![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > simpgnsgd | Structured version Visualization version GIF version |
Description: The only normal subgroups of a simple group are the group itself and the trivial group. (Contributed by Rohan Ridenour, 3-Aug-2023.) |
Ref | Expression |
---|---|
simpgnsgd.1 | ⊢ 𝐵 = (Base‘𝐺) |
simpgnsgd.2 | ⊢ 0 = (0g‘𝐺) |
simpgnsgd.3 | ⊢ (𝜑 → 𝐺 ∈ SimpGrp) |
Ref | Expression |
---|---|
simpgnsgd | ⊢ (𝜑 → (NrmSGrp‘𝐺) = {{ 0 }, 𝐵}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2onn 8643 | . . . . 5 ⊢ 2o ∈ ω | |
2 | 1 | a1i 11 | . . . 4 ⊢ (𝜑 → 2o ∈ ω) |
3 | nnfi 9169 | . . . 4 ⊢ (2o ∈ ω → 2o ∈ Fin) | |
4 | 2, 3 | syl 17 | . . 3 ⊢ (𝜑 → 2o ∈ Fin) |
5 | simpgnsgd.3 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ SimpGrp) | |
6 | simpg2nsg 20018 | . . . 4 ⊢ (𝐺 ∈ SimpGrp → (NrmSGrp‘𝐺) ≈ 2o) | |
7 | 5, 6 | syl 17 | . . 3 ⊢ (𝜑 → (NrmSGrp‘𝐺) ≈ 2o) |
8 | enfii 9191 | . . 3 ⊢ ((2o ∈ Fin ∧ (NrmSGrp‘𝐺) ≈ 2o) → (NrmSGrp‘𝐺) ∈ Fin) | |
9 | 4, 7, 8 | syl2anc 583 | . 2 ⊢ (𝜑 → (NrmSGrp‘𝐺) ∈ Fin) |
10 | simpgnsgd.1 | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
11 | simpgnsgd.2 | . . 3 ⊢ 0 = (0g‘𝐺) | |
12 | 5 | simpggrpd 20017 | . . 3 ⊢ (𝜑 → 𝐺 ∈ Grp) |
13 | 10, 11, 12 | 0idnsgd 19098 | . 2 ⊢ (𝜑 → {{ 0 }, 𝐵} ⊆ (NrmSGrp‘𝐺)) |
14 | snex 5424 | . . . . . 6 ⊢ { 0 } ∈ V | |
15 | 14 | a1i 11 | . . . . 5 ⊢ (𝜑 → { 0 } ∈ V) |
16 | 10 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 𝐵 = (Base‘𝐺)) |
17 | fvex 6898 | . . . . . 6 ⊢ (Base‘𝐺) ∈ V | |
18 | 16, 17 | eqeltrdi 2835 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ V) |
19 | 10, 11, 5 | simpgntrivd 20020 | . . . . . 6 ⊢ (𝜑 → ¬ 𝐵 = { 0 }) |
20 | 19 | neqcomd 2736 | . . . . 5 ⊢ (𝜑 → ¬ { 0 } = 𝐵) |
21 | 15, 18, 20 | enpr2d 9051 | . . . 4 ⊢ (𝜑 → {{ 0 }, 𝐵} ≈ 2o) |
22 | 21 | ensymd 9003 | . . 3 ⊢ (𝜑 → 2o ≈ {{ 0 }, 𝐵}) |
23 | entr 9004 | . . 3 ⊢ (((NrmSGrp‘𝐺) ≈ 2o ∧ 2o ≈ {{ 0 }, 𝐵}) → (NrmSGrp‘𝐺) ≈ {{ 0 }, 𝐵}) | |
24 | 7, 22, 23 | syl2anc 583 | . 2 ⊢ (𝜑 → (NrmSGrp‘𝐺) ≈ {{ 0 }, 𝐵}) |
25 | 9, 13, 24 | phpeqd 9217 | 1 ⊢ (𝜑 → (NrmSGrp‘𝐺) = {{ 0 }, 𝐵}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 Vcvv 3468 {csn 4623 {cpr 4625 class class class wbr 5141 ‘cfv 6537 ωcom 7852 2oc2o 8461 ≈ cen 8938 Fincfn 8941 Basecbs 17153 0gc0g 17394 NrmSGrpcnsg 19048 SimpGrpcsimpg 20012 |
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 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-2o 8468 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-0g 17396 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-submnd 18714 df-grp 18866 df-minusg 18867 df-sbg 18868 df-subg 19050 df-nsg 19051 df-simpg 20013 |
This theorem is referenced by: simpgnsgeqd 20023 simpgnsgbid 20025 |
Copyright terms: Public domain | W3C validator |