mpllsslem - Metamath Proof Explorer

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

Theorem mpllsslem 19637

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 19591	. 2 ⊢ (𝜑 → 𝑅 = (Scalar‘𝑆))
5		eqidd 2761	. 2 ⊢ (𝜑 → (Base‘𝑅) = (Base‘𝑅))
6		mplsubglem.b	. . 3 ⊢ 𝐵 = (Base‘𝑆)
7	6	a1i 11	. 2 ⊢ (𝜑 → 𝐵 = (Base‘𝑆))
8		eqidd 2761	. 2 ⊢ (𝜑 → (+_g‘𝑆) = (+_g‘𝑆))
9		eqidd 2761	. 2 ⊢ (𝜑 → ( ·_𝑠 ‘𝑆) = ( ·_𝑠 ‘𝑆))
10		eqidd 2761	. 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 18752	. . . . 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 19636	. . 3 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝑆))
20	6	subgss 17796	. . 3 ⊢ (𝑈 ∈ (SubGrp‘𝑆) → 𝑈 ⊆ 𝐵)
21	19, 20	syl 17	. 2 ⊢ (𝜑 → 𝑈 ⊆ 𝐵)
22		eqid 2760	. . . 4 ⊢ (0_g‘𝑆) = (0_g‘𝑆)
23	22	subg0cl 17803	. . 3 ⊢ (𝑈 ∈ (SubGrp‘𝑆) → (0_g‘𝑆) ∈ 𝑈)
24		ne0i 4064	. . 3 ⊢ ((0_g‘𝑆) ∈ 𝑈 → 𝑈 ≠ ∅)
25	19, 23, 24	3syl 18	. 2 ⊢ (𝜑 → 𝑈 ≠ ∅)
26	19	adantr 472	. . 3 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈 ∧ 𝑤 ∈ 𝑈)) → 𝑈 ∈ (SubGrp‘𝑆))
27		eqid 2760	. . . . . 6 ⊢ ( ·_𝑠 ‘𝑆) = ( ·_𝑠 ‘𝑆)
28		eqid 2760	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
29	3	adantr 472	. . . . . 6 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) → 𝑅 ∈ Ring)
30		simprl 811	. . . . . 6 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) → 𝑢 ∈ (Base‘𝑅))
31		simprr 813	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) → 𝑣 ∈ 𝑈)
32	16	adantr 472	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) → 𝑈 = {𝑔 ∈ 𝐵 ∣ (𝑔 supp 0 ) ∈ 𝐴})
33	32	eleq2d 2825	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) → (𝑣 ∈ 𝑈 ↔ 𝑣 ∈ {𝑔 ∈ 𝐵 ∣ (𝑔 supp 0 ) ∈ 𝐴}))
34		oveq1 6820	. . . . . . . . . . 11 ⊢ (𝑔 = 𝑣 → (𝑔 supp 0 ) = (𝑣 supp 0 ))
35	34	eleq1d 2824	. . . . . . . . . 10 ⊢ (𝑔 = 𝑣 → ((𝑔 supp 0 ) ∈ 𝐴 ↔ (𝑣 supp 0 ) ∈ 𝐴))
36	35	elrab 3504	. . . . . . . . 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 477	. . . . . 6 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) → 𝑣 ∈ 𝐵)
40	1, 27, 28, 6, 29, 30, 39	psrvscacl 19595	. . . . 5 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) → (𝑢( ·_𝑠 ‘𝑆)𝑣) ∈ 𝐵)
41		ovex 6841	. . . . . . 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 3768	. . . . . . . . 9 ⊢ (𝑥 = (𝑣 supp 0 ) → (𝑦 ⊆ 𝑥 ↔ 𝑦 ⊆ (𝑣 supp 0 )))
44	43	imbi1d 330	. . . . . . . 8 ⊢ (𝑥 = (𝑣 supp 0 ) → ((𝑦 ⊆ 𝑥 → 𝑦 ∈ 𝐴) ↔ (𝑦 ⊆ (𝑣 supp 0 ) → 𝑦 ∈ 𝐴)))
45	44	albidv 1998	. . . . . . 7 ⊢ (𝑥 = (𝑣 supp 0 ) → (∀𝑦(𝑦 ⊆ 𝑥 → 𝑦 ∈ 𝐴) ↔ ∀𝑦(𝑦 ⊆ (𝑣 supp 0 ) → 𝑦 ∈ 𝐴)))
46	15	expr 644	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑦 ⊆ 𝑥 → 𝑦 ∈ 𝐴))
47	46	alrimiv 2004	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑦(𝑦 ⊆ 𝑥 → 𝑦 ∈ 𝐴))
48	47	ralrimiva 3104	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦(𝑦 ⊆ 𝑥 → 𝑦 ∈ 𝐴))
49	48	adantr 472	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) → ∀𝑥 ∈ 𝐴 ∀𝑦(𝑦 ⊆ 𝑥 → 𝑦 ∈ 𝐴))
50	38	simprd 482	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) → (𝑣 supp 0 ) ∈ 𝐴)
51	45, 49, 50	rspcdva 3455	. . . . . 6 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) → ∀𝑦(𝑦 ⊆ (𝑣 supp 0 ) → 𝑦 ∈ 𝐴))
52	1, 28, 12, 6, 40	psrelbas 19581	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) → (𝑢( ·_𝑠 ‘𝑆)𝑣):𝐷⟶(Base‘𝑅))
53		eqid 2760	. . . . . . . . 9 ⊢ (._r‘𝑅) = (._r‘𝑅)
54	30	adantr 472	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) ∧ 𝑘 ∈ (𝐷 ∖ (𝑣 supp 0 ))) → 𝑢 ∈ (Base‘𝑅))
55	39	adantr 472	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) ∧ 𝑘 ∈ (𝐷 ∖ (𝑣 supp 0 ))) → 𝑣 ∈ 𝐵)
56		eldifi 3875	. . . . . . . . . 10 ⊢ (𝑘 ∈ (𝐷 ∖ (𝑣 supp 0 )) → 𝑘 ∈ 𝐷)
57	56	adantl 473	. . . . . . . . 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 19594	. . . . . . . 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 19581	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) → 𝑣:𝐷⟶(Base‘𝑅))
