Theorem lsmsubg 19583

Description: The sum of two commuting subgroups is a subgroup. (Contributed by Mario Carneiro, 19-Apr-2016.)

Hypotheses

Ref	Expression
lsmsubg.p	⊢ ⊕ = (LSSum‘𝐺)
lsmsubg.z	⊢ 𝑍 = (Cntz‘𝐺)

Assertion

Ref	Expression
lsmsubg	⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) → (𝑇 ⊕ 𝑈) ∈ (SubGrp‘𝐺))

Proof of Theorem lsmsubg

Dummy variables 𝑎 𝑏 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1 1136	. . . 4 ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) → 𝑇 ∈ (SubGrp‘𝐺))
2		subgsubm 19078	. . . 4 ⊢ (𝑇 ∈ (SubGrp‘𝐺) → 𝑇 ∈ (SubMnd‘𝐺))
3	1, 2	syl 17	. . 3 ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) → 𝑇 ∈ (SubMnd‘𝐺))
4		simp2 1137	. . . 4 ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) → 𝑈 ∈ (SubGrp‘𝐺))
5		subgsubm 19078	. . . 4 ⊢ (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 ∈ (SubMnd‘𝐺))
6	4, 5	syl 17	. . 3 ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) → 𝑈 ∈ (SubMnd‘𝐺))
7		simp3 1138	. . 3 ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) → 𝑇 ⊆ (𝑍‘𝑈))
8		lsmsubg.p	. . . 4 ⊢ ⊕ = (LSSum‘𝐺)
9		lsmsubg.z	. . . 4 ⊢ 𝑍 = (Cntz‘𝐺)
10	8, 9	lsmsubm 19582	. . 3 ⊢ ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) → (𝑇 ⊕ 𝑈) ∈ (SubMnd‘𝐺))
11	3, 6, 7, 10	syl3anc 1373	. 2 ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) → (𝑇 ⊕ 𝑈) ∈ (SubMnd‘𝐺))
12		eqid 2736	. . . . . 6 ⊢ (+g‘𝐺) = (+g‘𝐺)
13	12, 8	lsmelval 19578	. . . . 5 ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (𝑥 ∈ (𝑇 ⊕ 𝑈) ↔ ∃𝑎 ∈ 𝑇 ∃𝑏 ∈ 𝑈 𝑥 = (𝑎(+g‘𝐺)𝑏)))
14	13	3adant3 1132	. . . 4 ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) → (𝑥 ∈ (𝑇 ⊕ 𝑈) ↔ ∃𝑎 ∈ 𝑇 ∃𝑏 ∈ 𝑈 𝑥 = (𝑎(+g‘𝐺)𝑏)))
15	1	adantr 480	. . . . . . . . . 10 ⊢ (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ (𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑈)) → 𝑇 ∈ (SubGrp‘𝐺))
16		subgrcl 19061	. . . . . . . . . 10 ⊢ (𝑇 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
17	15, 16	syl 17	. . . . . . . . 9 ⊢ (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ (𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑈)) → 𝐺 ∈ Grp)
18		eqid 2736	. . . . . . . . . . . 12 ⊢ (Base‘𝐺) = (Base‘𝐺)
19	18	subgss 19057	. . . . . . . . . . 11 ⊢ (𝑇 ∈ (SubGrp‘𝐺) → 𝑇 ⊆ (Base‘𝐺))
20	15, 19	syl 17	. . . . . . . . . 10 ⊢ (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ (𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑈)) → 𝑇 ⊆ (Base‘𝐺))
21		simprl 770	. . . . . . . . . 10 ⊢ (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ (𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑈)) → 𝑎 ∈ 𝑇)
22	20, 21	sseldd 3934	. . . . . . . . 9 ⊢ (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ (𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑈)) → 𝑎 ∈ (Base‘𝐺))
23	4	adantr 480	. . . . . . . . . . 11 ⊢ (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ (𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑈)) → 𝑈 ∈ (SubGrp‘𝐺))
