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| Description: The whole group is a normal subgroup of itself. (Contributed by Mario Carneiro, 4-Feb-2015.) | 
| Ref | Expression | 
|---|---|
| nsgid.z | ⊢ 𝐵 = (Base‘𝐺) | 
| Ref | Expression | 
|---|---|
| nsgid | ⊢ (𝐺 ∈ Grp → 𝐵 ∈ (NrmSGrp‘𝐺)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | nsgid.z | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
| 2 | 1 | subgid 19146 | . 2 ⊢ (𝐺 ∈ Grp → 𝐵 ∈ (SubGrp‘𝐺)) | 
| 3 | simp1 1137 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → 𝐺 ∈ Grp) | |
| 4 | eqid 2737 | . . . . . 6 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
| 5 | 1, 4 | grpcl 18959 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥(+g‘𝐺)𝑦) ∈ 𝐵) | 
| 6 | simp2 1138 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → 𝑥 ∈ 𝐵) | |
| 7 | eqid 2737 | . . . . . 6 ⊢ (-g‘𝐺) = (-g‘𝐺) | |
| 8 | 1, 7 | grpsubcl 19038 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ (𝑥(+g‘𝐺)𝑦) ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥) ∈ 𝐵) | 
| 9 | 3, 5, 6, 8 | syl3anc 1373 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥) ∈ 𝐵) | 
| 10 | 9 | 3expb 1121 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥) ∈ 𝐵) | 
| 11 | 10 | ralrimivva 3202 | . 2 ⊢ (𝐺 ∈ Grp → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥) ∈ 𝐵) | 
| 12 | 1, 4, 7 | isnsg3 19178 | . 2 ⊢ (𝐵 ∈ (NrmSGrp‘𝐺) ↔ (𝐵 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥) ∈ 𝐵)) | 
| 13 | 2, 11, 12 | sylanbrc 583 | 1 ⊢ (𝐺 ∈ Grp → 𝐵 ∈ (NrmSGrp‘𝐺)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 +gcplusg 17297 Grpcgrp 18951 -gcsg 18953 SubGrpcsubg 19138 NrmSGrpcnsg 19139 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-ress 17275 df-0g 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-minusg 18955 df-sbg 18956 df-subg 19141 df-nsg 19142 | 
| This theorem is referenced by: 0idnsgd 19189 trivnsgd 19190 1nsgtrivd 19192 2nsgsimpgd 20122 | 
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