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 19217 | . . . . . 6 ⊢ (𝜑 → 𝐺 ∈ Grp) |
5 | simpcntrab.c | . . . . . . 7 ⊢ 𝑍 = (Cntr‘𝐺) | |
6 | 5 | cntrnsg 18472 | . . . . . 6 ⊢ (𝐺 ∈ Grp → 𝑍 ∈ (NrmSGrp‘𝐺)) |
7 | 4, 6 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑍 ∈ (NrmSGrp‘𝐺)) |
8 | 1, 2, 3, 7 | simpgnsgeqd 19223 | . . . 4 ⊢ (𝜑 → (𝑍 = { 0 } ∨ 𝑍 = 𝐵)) |
9 | 8 | ancli 551 | . . 3 ⊢ (𝜑 → (𝜑 ∧ (𝑍 = { 0 } ∨ 𝑍 = 𝐵))) |
10 | andi 1004 | . . . 4 ⊢ ((𝜑 ∧ (𝑍 = { 0 } ∨ 𝑍 = 𝐵)) ↔ ((𝜑 ∧ 𝑍 = { 0 }) ∨ (𝜑 ∧ 𝑍 = 𝐵))) | |
11 | 10 | biimpi 218 | . . 3 ⊢ ((𝜑 ∧ (𝑍 = { 0 } ∨ 𝑍 = 𝐵)) → ((𝜑 ∧ 𝑍 = { 0 }) ∨ (𝜑 ∧ 𝑍 = 𝐵))) |
12 | simpr 487 | . . . 4 ⊢ ((𝜑 ∧ 𝑍 = { 0 }) → 𝑍 = { 0 }) | |
13 | 12 | orim1i 906 | . . 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 | 5 | oveq2i 7167 | . . . . . . 7 ⊢ (𝐺 ↾s 𝑍) = (𝐺 ↾s (Cntr‘𝐺)) |
16 | oveq2 7164 | . . . . . . 7 ⊢ (𝑍 = 𝐵 → (𝐺 ↾s 𝑍) = (𝐺 ↾s 𝐵)) | |
17 | 15, 16 | syl5reqr 2871 | . . . . . 6 ⊢ (𝑍 = 𝐵 → (𝐺 ↾s 𝐵) = (𝐺 ↾s (Cntr‘𝐺))) |
18 | 17 | adantl 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑍 = 𝐵) → (𝐺 ↾s 𝐵) = (𝐺 ↾s (Cntr‘𝐺))) |
19 | 1 | ressid 16559 | . . . . . . 7 ⊢ (𝐺 ∈ Grp → (𝐺 ↾s 𝐵) = 𝐺) |
20 | 4, 19 | syl 17 | . . . . . 6 ⊢ (𝜑 → (𝐺 ↾s 𝐵) = 𝐺) |
21 | 20 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝑍 = 𝐵) → (𝐺 ↾s 𝐵) = 𝐺) |
22 | 18, 21 | eqtr3d 2858 | . . . 4 ⊢ ((𝜑 ∧ 𝑍 = 𝐵) → (𝐺 ↾s (Cntr‘𝐺)) = 𝐺) |
23 | eqid 2821 | . . . . . . 7 ⊢ (𝐺 ↾s (Cntr‘𝐺)) = (𝐺 ↾s (Cntr‘𝐺)) | |
24 | 23 | cntrabl 18963 | . . . . . 6 ⊢ (𝐺 ∈ Grp → (𝐺 ↾s (Cntr‘𝐺)) ∈ Abel) |
25 | 4, 24 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝐺 ↾s (Cntr‘𝐺)) ∈ Abel) |
26 | 25 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑍 = 𝐵) → (𝐺 ↾s (Cntr‘𝐺)) ∈ Abel) |
27 | 22, 26 | eqeltrrd 2914 | . . 3 ⊢ ((𝜑 ∧ 𝑍 = 𝐵) → 𝐺 ∈ Abel) |
28 | 27 | orim2i 907 | . 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 398 ∨ wo 843 = wceq 1537 ∈ wcel 2114 {csn 4567 ‘cfv 6355 (class class class)co 7156 Basecbs 16483 ↾s cress 16484 0gc0g 16713 Grpcgrp 18103 NrmSGrpcnsg 18274 Cntrccntr 18446 Abelcabl 18907 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-rep 5190 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-submnd 17957 df-grp 18106 df-minusg 18107 df-sbg 18108 df-subg 18276 df-nsg 18277 df-cntz 18447 df-cntr 18448 df-cmn 18908 df-abl 18909 df-simpg 19213 |
This theorem is referenced by: (None) |
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