24	18	subgss 19057	. . . . . . . . . . 11 ⊢ (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 ⊆ (Base‘𝐺))
25	23, 24	syl 17	. . . . . . . . . 10 ⊢ (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ (𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑈)) → 𝑈 ⊆ (Base‘𝐺))
26		simprr 772	. . . . . . . . . 10 ⊢ (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ (𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑈)) → 𝑏 ∈ 𝑈)
27	25, 26	sseldd 3934	. . . . . . . . 9 ⊢ (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ (𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑈)) → 𝑏 ∈ (Base‘𝐺))
28		eqid 2736	. . . . . . . . . 10 ⊢ (invg‘𝐺) = (invg‘𝐺)
29	18, 12, 28	grpinvadd 18948	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ 𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → ((invg‘𝐺)‘(𝑎(+g‘𝐺)𝑏)) = (((invg‘𝐺)‘𝑏)(+g‘𝐺)((invg‘𝐺)‘𝑎)))
30	17, 22, 27, 29	syl3anc 1373	. . . . . . . 8 ⊢ (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ (𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑈)) → ((invg‘𝐺)‘(𝑎(+g‘𝐺)𝑏)) = (((invg‘𝐺)‘𝑏)(+g‘𝐺)((invg‘𝐺)‘𝑎)))
31	7	adantr 480	. . . . . . . . . 10 ⊢ (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ (𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑈)) → 𝑇 ⊆ (𝑍‘𝑈))
32	28	subginvcl 19065	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑎 ∈ 𝑇) → ((invg‘𝐺)‘𝑎) ∈ 𝑇)
33	15, 21, 32	syl2anc 584	. . . . . . . . . 10 ⊢ (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ (𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑈)) → ((invg‘𝐺)‘𝑎) ∈ 𝑇)
34	31, 33	sseldd 3934	. . . . . . . . 9 ⊢ (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ (𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑈)) → ((invg‘𝐺)‘𝑎) ∈ (𝑍‘𝑈))
35	28	subginvcl 19065	. . . . . . . . . 10 ⊢ ((𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑏 ∈ 𝑈) → ((invg‘𝐺)‘𝑏) ∈ 𝑈)
36	23, 26, 35	syl2anc 584	. . . . . . . . 9 ⊢ (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ (𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑈)) → ((invg‘𝐺)‘𝑏) ∈ 𝑈)
37	12, 9	cntzi 19258	. . . . . . . . 9 ⊢ ((((invg‘𝐺)‘𝑎) ∈ (𝑍‘𝑈) ∧ ((invg‘𝐺)‘𝑏) ∈ 𝑈) → (((invg‘𝐺)‘𝑎)(+g‘𝐺)((invg‘𝐺)‘𝑏)) = (((invg‘𝐺)‘𝑏)(+g‘𝐺)((invg‘𝐺)‘𝑎)))
38	34, 36, 37	syl2anc 584	. . . . . . . 8 ⊢ (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ (𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑈)) → (((invg‘𝐺)‘𝑎)(+g‘𝐺)((invg‘𝐺)‘𝑏)) = (((invg‘𝐺)‘𝑏)(+g‘𝐺)((invg‘𝐺)‘𝑎)))
39	30, 38	eqtr4d 2774	. . . . . . 7 ⊢ (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ (𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑈)) → ((invg‘𝐺)‘(𝑎(+g‘𝐺)𝑏)) = (((invg‘𝐺)‘𝑎)(+g‘𝐺)((invg‘𝐺)‘𝑏)))
40	12, 8	lsmelvali 19579	. . . . . . . 8 ⊢ (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ (((invg‘𝐺)‘𝑎) ∈ 𝑇 ∧ ((invg‘𝐺)‘𝑏) ∈ 𝑈)) → (((invg‘𝐺)‘𝑎)(+g‘𝐺)((invg‘𝐺)‘𝑏)) ∈ (𝑇 ⊕ 𝑈))
