Theorem lsmdisj2 18095

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	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ (SubGrp‘𝐺))
2		lsmcntz.u	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺))
3		eqid 2622	. . . . . . . . 9 ⊢ (+g‘𝐺) = (+g‘𝐺)
4		lsmcntz.p	. . . . . . . . 9 ⊢ ⊕ = (LSSum‘𝐺)
5	3, 4	lsmelval 18064	. . . . . . . 8 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (𝑥 ∈ (𝑆 ⊕ 𝑈) ↔ ∃𝑠 ∈ 𝑆 ∃𝑢 ∈ 𝑈 𝑥 = (𝑠(+g‘𝐺)𝑢)))
6	1, 2, 5	syl2anc 693	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (𝑆 ⊕ 𝑈) ↔ ∃𝑠 ∈ 𝑆 ∃𝑢 ∈ 𝑈 𝑥 = (𝑠(+g‘𝐺)𝑢)))
7		simplrl 800	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑢 ∈ 𝑈)) ∧ (𝑠(+g‘𝐺)𝑢) ∈ 𝑇) → 𝑠 ∈ 𝑆)
8		subgrcl 17599	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
9	1, 8	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → 𝐺 ∈ Grp)
10	9	ad2antrr 762	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑢 ∈ 𝑈)) ∧ (𝑠(+g‘𝐺)𝑢) ∈ 𝑇) → 𝐺 ∈ Grp)
11	1	ad2antrr 762	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑢 ∈ 𝑈)) ∧ (𝑠(+g‘𝐺)𝑢) ∈ 𝑇) → 𝑆 ∈ (SubGrp‘𝐺))
12		eqid 2622	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (Base‘𝐺) = (Base‘𝐺)
13	12	subgss 17595	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ (Base‘𝐺))
14	11, 13	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑢 ∈ 𝑈)) ∧ (𝑠(+g‘𝐺)𝑢) ∈ 𝑇) → 𝑆 ⊆ (Base‘𝐺))
15	14, 7	sseldd 3604	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑢 ∈ 𝑈)) ∧ (𝑠(+g‘𝐺)𝑢) ∈ 𝑇) → 𝑠 ∈ (Base‘𝐺))
16		lsmdisj.o	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ 0 = (0g‘𝐺)
17		eqid 2622	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (invg‘𝐺) = (invg‘𝐺)
18	12, 3, 16, 17	grplinv 17468	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐺 ∈ Grp ∧ 𝑠 ∈ (Base‘𝐺)) → (((invg‘𝐺)‘𝑠)(+g‘𝐺)𝑠) = 0 )
19	10, 15, 18	syl2anc 693	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑢 ∈ 𝑈)) ∧ (𝑠(+g‘𝐺)𝑢) ∈ 𝑇) → (((invg‘𝐺)‘𝑠)(+g‘𝐺)𝑠) = 0 )
20	19	oveq1d 6665	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((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 17603	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑠 ∈ 𝑆) → ((invg‘𝐺)‘𝑠) ∈ 𝑆)
22	11, 7, 21	syl2anc 693	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑢 ∈ 𝑈)) ∧ (𝑠(+g‘𝐺)𝑢) ∈ 𝑇) → ((invg‘𝐺)‘𝑠) ∈ 𝑆)
23	14, 22	sseldd 3604	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑢 ∈ 𝑈)) ∧ (𝑠(+g‘𝐺)𝑢) ∈ 𝑇) → ((invg‘𝐺)‘𝑠) ∈ (Base‘𝐺))
24	2	ad2antrr 762	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑢 ∈ 𝑈)) ∧ (𝑠(+g‘𝐺)𝑢) ∈ 𝑇) → 𝑈 ∈ (SubGrp‘𝐺))
25	12	subgss 17595	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 ⊆ (Base‘𝐺))
26	24, 25	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑢 ∈ 𝑈)) ∧ (𝑠(+g‘𝐺)𝑢) ∈ 𝑇) → 𝑈 ⊆ (Base‘𝐺))
27		simplrr 801	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑢 ∈ 𝑈)) ∧ (𝑠(+g‘𝐺)𝑢) ∈ 𝑇) → 𝑢 ∈ 𝑈)
28	26, 27	sseldd 3604	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑢 ∈ 𝑈)) ∧ (𝑠(+g‘𝐺)𝑢) ∈ 𝑇) → 𝑢 ∈ (Base‘𝐺))
29	12, 3	grpass 17431	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((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 1328	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((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 17452	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐺 ∈ Grp ∧ 𝑢 ∈ (Base‘𝐺)) → ( 0 (+g‘𝐺)𝑢) = 𝑢)
