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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 | . . . 4 ⊢ 𝐵 = (Base‘𝐺) | |
2 | simpcntrab.b | . . . 4 ⊢ 0 = (0g‘𝐺) | |
3 | simpcntrab.d | . . . 4 ⊢ (𝜑 → 𝐺 ∈ SimpGrp) | |
4 | 3 | simpggrpd 20130 | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ Grp) |
5 | simpcntrab.c | . . . . . 6 ⊢ 𝑍 = (Cntr‘𝐺) | |
6 | 5 | cntrnsg 19375 | . . . . 5 ⊢ (𝐺 ∈ Grp → 𝑍 ∈ (NrmSGrp‘𝐺)) |
7 | 4, 6 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑍 ∈ (NrmSGrp‘𝐺)) |
8 | 1, 2, 3, 7 | simpgnsgeqd 20136 | . . 3 ⊢ (𝜑 → (𝑍 = { 0 } ∨ 𝑍 = 𝐵)) |
9 | 8 | ancli 548 | . 2 ⊢ (𝜑 → (𝜑 ∧ (𝑍 = { 0 } ∨ 𝑍 = 𝐵))) |
10 | andi 1009 | . . 3 ⊢ ((𝜑 ∧ (𝑍 = { 0 } ∨ 𝑍 = 𝐵)) ↔ ((𝜑 ∧ 𝑍 = { 0 }) ∨ (𝜑 ∧ 𝑍 = 𝐵))) | |
11 | 10 | biimpi 216 | . 2 ⊢ ((𝜑 ∧ (𝑍 = { 0 } ∨ 𝑍 = 𝐵)) → ((𝜑 ∧ 𝑍 = { 0 }) ∨ (𝜑 ∧ 𝑍 = 𝐵))) |
12 | simpr 484 | . . 3 ⊢ ((𝜑 ∧ 𝑍 = { 0 }) → 𝑍 = { 0 }) | |
13 | 12 | orim1i 909 | . 2 ⊢ (((𝜑 ∧ 𝑍 = { 0 }) ∨ (𝜑 ∧ 𝑍 = 𝐵)) → (𝑍 = { 0 } ∨ (𝜑 ∧ 𝑍 = 𝐵))) |
14 | oveq2 7439 | . . . . . . 7 ⊢ (𝑍 = 𝐵 → (𝐺 ↾s 𝑍) = (𝐺 ↾s 𝐵)) | |
15 | 5 | oveq2i 7442 | . . . . . . 7 ⊢ (𝐺 ↾s 𝑍) = (𝐺 ↾s (Cntr‘𝐺)) |
16 | 14, 15 | eqtr3di 2790 | . . . . . 6 ⊢ (𝑍 = 𝐵 → (𝐺 ↾s 𝐵) = (𝐺 ↾s (Cntr‘𝐺))) |
17 | 16 | adantl 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑍 = 𝐵) → (𝐺 ↾s 𝐵) = (𝐺 ↾s (Cntr‘𝐺))) |
18 | 1 | ressid 17290 | . . . . . . 7 ⊢ (𝐺 ∈ Grp → (𝐺 ↾s 𝐵) = 𝐺) |
19 | 4, 18 | syl 17 | . . . . . 6 ⊢ (𝜑 → (𝐺 ↾s 𝐵) = 𝐺) |
20 | 19 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑍 = 𝐵) → (𝐺 ↾s 𝐵) = 𝐺) |
21 | 17, 20 | eqtr3d 2777 | . . . 4 ⊢ ((𝜑 ∧ 𝑍 = 𝐵) → (𝐺 ↾s (Cntr‘𝐺)) = 𝐺) |
22 | eqid 2735 | . . . . . . 7 ⊢ (𝐺 ↾s (Cntr‘𝐺)) = (𝐺 ↾s (Cntr‘𝐺)) | |
23 | 22 | cntrabl 19876 | . . . . . 6 ⊢ (𝐺 ∈ Grp → (𝐺 ↾s (Cntr‘𝐺)) ∈ Abel) |
24 | 4, 23 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝐺 ↾s (Cntr‘𝐺)) ∈ Abel) |
25 | 24 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑍 = 𝐵) → (𝐺 ↾s (Cntr‘𝐺)) ∈ Abel) |
26 | 21, 25 | eqeltrrd 2840 | . . 3 ⊢ ((𝜑 ∧ 𝑍 = 𝐵) → 𝐺 ∈ Abel) |
27 | 26 | orim2i 910 | . 2 ⊢ ((𝑍 = { 0 } ∨ (𝜑 ∧ 𝑍 = 𝐵)) → (𝑍 = { 0 } ∨ 𝐺 ∈ Abel)) |
28 | 9, 11, 13, 27 | 4syl 19 | 1 ⊢ (𝜑 → (𝑍 = { 0 } ∨ 𝐺 ∈ Abel)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∨ wo 847 = wceq 1537 ∈ wcel 2106 {csn 4631 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 ↾s cress 17274 0gc0g 17486 Grpcgrp 18964 NrmSGrpcnsg 19152 Cntrccntr 19347 Abelcabl 19814 SimpGrpcsimpg 20125 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-0g 17488 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-submnd 18810 df-grp 18967 df-minusg 18968 df-sbg 18969 df-subg 19154 df-nsg 19155 df-cntz 19348 df-cntr 19349 df-cmn 19815 df-abl 19816 df-simpg 20126 |
This theorem is referenced by: (None) |
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