![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 4644 | . . . . . . 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 19094 | . . . . . . . . . 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 2827 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = { 0 }) → 𝑥 ∈ (NrmSGrp‘𝐺)) |
12 | 11 | adantlr 712 | . . . . . 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 19095 | . . . . . . . . . 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 2827 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → 𝑥 ∈ (NrmSGrp‘𝐺)) |
19 | 18 | adantlr 712 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ {{ 0 }, 𝐵}) ∧ 𝑥 = 𝐵) → 𝑥 ∈ (NrmSGrp‘𝐺)) |
20 | elpri 4645 | . . . . . . 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 955 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ {{ 0 }, 𝐵}) → 𝑥 ∈ (NrmSGrp‘𝐺)) |
23 | 5, 22 | impbida 798 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ (NrmSGrp‘𝐺) ↔ 𝑥 ∈ {{ 0 }, 𝐵})) |
24 | 23 | eqrdv 2724 | . . 3 ⊢ (𝜑 → (NrmSGrp‘𝐺) = {{ 0 }, 𝐵}) |
25 | snex 5424 | . . . . 5 ⊢ { 0 } ∈ V | |
26 | 25 | a1i 11 | . . . 4 ⊢ (𝜑 → { 0 } ∈ V) |
27 | 14 | fvexi 6898 | . . . . 5 ⊢ 𝐵 ∈ V |
28 | 27 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ V) |
29 | 2nsgsimpgd.4 | . . . 4 ⊢ (𝜑 → ¬ { 0 } = 𝐵) | |
30 | 26, 28, 29 | enpr2d 9048 | . . 3 ⊢ (𝜑 → {{ 0 }, 𝐵} ≈ 2o) |
31 | 24, 30 | eqbrtrd 5163 | . 2 ⊢ (𝜑 → (NrmSGrp‘𝐺) ≈ 2o) |
32 | 1, 31 | issimpgd 20013 | 1 ⊢ (𝜑 → 𝐺 ∈ SimpGrp) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∨ wo 844 = wceq 1533 ∈ wcel 2098 Vcvv 3468 {csn 4623 {cpr 4625 ‘cfv 6536 2oc2o 8458 ≈ cen 8935 Basecbs 17151 0gc0g 17392 Grpcgrp 18861 NrmSGrpcnsg 19046 SimpGrpcsimpg 20010 |
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 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 |
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 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-2o 8465 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-nn 12214 df-2 12276 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-0g 17394 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-submnd 18712 df-grp 18864 df-minusg 18865 df-sbg 18866 df-subg 19048 df-nsg 19049 df-simpg 20011 |
This theorem is referenced by: simpgnsgbid 20023 prmgrpsimpgd 20034 |
Copyright terms: Public domain | W3C validator |