60		ssid 3765	. . . . . . . . . . 11 ⊢ (𝑣 supp 0 ) ⊆ (𝑣 supp 0 )
61	60	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) → (𝑣 supp 0 ) ⊆ (𝑣 supp 0 ))
62		ovex 6841	. . . . . . . . . . . 12 ⊢ (ℕ₀ ↑_𝑚 𝐼) ∈ V
63	12, 62	rabex2 4966	. . . . . . . . . . 11 ⊢ 𝐷 ∈ V
64	63	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) → 𝐷 ∈ V)
65		fvex 6362	. . . . . . . . . . . 12 ⊢ (0_g‘𝑅) ∈ V
66	11, 65	eqeltri 2835	. . . . . . . . . . 11 ⊢ 0 ∈ V
67	66	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) → 0 ∈ V)
68	59, 61, 64, 67	suppssr 7495	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) ∧ 𝑘 ∈ (𝐷 ∖ (𝑣 supp 0 ))) → (𝑣‘𝑘) = 0 )
69	68	oveq2d 6829	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) ∧ 𝑘 ∈ (𝐷 ∖ (𝑣 supp 0 ))) → (𝑢(._r‘𝑅)(𝑣‘𝑘)) = (𝑢(._r‘𝑅) 0 ))
70	28, 53, 11	ringrz 18788	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝑢 ∈ (Base‘𝑅)) → (𝑢(._r‘𝑅) 0 ) = 0 )
71	3, 30, 70	syl2an2r 911	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) → (𝑢(._r‘𝑅) 0 ) = 0 )
72	71	adantr 472	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) ∧ 𝑘 ∈ (𝐷 ∖ (𝑣 supp 0 ))) → (𝑢(._r‘𝑅) 0 ) = 0 )
73	58, 69, 72	3eqtrd 2798	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) ∧ 𝑘 ∈ (𝐷 ∖ (𝑣 supp 0 ))) → ((𝑢( ·_𝑠 ‘𝑆)𝑣)‘𝑘) = 0 )
74	52, 73	suppss 7494	. . . . . 6 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) → ((𝑢( ·_𝑠 ‘𝑆)𝑣) supp 0 ) ⊆ (𝑣 supp 0 ))
75		sseq1 3767	. . . . . . . 8 ⊢ (𝑦 = ((𝑢( ·_𝑠 ‘𝑆)𝑣) supp 0 ) → (𝑦 ⊆ (𝑣 supp 0 ) ↔ ((𝑢( ·_𝑠 ‘𝑆)𝑣) supp 0 ) ⊆ (𝑣 supp 0 )))
76		eleq1 2827	. . . . . . . 8 ⊢ (𝑦 = ((𝑢( ·_𝑠 ‘𝑆)𝑣) supp 0 ) → (𝑦 ∈ 𝐴 ↔ ((𝑢( ·_𝑠 ‘𝑆)𝑣) supp 0 ) ∈ 𝐴))
77	75, 76	imbi12d 333	. . . . . . 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)))
78	77	spcgv 3433	. . . . . 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)))
79	42, 51, 74, 78	syl3c 66	. . . . 5 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) → ((𝑢( ·_𝑠 ‘𝑆)𝑣) supp 0 ) ∈ 𝐴)
80	32	eleq2d 2825	. . . . . 6 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) → ((𝑢( ·_𝑠 ‘𝑆)𝑣) ∈ 𝑈 ↔ (𝑢( ·_𝑠 ‘𝑆)𝑣) ∈ {𝑔 ∈ 𝐵 ∣ (𝑔 supp 0 ) ∈ 𝐴}))
81		oveq1 6820	. . . . . . . 8 ⊢ (𝑔 = (𝑢( ·_𝑠 ‘𝑆)𝑣) → (𝑔 supp 0 ) = ((𝑢( ·_𝑠 ‘𝑆)𝑣) supp 0 ))
82	81	eleq1d 2824	. . . . . . 7 ⊢ (𝑔 = (𝑢( ·_𝑠 ‘𝑆)𝑣) → ((𝑔 supp 0 ) ∈ 𝐴 ↔ ((𝑢( ·_𝑠 ‘𝑆)𝑣) supp 0 ) ∈ 𝐴))
83	82	elrab 3504	. . . . . 6 ⊢ ((𝑢( ·_𝑠 ‘𝑆)𝑣) ∈ {𝑔 ∈ 𝐵 ∣ (𝑔 supp 0 ) ∈ 𝐴} ↔ ((𝑢( ·_𝑠 ‘𝑆)𝑣) ∈ 𝐵 ∧ ((𝑢( ·_𝑠 ‘𝑆)𝑣) supp 0 ) ∈ 𝐴))
84	80, 83	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)))
85	40, 79, 84	mpbir2and 995	. . . 4 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈)) → (𝑢( ·_𝑠 ‘𝑆)𝑣) ∈ 𝑈)
86	85	3adantr3 1177	. . 3 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈 ∧ 𝑤 ∈ 𝑈)) → (𝑢( ·_𝑠 ‘𝑆)𝑣) ∈ 𝑈)
87		simpr3 1238	. . 3 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈 ∧ 𝑤 ∈ 𝑈)) → 𝑤 ∈ 𝑈)
88		eqid 2760	. . . 4 ⊢ (+_g‘𝑆) = (+_g‘𝑆)
89	88	subgcl 17805	. . 3 ⊢ ((𝑈 ∈ (SubGrp‘𝑆) ∧ (𝑢( ·_𝑠 ‘𝑆)𝑣) ∈ 𝑈 ∧ 𝑤 ∈ 𝑈) → ((𝑢( ·_𝑠 ‘𝑆)𝑣)(+_g‘𝑆)𝑤) ∈ 𝑈)
90	26, 86, 87, 89	syl3anc 1477	. 2 ⊢ ((𝜑 ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ 𝑈 ∧ 𝑤 ∈ 𝑈)) → ((𝑢( ·_𝑠 ‘𝑆)𝑣)(+_g‘𝑆)𝑤) ∈ 𝑈)
91	4, 5, 7, 8, 9, 10, 21, 25, 90	islssd 19138	1 ⊢ (𝜑 → 𝑈 ∈ (LSubSp‘𝑆))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1072 ∀wal 1630 = wceq 1632 ∈ wcel 2139 ≠ wne 2932 ∀wral 3050 {crab 3054 Vcvv 3340 ∖ cdif 3712 ∪ cun 3713 ⊆ wss 3715 ∅c0 4058 ^◡ccnv 5265 “ cima 5269 ‘cfv 6049 (class class class)co 6813 supp csupp 7463 ↑_𝑚 cmap 8023 Fincfn 8121 ℕcn 11212 ℕ₀cn0 11484 Basecbs 16059 +_gcplusg 16143 ._rcmulr 16144 ·_𝑠 cvsca 16147 0_gc0g 16302 Grpcgrp 17623 SubGrpcsubg 17789 Ringcrg 18747 LSubSpclss 19134 mPwSer cmps 19553

