Theorem lsmsubg 18773

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 1132	. . . 4 ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) → 𝑇 ∈ (SubGrp‘𝐺))
2		subgsubm 18295	. . . 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 1133	. . . 4 ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) → 𝑈 ∈ (SubGrp‘𝐺))
5		subgsubm 18295	. . . 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 1134	. . 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 18772	. . 3 ⊢ ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) → (𝑇 ⊕ 𝑈) ∈ (SubMnd‘𝐺))
11	3, 6, 7, 10	syl3anc 1367	. 2 ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) → (𝑇 ⊕ 𝑈) ∈ (SubMnd‘𝐺))
12		eqid 2821	. . . . . 6 ⊢ (+g‘𝐺) = (+g‘𝐺)
13	12, 8	lsmelval 18768	. . . . 5 ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (𝑥 ∈ (𝑇 ⊕ 𝑈) ↔ ∃𝑎 ∈ 𝑇 ∃𝑏 ∈ 𝑈 𝑥 = (𝑎(+g‘𝐺)𝑏)))
14	13	3adant3 1128	. . . 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 483	. . . . . . . . . 10 ⊢ (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ (𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑈)) → 𝑇 ∈ (SubGrp‘𝐺))
16		subgrcl 18278	. . . . . . . . . 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 2821	. . . . . . . . . . . 12 ⊢ (Base‘𝐺) = (Base‘𝐺)
19	18	subgss 18274	. . . . . . . . . . 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 769	. . . . . . . . . 10 ⊢ (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ (𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑈)) → 𝑎 ∈ 𝑇)
22	20, 21	sseldd 3967	. . . . . . . . 9 ⊢ (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ (𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑈)) → 𝑎 ∈ (Base‘𝐺))
23	4	adantr 483	. . . . . . . . . . 11 ⊢ (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ (𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑈)) → 𝑈 ∈ (SubGrp‘𝐺))
24	18	subgss 18274	. . . . . . . . . . 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 771	. . . . . . . . . 10 ⊢ (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ (𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑈)) → 𝑏 ∈ 𝑈)
27	25, 26	sseldd 3967	. . . . . . . . 9 ⊢ (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ (𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑈)) → 𝑏 ∈ (Base‘𝐺))
28		eqid 2821	. . . . . . . . . 10 ⊢ (invg‘𝐺) = (invg‘𝐺)
29	18, 12, 28	grpinvadd 18171	. . . . . . . . 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 1367	. . . . . . . 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 483	. . . . . . . . . 10 ⊢ (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ (𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑈)) → 𝑇 ⊆ (𝑍‘𝑈))
32	28	subginvcl 18282	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑎 ∈ 𝑇) → ((invg‘𝐺)‘𝑎) ∈ 𝑇)
33	15, 21, 32	syl2anc 586	. . . . . . . . . 10 ⊢ (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ (𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑈)) → ((invg‘𝐺)‘𝑎) ∈ 𝑇)
34	31, 33	sseldd 3967	. . . . . . . . 9 ⊢ (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ (𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑈)) → ((invg‘𝐺)‘𝑎) ∈ (𝑍‘𝑈))
35	28	subginvcl 18282	. . . . . . . . . 10 ⊢ ((𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑏 ∈ 𝑈) → ((invg‘𝐺)‘𝑏) ∈ 𝑈)
36	23, 26, 35	syl2anc 586	. . . . . . . . 9 ⊢ (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ (𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑈)) → ((invg‘𝐺)‘𝑏) ∈ 𝑈)
37	12, 9	cntzi 18453	. . . . . . . . 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 586	. . . . . . . 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 2859	. . . . . . 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 18769	. . . . . . . 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 836	. . . . . . 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 2913	. . . . . 6 ⊢ (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ (𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑈)) → ((invg‘𝐺)‘(𝑎(+g‘𝐺)𝑏)) ∈ (𝑇 ⊕ 𝑈))
43		fveq2 6664	. . . . . . 7 ⊢ (𝑥 = (𝑎(+g‘𝐺)𝑏) → ((invg‘𝐺)‘𝑥) = ((invg‘𝐺)‘(𝑎(+g‘𝐺)𝑏)))
44	43	eleq1d 2897	. . . . . 6 ⊢ (𝑥 = (𝑎(+g‘𝐺)𝑏) → (((invg‘𝐺)‘𝑥) ∈ (𝑇 ⊕ 𝑈) ↔ ((invg‘𝐺)‘(𝑎(+g‘𝐺)𝑏)) ∈ (𝑇 ⊕ 𝑈)))
45	42, 44	syl5ibrcom 249	. . . . 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 3294	. . . 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 242	. . 3 ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) → (𝑥 ∈ (𝑇 ⊕ 𝑈) → ((invg‘𝐺)‘𝑥) ∈ (𝑇 ⊕ 𝑈)))
48	47	ralrimiv 3181	. 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 18291	. . 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 711	1 ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) → (𝑇 ⊕ 𝑈) ∈ (SubGrp‘𝐺))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110  ∀wral 3138  ∃wrex 3139   ⊆ wss 3935  ‘cfv 6349  (class class class)co 7150  Basecbs 16477  +_gcplusg 16559  SubMndcsubmnd 17949  Grpcgrp 18097  inv_gcminusg 18098  SubGrpcsubg 18267  Cntzccntz 18439  LSSumclsm 18753

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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-tp 4565  df-op 4567  df-uni 4832  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-pred 6142  df-ord 6188  df-on 6189  df-lim 6190  df-suc 6191  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7575  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-er 8283  df-en 8504  df-dom 8505  df-sdom 8506  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-nn 11633  df-2 11694  df-ndx 16480  df-slot 16481  df-base 16483  df-sets 16484  df-ress 16485  df-plusg 16572  df-0g 16709  df-mgm 17846  df-sgrp 17895  df-mnd 17906  df-submnd 17951  df-grp 18100  df-minusg 18101  df-subg 18270  df-cntz 18441  df-lsm 18755

This theorem is referenced by:  pj1ghm  18823  lsmsubg2  18973  dprd2da  19158  dmdprdsplit2lem  19161  dprdsplit  19164