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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 4585 | . . . . . . 7 ⊢ (𝑥 ∈ (NrmSGrp‘𝐺) → (𝑥 ∈ {{ 0 }, 𝐵} ↔ (𝑥 = { 0 } ∨ 𝑥 = 𝐵))) | |
4 | 3 | adantl 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (NrmSGrp‘𝐺)) → (𝑥 ∈ {{ 0 }, 𝐵} ↔ (𝑥 = { 0 } ∨ 𝑥 = 𝐵))) |
5 | 2, 4 | mpbird 259 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (NrmSGrp‘𝐺)) → 𝑥 ∈ {{ 0 }, 𝐵}) |
6 | simpr 487 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = { 0 }) → 𝑥 = { 0 }) | |
7 | 2nsgsimpgd.2 | . . . . . . . . . . 11 ⊢ 0 = (0g‘𝐺) | |
8 | 7 | 0nsg 18317 | . . . . . . . . . 10 ⊢ (𝐺 ∈ Grp → { 0 } ∈ (NrmSGrp‘𝐺)) |
9 | 1, 8 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → { 0 } ∈ (NrmSGrp‘𝐺)) |
10 | 9 | adantr 483 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = { 0 }) → { 0 } ∈ (NrmSGrp‘𝐺)) |
11 | 6, 10 | eqeltrd 2912 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = { 0 }) → 𝑥 ∈ (NrmSGrp‘𝐺)) |
12 | 11 | adantlr 713 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ {{ 0 }, 𝐵}) ∧ 𝑥 = { 0 }) → 𝑥 ∈ (NrmSGrp‘𝐺)) |
13 | simpr 487 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → 𝑥 = 𝐵) | |
14 | 2nsgsimpgd.1 | . . . . . . . . . . 11 ⊢ 𝐵 = (Base‘𝐺) | |
15 | 14 | nsgid 18318 | . . . . . . . . . 10 ⊢ (𝐺 ∈ Grp → 𝐵 ∈ (NrmSGrp‘𝐺)) |
16 | 1, 15 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ (NrmSGrp‘𝐺)) |
17 | 16 | adantr 483 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → 𝐵 ∈ (NrmSGrp‘𝐺)) |
18 | 13, 17 | eqeltrd 2912 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → 𝑥 ∈ (NrmSGrp‘𝐺)) |
19 | 18 | adantlr 713 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ {{ 0 }, 𝐵}) ∧ 𝑥 = 𝐵) → 𝑥 ∈ (NrmSGrp‘𝐺)) |
20 | elpri 4586 | . . . . . . 7 ⊢ (𝑥 ∈ {{ 0 }, 𝐵} → (𝑥 = { 0 } ∨ 𝑥 = 𝐵)) | |
21 | 20 | adantl 484 | . . . . . 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 799 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ (NrmSGrp‘𝐺) ↔ 𝑥 ∈ {{ 0 }, 𝐵})) |
24 | 23 | eqrdv 2818 | . . 3 ⊢ (𝜑 → (NrmSGrp‘𝐺) = {{ 0 }, 𝐵}) |
25 | snex 5329 | . . . . 5 ⊢ { 0 } ∈ V | |
26 | 25 | a1i 11 | . . . 4 ⊢ (𝜑 → { 0 } ∈ V) |
27 | 14 | fvexi 6681 | . . . . 5 ⊢ 𝐵 ∈ V |
28 | 27 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ V) |
29 | 2nsgsimpgd.4 | . . . 4 ⊢ (𝜑 → ¬ { 0 } = 𝐵) | |
30 | 26, 28, 29 | enpr2d 8594 | . . 3 ⊢ (𝜑 → {{ 0 }, 𝐵} ≈ 2o) |
31 | 24, 30 | eqbrtrd 5085 | . 2 ⊢ (𝜑 → (NrmSGrp‘𝐺) ≈ 2o) |
32 | 1, 31 | issimpgd 19211 | 1 ⊢ (𝜑 → 𝐺 ∈ SimpGrp) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∨ wo 843 = wceq 1536 ∈ wcel 2113 Vcvv 3493 {csn 4564 {cpr 4566 ‘cfv 6352 2oc2o 8093 ≈ cen 8503 Basecbs 16479 0gc0g 16709 Grpcgrp 18099 NrmSGrpcnsg 18270 SimpGrpcsimpg 19208 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5200 ax-nul 5207 ax-pow 5263 ax-pr 5327 ax-un 7458 ax-cnex 10590 ax-resscn 10591 ax-1cn 10592 ax-icn 10593 ax-addcl 10594 ax-addrcl 10595 ax-mulcl 10596 ax-mulrcl 10597 ax-mulcom 10598 ax-addass 10599 ax-mulass 10600 ax-distr 10601 ax-i2m1 10602 ax-1ne0 10603 ax-1rid 10604 ax-rnegex 10605 ax-rrecex 10606 ax-cnre 10607 ax-pre-lttri 10608 ax-pre-lttrn 10609 ax-pre-ltadd 10610 ax-pre-mulgt0 10611 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3495 df-sbc 3771 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4465 df-pw 4538 df-sn 4565 df-pr 4567 df-tp 4569 df-op 4571 df-uni 4836 df-iun 4918 df-br 5064 df-opab 5126 df-mpt 5144 df-tr 5170 df-id 5457 df-eprel 5462 df-po 5471 df-so 5472 df-fr 5511 df-we 5513 df-xp 5558 df-rel 5559 df-cnv 5560 df-co 5561 df-dm 5562 df-rn 5563 df-res 5564 df-ima 5565 df-pred 6145 df-ord 6191 df-on 6192 df-lim 6193 df-suc 6194 df-iota 6311 df-fun 6354 df-fn 6355 df-f 6356 df-f1 6357 df-fo 6358 df-f1o 6359 df-fv 6360 df-riota 7111 df-ov 7156 df-oprab 7157 df-mpo 7158 df-om 7578 df-1st 7686 df-2nd 7687 df-wrecs 7944 df-recs 8005 df-rdg 8043 df-1o 8099 df-2o 8100 df-er 8286 df-en 8507 df-dom 8508 df-sdom 8509 df-pnf 10674 df-mnf 10675 df-xr 10676 df-ltxr 10677 df-le 10678 df-sub 10869 df-neg 10870 df-nn 11636 df-2 11698 df-ndx 16482 df-slot 16483 df-base 16485 df-sets 16486 df-ress 16487 df-plusg 16574 df-0g 16711 df-mgm 17848 df-sgrp 17897 df-mnd 17908 df-grp 18102 df-minusg 18103 df-sbg 18104 df-subg 18272 df-nsg 18273 df-simpg 19209 |
This theorem is referenced by: simpgnsgbid 19221 prmgrpsimpgd 19232 |
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