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Mirrors > Home > MPE Home > Th. List > 1nsgtrivd | Structured version Visualization version GIF version |
Description: A group with exactly one normal subgroup is trivial. (Contributed by Rohan Ridenour, 3-Aug-2023.) |
Ref | Expression |
---|---|
1nsgtrivd.1 | ⊢ 𝐵 = (Base‘𝐺) |
1nsgtrivd.2 | ⊢ 0 = (0g‘𝐺) |
1nsgtrivd.3 | ⊢ (𝜑 → 𝐺 ∈ Grp) |
1nsgtrivd.4 | ⊢ (𝜑 → (NrmSGrp‘𝐺) ≈ 1o) |
Ref | Expression |
---|---|
1nsgtrivd | ⊢ (𝜑 → 𝐵 = { 0 }) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1nsgtrivd.3 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ Grp) | |
2 | 1nsgtrivd.1 | . . . . 5 ⊢ 𝐵 = (Base‘𝐺) | |
3 | 2 | nsgid 18442 | . . . 4 ⊢ (𝐺 ∈ Grp → 𝐵 ∈ (NrmSGrp‘𝐺)) |
4 | 1, 3 | syl 17 | . . 3 ⊢ (𝜑 → 𝐵 ∈ (NrmSGrp‘𝐺)) |
5 | 1nsgtrivd.2 | . . . . . 6 ⊢ 0 = (0g‘𝐺) | |
6 | 5 | 0nsg 18441 | . . . . 5 ⊢ (𝐺 ∈ Grp → { 0 } ∈ (NrmSGrp‘𝐺)) |
7 | 1, 6 | syl 17 | . . . 4 ⊢ (𝜑 → { 0 } ∈ (NrmSGrp‘𝐺)) |
8 | 1nsgtrivd.4 | . . . 4 ⊢ (𝜑 → (NrmSGrp‘𝐺) ≈ 1o) | |
9 | en1eqsn 8827 | . . . 4 ⊢ (({ 0 } ∈ (NrmSGrp‘𝐺) ∧ (NrmSGrp‘𝐺) ≈ 1o) → (NrmSGrp‘𝐺) = {{ 0 }}) | |
10 | 7, 8, 9 | syl2anc 587 | . . 3 ⊢ (𝜑 → (NrmSGrp‘𝐺) = {{ 0 }}) |
11 | 4, 10 | eleqtrd 2835 | . 2 ⊢ (𝜑 → 𝐵 ∈ {{ 0 }}) |
12 | snex 5298 | . . 3 ⊢ { 0 } ∈ V | |
13 | elsn2g 4554 | . . 3 ⊢ ({ 0 } ∈ V → (𝐵 ∈ {{ 0 }} ↔ 𝐵 = { 0 })) | |
14 | 12, 13 | mp1i 13 | . 2 ⊢ (𝜑 → (𝐵 ∈ {{ 0 }} ↔ 𝐵 = { 0 })) |
15 | 11, 14 | mpbid 235 | 1 ⊢ (𝜑 → 𝐵 = { 0 }) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1542 ∈ wcel 2114 Vcvv 3398 {csn 4516 class class class wbr 5030 ‘cfv 6339 1oc1o 8126 ≈ cen 8554 Basecbs 16588 0gc0g 16818 Grpcgrp 18221 NrmSGrpcnsg 18394 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7481 ax-cnex 10673 ax-resscn 10674 ax-1cn 10675 ax-icn 10676 ax-addcl 10677 ax-addrcl 10678 ax-mulcl 10679 ax-mulrcl 10680 ax-mulcom 10681 ax-addass 10682 ax-mulass 10683 ax-distr 10684 ax-i2m1 10685 ax-1ne0 10686 ax-1rid 10687 ax-rnegex 10688 ax-rrecex 10689 ax-cnre 10690 ax-pre-lttri 10691 ax-pre-lttrn 10692 ax-pre-ltadd 10693 ax-pre-mulgt0 10694 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7129 df-ov 7175 df-oprab 7176 df-mpo 7177 df-om 7602 df-1st 7716 df-2nd 7717 df-wrecs 7978 df-recs 8039 df-rdg 8077 df-1o 8133 df-er 8322 df-en 8558 df-dom 8559 df-sdom 8560 df-fin 8561 df-pnf 10757 df-mnf 10758 df-xr 10759 df-ltxr 10760 df-le 10761 df-sub 10952 df-neg 10953 df-nn 11719 df-2 11781 df-ndx 16591 df-slot 16592 df-base 16594 df-sets 16595 df-ress 16596 df-plusg 16683 df-0g 16820 df-mgm 17970 df-sgrp 18019 df-mnd 18030 df-grp 18224 df-minusg 18225 df-sbg 18226 df-subg 18396 df-nsg 18397 |
This theorem is referenced by: (None) |
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