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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 8249 | . . . . 5 ⊢ 2o ∈ ω | |
2 | 1 | a1i 11 | . . . 4 ⊢ (𝜑 → 2o ∈ ω) |
3 | nnfi 8696 | . . . 4 ⊢ (2o ∈ ω → 2o ∈ Fin) | |
4 | 2, 3 | syl 17 | . . 3 ⊢ (𝜑 → 2o ∈ Fin) |
5 | simpgnsgd.3 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ SimpGrp) | |
6 | simpg2nsg 19211 | . . . 4 ⊢ (𝐺 ∈ SimpGrp → (NrmSGrp‘𝐺) ≈ 2o) | |
7 | 5, 6 | syl 17 | . . 3 ⊢ (𝜑 → (NrmSGrp‘𝐺) ≈ 2o) |
8 | enfii 8719 | . . 3 ⊢ ((2o ∈ Fin ∧ (NrmSGrp‘𝐺) ≈ 2o) → (NrmSGrp‘𝐺) ∈ Fin) | |
9 | 4, 7, 8 | syl2anc 587 | . 2 ⊢ (𝜑 → (NrmSGrp‘𝐺) ∈ Fin) |
10 | simpgnsgd.1 | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
11 | simpgnsgd.2 | . . 3 ⊢ 0 = (0g‘𝐺) | |
12 | 5 | simpggrpd 19210 | . . 3 ⊢ (𝜑 → 𝐺 ∈ Grp) |
13 | 10, 11, 12 | 0idnsgd 18315 | . 2 ⊢ (𝜑 → {{ 0 }, 𝐵} ⊆ (NrmSGrp‘𝐺)) |
14 | snex 5297 | . . . . . 6 ⊢ { 0 } ∈ V | |
15 | 14 | a1i 11 | . . . . 5 ⊢ (𝜑 → { 0 } ∈ V) |
16 | 10 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 𝐵 = (Base‘𝐺)) |
17 | fvex 6658 | . . . . . 6 ⊢ (Base‘𝐺) ∈ V | |
18 | 16, 17 | eqeltrdi 2898 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ V) |
19 | 10, 11, 5 | simpgntrivd 19213 | . . . . . 6 ⊢ (𝜑 → ¬ 𝐵 = { 0 }) |
20 | 19 | neqcomd 2808 | . . . . 5 ⊢ (𝜑 → ¬ { 0 } = 𝐵) |
21 | 15, 18, 20 | enpr2d 8580 | . . . 4 ⊢ (𝜑 → {{ 0 }, 𝐵} ≈ 2o) |
22 | 21 | ensymd 8543 | . . 3 ⊢ (𝜑 → 2o ≈ {{ 0 }, 𝐵}) |
23 | entr 8544 | . . 3 ⊢ (((NrmSGrp‘𝐺) ≈ 2o ∧ 2o ≈ {{ 0 }, 𝐵}) → (NrmSGrp‘𝐺) ≈ {{ 0 }, 𝐵}) | |
24 | 7, 22, 23 | syl2anc 587 | . 2 ⊢ (𝜑 → (NrmSGrp‘𝐺) ≈ {{ 0 }, 𝐵}) |
25 | 9, 13, 24 | phpeqd 8690 | 1 ⊢ (𝜑 → (NrmSGrp‘𝐺) = {{ 0 }, 𝐵}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 Vcvv 3441 {csn 4525 {cpr 4527 class class class wbr 5030 ‘cfv 6324 ωcom 7560 2oc2o 8079 ≈ cen 8489 Fincfn 8492 Basecbs 16475 0gc0g 16705 NrmSGrpcnsg 18266 SimpGrpcsimpg 19205 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-0g 16707 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-grp 18098 df-minusg 18099 df-sbg 18100 df-subg 18268 df-nsg 18269 df-simpg 19206 |
This theorem is referenced by: simpgnsgeqd 19216 simpgnsgbid 19218 |
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