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Mirrors > Home > MPE Home > Th. List > 2nsgsimpgd | Structured version Visualization version GIF version |
Description: If any normal subgroup of a nontrivial group is either the trivial subgroup or the whole group, the group is simple. (Contributed by Rohan Ridenour, 3-Aug-2023.) |
Ref | Expression |
---|---|
2nsgsimpgd.1 | ⊢ 𝐵 = (Base‘𝐺) |
2nsgsimpgd.2 | ⊢ 0 = (0g‘𝐺) |
2nsgsimpgd.3 | ⊢ (𝜑 → 𝐺 ∈ Grp) |
2nsgsimpgd.4 | ⊢ (𝜑 → ¬ { 0 } = 𝐵) |
2nsgsimpgd.5 | ⊢ ((𝜑 ∧ 𝑥 ∈ (NrmSGrp‘𝐺)) → (𝑥 = { 0 } ∨ 𝑥 = 𝐵)) |
Ref | Expression |
---|---|
2nsgsimpgd | ⊢ (𝜑 → 𝐺 ∈ SimpGrp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2nsgsimpgd.3 | . 2 ⊢ (𝜑 → 𝐺 ∈ Grp) | |
2 | 2nsgsimpgd.5 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (NrmSGrp‘𝐺)) → (𝑥 = { 0 } ∨ 𝑥 = 𝐵)) | |
3 | elprg 4646 | . . . . . . 7 ⊢ (𝑥 ∈ (NrmSGrp‘𝐺) → (𝑥 ∈ {{ 0 }, 𝐵} ↔ (𝑥 = { 0 } ∨ 𝑥 = 𝐵))) | |
4 | 3 | adantl 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (NrmSGrp‘𝐺)) → (𝑥 ∈ {{ 0 }, 𝐵} ↔ (𝑥 = { 0 } ∨ 𝑥 = 𝐵))) |
5 | 2, 4 | mpbird 257 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (NrmSGrp‘𝐺)) → 𝑥 ∈ {{ 0 }, 𝐵}) |
6 | simpr 484 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = { 0 }) → 𝑥 = { 0 }) | |
7 | 2nsgsimpgd.2 | . . . . . . . . . . 11 ⊢ 0 = (0g‘𝐺) | |
8 | 7 | 0nsg 19118 | . . . . . . . . . 10 ⊢ (𝐺 ∈ Grp → { 0 } ∈ (NrmSGrp‘𝐺)) |
9 | 1, 8 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → { 0 } ∈ (NrmSGrp‘𝐺)) |
10 | 9 | adantr 480 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = { 0 }) → { 0 } ∈ (NrmSGrp‘𝐺)) |
11 | 6, 10 | eqeltrd 2829 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = { 0 }) → 𝑥 ∈ (NrmSGrp‘𝐺)) |
12 | 11 | adantlr 714 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ {{ 0 }, 𝐵}) ∧ 𝑥 = { 0 }) → 𝑥 ∈ (NrmSGrp‘𝐺)) |
13 | simpr 484 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → 𝑥 = 𝐵) | |
14 | 2nsgsimpgd.1 | . . . . . . . . . . 11 ⊢ 𝐵 = (Base‘𝐺) | |
15 | 14 | nsgid 19119 | . . . . . . . . . 10 ⊢ (𝐺 ∈ Grp → 𝐵 ∈ (NrmSGrp‘𝐺)) |
16 | 1, 15 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ (NrmSGrp‘𝐺)) |
17 | 16 | adantr 480 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → 𝐵 ∈ (NrmSGrp‘𝐺)) |
18 | 13, 17 | eqeltrd 2829 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → 𝑥 ∈ (NrmSGrp‘𝐺)) |
19 | 18 | adantlr 714 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ {{ 0 }, 𝐵}) ∧ 𝑥 = 𝐵) → 𝑥 ∈ (NrmSGrp‘𝐺)) |
20 | elpri 4647 | . . . . . . 7 ⊢ (𝑥 ∈ {{ 0 }, 𝐵} → (𝑥 = { 0 } ∨ 𝑥 = 𝐵)) | |
21 | 20 | adantl 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ {{ 0 }, 𝐵}) → (𝑥 = { 0 } ∨ 𝑥 = 𝐵)) |
22 | 12, 19, 21 | mpjaodan 957 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ {{ 0 }, 𝐵}) → 𝑥 ∈ (NrmSGrp‘𝐺)) |
23 | 5, 22 | impbida 800 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ (NrmSGrp‘𝐺) ↔ 𝑥 ∈ {{ 0 }, 𝐵})) |
24 | 23 | eqrdv 2726 | . . 3 ⊢ (𝜑 → (NrmSGrp‘𝐺) = {{ 0 }, 𝐵}) |
25 | snex 5428 | . . . . 5 ⊢ { 0 } ∈ V | |
26 | 25 | a1i 11 | . . . 4 ⊢ (𝜑 → { 0 } ∈ V) |
27 | 14 | fvexi 6906 | . . . . 5 ⊢ 𝐵 ∈ V |
28 | 27 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ V) |
29 | 2nsgsimpgd.4 | . . . 4 ⊢ (𝜑 → ¬ { 0 } = 𝐵) | |
30 | 26, 28, 29 | enpr2d 9068 | . . 3 ⊢ (𝜑 → {{ 0 }, 𝐵} ≈ 2o) |
31 | 24, 30 | eqbrtrd 5165 | . 2 ⊢ (𝜑 → (NrmSGrp‘𝐺) ≈ 2o) |
32 | 1, 31 | issimpgd 20044 | 1 ⊢ (𝜑 → 𝐺 ∈ SimpGrp) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∨ wo 846 = wceq 1534 ∈ wcel 2099 Vcvv 3470 {csn 4625 {cpr 4627 ‘cfv 6543 2oc2o 8475 ≈ cen 8955 Basecbs 17174 0gc0g 17415 Grpcgrp 18884 NrmSGrpcnsg 19070 SimpGrpcsimpg 20041 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5294 ax-nul 5301 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-iun 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7866 df-1st 7988 df-2nd 7989 df-frecs 8281 df-wrecs 8312 df-recs 8386 df-rdg 8425 df-1o 8481 df-2o 8482 df-er 8719 df-en 8959 df-dom 8960 df-sdom 8961 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-nn 12238 df-2 12300 df-sets 17127 df-slot 17145 df-ndx 17157 df-base 17175 df-ress 17204 df-plusg 17240 df-0g 17417 df-mgm 18594 df-sgrp 18673 df-mnd 18689 df-submnd 18735 df-grp 18887 df-minusg 18888 df-sbg 18889 df-subg 19072 df-nsg 19073 df-simpg 20042 |
This theorem is referenced by: simpgnsgbid 20054 prmgrpsimpgd 20065 |
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