Theorem subgabl 13586

Description: A subgroup of an abelian group is also abelian. (Contributed by Mario Carneiro, 3-Dec-2014.)

Hypothesis

Ref	Expression
subgabl.h	⊢ 𝐻 = (𝐺 ↾s 𝑆)

Assertion

Ref	Expression
subgabl	⊢ ((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝐻 ∈ Abel)

Proof of Theorem subgabl

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		subgabl.h	. . . 4 ⊢ 𝐻 = (𝐺 ↾s 𝑆)
2	1	subgbas 13432	. . 3 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 = (Base‘𝐻))
3	2	adantl 277	. 2 ⊢ ((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝑆 = (Base‘𝐻))
4	1	a1i 9	. . 3 ⊢ ((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝐻 = (𝐺 ↾s 𝑆))
5		eqid 2204	. . . 4 ⊢ (+g‘𝐺) = (+g‘𝐺)
6	5	a1i 9	. . 3 ⊢ ((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (+g‘𝐺) = (+g‘𝐺))
7		simpr 110	. . 3 ⊢ ((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝑆 ∈ (SubGrp‘𝐺))
8		simpl 109	. . 3 ⊢ ((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝐺 ∈ Abel)
9	4, 6, 7, 8	ressplusgd 12879	. 2 ⊢ ((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (+g‘𝐺) = (+g‘𝐻))
10	1	subggrp 13431	. . 3 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐻 ∈ Grp)
11	10	adantl 277	. 2 ⊢ ((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝐻 ∈ Grp)
12		simp1l 1023	. . 3 ⊢ (((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → 𝐺 ∈ Abel)
13		simp1r 1024	. . . . 5 ⊢ (((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → 𝑆 ∈ (SubGrp‘𝐺))
14		eqid 2204	. . . . . 6 ⊢ (Base‘𝐺) = (Base‘𝐺)
15	14	subgss 13428	. . . . 5 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ (Base‘𝐺))
16	13, 15	syl 14	. . . 4 ⊢ (((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → 𝑆 ⊆ (Base‘𝐺))
17		simp2 1000	. . . 4 ⊢ (((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → 𝑥 ∈ 𝑆)
18	16, 17	sseldd 3193	. . 3 ⊢ (((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → 𝑥 ∈ (Base‘𝐺))
19		simp3 1001	. . . 4 ⊢ (((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → 𝑦 ∈ 𝑆)
20	16, 19	sseldd 3193	. . 3 ⊢ (((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → 𝑦 ∈ (Base‘𝐺))
21	14, 5	ablcom 13557	. . 3 ⊢ ((𝐺 ∈ Abel ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → (𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥))
22	12, 18, 20, 21	syl3anc 1249	. 2 ⊢ (((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥))
23	3, 9, 11, 22	isabld 13553	1 ⊢ ((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝐻 ∈ Abel)

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 980   = wceq 1372   ∈ wcel 2175   ⊆ wss 3165  ‘cfv 5268  (class class class)co 5934  Basecbs 12751   ↾_s cress 12752  +_gcplusg 12828  Grpcgrp 13250  SubGrpcsubg 13421  Abelcabl 13539

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4478  ax-setind 4583  ax-cnex 7998  ax-resscn 7999  ax-1cn 8000  ax-1re 8001  ax-icn 8002  ax-addcl 8003  ax-addrcl 8004  ax-mulcl 8005  ax-addcom 8007  ax-addass 8009  ax-i2m1 8012  ax-0lt1 8013  ax-0id 8015  ax-rnegex 8016  ax-pre-ltirr 8019  ax-pre-ltadd 8023

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-nel 2471  df-ral 2488  df-rex 2489  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4338  df-xp 4679  df-rel 4680  df-cnv 4681  df-co 4682  df-dm 4683  df-rn 4684  df-res 4685  df-ima 4686  df-iota 5229  df-fun 5270  df-fn 5271  df-fv 5276  df-ov 5937  df-oprab 5938  df-mpo 5939  df-pnf 8091  df-mnf 8092  df-ltxr 8094  df-inn 9019  df-2 9077  df-ndx 12754  df-slot 12755  df-base 12757  df-sets 12758  df-iress 12759  df-plusg 12841  df-grp 13253  df-subg 13424  df-cmn 13540  df-abl 13541

This theorem is referenced by:  issubrng2  13890  rnglidlrng  14178