Theorem subg0 13936

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 2234	. . . . . 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 13935	. . . . 5 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
7	2, 4, 5, 6	ressplusgd 13429	. . . 4 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (+g‘𝐺) = (+g‘𝐻))
8	7	oveqd 6075	. . 3 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → ((0g‘𝐻)(+g‘𝐺)(0g‘𝐻)) = ((0g‘𝐻)(+g‘𝐻)(0g‘𝐻)))
9	1	subggrp 13933	. . . 4 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐻 ∈ Grp)
10		eqid 2234	. . . . . 6 ⊢ (Base‘𝐻) = (Base‘𝐻)
11		eqid 2234	. . . . . 6 ⊢ (0g‘𝐻) = (0g‘𝐻)
12	10, 11	grpidcl 13787	. . . . 5 ⊢ (𝐻 ∈ Grp → (0g‘𝐻) ∈ (Base‘𝐻))
13	9, 12	syl 14	. . . 4 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (0g‘𝐻) ∈ (Base‘𝐻))
14		eqid 2234	. . . . 5 ⊢ (+g‘𝐻) = (+g‘𝐻)
15	10, 14, 11	grplid 13789	. . . 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 2267	. 2 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → ((0g‘𝐻)(+g‘𝐺)(0g‘𝐻)) = (0g‘𝐻))
18		eqid 2234	. . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺)
19	18	subgss 13930	. . . 4 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ (Base‘𝐺))
20	1	subgbas 13934	. . . . 5 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 = (Base‘𝐻))
21	13, 20	eleqtrrd 2314	. . . 4 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (0g‘𝐻) ∈ 𝑆)
22	19, 21	sseldd 3243	. . 3 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (0g‘𝐻) ∈ (Base‘𝐺))
23		subg0.i	. . . 4 ⊢ 0 = (0g‘𝐺)
24	18, 3, 23	grpid 13797	. . 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 2205  ‘cfv 5357  (class class class)co 6058  Basecbs 13299   ↾_s cress 13300  +_gcplusg 13377  0_gc0g 13556  Grpcgrp 13758  SubGrpcsubg 13923

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 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-pre-ltirr 8255  ax-pre-ltadd 8259

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 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-pnf 8326  df-mnf 8327  df-ltxr 8329  df-inn 9258  df-2 9316  df-ndx 13302  df-slot 13303  df-base 13305  df-sets 13306  df-iress 13307  df-plusg 13390  df-0g 13558  df-mgm 13622  df-sgrp 13668  df-mnd 13681  df-grp 13761  df-subg 13926

This theorem is referenced by:  subginv  13937  subg0cl  13938  subgmulg  13944  subrng0  14456  subrg0  14477  mpl0fi  14986