32	10, 28, 31	syl2anc 693	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑢 ∈ 𝑈)) ∧ (𝑠(+g‘𝐺)𝑢) ∈ 𝑇) → ( 0 (+g‘𝐺)𝑢) = 𝑢)
33	20, 30, 32	3eqtr3d 2664	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑢 ∈ 𝑈)) ∧ (𝑠(+g‘𝐺)𝑢) ∈ 𝑇) → (((invg‘𝐺)‘𝑠)(+g‘𝐺)(𝑠(+g‘𝐺)𝑢)) = 𝑢)
34		lsmcntz.t	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺))
35	34	ad2antrr 762	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑢 ∈ 𝑈)) ∧ (𝑠(+g‘𝐺)𝑢) ∈ 𝑇) → 𝑇 ∈ (SubGrp‘𝐺))
36		simpr 477	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑢 ∈ 𝑈)) ∧ (𝑠(+g‘𝐺)𝑢) ∈ 𝑇) → (𝑠(+g‘𝐺)𝑢) ∈ 𝑇)
37	3, 4	lsmelvali 18065	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((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 1327	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((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 2702	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑢 ∈ 𝑈)) ∧ (𝑠(+g‘𝐺)𝑢) ∈ 𝑇) → 𝑢 ∈ (𝑆 ⊕ 𝑇))
40	39, 27	elind 3798	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑢 ∈ 𝑈)) ∧ (𝑠(+g‘𝐺)𝑢) ∈ 𝑇) → 𝑢 ∈ ((𝑆 ⊕ 𝑇) ∩ 𝑈))
41		lsmdisj.i	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ((𝑆 ⊕ 𝑇) ∩ 𝑈) = { 0 })
42	41	ad2antrr 762	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑢 ∈ 𝑈)) ∧ (𝑠(+g‘𝐺)𝑢) ∈ 𝑇) → ((𝑆 ⊕ 𝑇) ∩ 𝑈) = { 0 })
43	40, 42	eleqtrd 2703	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑢 ∈ 𝑈)) ∧ (𝑠(+g‘𝐺)𝑢) ∈ 𝑇) → 𝑢 ∈ { 0 })
44		elsni 4194	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑢 ∈ { 0 } → 𝑢 = 0 )
45	43, 44	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑢 ∈ 𝑈)) ∧ (𝑠(+g‘𝐺)𝑢) ∈ 𝑇) → 𝑢 = 0 )
46	45	oveq2d 6666	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑢 ∈ 𝑈)) ∧ (𝑠(+g‘𝐺)𝑢) ∈ 𝑇) → (𝑠(+g‘𝐺)𝑢) = (𝑠(+g‘𝐺) 0 ))
47	12, 3, 16	grprid 17453	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐺 ∈ Grp ∧ 𝑠 ∈ (Base‘𝐺)) → (𝑠(+g‘𝐺) 0 ) = 𝑠)
48	10, 15, 47	syl2anc 693	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑢 ∈ 𝑈)) ∧ (𝑠(+g‘𝐺)𝑢) ∈ 𝑇) → (𝑠(+g‘𝐺) 0 ) = 𝑠)
49	46, 48	eqtrd 2656	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑢 ∈ 𝑈)) ∧ (𝑠(+g‘𝐺)𝑢) ∈ 𝑇) → (𝑠(+g‘𝐺)𝑢) = 𝑠)
50	49, 36	eqeltrrd 2702	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑢 ∈ 𝑈)) ∧ (𝑠(+g‘𝐺)𝑢) ∈ 𝑇) → 𝑠 ∈ 𝑇)
51	7, 50	elind 3798	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑢 ∈ 𝑈)) ∧ (𝑠(+g‘𝐺)𝑢) ∈ 𝑇) → 𝑠 ∈ (𝑆 ∩ 𝑇))
52		lsmdisj2.i	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑆 ∩ 𝑇) = { 0 })
53	52	ad2antrr 762	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑢 ∈ 𝑈)) ∧ (𝑠(+g‘𝐺)𝑢) ∈ 𝑇) → (𝑆 ∩ 𝑇) = { 0 })
54	51, 53	eleqtrd 2703	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑢 ∈ 𝑈)) ∧ (𝑠(+g‘𝐺)𝑢) ∈ 𝑇) → 𝑠 ∈ { 0 })
55		elsni 4194	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ { 0 } → 𝑠 = 0 )
56	54, 55	syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑢 ∈ 𝑈)) ∧ (𝑠(+g‘𝐺)𝑢) ∈ 𝑇) → 𝑠 = 0 )
57	56, 45	oveq12d 6668	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑢 ∈ 𝑈)) ∧ (𝑠(+g‘𝐺)𝑢) ∈ 𝑇) → (𝑠(+g‘𝐺)𝑢) = ( 0 (+g‘𝐺) 0 ))
