Theorem lsmdisj2 19783

Description: Association of the disjointness constraint in a subgroup sum. (Contributed by Mario Carneiro, 22-Apr-2016.)

Hypotheses

Ref	Expression
lsmcntz.p	⊢ ⊕ = (LSSum‘𝐺)
lsmcntz.s	⊢ (𝜑 → 𝑆 ∈ (SubGrp‘𝐺))
lsmcntz.t	⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺))
lsmcntz.u	⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺))
lsmdisj.o	⊢ 0 = (0g‘𝐺)
lsmdisj.i	⊢ (𝜑 → ((𝑆 ⊕ 𝑇) ∩ 𝑈) = { 0 })
lsmdisj2.i	⊢ (𝜑 → (𝑆 ∩ 𝑇) = { 0 })

Assertion

Ref	Expression
lsmdisj2	⊢ (𝜑 → (𝑇 ∩ (𝑆 ⊕ 𝑈)) = { 0 })

Proof of Theorem lsmdisj2

Dummy variables 𝑥 𝑢 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lsmcntz.s	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ (SubGrp‘𝐺))
2		lsmcntz.u	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺))
3		eqid 2766	. . . . . . . 8 ⊢ (+g‘𝐺) = (+g‘𝐺)
4		lsmcntz.p	. . . . . . . 8 ⊢ ⊕ = (LSSum‘𝐺)
5	3, 4	lsmelval 19750	. . . . . . 7 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (𝑥 ∈ (𝑆 ⊕ 𝑈) ↔ ∃𝑠 ∈ 𝑆 ∃𝑢 ∈ 𝑈 𝑥 = (𝑠(+g‘𝐺)𝑢)))
6	1, 2, 5	syl2anc 596	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (𝑆 ⊕ 𝑈) ↔ ∃𝑠 ∈ 𝑆 ∃𝑢 ∈ 𝑈 𝑥 = (𝑠(+g‘𝐺)𝑢)))
7		simplrl 789	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑢 ∈ 𝑈)) ∧ (𝑠(+g‘𝐺)𝑢) ∈ 𝑇) → 𝑠 ∈ 𝑆)
8		subgrcl 19228	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
9	1, 8	syl 18	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → 𝐺 ∈ Grp)
10	9	ad2antrr 739	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑢 ∈ 𝑈)) ∧ (𝑠(+g‘𝐺)𝑢) ∈ 𝑇) → 𝐺 ∈ Grp)
11	1	ad2antrr 739	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑢 ∈ 𝑈)) ∧ (𝑠(+g‘𝐺)𝑢) ∈ 𝑇) → 𝑆 ∈ (SubGrp‘𝐺))
12		eqid 2766	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (Base‘𝐺) = (Base‘𝐺)
13	12	subgss 19224	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ (Base‘𝐺))
14	11, 13	syl 18	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑢 ∈ 𝑈)) ∧ (𝑠(+g‘𝐺)𝑢) ∈ 𝑇) → 𝑆 ⊆ (Base‘𝐺))
15	14, 7	sseldd 3941	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑢 ∈ 𝑈)) ∧ (𝑠(+g‘𝐺)𝑢) ∈ 𝑇) → 𝑠 ∈ (Base‘𝐺))
16		lsmdisj.o	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 0 = (0g‘𝐺)
17		eqid 2766	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (invg‘𝐺) = (invg‘𝐺)
18	12, 3, 16, 17	grplinv 19087	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐺 ∈ Grp ∧ 𝑠 ∈ (Base‘𝐺)) → (((invg‘𝐺)‘𝑠)(+g‘𝐺)𝑠) = 0 )
