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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 18931 | . 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 18757 | . . . . 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 18828 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ (𝑥(+g‘𝐺)𝑦) ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥) ∈ 𝐵) |
9 | 3, 5, 6, 8 | syl3anc 1372 | . . . 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 3198 | . 2 ⊢ (𝐺 ∈ Grp → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥) ∈ 𝐵) |
12 | 1, 4, 7 | isnsg3 18963 | . 2 ⊢ (𝐵 ∈ (NrmSGrp‘𝐺) ↔ (𝐵 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥) ∈ 𝐵)) |
13 | 2, 11, 12 | sylanbrc 584 | 1 ⊢ (𝐺 ∈ Grp → 𝐵 ∈ (NrmSGrp‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ∀wral 3065 ‘cfv 6497 (class class class)co 7358 Basecbs 17084 +gcplusg 17134 Grpcgrp 18749 -gcsg 18751 SubGrpcsubg 18923 NrmSGrpcnsg 18924 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rmo 3354 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-1st 7922 df-2nd 7923 df-ress 17114 df-0g 17324 df-mgm 18498 df-sgrp 18547 df-mnd 18558 df-grp 18752 df-minusg 18753 df-sbg 18754 df-subg 18926 df-nsg 18927 |
This theorem is referenced by: 0idnsgd 18974 trivnsgd 18975 1nsgtrivd 18977 2nsgsimpgd 19882 |
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