Theorem cntrval2 33126

Description: Express the center 𝑍 of a group 𝑀 as the set of fixed points of the conjugation operation ⊕. (Contributed by Thierry Arnoux, 18-Nov-2025.)

Hypotheses

Ref	Expression
cntrval2.1	⊢ 𝐵 = (Base‘𝑀)
cntrval2.2	⊢ + = (+g‘𝑀)
cntrval2.3	⊢ − = (-g‘𝑀)
cntrval2.4	⊢ ⊕ = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥 + 𝑦) − 𝑥))
cntrval2.5	⊢ 𝑍 = (Cntr‘𝑀)

Assertion

Ref	Expression
cntrval2	⊢ (𝑀 ∈ Grp → 𝑍 = (𝐵FixPts ⊕ ))

Distinct variable groups:   𝑥, ⊕ ,𝑦   𝑥, + ,𝑦   𝑥, − ,𝑦   𝑥,𝐵,𝑦   𝑥,𝑀,𝑦

Allowed substitution hints:   𝑍(𝑥,𝑦)

Proof of Theorem cntrval2

Dummy variables 𝑝 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpll 766	. . . . . . . 8 ⊢ (((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑝 ∈ 𝐵) → 𝑀 ∈ Grp)
2		cntrval2.1	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝑀)
3		cntrval2.3	. . . . . . . . 9 ⊢ − = (-g‘𝑀)
4		cntrval2.2	. . . . . . . . . 10 ⊢ + = (+g‘𝑀)
5		simpr 484	. . . . . . . . . 10 ⊢ (((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑝 ∈ 𝐵) → 𝑝 ∈ 𝐵)
6		simplr 768	. . . . . . . . . 10 ⊢ (((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑝 ∈ 𝐵) → 𝑧 ∈ 𝐵)
7	2, 4, 1, 5, 6	grpcld 18844	. . . . . . . . 9 ⊢ (((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑝 ∈ 𝐵) → (𝑝 + 𝑧) ∈ 𝐵)
8	2, 3, 1, 7, 5	grpsubcld 33007	. . . . . . . 8 ⊢ (((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑝 ∈ 𝐵) → ((𝑝 + 𝑧) − 𝑝) ∈ 𝐵)
9	2, 4	grprcan 18870	. . . . . . . 8 ⊢ ((𝑀 ∈ Grp ∧ (((𝑝 + 𝑧) − 𝑝) ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑝 ∈ 𝐵)) → ((((𝑝 + 𝑧) − 𝑝) + 𝑝) = (𝑧 + 𝑝) ↔ ((𝑝 + 𝑧) − 𝑝) = 𝑧))
10	1, 8, 6, 5, 9	syl13anc 1374	. . . . . . 7 ⊢ (((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑝 ∈ 𝐵) → ((((𝑝 + 𝑧) − 𝑝) + 𝑝) = (𝑧 + 𝑝) ↔ ((𝑝 + 𝑧) − 𝑝) = 𝑧))
11	2, 4, 3	grpnpcan 18929	. . . . . . . . . 10 ⊢ ((𝑀 ∈ Grp ∧ (𝑝 + 𝑧) ∈ 𝐵 ∧ 𝑝 ∈ 𝐵) → (((𝑝 + 𝑧) − 𝑝) + 𝑝) = (𝑝 + 𝑧))
12	1, 7, 5, 11	syl3anc 1373	. . . . . . . . 9 ⊢ (((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑝 ∈ 𝐵) → (((𝑝 + 𝑧) − 𝑝) + 𝑝) = (𝑝 + 𝑧))
13	12	eqeq2d 2740	. . . . . . . 8 ⊢ (((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑝 ∈ 𝐵) → ((𝑧 + 𝑝) = (((𝑝 + 𝑧) − 𝑝) + 𝑝) ↔ (𝑧 + 𝑝) = (𝑝 + 𝑧)))
14		eqcom 2736	. . . . . . . 8 ⊢ ((𝑧 + 𝑝) = (((𝑝 + 𝑧) − 𝑝) + 𝑝) ↔ (((𝑝 + 𝑧) − 𝑝) + 𝑝) = (𝑧 + 𝑝))
15	13, 14	bitr3di 286	. . . . . . 7 ⊢ (((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑝 ∈ 𝐵) → ((𝑧 + 𝑝) = (𝑝 + 𝑧) ↔ (((𝑝 + 𝑧) − 𝑝) + 𝑝) = (𝑧 + 𝑝)))
16		cntrval2.4	. . . . . . . . . 10 ⊢ ⊕ = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥 + 𝑦) − 𝑥))
17	16	a1i 11	. . . . . . . . 9 ⊢ (((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑝 ∈ 𝐵) → ⊕ = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥 + 𝑦) − 𝑥)))
