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Mirrors > Home > MPE Home > Th. List > submcmn2 | Structured version Visualization version GIF version |
Description: A submonoid is commutative iff it is a subset of its own centralizer. (Contributed by Mario Carneiro, 24-Apr-2016.) |
Ref | Expression |
---|---|
subgabl.h | ⊢ 𝐻 = (𝐺 ↾s 𝑆) |
submcmn2.z | ⊢ 𝑍 = (Cntz‘𝐺) |
Ref | Expression |
---|---|
submcmn2 | ⊢ (𝑆 ∈ (SubMnd‘𝐺) → (𝐻 ∈ CMnd ↔ 𝑆 ⊆ (𝑍‘𝑆))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | subgabl.h | . . . 4 ⊢ 𝐻 = (𝐺 ↾s 𝑆) | |
2 | 1 | submbas 18088 | . . 3 ⊢ (𝑆 ∈ (SubMnd‘𝐺) → 𝑆 = (Base‘𝐻)) |
3 | eqid 2738 | . . . . . . 7 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
4 | 1, 3 | ressplusg 16708 | . . . . . 6 ⊢ (𝑆 ∈ (SubMnd‘𝐺) → (+g‘𝐺) = (+g‘𝐻)) |
5 | 4 | oveqd 7181 | . . . . 5 ⊢ (𝑆 ∈ (SubMnd‘𝐺) → (𝑥(+g‘𝐺)𝑦) = (𝑥(+g‘𝐻)𝑦)) |
6 | 4 | oveqd 7181 | . . . . 5 ⊢ (𝑆 ∈ (SubMnd‘𝐺) → (𝑦(+g‘𝐺)𝑥) = (𝑦(+g‘𝐻)𝑥)) |
7 | 5, 6 | eqeq12d 2754 | . . . 4 ⊢ (𝑆 ∈ (SubMnd‘𝐺) → ((𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥) ↔ (𝑥(+g‘𝐻)𝑦) = (𝑦(+g‘𝐻)𝑥))) |
8 | 2, 7 | raleqbidv 3303 | . . 3 ⊢ (𝑆 ∈ (SubMnd‘𝐺) → (∀𝑦 ∈ 𝑆 (𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥) ↔ ∀𝑦 ∈ (Base‘𝐻)(𝑥(+g‘𝐻)𝑦) = (𝑦(+g‘𝐻)𝑥))) |
9 | 2, 8 | raleqbidv 3303 | . 2 ⊢ (𝑆 ∈ (SubMnd‘𝐺) → (∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥) ↔ ∀𝑥 ∈ (Base‘𝐻)∀𝑦 ∈ (Base‘𝐻)(𝑥(+g‘𝐻)𝑦) = (𝑦(+g‘𝐻)𝑥))) |
10 | eqid 2738 | . . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
11 | 10 | submss 18083 | . . 3 ⊢ (𝑆 ∈ (SubMnd‘𝐺) → 𝑆 ⊆ (Base‘𝐺)) |
12 | submcmn2.z | . . . 4 ⊢ 𝑍 = (Cntz‘𝐺) | |
13 | 10, 3, 12 | sscntz 18567 | . . 3 ⊢ ((𝑆 ⊆ (Base‘𝐺) ∧ 𝑆 ⊆ (Base‘𝐺)) → (𝑆 ⊆ (𝑍‘𝑆) ↔ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥))) |
14 | 11, 11, 13 | syl2anc 587 | . 2 ⊢ (𝑆 ∈ (SubMnd‘𝐺) → (𝑆 ⊆ (𝑍‘𝑆) ↔ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥))) |
15 | 1 | submmnd 18087 | . . 3 ⊢ (𝑆 ∈ (SubMnd‘𝐺) → 𝐻 ∈ Mnd) |
16 | eqid 2738 | . . . . 5 ⊢ (Base‘𝐻) = (Base‘𝐻) | |
17 | eqid 2738 | . . . . 5 ⊢ (+g‘𝐻) = (+g‘𝐻) | |
18 | 16, 17 | iscmn 19025 | . . . 4 ⊢ (𝐻 ∈ CMnd ↔ (𝐻 ∈ Mnd ∧ ∀𝑥 ∈ (Base‘𝐻)∀𝑦 ∈ (Base‘𝐻)(𝑥(+g‘𝐻)𝑦) = (𝑦(+g‘𝐻)𝑥))) |
19 | 18 | baib 539 | . . 3 ⊢ (𝐻 ∈ Mnd → (𝐻 ∈ CMnd ↔ ∀𝑥 ∈ (Base‘𝐻)∀𝑦 ∈ (Base‘𝐻)(𝑥(+g‘𝐻)𝑦) = (𝑦(+g‘𝐻)𝑥))) |
20 | 15, 19 | syl 17 | . 2 ⊢ (𝑆 ∈ (SubMnd‘𝐺) → (𝐻 ∈ CMnd ↔ ∀𝑥 ∈ (Base‘𝐻)∀𝑦 ∈ (Base‘𝐻)(𝑥(+g‘𝐻)𝑦) = (𝑦(+g‘𝐻)𝑥))) |
21 | 9, 14, 20 | 3bitr4rd 315 | 1 ⊢ (𝑆 ∈ (SubMnd‘𝐺) → (𝐻 ∈ CMnd ↔ 𝑆 ⊆ (𝑍‘𝑆))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1542 ∈ wcel 2113 ∀wral 3053 ⊆ wss 3841 ‘cfv 6333 (class class class)co 7164 Basecbs 16579 ↾s cress 16580 +gcplusg 16661 Mndcmnd 18020 SubMndcsubmnd 18064 Cntzccntz 18556 CMndccmn 19017 |
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 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-rep 5151 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 ax-cnex 10664 ax-resscn 10665 ax-1cn 10666 ax-icn 10667 ax-addcl 10668 ax-addrcl 10669 ax-mulcl 10670 ax-mulrcl 10671 ax-mulcom 10672 ax-addass 10673 ax-mulass 10674 ax-distr 10675 ax-i2m1 10676 ax-1ne0 10677 ax-1rid 10678 ax-rnegex 10679 ax-rrecex 10680 ax-cnre 10681 ax-pre-lttri 10682 ax-pre-lttrn 10683 ax-pre-ltadd 10684 ax-pre-mulgt0 10685 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-pss 3860 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-tp 4518 df-op 4520 df-uni 4794 df-iun 4880 df-br 5028 df-opab 5090 df-mpt 5108 df-tr 5134 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6123 df-ord 6169 df-on 6170 df-lim 6171 df-suc 6172 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-riota 7121 df-ov 7167 df-oprab 7168 df-mpo 7169 df-om 7594 df-wrecs 7969 df-recs 8030 df-rdg 8068 df-er 8313 df-en 8549 df-dom 8550 df-sdom 8551 df-pnf 10748 df-mnf 10749 df-xr 10750 df-ltxr 10751 df-le 10752 df-sub 10943 df-neg 10944 df-nn 11710 df-2 11772 df-ndx 16582 df-slot 16583 df-base 16585 df-sets 16586 df-ress 16587 df-plusg 16674 df-0g 16811 df-mgm 17961 df-sgrp 18010 df-mnd 18021 df-submnd 18066 df-cntz 18558 df-cmn 19019 |
This theorem is referenced by: cntzspan 19076 gsumzsplit 19159 gsumzoppg 19176 gsumpt 19194 |
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