝑢 ∈ (Base‘𝑅))
55	39	adantr 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) ∧ 𝑘 ∈ (𝐷 ∖ (𝑣 supp 0 ))) → 𝑣 ∈ 𝐵)
56		eldifi 4086	. . . . . . . . . 10 ⊢ (𝑘 ∈ (𝐷 ∖ (𝑣 supp 0 )) → 𝑘 ∈ 𝐷)
57	56	adantl 482	. . . . . . . . 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 21360	. . . . . . . 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 21347	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) → 𝑣:𝐷⟶(Base‘𝑅))
60		ssidd 3967	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) → (𝑣 supp 0 ) ⊆ (𝑣 supp 0 ))
61		ovex 7390	. . . . . . . . . . . 12 ⊢ (ℕ₀ ↑_m 𝐼) ∈ V
62	12, 61	rabex2 5291	. . . . . . . . . . 11 ⊢ 𝐷 ∈ V
63	62	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) → 𝐷 ∈ V)
64	11	fvexi 6856	. . . . . . . . . . 11 ⊢ 0 ∈ V
65	64	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) → 0 ∈ V)
66	59, 60, 63, 65	suppssr 8127	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) ∧ 𝑘 ∈ (𝐷 ∖ (𝑣 supp 0 ))) → (𝑣‘𝑘) = 0 )
67	66	oveq2d 7373	. . . . . . . 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 20012	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝑢 ∈ (Base‘𝑅)) → (𝑢(._r‘𝑅) 0 ) = 0 )
69	3, 30, 68	syl2an2r 683	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) → (𝑢(._r‘𝑅) 0 ) = 0 )
70	69	adantr 481	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) ∧ 𝑘 ∈ (𝐷 ∖ (𝑣 supp 0 ))) → (𝑢(._r‘𝑅) 0 ) = 0 )
71	58, 67, 70	3eqtrd 2780	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) ∧ 𝑘 ∈ (𝐷 ∖ (𝑣 supp 0 ))) → ((𝑢( ·_𝑠 ‘𝑆)𝑣)‘𝑘) = 0 )
72	52, 71	suppss 8125	. . . . . 6 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) → ((𝑢( ·_𝑠 ‘𝑆)𝑣) supp 0 ) ⊆ (𝑣 supp 0 ))
73		sseq1 3969	. . . . . . . 8 ⊢ (𝑦 = ((𝑢( ·_𝑠 ‘𝑆)𝑣) supp 0 ) → (𝑦 ⊆ (𝑣 supp 0 ) ↔ ((𝑢( ·_𝑠 ‘𝑆)𝑣) supp 0 ) ⊆ (𝑣 supp 0 )))
74		eleq1 2825	. . . . . . . 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 3555	. . . . . 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 2823	. . . . . 6 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) → ((𝑢( ·_𝑠 ‘𝑆)𝑣) ∈ 𝑈 ↔ (𝑢( ·_𝑠 ‘𝑆)𝑣) ∈ {𝑔 ∈ 𝐵 ∣ (𝑔 supp 0 ) ∈ 𝐴}))
79		oveq1 7364	. . . . . . . 8 ⊢ (𝑔 = (𝑢( ·_𝑠 ‘𝑆)𝑣) → (𝑔 supp 0 ) = ((𝑢( ·_𝑠 ‘𝑆)𝑣) supp 0 ))
80	79	eleq1d 2822	. . . . . . 7 ⊢ (𝑔 = (𝑢( ·_𝑠 ‘𝑆)𝑣) → ((𝑔 supp 0 ) ∈ 𝐴 ↔ ((𝑢( ·_𝑠 ‘𝑆)𝑣) supp 0 ) ∈ 𝐴))
81	80	elrab 3645	. . . . . 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 286	. . . . 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 711	. . . 4 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) → (𝑢( ·_𝑠 ‘𝑆)𝑣) ∈ 𝑈)
84	83	3adantr3 1171	. . 3 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈 ∧ 𝑤 ∈ 𝑈)) → (𝑢( ·_𝑠 ‘𝑆)𝑣) ∈ 𝑈)
85		simpr3 1196	. . 3 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈 ∧ 𝑤 ∈ 𝑈)) → 𝑤 ∈ 𝑈)
86		eqid 2736	. . . 4 ⊢ (+_g‘𝑆) = (+_g‘𝑆)
87	86	subgcl 18938	. . 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 1371	. 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 20396	1 ⊢ (𝜑 → 𝑈 ∈ (LSubSp‘𝑆))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1087 ∀wal 1539 = wceq 1541 ∈ wcel 2106 ≠ wne 2943 ∀wral 3064 {crab 3407 Vcvv 3445 ∖ cdif 3907 ∪ cun 3908 ⊆ wss 3910 ∅c0 4282 ^◡ccnv 5632 “ cima 5636 ‘cfv 6496 (class class class)co 7357 supp csupp 8092 ↑_m cmap 8765 Fincfn 8883 ℕcn 12153 ℕ₀cn0 12413 Basecbs 17083 +_gcplusg 17133 ._rcmulr 17134 ·_𝑠 cvsca 17137 0_gc0g 17321 Grpcgrp 18748 SubGrpcsubg 18922 Ringcrg 19964 LSubSpclss 20392 mPwSer cmps 21306

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 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128

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 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7617 df-om 7803 df-1st 7921 df-2nd 7922 df-supp 8093 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-er 8648 df-map 8767 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9306 df-sup 9378 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-z 12500 df-dec 12619 df-uz 12764 df-fz 13425 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-sca 17149 df-vsca 17150 df-ip 17151 df-tset 17152 df-ple 17153 df-ds 17155 df-hom 17157 df-cco 17158 df-0g 17323 df-prds 17329 df-pws 17331 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-grp 18751 df-minusg 18752 df-subg 18925 df-mgp 19897 df-ring 19966 df-lss 20393 df-psr 21311

This theorem is referenced by: mpllss 21409

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