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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 19047 | . 2 ⊢ (𝐺 ∈ Grp → 𝐵 ∈ (SubGrp‘𝐺)) |
3 | simp1 1133 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → 𝐺 ∈ Grp) | |
4 | eqid 2724 | . . . . . 6 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
5 | 1, 4 | grpcl 18863 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥(+g‘𝐺)𝑦) ∈ 𝐵) |
6 | simp2 1134 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → 𝑥 ∈ 𝐵) | |
7 | eqid 2724 | . . . . . 6 ⊢ (-g‘𝐺) = (-g‘𝐺) | |
8 | 1, 7 | grpsubcl 18940 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ (𝑥(+g‘𝐺)𝑦) ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥) ∈ 𝐵) |
9 | 3, 5, 6, 8 | syl3anc 1368 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥) ∈ 𝐵) |
10 | 9 | 3expb 1117 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥) ∈ 𝐵) |
11 | 10 | ralrimivva 3192 | . 2 ⊢ (𝐺 ∈ Grp → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥) ∈ 𝐵) |
12 | 1, 4, 7 | isnsg3 19079 | . 2 ⊢ (𝐵 ∈ (NrmSGrp‘𝐺) ↔ (𝐵 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥) ∈ 𝐵)) |
13 | 2, 11, 12 | sylanbrc 582 | 1 ⊢ (𝐺 ∈ Grp → 𝐵 ∈ (NrmSGrp‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∀wral 3053 ‘cfv 6534 (class class class)co 7402 Basecbs 17145 +gcplusg 17198 Grpcgrp 18855 -gcsg 18857 SubGrpcsubg 19039 NrmSGrpcnsg 19040 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-1st 7969 df-2nd 7970 df-ress 17175 df-0g 17388 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-grp 18858 df-minusg 18859 df-sbg 18860 df-subg 19042 df-nsg 19043 |
This theorem is referenced by: 0idnsgd 19090 trivnsgd 19091 1nsgtrivd 19093 2nsgsimpgd 20016 |
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