Theorem subg0 13886

Description: A subgroup of a group must have the same identity as the group. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)

Hypotheses

Ref	Expression
subg0.h	⊢ 𝐻 = (𝐺 ↾s 𝑆)
subg0.i	⊢ 0 = (0g‘𝐺)

Assertion

Ref	Expression
subg0	⊢ (𝑆 ∈ (SubGrp‘𝐺) → 0 = (0g‘𝐻))

Proof of Theorem subg0

Step	Hyp	Ref	Expression
1		subg0.h	. . . . . 6 ⊢ 𝐻 = (𝐺 ↾s 𝑆)
2	1	a1i 9	. . . . 5 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐻 = (𝐺 ↾s 𝑆))
3		eqid 2232	. . . . . 6 ⊢ (+g‘𝐺) = (+g‘𝐺)
4	3	a1i 9	. . . . 5 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (+g‘𝐺) = (+g‘𝐺))
5		id 19	. . . . 5 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ∈ (SubGrp‘𝐺))
6		subgrcl 13885	. . . . 5 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
7	2, 4, 5, 6	ressplusgd 13331	. . . 4 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (+g‘𝐺) = (+g‘𝐻))
8	7	oveqd 6066	. . 3 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → ((0g‘𝐻)(+g‘𝐺)(0g‘𝐻)) = ((0g‘𝐻)(+g‘𝐻)(0g‘𝐻)))
9	1	subggrp 13883	. . . 4 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐻 ∈ Grp)
10		eqid 2232	. . . . . 6 ⊢ (Base‘𝐻) = (Base‘𝐻)
11		eqid 2232	. . . . . 6 ⊢ (0g‘𝐻) = (0g‘𝐻)
12	10, 11	grpidcl 13731	. . . . 5 ⊢ (𝐻 ∈ Grp → (0g‘𝐻) ∈ (Base‘𝐻))
13	9, 12	syl 14	. . . 4 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (0g‘𝐻) ∈ (Base‘𝐻))
14		eqid 2232	. . . . 5 ⊢ (+g‘𝐻) = (+g‘𝐻)
15	10, 14, 11	grplid 13733	. . . 4 ⊢ ((𝐻 ∈ Grp ∧ (0g‘𝐻) ∈ (Base‘𝐻)) → ((0g‘𝐻)(+g‘𝐻)(0g‘𝐻)) = (0g‘𝐻))
16	9, 13, 15	syl2anc 411	. . 3 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → ((0g‘𝐻)(+g‘𝐻)(0g‘𝐻)) = (0g‘𝐻))
17	8, 16	eqtrd 2265	. 2 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → ((0g‘𝐻)(+g‘𝐺)(0g‘𝐻)) = (0g‘𝐻))
18		eqid 2232	. . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺)
19	18	subgss 13880	. . . 4 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ (Base‘𝐺))
20	1	subgbas 13884	. . . . 5 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 = (Base‘𝐻))
21	13, 20	eleqtrrd 2312	. . . 4 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (0g‘𝐻) ∈ 𝑆)
22	19, 21	sseldd 3238	. . 3 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (0g‘𝐻) ∈ (Base‘𝐺))
23		subg0.i	. . . 4 ⊢ 0 = (0g‘𝐺)
24	18, 3, 23	grpid 13741	. . 3 ⊢ ((𝐺 ∈ Grp ∧ (0g‘𝐻) ∈ (Base‘𝐺)) → (((0g‘𝐻)(+g‘𝐺)(0g‘𝐻)) = (0g‘𝐻) ↔ 0 = (0g‘𝐻)))
25	6, 22, 24	syl2anc 411	. 2 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (((0g‘𝐻)(+g‘𝐺)(0g‘𝐻)) = (0g‘𝐻) ↔ 0 = (0g‘𝐻)))
26	17, 25	mpbid 147	1 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 0 = (0g‘𝐻))

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1398   ∈ wcel 2203  ‘cfv 5351  (class class class)co 6049  Basecbs 13201   ↾_s cress 13202  +_gcplusg 13279  0_gc0g 13458  Grpcgrp 13702  SubGrpcsubg 13873

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-cnex 8214  ax-resscn 8215  ax-1cn 8216  ax-1re 8217  ax-icn 8218  ax-addcl 8219  ax-addrcl 8220  ax-mulcl 8221  ax-addcom 8223  ax-addass 8225  ax-i2m1 8228  ax-0lt1 8229  ax-0id 8231  ax-rnegex 8232  ax-pre-ltirr 8235  ax-pre-ltadd 8239

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-br 4109  df-opab 4171  df-mpt 4172  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-fv 5359  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-pnf 8306  df-mnf 8307  df-ltxr 8309  df-inn 9234  df-2 9292  df-ndx 13204  df-slot 13205  df-base 13207  df-sets 13208  df-iress 13209  df-plusg 13292  df-0g 13460  df-mgm 13558  df-sgrp 13604  df-mnd 13619  df-grp 13705  df-subg 13876

This theorem is referenced by:  subginv  13887  subg0cl  13888  subgmulg  13894  subrng0  14341  subrg0  14362  mpl0fi  14844