41	15, 23, 33, 36, 40	syl22anc 838	. . . . . . 7 ⊢ (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ (𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑈)) → (((invg‘𝐺)‘𝑎)(+g‘𝐺)((invg‘𝐺)‘𝑏)) ∈ (𝑇 ⊕ 𝑈))
42	39, 41	eqeltrd 2836	. . . . . 6 ⊢ (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ (𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑈)) → ((invg‘𝐺)‘(𝑎(+g‘𝐺)𝑏)) ∈ (𝑇 ⊕ 𝑈))
43		fveq2 6834	. . . . . . 7 ⊢ (𝑥 = (𝑎(+g‘𝐺)𝑏) → ((invg‘𝐺)‘𝑥) = ((invg‘𝐺)‘(𝑎(+g‘𝐺)𝑏)))
44	43	eleq1d 2821	. . . . . 6 ⊢ (𝑥 = (𝑎(+g‘𝐺)𝑏) → (((invg‘𝐺)‘𝑥) ∈ (𝑇 ⊕ 𝑈) ↔ ((invg‘𝐺)‘(𝑎(+g‘𝐺)𝑏)) ∈ (𝑇 ⊕ 𝑈)))
45	42, 44	syl5ibrcom 247	. . . . 5 ⊢ (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ (𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑈)) → (𝑥 = (𝑎(+g‘𝐺)𝑏) → ((invg‘𝐺)‘𝑥) ∈ (𝑇 ⊕ 𝑈)))
46	45	rexlimdvva 3193	. . . 4 ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) → (∃𝑎 ∈ 𝑇 ∃𝑏 ∈ 𝑈 𝑥 = (𝑎(+g‘𝐺)𝑏) → ((invg‘𝐺)‘𝑥) ∈ (𝑇 ⊕ 𝑈)))
47	14, 46	sylbid 240	. . 3 ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) → (𝑥 ∈ (𝑇 ⊕ 𝑈) → ((invg‘𝐺)‘𝑥) ∈ (𝑇 ⊕ 𝑈)))
48	47	ralrimiv 3127	. 2 ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) → ∀𝑥 ∈ (𝑇 ⊕ 𝑈)((invg‘𝐺)‘𝑥) ∈ (𝑇 ⊕ 𝑈))
49	1, 16	syl 17	. . 3 ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) → 𝐺 ∈ Grp)
50	28	issubg3 19074	. . 3 ⊢ (𝐺 ∈ Grp → ((𝑇 ⊕ 𝑈) ∈ (SubGrp‘𝐺) ↔ ((𝑇 ⊕ 𝑈) ∈ (SubMnd‘𝐺) ∧ ∀𝑥 ∈ (𝑇 ⊕ 𝑈)((invg‘𝐺)‘𝑥) ∈ (𝑇 ⊕ 𝑈))))
51	49, 50	syl 17	. 2 ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) → ((𝑇 ⊕ 𝑈) ∈ (SubGrp‘𝐺) ↔ ((𝑇 ⊕ 𝑈) ∈ (SubMnd‘𝐺) ∧ ∀𝑥 ∈ (𝑇 ⊕ 𝑈)((invg‘𝐺)‘𝑥) ∈ (𝑇 ⊕ 𝑈))))
52	11, 48, 51	mpbir2and 713	1 ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) → (𝑇 ⊕ 𝑈) ∈ (SubGrp‘𝐺))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  ∀wral 3051  ∃wrex 3060   ⊆ wss 3901  ‘cfv 6492  (class class class)co 7358  Basecbs 17136  +_gcplusg 17177  SubMndcsubmnd 18707  Grpcgrp 18863  inv_gcminusg 18864  SubGrpcsubg 19050  Cntzccntz 19244  LSSumclsm 19563

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-cnex 11082  ax-resscn 11083  ax-1cn 11084  ax-icn 11085  ax-addcl 11086  ax-addrcl 11087  ax-mulcl 11088  ax-mulrcl 11089  ax-mulcom 11090  ax-addass 11091  ax-mulass 11092  ax-distr 11093  ax-i2m1 11094  ax-1ne0 11095  ax-1rid 11096  ax-rnegex 11097  ax-rrecex 11098  ax-cnre 11099  ax-pre-lttri 11100  ax-pre-lttrn 11101  ax-pre-ltadd 11102  ax-pre-mulgt0 11103

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-er 8635  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11168  df-mnf 11169  df-xr 11170  df-ltxr 11171  df-le 11172  df-sub 11366  df-neg 11367  df-nn 12146  df-2 12208  df-sets 17091  df-slot 17109  df-ndx 17121  df-base 17137  df-ress 17158  df-plusg 17190  df-0g 17361  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-submnd 18709  df-grp 18866  df-minusg 18867  df-subg 19053  df-cntz 19246  df-lsm 19565

This theorem is referenced by:  pj1ghm  19632  lsmsubg2  19788  dprd2da  19973  dmdprdsplit2lem  19976  dprdsplit  19979