18		simprl 770	. . . . . . . . . . 11 ⊢ ((((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑝 ∈ 𝐵) ∧ (𝑥 = 𝑝 ∧ 𝑦 = 𝑧)) → 𝑥 = 𝑝)
19		simprr 772	. . . . . . . . . . 11 ⊢ ((((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑝 ∈ 𝐵) ∧ (𝑥 = 𝑝 ∧ 𝑦 = 𝑧)) → 𝑦 = 𝑧)
20	18, 19	oveq12d 7371	. . . . . . . . . 10 ⊢ ((((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑝 ∈ 𝐵) ∧ (𝑥 = 𝑝 ∧ 𝑦 = 𝑧)) → (𝑥 + 𝑦) = (𝑝 + 𝑧))
21	20, 18	oveq12d 7371	. . . . . . . . 9 ⊢ ((((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑝 ∈ 𝐵) ∧ (𝑥 = 𝑝 ∧ 𝑦 = 𝑧)) → ((𝑥 + 𝑦) − 𝑥) = ((𝑝 + 𝑧) − 𝑝))
22		ovexd 7388	. . . . . . . . 9 ⊢ (((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑝 ∈ 𝐵) → ((𝑝 + 𝑧) − 𝑝) ∈ V)
23	17, 21, 5, 6, 22	ovmpod 7505	. . . . . . . 8 ⊢ (((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑝 ∈ 𝐵) → (𝑝 ⊕ 𝑧) = ((𝑝 + 𝑧) − 𝑝))
24	23	eqeq1d 2731	. . . . . . 7 ⊢ (((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑝 ∈ 𝐵) → ((𝑝 ⊕ 𝑧) = 𝑧 ↔ ((𝑝 + 𝑧) − 𝑝) = 𝑧))
25	10, 15, 24	3bitr4d 311	. . . . . 6 ⊢ (((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑝 ∈ 𝐵) → ((𝑧 + 𝑝) = (𝑝 + 𝑧) ↔ (𝑝 ⊕ 𝑧) = 𝑧))
26	25	ralbidva 3150	. . . . 5 ⊢ ((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) → (∀𝑝 ∈ 𝐵 (𝑧 + 𝑝) = (𝑝 + 𝑧) ↔ ∀𝑝 ∈ 𝐵 (𝑝 ⊕ 𝑧) = 𝑧))
27	26	pm5.32da 579	. . . 4 ⊢ (𝑀 ∈ Grp → ((𝑧 ∈ 𝐵 ∧ ∀𝑝 ∈ 𝐵 (𝑧 + 𝑝) = (𝑝 + 𝑧)) ↔ (𝑧 ∈ 𝐵 ∧ ∀𝑝 ∈ 𝐵 (𝑝 ⊕ 𝑧) = 𝑧)))
28		cntrval2.5	. . . . 5 ⊢ 𝑍 = (Cntr‘𝑀)
29	2, 4, 28	elcntr 19227	. . . 4 ⊢ (𝑧 ∈ 𝑍 ↔ (𝑧 ∈ 𝐵 ∧ ∀𝑝 ∈ 𝐵 (𝑧 + 𝑝) = (𝑝 + 𝑧)))
30		rabid 3418	. . . 4 ⊢ (𝑧 ∈ {𝑧 ∈ 𝐵 ∣ ∀𝑝 ∈ 𝐵 (𝑝 ⊕ 𝑧) = 𝑧} ↔ (𝑧 ∈ 𝐵 ∧ ∀𝑝 ∈ 𝐵 (𝑝 ⊕ 𝑧) = 𝑧))
31	27, 29, 30	3bitr4g 314	. . 3 ⊢ (𝑀 ∈ Grp → (𝑧 ∈ 𝑍 ↔ 𝑧 ∈ {𝑧 ∈ 𝐵 ∣ ∀𝑝 ∈ 𝐵 (𝑝 ⊕ 𝑧) = 𝑧}))
32	2, 4, 3, 16	conjga 33125	. . . . 5 ⊢ (𝑀 ∈ Grp → ⊕ ∈ (𝑀 GrpAct 𝐵))
33	2, 32	fxpgaval 33122	. . . 4 ⊢ (𝑀 ∈ Grp → (𝐵FixPts ⊕ ) = {𝑧 ∈ 𝐵 ∣ ∀𝑝 ∈ 𝐵 (𝑝 ⊕ 𝑧) = 𝑧})
34	33	eleq2d 2814	. . 3 ⊢ (𝑀 ∈ Grp → (𝑧 ∈ (𝐵FixPts ⊕ ) ↔ 𝑧 ∈ {𝑧 ∈ 𝐵 ∣ ∀𝑝 ∈ 𝐵 (𝑝 ⊕ 𝑧) = 𝑧}))
35	31, 34	bitr4d 282	. 2 ⊢ (𝑀 ∈ Grp → (𝑧 ∈ 𝑍 ↔ 𝑧 ∈ (𝐵FixPts ⊕ )))
36	35	eqrdv 2727	1 ⊢ (𝑀 ∈ Grp → 𝑍 = (𝐵FixPts ⊕ ))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  {crab 3396  Vcvv 3438  ‘cfv 6486  (class class class)co 7353   ∈ cmpo 7355  Basecbs 17138  +_gcplusg 17179  Grpcgrp 18830  -_gcsg 18832  Cntrccntr 19213  FixPtscfxp 33118

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3345  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7310  df-ov 7356  df-oprab 7357  df-mpo 7358  df-1st 7931  df-2nd 7932  df-map 8762  df-0g 17363  df-mgm 18532  df-sgrp 18611  df-mnd 18627  df-grp 18833  df-minusg 18834  df-sbg 18835  df-ga 19187  df-cntz 19214  df-cntr 19215  df-fxp 33119

This theorem is referenced by: (None)