Mathbox for Saveliy Skresanov |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > simpcntrab | Structured version Visualization version GIF version |
Description: The center of a simple group is trivial or the group is abelian. (Contributed by SS, 3-Jan-2024.) |
Ref | Expression |
---|---|
simpcntrab.a | ⊢ 𝐵 = (Base‘𝐺) |
simpcntrab.b | ⊢ 0 = (0g‘𝐺) |
simpcntrab.c | ⊢ 𝑍 = (Cntr‘𝐺) |
simpcntrab.d | ⊢ (𝜑 → 𝐺 ∈ SimpGrp) |
Ref | Expression |
---|---|
simpcntrab | ⊢ (𝜑 → (𝑍 = { 0 } ∨ 𝐺 ∈ Abel)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpcntrab.a | . . . . 5 ⊢ 𝐵 = (Base‘𝐺) | |
2 | simpcntrab.b | . . . . 5 ⊢ 0 = (0g‘𝐺) | |
3 | simpcntrab.d | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ SimpGrp) | |
4 | 3 | simpggrpd 19482 | . . . . . 6 ⊢ (𝜑 → 𝐺 ∈ Grp) |
5 | simpcntrab.c | . . . . . . 7 ⊢ 𝑍 = (Cntr‘𝐺) | |
6 | 5 | cntrnsg 18736 | . . . . . 6 ⊢ (𝐺 ∈ Grp → 𝑍 ∈ (NrmSGrp‘𝐺)) |
7 | 4, 6 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑍 ∈ (NrmSGrp‘𝐺)) |
8 | 1, 2, 3, 7 | simpgnsgeqd 19488 | . . . 4 ⊢ (𝜑 → (𝑍 = { 0 } ∨ 𝑍 = 𝐵)) |
9 | 8 | ancli 552 | . . 3 ⊢ (𝜑 → (𝜑 ∧ (𝑍 = { 0 } ∨ 𝑍 = 𝐵))) |
10 | andi 1008 | . . . 4 ⊢ ((𝜑 ∧ (𝑍 = { 0 } ∨ 𝑍 = 𝐵)) ↔ ((𝜑 ∧ 𝑍 = { 0 }) ∨ (𝜑 ∧ 𝑍 = 𝐵))) | |
11 | 10 | biimpi 219 | . . 3 ⊢ ((𝜑 ∧ (𝑍 = { 0 } ∨ 𝑍 = 𝐵)) → ((𝜑 ∧ 𝑍 = { 0 }) ∨ (𝜑 ∧ 𝑍 = 𝐵))) |
12 | simpr 488 | . . . 4 ⊢ ((𝜑 ∧ 𝑍 = { 0 }) → 𝑍 = { 0 }) | |
13 | 12 | orim1i 910 | . . 3 ⊢ (((𝜑 ∧ 𝑍 = { 0 }) ∨ (𝜑 ∧ 𝑍 = 𝐵)) → (𝑍 = { 0 } ∨ (𝜑 ∧ 𝑍 = 𝐵))) |
14 | 9, 11, 13 | 3syl 18 | . 2 ⊢ (𝜑 → (𝑍 = { 0 } ∨ (𝜑 ∧ 𝑍 = 𝐵))) |
15 | oveq2 7221 | . . . . . . 7 ⊢ (𝑍 = 𝐵 → (𝐺 ↾s 𝑍) = (𝐺 ↾s 𝐵)) | |
16 | 5 | oveq2i 7224 | . . . . . . 7 ⊢ (𝐺 ↾s 𝑍) = (𝐺 ↾s (Cntr‘𝐺)) |
17 | 15, 16 | eqtr3di 2793 | . . . . . 6 ⊢ (𝑍 = 𝐵 → (𝐺 ↾s 𝐵) = (𝐺 ↾s (Cntr‘𝐺))) |
18 | 17 | adantl 485 | . . . . 5 ⊢ ((𝜑 ∧ 𝑍 = 𝐵) → (𝐺 ↾s 𝐵) = (𝐺 ↾s (Cntr‘𝐺))) |
19 | 1 | ressid 16796 | . . . . . . 7 ⊢ (𝐺 ∈ Grp → (𝐺 ↾s 𝐵) = 𝐺) |
20 | 4, 19 | syl 17 | . . . . . 6 ⊢ (𝜑 → (𝐺 ↾s 𝐵) = 𝐺) |
21 | 20 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑍 = 𝐵) → (𝐺 ↾s 𝐵) = 𝐺) |
22 | 18, 21 | eqtr3d 2779 | . . . 4 ⊢ ((𝜑 ∧ 𝑍 = 𝐵) → (𝐺 ↾s (Cntr‘𝐺)) = 𝐺) |
23 | eqid 2737 | . . . . . . 7 ⊢ (𝐺 ↾s (Cntr‘𝐺)) = (𝐺 ↾s (Cntr‘𝐺)) | |
24 | 23 | cntrabl 19228 | . . . . . 6 ⊢ (𝐺 ∈ Grp → (𝐺 ↾s (Cntr‘𝐺)) ∈ Abel) |
25 | 4, 24 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝐺 ↾s (Cntr‘𝐺)) ∈ Abel) |
26 | 25 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑍 = 𝐵) → (𝐺 ↾s (Cntr‘𝐺)) ∈ Abel) |
27 | 22, 26 | eqeltrrd 2839 | . . 3 ⊢ ((𝜑 ∧ 𝑍 = 𝐵) → 𝐺 ∈ Abel) |
28 | 27 | orim2i 911 | . 2 ⊢ ((𝑍 = { 0 } ∨ (𝜑 ∧ 𝑍 = 𝐵)) → (𝑍 = { 0 } ∨ 𝐺 ∈ Abel)) |
29 | 14, 28 | syl 17 | 1 ⊢ (𝜑 → (𝑍 = { 0 } ∨ 𝐺 ∈ Abel)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∨ wo 847 = wceq 1543 ∈ wcel 2110 {csn 4541 ‘cfv 6380 (class class class)co 7213 Basecbs 16760 ↾s cress 16784 0gc0g 16944 Grpcgrp 18365 NrmSGrpcnsg 18538 Cntrccntr 18710 Abelcabl 19171 SimpGrpcsimpg 19477 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-1st 7761 df-2nd 7762 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-2o 8203 df-er 8391 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-nn 11831 df-2 11893 df-sets 16717 df-slot 16735 df-ndx 16745 df-base 16761 df-ress 16785 df-plusg 16815 df-0g 16946 df-mgm 18114 df-sgrp 18163 df-mnd 18174 df-submnd 18219 df-grp 18368 df-minusg 18369 df-sbg 18370 df-subg 18540 df-nsg 18541 df-cntz 18711 df-cntr 18712 df-cmn 19172 df-abl 19173 df-simpg 19478 |
This theorem is referenced by: (None) |
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