Theorem lsmsubg 19687

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 1135	. . . 4 ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) → 𝑇 ∈ (SubGrp‘𝐺))
2		subgsubm 19179	. . . 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 1136	. . . 4 ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) → 𝑈 ∈ (SubGrp‘𝐺))
5		subgsubm 19179	. . . 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 1137	. . 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 19686	. . 3 ⊢ ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) → (𝑇 ⊕ 𝑈) ∈ (SubMnd‘𝐺))
11	3, 6, 7, 10	syl3anc 1370	. 2 ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) → (𝑇 ⊕ 𝑈) ∈ (SubMnd‘𝐺))
12		eqid 2735	. . . . . 6 ⊢ (+g‘𝐺) = (+g‘𝐺)
13	12, 8	lsmelval 19682	. . . . 5 ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (𝑥 ∈ (𝑇 ⊕ 𝑈) ↔ ∃𝑎 ∈ 𝑇 ∃𝑏 ∈ 𝑈 𝑥 = (𝑎(+g‘𝐺)𝑏)))
14	13	3adant3 1131	. . . 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 19162	. . . . . . . . . 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 2735	. . . . . . . . . . . 12 ⊢ (Base‘𝐺) = (Base‘𝐺)
19	18	subgss 19158	. . . . . . . . . . 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 771	. . . . . . . . . 10 ⊢ (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ (𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑈)) → 𝑎 ∈ 𝑇)
22	20, 21	sseldd 3996	. . . . . . . . 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 19158	. . . . . . . . . . 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 773	. . . . . . . . . 10 ⊢ (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ (𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑈)) → 𝑏 ∈ 𝑈)
27	25, 26	sseldd 3996	. . . . . . . . 9 ⊢ (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ (𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑈)) → 𝑏 ∈ (Base‘𝐺))
28		eqid 2735	. . . . . . . . . 10 ⊢ (invg‘𝐺) = (invg‘𝐺)
29	18, 12, 28	grpinvadd 19049	. . . . . . . . 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 1370	. . . . . . . 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 19166	. . . . . . . . . . 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 3996	. . . . . . . . 9 ⊢ (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ (𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑈)) → ((invg‘𝐺)‘𝑎) ∈ (𝑍‘𝑈))
35	28	subginvcl 19166	. . . . . . . . . 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 19360	. . . . . . . . 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 2778	. . . . . . 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 19683	. . . . . . . 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 839	. . . . . . 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 2839	. . . . . 6 ⊢ (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ (𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑈)) → ((invg‘𝐺)‘(𝑎(+g‘𝐺)𝑏)) ∈ (𝑇 ⊕ 𝑈))
43		fveq2 6907	. . . . . . 7 ⊢ (𝑥 = (𝑎(+g‘𝐺)𝑏) → ((invg‘𝐺)‘𝑥) = ((invg‘𝐺)‘(𝑎(+g‘𝐺)𝑏)))
44	43	eleq1d 2824	. . . . . 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 3211	. . . 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 3143	. 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 19175	. . 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 1537   ∈ wcel 2106  ∀wral 3059  ∃wrex 3068   ⊆ wss 3963  ‘cfv 6563  (class class class)co 7431  Basecbs 17245  +_gcplusg 17298  SubMndcsubmnd 18808  Grpcgrp 18964  inv_gcminusg 18965  SubGrpcsubg 19151  Cntzccntz 19346  LSSumclsm 19667

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-0g 17488  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-submnd 18810  df-grp 18967  df-minusg 18968  df-subg 19154  df-cntz 19348  df-lsm 19669

This theorem is referenced by:  pj1ghm  19736  lsmsubg2  19892  dprd2da  20077  dmdprdsplit2lem  20080  dprdsplit  20083