Theorem lsmsubg 19595

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 1137	. . . 4 ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) → 𝑇 ∈ (SubGrp‘𝐺))
2		subgsubm 19090	. . . 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 1138	. . . 4 ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) → 𝑈 ∈ (SubGrp‘𝐺))
5		subgsubm 19090	. . . 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 1139	. . 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 19594	. . 3 ⊢ ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) → (𝑇 ⊕ 𝑈) ∈ (SubMnd‘𝐺))
11	3, 6, 7, 10	syl3anc 1374	. 2 ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) → (𝑇 ⊕ 𝑈) ∈ (SubMnd‘𝐺))
12		eqid 2737	. . . . . 6 ⊢ (+g‘𝐺) = (+g‘𝐺)
13	12, 8	lsmelval 19590	. . . . 5 ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (𝑥 ∈ (𝑇 ⊕ 𝑈) ↔ ∃𝑎 ∈ 𝑇 ∃𝑏 ∈ 𝑈 𝑥 = (𝑎(+g‘𝐺)𝑏)))
14	13	3adant3 1133	. . . 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 19073	. . . . . . . . . 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 2737	. . . . . . . . . . . 12 ⊢ (Base‘𝐺) = (Base‘𝐺)
19	18	subgss 19069	. . . . . . . . . . 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 3936	. . . . . . . . 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 19069	. . . . . . . . . . 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 3936	. . . . . . . . 9 ⊢ (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ (𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑈)) → 𝑏 ∈ (Base‘𝐺))
28		eqid 2737	. . . . . . . . . 10 ⊢ (invg‘𝐺) = (invg‘𝐺)
29	18, 12, 28	grpinvadd 18960	. . . . . . . . 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 1374	. . . . . . . 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 19077	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑎 ∈ 𝑇) → ((invg‘𝐺)‘𝑎) ∈ 𝑇)
33	15, 21, 32	syl2anc 585	. . . . . . . . . 10 ⊢ (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ (𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑈)) → ((invg‘𝐺)‘𝑎) ∈ 𝑇)
34	31, 33	sseldd 3936	. . . . . . . . 9 ⊢ (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ (𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑈)) → ((invg‘𝐺)‘𝑎) ∈ (𝑍‘𝑈))
35	28	subginvcl 19077	. . . . . . . . . 10 ⊢ ((𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑏 ∈ 𝑈) → ((invg‘𝐺)‘𝑏) ∈ 𝑈)
36	23, 26, 35	syl2anc 585	. . . . . . . . 9 ⊢ (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ (𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑈)) → ((invg‘𝐺)‘𝑏) ∈ 𝑈)
37	12, 9	cntzi 19270	. . . . . . . . 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 585	. . . . . . . 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 2775	. . . . . . 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 19591	. . . . . . . 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 2837	. . . . . 6 ⊢ (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ (𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑈)) → ((invg‘𝐺)‘(𝑎(+g‘𝐺)𝑏)) ∈ (𝑇 ⊕ 𝑈))
43		fveq2 6842	. . . . . . 7 ⊢ (𝑥 = (𝑎(+g‘𝐺)𝑏) → ((invg‘𝐺)‘𝑥) = ((invg‘𝐺)‘(𝑎(+g‘𝐺)𝑏)))
44	43	eleq1d 2822	. . . . . 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 3195	. . . 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 3129	. 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 19086	. . 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 714	1 ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) → (𝑇 ⊕ 𝑈) ∈ (SubGrp‘𝐺))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062   ⊆ wss 3903  ‘cfv 6500  (class class class)co 7368  Basecbs 17148  +_gcplusg 17189  SubMndcsubmnd 18719  Grpcgrp 18875  inv_gcminusg 18876  SubGrpcsubg 19062  Cntzccntz 19256  LSSumclsm 19575

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-1st 7943  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-er 8645  df-en 8896  df-dom 8897  df-sdom 8898  df-pnf 11180  df-mnf 11181  df-xr 11182  df-ltxr 11183  df-le 11184  df-sub 11378  df-neg 11379  df-nn 12158  df-2 12220  df-sets 17103  df-slot 17121  df-ndx 17133  df-base 17149  df-ress 17170  df-plusg 17202  df-0g 17373  df-mgm 18577  df-sgrp 18656  df-mnd 18672  df-submnd 18721  df-grp 18878  df-minusg 18879  df-subg 19065  df-cntz 19258  df-lsm 19577

This theorem is referenced by:  pj1ghm  19644  lsmsubg2  19800  dprd2da  19985  dmdprdsplit2lem  19988  dprdsplit  19991