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Theorem nsgid 17952
Description: The whole group is a normal subgroup of itself. (Contributed by Mario Carneiro, 4-Feb-2015.)
Hypothesis
Ref Expression
nsgid.z 𝐵 = (Base‘𝐺)
Assertion
Ref Expression
nsgid (𝐺 ∈ Grp → 𝐵 ∈ (NrmSGrp‘𝐺))

Proof of Theorem nsgid
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nsgid.z . . 3 𝐵 = (Base‘𝐺)
21subgid 17908 . 2 (𝐺 ∈ Grp → 𝐵 ∈ (SubGrp‘𝐺))
3 simp1 1167 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑥𝐵𝑦𝐵) → 𝐺 ∈ Grp)
4 eqid 2800 . . . . . 6 (+g𝐺) = (+g𝐺)
51, 4grpcl 17745 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑥𝐵𝑦𝐵) → (𝑥(+g𝐺)𝑦) ∈ 𝐵)
6 simp2 1168 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑥𝐵𝑦𝐵) → 𝑥𝐵)
7 eqid 2800 . . . . . 6 (-g𝐺) = (-g𝐺)
81, 7grpsubcl 17810 . . . . 5 ((𝐺 ∈ Grp ∧ (𝑥(+g𝐺)𝑦) ∈ 𝐵𝑥𝐵) → ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥) ∈ 𝐵)
93, 5, 6, 8syl3anc 1491 . . . 4 ((𝐺 ∈ Grp ∧ 𝑥𝐵𝑦𝐵) → ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥) ∈ 𝐵)
1093expb 1150 . . 3 ((𝐺 ∈ Grp ∧ (𝑥𝐵𝑦𝐵)) → ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥) ∈ 𝐵)
1110ralrimivva 3153 . 2 (𝐺 ∈ Grp → ∀𝑥𝐵𝑦𝐵 ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥) ∈ 𝐵)
121, 4, 7isnsg3 17940 . 2 (𝐵 ∈ (NrmSGrp‘𝐺) ↔ (𝐵 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝐵𝑦𝐵 ((𝑥(+g𝐺)𝑦)(-g𝐺)𝑥) ∈ 𝐵))
132, 11, 12sylanbrc 579 1 (𝐺 ∈ Grp → 𝐵 ∈ (NrmSGrp‘𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1108   = wceq 1653  wcel 2157  wral 3090  cfv 6102  (class class class)co 6879  Basecbs 16183  +gcplusg 16266  Grpcgrp 17737  -gcsg 17739  SubGrpcsubg 17900  NrmSGrpcnsg 17901
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778  ax-rep 4965  ax-sep 4976  ax-nul 4984  ax-pow 5036  ax-pr 5098  ax-un 7184
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2592  df-eu 2610  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ne 2973  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3388  df-sbc 3635  df-csb 3730  df-dif 3773  df-un 3775  df-in 3777  df-ss 3784  df-nul 4117  df-if 4279  df-pw 4352  df-sn 4370  df-pr 4372  df-op 4376  df-uni 4630  df-iun 4713  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5221  df-xp 5319  df-rel 5320  df-cnv 5321  df-co 5322  df-dm 5323  df-rn 5324  df-res 5325  df-ima 5326  df-iota 6065  df-fun 6104  df-fn 6105  df-f 6106  df-f1 6107  df-fo 6108  df-f1o 6109  df-fv 6110  df-riota 6840  df-ov 6882  df-oprab 6883  df-mpt2 6884  df-1st 7402  df-2nd 7403  df-ress 16191  df-0g 16416  df-mgm 17556  df-sgrp 17598  df-mnd 17609  df-grp 17740  df-minusg 17741  df-sbg 17742  df-subg 17903  df-nsg 17904
This theorem is referenced by: (None)
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