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Mirrors > Home > MPE Home > Th. List > cntrabl | Structured version Visualization version GIF version |
Description: The center of a group is an abelian group. (Contributed by Thierry Arnoux, 21-Aug-2023.) |
Ref | Expression |
---|---|
cntrcmnd.z | ⊢ 𝑍 = (𝑀 ↾s (Cntr‘𝑀)) |
Ref | Expression |
---|---|
cntrabl | ⊢ (𝑀 ∈ Grp → 𝑍 ∈ Abel) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2740 | . . . . 5 ⊢ (Base‘𝑀) = (Base‘𝑀) | |
2 | eqid 2740 | . . . . 5 ⊢ (Cntz‘𝑀) = (Cntz‘𝑀) | |
3 | 1, 2 | cntrval 18921 | . . . 4 ⊢ ((Cntz‘𝑀)‘(Base‘𝑀)) = (Cntr‘𝑀) |
4 | ssid 3948 | . . . . 5 ⊢ (Base‘𝑀) ⊆ (Base‘𝑀) | |
5 | 1, 2 | cntzsubg 18939 | . . . . 5 ⊢ ((𝑀 ∈ Grp ∧ (Base‘𝑀) ⊆ (Base‘𝑀)) → ((Cntz‘𝑀)‘(Base‘𝑀)) ∈ (SubGrp‘𝑀)) |
6 | 4, 5 | mpan2 688 | . . . 4 ⊢ (𝑀 ∈ Grp → ((Cntz‘𝑀)‘(Base‘𝑀)) ∈ (SubGrp‘𝑀)) |
7 | 3, 6 | eqeltrrid 2846 | . . 3 ⊢ (𝑀 ∈ Grp → (Cntr‘𝑀) ∈ (SubGrp‘𝑀)) |
8 | cntrcmnd.z | . . . 4 ⊢ 𝑍 = (𝑀 ↾s (Cntr‘𝑀)) | |
9 | 8 | subggrp 18754 | . . 3 ⊢ ((Cntr‘𝑀) ∈ (SubGrp‘𝑀) → 𝑍 ∈ Grp) |
10 | 7, 9 | syl 17 | . 2 ⊢ (𝑀 ∈ Grp → 𝑍 ∈ Grp) |
11 | grpmnd 18580 | . . 3 ⊢ (𝑀 ∈ Grp → 𝑀 ∈ Mnd) | |
12 | 8 | cntrcmnd 19439 | . . 3 ⊢ (𝑀 ∈ Mnd → 𝑍 ∈ CMnd) |
13 | 11, 12 | syl 17 | . 2 ⊢ (𝑀 ∈ Grp → 𝑍 ∈ CMnd) |
14 | isabl 19386 | . 2 ⊢ (𝑍 ∈ Abel ↔ (𝑍 ∈ Grp ∧ 𝑍 ∈ CMnd)) | |
15 | 10, 13, 14 | sylanbrc 583 | 1 ⊢ (𝑀 ∈ Grp → 𝑍 ∈ Abel) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2110 ⊆ wss 3892 ‘cfv 6431 (class class class)co 7269 Basecbs 16908 ↾s cress 16937 Mndcmnd 18381 Grpcgrp 18573 SubGrpcsubg 18745 Cntzccntz 18917 Cntrccntr 18918 CMndccmn 19382 Abelcabl 19383 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7580 ax-cnex 10926 ax-resscn 10927 ax-1cn 10928 ax-icn 10929 ax-addcl 10930 ax-addrcl 10931 ax-mulcl 10932 ax-mulrcl 10933 ax-mulcom 10934 ax-addass 10935 ax-mulass 10936 ax-distr 10937 ax-i2m1 10938 ax-1ne0 10939 ax-1rid 10940 ax-rnegex 10941 ax-rrecex 10942 ax-cnre 10943 ax-pre-lttri 10944 ax-pre-lttrn 10945 ax-pre-ltadd 10946 ax-pre-mulgt0 10947 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6200 df-ord 6267 df-on 6268 df-lim 6269 df-suc 6270 df-iota 6389 df-fun 6433 df-fn 6434 df-f 6435 df-f1 6436 df-fo 6437 df-f1o 6438 df-fv 6439 df-riota 7226 df-ov 7272 df-oprab 7273 df-mpo 7274 df-om 7705 df-2nd 7823 df-frecs 8086 df-wrecs 8117 df-recs 8191 df-rdg 8230 df-er 8479 df-en 8715 df-dom 8716 df-sdom 8717 df-pnf 11010 df-mnf 11011 df-xr 11012 df-ltxr 11013 df-le 11014 df-sub 11205 df-neg 11206 df-nn 11972 df-2 12034 df-sets 16861 df-slot 16879 df-ndx 16891 df-base 16909 df-ress 16938 df-plusg 16971 df-0g 17148 df-mgm 18322 df-sgrp 18371 df-mnd 18382 df-submnd 18427 df-grp 18576 df-minusg 18577 df-subg 18748 df-cntz 18919 df-cntr 18920 df-cmn 19384 df-abl 19385 |
This theorem is referenced by: simpcntrab 44352 |
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