58	12, 16	grpidcl 17450	. . . . . . . . . . . . . 14 ⊢ (𝐺 ∈ Grp → 0 ∈ (Base‘𝐺))
59	9, 58	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 ∈ (Base‘𝐺))
60	12, 3, 16	grplid 17452	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ Grp ∧ 0 ∈ (Base‘𝐺)) → ( 0 (+g‘𝐺) 0 ) = 0 )
61	9, 59, 60	syl2anc 693	. . . . . . . . . . . 12 ⊢ (𝜑 → ( 0 (+g‘𝐺) 0 ) = 0 )
62	61	ad2antrr 762	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑢 ∈ 𝑈)) ∧ (𝑠(+g‘𝐺)𝑢) ∈ 𝑇) → ( 0 (+g‘𝐺) 0 ) = 0 )
63	57, 62	eqtrd 2656	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑢 ∈ 𝑈)) ∧ (𝑠(+g‘𝐺)𝑢) ∈ 𝑇) → (𝑠(+g‘𝐺)𝑢) = 0 )
64	63	ex 450	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑢 ∈ 𝑈)) → ((𝑠(+g‘𝐺)𝑢) ∈ 𝑇 → (𝑠(+g‘𝐺)𝑢) = 0 ))
65		eleq1 2689	. . . . . . . . . 10 ⊢ (𝑥 = (𝑠(+g‘𝐺)𝑢) → (𝑥 ∈ 𝑇 ↔ (𝑠(+g‘𝐺)𝑢) ∈ 𝑇))
66		eqeq1 2626	. . . . . . . . . 10 ⊢ (𝑥 = (𝑠(+g‘𝐺)𝑢) → (𝑥 = 0 ↔ (𝑠(+g‘𝐺)𝑢) = 0 ))
67	65, 66	imbi12d 334	. . . . . . . . 9 ⊢ (𝑥 = (𝑠(+g‘𝐺)𝑢) → ((𝑥 ∈ 𝑇 → 𝑥 = 0 ) ↔ ((𝑠(+g‘𝐺)𝑢) ∈ 𝑇 → (𝑠(+g‘𝐺)𝑢) = 0 )))
68	64, 67	syl5ibrcom 237	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑢 ∈ 𝑈)) → (𝑥 = (𝑠(+g‘𝐺)𝑢) → (𝑥 ∈ 𝑇 → 𝑥 = 0 )))
69	68	rexlimdvva 3038	. . . . . . 7 ⊢ (𝜑 → (∃𝑠 ∈ 𝑆 ∃𝑢 ∈ 𝑈 𝑥 = (𝑠(+g‘𝐺)𝑢) → (𝑥 ∈ 𝑇 → 𝑥 = 0 )))
70	6, 69	sylbid 230	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (𝑆 ⊕ 𝑈) → (𝑥 ∈ 𝑇 → 𝑥 = 0 )))
71	70	com23 86	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑇 → (𝑥 ∈ (𝑆 ⊕ 𝑈) → 𝑥 = 0 )))
72	71	impd 447	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝑇 ∧ 𝑥 ∈ (𝑆 ⊕ 𝑈)) → 𝑥 = 0 ))
73		elin 3796	. . . 4 ⊢ (𝑥 ∈ (𝑇 ∩ (𝑆 ⊕ 𝑈)) ↔ (𝑥 ∈ 𝑇 ∧ 𝑥 ∈ (𝑆 ⊕ 𝑈)))
74		velsn 4193	. . . 4 ⊢ (𝑥 ∈ { 0 } ↔ 𝑥 = 0 )
75	72, 73, 74	3imtr4g 285	. . 3 ⊢ (𝜑 → (𝑥 ∈ (𝑇 ∩ (𝑆 ⊕ 𝑈)) → 𝑥 ∈ { 0 }))
76	75	ssrdv 3609	. 2 ⊢ (𝜑 → (𝑇 ∩ (𝑆 ⊕ 𝑈)) ⊆ { 0 })
77	16	subg0cl 17602	. . . . 5 ⊢ (𝑇 ∈ (SubGrp‘𝐺) → 0 ∈ 𝑇)
78	34, 77	syl 17	. . . 4 ⊢ (𝜑 → 0 ∈ 𝑇)
79	4	lsmub1 18071	. . . . . 6 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → 𝑆 ⊆ (𝑆 ⊕ 𝑈))
80	1, 2, 79	syl2anc 693	. . . . 5 ⊢ (𝜑 → 𝑆 ⊆ (𝑆 ⊕ 𝑈))
81	16	subg0cl 17602	. . . . . 6 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 0 ∈ 𝑆)
82	1, 81	syl 17	. . . . 5 ⊢ (𝜑 → 0 ∈ 𝑆)
83	80, 82	sseldd 3604	. . . 4 ⊢ (𝜑 → 0 ∈ (𝑆 ⊕ 𝑈))
84	78, 83	elind 3798	. . 3 ⊢ (𝜑 → 0 ∈ (𝑇 ∩ (𝑆 ⊕ 𝑈)))
85	84	snssd 4340	. 2 ⊢ (𝜑 → { 0 } ⊆ (𝑇 ∩ (𝑆 ⊕ 𝑈)))
86	76, 85	eqssd 3620	1 ⊢ (𝜑 → (𝑇 ∩ (𝑆 ⊕ 𝑈)) = { 0 })

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∃wrex 2913   ∩ cin 3573   ⊆ wss 3574  {csn 4177  ‘cfv 5888  (class class class)co 6650  Basecbs 15857  +_gcplusg 15941  0_gc0g 16100  Grpcgrp 17422  inv_gcminusg 17423  SubGrpcsubg 17588  LSSumclsm 18049

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-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-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  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-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-grp 17425  df-minusg 17426  df-subg 17591  df-lsm 18051

This theorem is referenced by:  lsmdisj3  18096  lsmdisj2r  18098  lsmdisj2a  18100  dprd2da  18441