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Theorem subg0 12971
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 = (𝐺s 𝑆)
subg0.i 0 = (0g𝐺)
Assertion
Ref Expression
subg0 (𝑆 ∈ (SubGrp‘𝐺) → 0 = (0g𝐻))

Proof of Theorem subg0
StepHypRef Expression
1 subg0.h . . . . . 6 𝐻 = (𝐺s 𝑆)
21a1i 9 . . . . 5 (𝑆 ∈ (SubGrp‘𝐺) → 𝐻 = (𝐺s 𝑆))
3 eqid 2177 . . . . . 6 (+g𝐺) = (+g𝐺)
43a1i 9 . . . . 5 (𝑆 ∈ (SubGrp‘𝐺) → (+g𝐺) = (+g𝐺))
5 id 19 . . . . 5 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ∈ (SubGrp‘𝐺))
6 subgrcl 12970 . . . . 5 (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
72, 4, 5, 6ressplusgd 12579 . . . 4 (𝑆 ∈ (SubGrp‘𝐺) → (+g𝐺) = (+g𝐻))
87oveqd 5889 . . 3 (𝑆 ∈ (SubGrp‘𝐺) → ((0g𝐻)(+g𝐺)(0g𝐻)) = ((0g𝐻)(+g𝐻)(0g𝐻)))
91subggrp 12968 . . . 4 (𝑆 ∈ (SubGrp‘𝐺) → 𝐻 ∈ Grp)
10 eqid 2177 . . . . . 6 (Base‘𝐻) = (Base‘𝐻)
11 eqid 2177 . . . . . 6 (0g𝐻) = (0g𝐻)
1210, 11grpidcl 12836 . . . . 5 (𝐻 ∈ Grp → (0g𝐻) ∈ (Base‘𝐻))
139, 12syl 14 . . . 4 (𝑆 ∈ (SubGrp‘𝐺) → (0g𝐻) ∈ (Base‘𝐻))
14 eqid 2177 . . . . 5 (+g𝐻) = (+g𝐻)
1510, 14, 11grplid 12838 . . . 4 ((𝐻 ∈ Grp ∧ (0g𝐻) ∈ (Base‘𝐻)) → ((0g𝐻)(+g𝐻)(0g𝐻)) = (0g𝐻))
169, 13, 15syl2anc 411 . . 3 (𝑆 ∈ (SubGrp‘𝐺) → ((0g𝐻)(+g𝐻)(0g𝐻)) = (0g𝐻))
178, 16eqtrd 2210 . 2 (𝑆 ∈ (SubGrp‘𝐺) → ((0g𝐻)(+g𝐺)(0g𝐻)) = (0g𝐻))
18 eqid 2177 . . . . 5 (Base‘𝐺) = (Base‘𝐺)
1918subgss 12965 . . . 4 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ (Base‘𝐺))
201subgbas 12969 . . . . 5 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 = (Base‘𝐻))
2113, 20eleqtrrd 2257 . . . 4 (𝑆 ∈ (SubGrp‘𝐺) → (0g𝐻) ∈ 𝑆)
2219, 21sseldd 3156 . . 3 (𝑆 ∈ (SubGrp‘𝐺) → (0g𝐻) ∈ (Base‘𝐺))
23 subg0.i . . . 4 0 = (0g𝐺)
2418, 3, 23grpid 12844 . . 3 ((𝐺 ∈ Grp ∧ (0g𝐻) ∈ (Base‘𝐺)) → (((0g𝐻)(+g𝐺)(0g𝐻)) = (0g𝐻) ↔ 0 = (0g𝐻)))
256, 22, 24syl2anc 411 . 2 (𝑆 ∈ (SubGrp‘𝐺) → (((0g𝐻)(+g𝐺)(0g𝐻)) = (0g𝐻) ↔ 0 = (0g𝐻)))
2617, 25mpbid 147 1 (𝑆 ∈ (SubGrp‘𝐺) → 0 = (0g𝐻))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105   = wceq 1353  wcel 2148  cfv 5215  (class class class)co 5872  Basecbs 12454  s cress 12455  +gcplusg 12528  0gc0g 12693  Grpcgrp 12809  SubGrpcsubg 12958
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4120  ax-pow 4173  ax-pr 4208  ax-un 4432  ax-setind 4535  ax-cnex 7899  ax-resscn 7900  ax-1cn 7901  ax-1re 7902  ax-icn 7903  ax-addcl 7904  ax-addrcl 7905  ax-mulcl 7906  ax-addcom 7908  ax-addass 7910  ax-i2m1 7913  ax-0lt1 7914  ax-0id 7916  ax-rnegex 7917  ax-pre-ltirr 7920  ax-pre-ltadd 7924
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-br 4003  df-opab 4064  df-mpt 4065  df-id 4292  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-ima 4638  df-iota 5177  df-fun 5217  df-fn 5218  df-fv 5223  df-riota 5828  df-ov 5875  df-oprab 5876  df-mpo 5877  df-pnf 7990  df-mnf 7991  df-ltxr 7993  df-inn 8916  df-2 8974  df-ndx 12457  df-slot 12458  df-base 12460  df-sets 12461  df-iress 12462  df-plusg 12541  df-0g 12695  df-mgm 12707  df-sgrp 12740  df-mnd 12750  df-grp 12812  df-subg 12961
This theorem is referenced by:  subginv  12972  subg0cl  12973  subgmulg  12979  subrg0  13287
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