19	10, 15, 18	syl2anc 596	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑢 ∈ 𝑈)) ∧ (𝑠(+g‘𝐺)𝑢) ∈ 𝑇) → (((invg‘𝐺)‘𝑠)(+g‘𝐺)𝑠) = 0 )
20	19	oveq1d 7438	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑢 ∈ 𝑈)) ∧ (𝑠(+g‘𝐺)𝑢) ∈ 𝑇) → ((((invg‘𝐺)‘𝑠)(+g‘𝐺)𝑠)(+g‘𝐺)𝑢) = ( 0 (+g‘𝐺)𝑢))
21	17	subginvcl 19232	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑠 ∈ 𝑆) → ((invg‘𝐺)‘𝑠) ∈ 𝑆)
22	11, 7, 21	syl2anc 596	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑢 ∈ 𝑈)) ∧ (𝑠(+g‘𝐺)𝑢) ∈ 𝑇) → ((invg‘𝐺)‘𝑠) ∈ 𝑆)
23	14, 22	sseldd 3941	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑢 ∈ 𝑈)) ∧ (𝑠(+g‘𝐺)𝑢) ∈ 𝑇) → ((invg‘𝐺)‘𝑠) ∈ (Base‘𝐺))
24	2	ad2antrr 739	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑢 ∈ 𝑈)) ∧ (𝑠(+g‘𝐺)𝑢) ∈ 𝑇) → 𝑈 ∈ (SubGrp‘𝐺))
25	12	subgss 19224	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 ⊆ (Base‘𝐺))
26	24, 25	syl 18	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑢 ∈ 𝑈)) ∧ (𝑠(+g‘𝐺)𝑢) ∈ 𝑇) → 𝑈 ⊆ (Base‘𝐺))
27		simplrr 790	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑢 ∈ 𝑈)) ∧ (𝑠(+g‘𝐺)𝑢) ∈ 𝑇) → 𝑢 ∈ 𝑈)
28	26, 27	sseldd 3941	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑢 ∈ 𝑈)) ∧ (𝑠(+g‘𝐺)𝑢) ∈ 𝑇) → 𝑢 ∈ (Base‘𝐺))
29	12, 3	grpass 19040	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐺 ∈ Grp ∧ (((invg‘𝐺)‘𝑠) ∈ (Base‘𝐺) ∧ 𝑠 ∈ (Base‘𝐺) ∧ 𝑢 ∈ (Base‘𝐺))) → ((((invg‘𝐺)‘𝑠)(+g‘𝐺)𝑠)(+g‘𝐺)𝑢) = (((invg‘𝐺)‘𝑠)(+g‘𝐺)(𝑠(+g‘𝐺)𝑢)))
30	10, 23, 15, 28, 29	syl13anc 1399	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑢 ∈ 𝑈)) ∧ (𝑠(+g‘𝐺)𝑢) ∈ 𝑇) → ((((invg‘𝐺)‘𝑠)(+g‘𝐺)𝑠)(+g‘𝐺)𝑢) = (((invg‘𝐺)‘𝑠)(+g‘𝐺)(𝑠(+g‘𝐺)𝑢)))
31	12, 3, 16	grplid 19065	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐺 ∈ Grp ∧ 𝑢 ∈ (Base‘𝐺)) → ( 0 (+g‘𝐺)𝑢) = 𝑢)
32	10, 28, 31	syl2anc 596	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑢 ∈ 𝑈)) ∧ (𝑠(+g‘𝐺)𝑢) ∈ 𝑇) → ( 0 (+g‘𝐺)𝑢) = 𝑢)
33	20, 30, 32	3eqtr3d 2809	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑢 ∈ 𝑈)) ∧ (𝑠(+g‘𝐺)𝑢) ∈ 𝑇) → (((invg‘𝐺)‘𝑠)(+g‘𝐺)(𝑠(+g‘𝐺)𝑢)) = 𝑢)
34		lsmcntz.t	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺))
35	34	ad2antrr 739	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑢 ∈ 𝑈)) ∧ (𝑠(+g‘𝐺)𝑢) ∈ 𝑇) → 𝑇 ∈ (SubGrp‘𝐺))
36		simpr 490	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑢 ∈ 𝑈)) ∧ (𝑠(+g‘𝐺)𝑢) ∈ 𝑇) → (𝑠(+g‘𝐺)𝑢) ∈ 𝑇)
