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Mirrors > Home > MPE Home > Th. List > nsgid | Structured version Visualization version GIF version |
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 17908 | . 2 ⊢ (𝐺 ∈ Grp → 𝐵 ∈ (SubGrp‘𝐺)) |
3 | simp1 1167 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → 𝐺 ∈ Grp) | |
4 | eqid 2800 | . . . . . 6 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
5 | 1, 4 | grpcl 17745 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥(+g‘𝐺)𝑦) ∈ 𝐵) |
6 | simp2 1168 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → 𝑥 ∈ 𝐵) | |
7 | eqid 2800 | . . . . . 6 ⊢ (-g‘𝐺) = (-g‘𝐺) | |
8 | 1, 7 | grpsubcl 17810 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ (𝑥(+g‘𝐺)𝑦) ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥) ∈ 𝐵) |
9 | 3, 5, 6, 8 | syl3anc 1491 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥) ∈ 𝐵) |
10 | 9 | 3expb 1150 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥) ∈ 𝐵) |
11 | 10 | ralrimivva 3153 | . 2 ⊢ (𝐺 ∈ Grp → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥) ∈ 𝐵) |
12 | 1, 4, 7 | isnsg3 17940 | . 2 ⊢ (𝐵 ∈ (NrmSGrp‘𝐺) ↔ (𝐵 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥) ∈ 𝐵)) |
13 | 2, 11, 12 | sylanbrc 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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