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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 19076 | . 2 ⊢ (𝐺 ∈ Grp → 𝐵 ∈ (SubGrp‘𝐺)) |
3 | simp1 1134 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → 𝐺 ∈ Grp) | |
4 | eqid 2728 | . . . . . 6 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
5 | 1, 4 | grpcl 18891 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥(+g‘𝐺)𝑦) ∈ 𝐵) |
6 | simp2 1135 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → 𝑥 ∈ 𝐵) | |
7 | eqid 2728 | . . . . . 6 ⊢ (-g‘𝐺) = (-g‘𝐺) | |
8 | 1, 7 | grpsubcl 18969 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ (𝑥(+g‘𝐺)𝑦) ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥) ∈ 𝐵) |
9 | 3, 5, 6, 8 | syl3anc 1369 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥) ∈ 𝐵) |
10 | 9 | 3expb 1118 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥) ∈ 𝐵) |
11 | 10 | ralrimivva 3196 | . 2 ⊢ (𝐺 ∈ Grp → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥) ∈ 𝐵) |
12 | 1, 4, 7 | isnsg3 19108 | . 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 1085 = wceq 1534 ∈ wcel 2099 ∀wral 3057 ‘cfv 6542 (class class class)co 7414 Basecbs 17173 +gcplusg 17226 Grpcgrp 18883 -gcsg 18885 SubGrpcsubg 19068 NrmSGrpcnsg 19069 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-1st 7987 df-2nd 7988 df-ress 17203 df-0g 17416 df-mgm 18593 df-sgrp 18672 df-mnd 18688 df-grp 18886 df-minusg 18887 df-sbg 18888 df-subg 19071 df-nsg 19072 |
This theorem is referenced by: 0idnsgd 19119 trivnsgd 19120 1nsgtrivd 19122 2nsgsimpgd 20052 |
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