37	3, 4	lsmelvali 19751	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ (SubGrp‘𝐺)) ∧ (((invg‘𝐺)‘𝑠) ∈ 𝑆 ∧ (𝑠(+g‘𝐺)𝑢) ∈ 𝑇)) → (((invg‘𝐺)‘𝑠)(+g‘𝐺)(𝑠(+g‘𝐺)𝑢)) ∈ (𝑆 ⊕ 𝑇))
38	11, 35, 22, 36, 37	syl22anc 852	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑢 ∈ 𝑈)) ∧ (𝑠(+g‘𝐺)𝑢) ∈ 𝑇) → (((invg‘𝐺)‘𝑠)(+g‘𝐺)(𝑠(+g‘𝐺)𝑢)) ∈ (𝑆 ⊕ 𝑇))
39	33, 38	eqeltrrd 2867	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑢 ∈ 𝑈)) ∧ (𝑠(+g‘𝐺)𝑢) ∈ 𝑇) → 𝑢 ∈ (𝑆 ⊕ 𝑇))
40	39, 27	elind 4156	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑢 ∈ 𝑈)) ∧ (𝑠(+g‘𝐺)𝑢) ∈ 𝑇) → 𝑢 ∈ ((𝑆 ⊕ 𝑇) ∩ 𝑈))
41		lsmdisj.i	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ((𝑆 ⊕ 𝑇) ∩ 𝑈) = { 0 })
42	41	ad2antrr 739	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑢 ∈ 𝑈)) ∧ (𝑠(+g‘𝐺)𝑢) ∈ 𝑇) → ((𝑆 ⊕ 𝑇) ∩ 𝑈) = { 0 })
43	40, 42	eleqtrd 2868	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑢 ∈ 𝑈)) ∧ (𝑠(+g‘𝐺)𝑢) ∈ 𝑇) → 𝑢 ∈ { 0 })
44		elsni 4611	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑢 ∈ { 0 } → 𝑢 = 0 )
45	43, 44	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑢 ∈ 𝑈)) ∧ (𝑠(+g‘𝐺)𝑢) ∈ 𝑇) → 𝑢 = 0 )
46	45	oveq2d 7439	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑢 ∈ 𝑈)) ∧ (𝑠(+g‘𝐺)𝑢) ∈ 𝑇) → (𝑠(+g‘𝐺)𝑢) = (𝑠(+g‘𝐺) 0 ))
47	12, 3, 16	grprid 19066	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐺 ∈ Grp ∧ 𝑠 ∈ (Base‘𝐺)) → (𝑠(+g‘𝐺) 0 ) = 𝑠)
48	10, 15, 47	syl2anc 596	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑢 ∈ 𝑈)) ∧ (𝑠(+g‘𝐺)𝑢) ∈ 𝑇) → (𝑠(+g‘𝐺) 0 ) = 𝑠)
49	46, 48	eqtrd 2801	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑢 ∈ 𝑈)) ∧ (𝑠(+g‘𝐺)𝑢) ∈ 𝑇) → (𝑠(+g‘𝐺)𝑢) = 𝑠)
50	49, 36	eqeltrrd 2867	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑢 ∈ 𝑈)) ∧ (𝑠(+g‘𝐺)𝑢) ∈ 𝑇) → 𝑠 ∈ 𝑇)
51	7, 50	elind 4156	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑢 ∈ 𝑈)) ∧ (𝑠(+g‘𝐺)𝑢) ∈ 𝑇) → 𝑠 ∈ (𝑆 ∩ 𝑇))
52		lsmdisj2.i	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑆 ∩ 𝑇) = { 0 })
53	52	ad2antrr 739	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑢 ∈ 𝑈)) ∧ (𝑠(+g‘𝐺)𝑢) ∈ 𝑇) → (𝑆 ∩ 𝑇) = { 0 })
54	51, 53	eleqtrd 2868	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑢 ∈ 𝑈)) ∧ (𝑠(+g‘𝐺)𝑢) ∈ 𝑇) → 𝑠 ∈ { 0 })
55		elsni 4611	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ { 0 } → 𝑠 = 0 )
56	54, 55	syl 18	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑢 ∈ 𝑈)) ∧ (𝑠(+g‘𝐺)𝑢) ∈ 𝑇) → 𝑠 = 0 )