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 ax-cnex 10184 ax-resscn 10185 ax-1cn 10186 ax-icn 10187 ax-addcl 10188 ax-addrcl 10189 ax-mulcl 10190 ax-mulrcl 10191 ax-mulcom 10192 ax-addass 10193 ax-mulass 10194 ax-distr 10195 ax-i2m1 10196 ax-1ne0 10197 ax-1rid 10198 ax-rnegex 10199 ax-rrecex 10200 ax-cnre 10201 ax-pre-lttri 10202 ax-pre-lttrn 10203 ax-pre-ltadd 10204 ax-pre-mulgt0 10205

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rmo 3058 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-int 4628 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-riota 6774 df-ov 6816 df-oprab 6817 df-mpt2 6818 df-of 7062 df-om 7231 df-1st 7333 df-2nd 7334 df-supp 7464 df-wrecs 7576 df-recs 7637 df-rdg 7675 df-1o 7729 df-oadd 7733 df-er 7911 df-map 8025 df-en 8122 df-dom 8123 df-sdom 8124 df-fin 8125 df-fsupp 8441 df-pnf 10268 df-mnf 10269 df-xr 10270 df-ltxr 10271 df-le 10272 df-sub 10460 df-neg 10461 df-nn 11213 df-2 11271 df-3 11272 df-4 11273 df-5 11274 df-6 11275 df-7 11276 df-8 11277 df-9 11278 df-n0 11485 df-z 11570 df-uz 11880 df-fz 12520 df-struct 16061 df-ndx 16062 df-slot 16063 df-base 16065 df-sets 16066 df-ress 16067 df-plusg 16156 df-mulr 16157 df-sca 16159 df-vsca 16160 df-tset 16162 df-0g 16304 df-mgm 17443 df-sgrp 17485 df-mnd 17496 df-grp 17626 df-minusg 17627 df-subg 17792 df-mgp 18690 df-ring 18749 df-lss 19135 df-psr 19558

This theorem is referenced by: mpllss 19640

W3C validator