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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 8266 | . . . . 5 ⊢ 2o ∈ ω | |
2 | 1 | a1i 11 | . . . 4 ⊢ (𝜑 → 2o ∈ ω) |
3 | nnfi 8711 | . . . 4 ⊢ (2o ∈ ω → 2o ∈ Fin) | |
4 | 2, 3 | syl 17 | . . 3 ⊢ (𝜑 → 2o ∈ Fin) |
5 | simpgnsgd.3 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ SimpGrp) | |
6 | simpg2nsg 19218 | . . . 4 ⊢ (𝐺 ∈ SimpGrp → (NrmSGrp‘𝐺) ≈ 2o) | |
7 | 5, 6 | syl 17 | . . 3 ⊢ (𝜑 → (NrmSGrp‘𝐺) ≈ 2o) |
8 | enfii 8735 | . . 3 ⊢ ((2o ∈ Fin ∧ (NrmSGrp‘𝐺) ≈ 2o) → (NrmSGrp‘𝐺) ∈ Fin) | |
9 | 4, 7, 8 | syl2anc 586 | . 2 ⊢ (𝜑 → (NrmSGrp‘𝐺) ∈ Fin) |
10 | simpgnsgd.1 | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
11 | simpgnsgd.2 | . . 3 ⊢ 0 = (0g‘𝐺) | |
12 | 5 | simpggrpd 19217 | . . 3 ⊢ (𝜑 → 𝐺 ∈ Grp) |
13 | 10, 11, 12 | 0idnsgd 18323 | . 2 ⊢ (𝜑 → {{ 0 }, 𝐵} ⊆ (NrmSGrp‘𝐺)) |
14 | snex 5332 | . . . . . 6 ⊢ { 0 } ∈ V | |
15 | 14 | a1i 11 | . . . . 5 ⊢ (𝜑 → { 0 } ∈ V) |
16 | 10 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 𝐵 = (Base‘𝐺)) |
17 | fvex 6683 | . . . . . 6 ⊢ (Base‘𝐺) ∈ V | |
18 | 16, 17 | eqeltrdi 2921 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ V) |
19 | 10, 11, 5 | simpgntrivd 19220 | . . . . . 6 ⊢ (𝜑 → ¬ 𝐵 = { 0 }) |
20 | 19 | neqcomd 2831 | . . . . 5 ⊢ (𝜑 → ¬ { 0 } = 𝐵) |
21 | 15, 18, 20 | enpr2d 8597 | . . . 4 ⊢ (𝜑 → {{ 0 }, 𝐵} ≈ 2o) |
22 | 21 | ensymd 8560 | . . 3 ⊢ (𝜑 → 2o ≈ {{ 0 }, 𝐵}) |
23 | entr 8561 | . . 3 ⊢ (((NrmSGrp‘𝐺) ≈ 2o ∧ 2o ≈ {{ 0 }, 𝐵}) → (NrmSGrp‘𝐺) ≈ {{ 0 }, 𝐵}) | |
24 | 7, 22, 23 | syl2anc 586 | . 2 ⊢ (𝜑 → (NrmSGrp‘𝐺) ≈ {{ 0 }, 𝐵}) |
25 | 9, 13, 24 | phpeqd 8706 | 1 ⊢ (𝜑 → (NrmSGrp‘𝐺) = {{ 0 }, 𝐵}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 Vcvv 3494 {csn 4567 {cpr 4569 class class class wbr 5066 ‘cfv 6355 ωcom 7580 2oc2o 8096 ≈ cen 8506 Fincfn 8509 Basecbs 16483 0gc0g 16713 NrmSGrpcnsg 18274 SimpGrpcsimpg 19212 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-0g 16715 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-grp 18106 df-minusg 18107 df-sbg 18108 df-subg 18276 df-nsg 18277 df-simpg 19213 |
This theorem is referenced by: simpgnsgeqd 19223 simpgnsgbid 19225 |
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