57	56, 45	oveq12d 7441	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑢 ∈ 𝑈)) ∧ (𝑠(+g‘𝐺)𝑢) ∈ 𝑇) → (𝑠(+g‘𝐺)𝑢) = ( 0 (+g‘𝐺) 0 ))
58	12, 16	grpidcl 19063	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ Grp → 0 ∈ (Base‘𝐺))
59	12, 3, 16	grplid 19065	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ Grp ∧ 0 ∈ (Base‘𝐺)) → ( 0 (+g‘𝐺) 0 ) = 0 )
60	9, 58, 59	syl2anc2 597	. . . . . . . . . . 11 ⊢ (𝜑 → ( 0 (+g‘𝐺) 0 ) = 0 )
61	60	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑢 ∈ 𝑈)) ∧ (𝑠(+g‘𝐺)𝑢) ∈ 𝑇) → ( 0 (+g‘𝐺) 0 ) = 0 )
62	57, 61	eqtrd 2801	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑢 ∈ 𝑈)) ∧ (𝑠(+g‘𝐺)𝑢) ∈ 𝑇) → (𝑠(+g‘𝐺)𝑢) = 0 )
63	62	ex 418	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑢 ∈ 𝑈)) → ((𝑠(+g‘𝐺)𝑢) ∈ 𝑇 → (𝑠(+g‘𝐺)𝑢) = 0 ))
64		eleq1 2854	. . . . . . . . 9 ⊢ (𝑥 = (𝑠(+g‘𝐺)𝑢) → (𝑥 ∈ 𝑇 ↔ (𝑠(+g‘𝐺)𝑢) ∈ 𝑇))
65		eqeq1 2770	. . . . . . . . 9 ⊢ (𝑥 = (𝑠(+g‘𝐺)𝑢) → (𝑥 = 0 ↔ (𝑠(+g‘𝐺)𝑢) = 0 ))
66	64, 65	imbi12d 347	. . . . . . . 8 ⊢ (𝑥 = (𝑠(+g‘𝐺)𝑢) → ((𝑥 ∈ 𝑇 → 𝑥 = 0 ) ↔ ((𝑠(+g‘𝐺)𝑢) ∈ 𝑇 → (𝑠(+g‘𝐺)𝑢) = 0 )))
67	63, 66	syl5ibrcom 250	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑢 ∈ 𝑈)) → (𝑥 = (𝑠(+g‘𝐺)𝑢) → (𝑥 ∈ 𝑇 → 𝑥 = 0 )))
68	67	rexlimdvva 3225	. . . . . 6 ⊢ (𝜑 → (∃𝑠 ∈ 𝑆 ∃𝑢 ∈ 𝑈 𝑥 = (𝑠(+g‘𝐺)𝑢) → (𝑥 ∈ 𝑇 → 𝑥 = 0 )))
69	6, 68	sylbid 243	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (𝑆 ⊕ 𝑈) → (𝑥 ∈ 𝑇 → 𝑥 = 0 )))
70	69	impcomd 417	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝑇 ∧ 𝑥 ∈ (𝑆 ⊕ 𝑈)) → 𝑥 = 0 ))
71		elin 3924	. . . 4 ⊢ (𝑥 ∈ (𝑇 ∩ (𝑆 ⊕ 𝑈)) ↔ (𝑥 ∈ 𝑇 ∧ 𝑥 ∈ (𝑆 ⊕ 𝑈)))
72		velsn 4610	. . . 4 ⊢ (𝑥 ∈ { 0 } ↔ 𝑥 = 0 )
73	70, 71, 72	3imtr4g 299	. . 3 ⊢ (𝜑 → (𝑥 ∈ (𝑇 ∩ (𝑆 ⊕ 𝑈)) → 𝑥 ∈ { 0 }))
74	73	ssrdv 3946	. 2 ⊢ (𝜑 → (𝑇 ∩ (𝑆 ⊕ 𝑈)) ⊆ { 0 })
75	16	subg0cl 19231	. . . . 5 ⊢ (𝑇 ∈ (SubGrp‘𝐺) → 0 ∈ 𝑇)
76	34, 75	syl 18	. . . 4 ⊢ (𝜑 → 0 ∈ 𝑇)
77	4	lsmub1 19758	. . . . . 6 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → 𝑆 ⊆ (𝑆 ⊕ 𝑈))
78	1, 2, 77	syl2anc 596	. . . . 5 ⊢ (𝜑 → 𝑆 ⊆ (𝑆 ⊕ 𝑈))
79	16	subg0cl 19231	. . . . . 6 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 0 ∈ 𝑆)
80	1, 79	syl 18	. . . . 5 ⊢ (𝜑 → 0 ∈ 𝑆)
81	78, 80	sseldd 3941	. . . 4 ⊢ (𝜑 → 0 ∈ (𝑆 ⊕ 𝑈))
82	76, 81	elind 4156	. . 3 ⊢ (𝜑 → 0 ∈ (𝑇 ∩ (𝑆 ⊕ 𝑈)))
83	82	snssd 4757	. 2 ⊢ (𝜑 → { 0 } ⊆ (𝑇 ∩ (𝑆 ⊕ 𝑈)))
84	74, 83	eqssd 3957	1 ⊢ (𝜑 → (𝑇 ∩ (𝑆 ⊕ 𝑈)) = { 0 })

