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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 18973 | . . . 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 18972 | . . . . 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 9219 | . . . 4 ⊢ (({ 0 } ∈ (NrmSGrp‘𝐺) ∧ (NrmSGrp‘𝐺) ≈ 1o) → (NrmSGrp‘𝐺) = {{ 0 }}) | |
10 | 7, 8, 9 | syl2anc 585 | . . 3 ⊢ (𝜑 → (NrmSGrp‘𝐺) = {{ 0 }}) |
11 | 4, 10 | eleqtrd 2840 | . 2 ⊢ (𝜑 → 𝐵 ∈ {{ 0 }}) |
12 | snex 5389 | . . 3 ⊢ { 0 } ∈ V | |
13 | elsn2g 4625 | . . 3 ⊢ ({ 0 } ∈ V → (𝐵 ∈ {{ 0 }} ↔ 𝐵 = { 0 })) | |
14 | 12, 13 | mp1i 13 | . 2 ⊢ (𝜑 → (𝐵 ∈ {{ 0 }} ↔ 𝐵 = { 0 })) |
15 | 11, 14 | mpbid 231 | 1 ⊢ (𝜑 → 𝐵 = { 0 }) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1542 ∈ wcel 2107 Vcvv 3446 {csn 4587 class class class wbr 5106 ‘cfv 6497 1oc1o 8406 ≈ cen 8881 Basecbs 17084 0gc0g 17322 Grpcgrp 18749 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 ax-cnex 11108 ax-resscn 11109 ax-1cn 11110 ax-icn 11111 ax-addcl 11112 ax-addrcl 11113 ax-mulcl 11114 ax-mulrcl 11115 ax-mulcom 11116 ax-addass 11117 ax-mulass 11118 ax-distr 11119 ax-i2m1 11120 ax-1ne0 11121 ax-1rid 11122 ax-rnegex 11123 ax-rrecex 11124 ax-cnre 11125 ax-pre-lttri 11126 ax-pre-lttrn 11127 ax-pre-ltadd 11128 ax-pre-mulgt0 11129 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 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-nel 3051 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-pss 3930 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-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 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-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8649 df-en 8885 df-dom 8886 df-sdom 8887 df-pnf 11192 df-mnf 11193 df-xr 11194 df-ltxr 11195 df-le 11196 df-sub 11388 df-neg 11389 df-nn 12155 df-2 12217 df-sets 17037 df-slot 17055 df-ndx 17067 df-base 17085 df-ress 17114 df-plusg 17147 df-0g 17324 df-mgm 18498 df-sgrp 18547 df-mnd 18558 df-submnd 18603 df-grp 18752 df-minusg 18753 df-sbg 18754 df-subg 18926 df-nsg 18927 |
This theorem is referenced by: (None) |
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