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∃wrex 3092   ∩ cin 3907   ⊆ wss 3908  {csn 4594  ‘cfv 6543  (class class class)co 7423  Basecbs 17294  +_gcplusg 17335  0_gc0g 17517  Grpcgrp 19031  inv_gcminusg 19032  SubGrpcsubg 19217  LSSumclsm 19735

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745  ax-cnex 11174  ax-resscn 11175  ax-1cn 11176  ax-icn 11177  ax-addcl 11178  ax-addrcl 11179  ax-mulcl 11180  ax-mulrcl 11181  ax-mulcom 11182  ax-addass 11183  ax-mulass 11184  ax-distr 11185  ax-i2m1 11186  ax-1ne0 11187  ax-1rid 11188  ax-rnegex 11189  ax-rrecex 11190  ax-cnre 11191  ax-pre-lttri 11192  ax-pre-lttrn 11193  ax-pre-ltadd 11194  ax-pre-mulgt0 11195

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-nel 3068  df-ral 3083  df-rex 3093  df-rmo 3372  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3928  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-tr 5224  df-id 5561  df-eprel 5566  df-po 5574  df-so 5575  df-fr 5619  df-we 5621  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6309  df-ord 6370  df-on 6371  df-lim 6372  df-suc 6373  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7380  df-ov 7426  df-oprab 7427  df-mpo 7428  df-om 7872  df-1st 7995  df-2nd 7996  df-frecs 8287  df-wrecs 8318  df-recs 8367  df-rdg 8406  df-er 8703  df-en 8953  df-dom 8954  df-sdom 8955  df-pnf 11263  df-mnf 11264  df-xr 11265  df-ltxr 11266  df-le 11267  df-sub 11461  df-neg 11462  df-nn 12252  df-2 12321  df-sets 17249  df-slot 17267  df-ndx 17279  df-base 17295  df-ress 17316  df-plusg 17348  df-0g 17519  df-mgm 18723  df-sgrp 18806  df-mnd 18822  df-submnd 18873  df-grp 19034  df-minusg 19035  df-subg 19220  df-lsm 19737

This theorem is used by:  lsmdisj3  19784  lsmdisj2r  19786  lsmdisj2a  19788  dprd2da  20145