Theorem cntrval2 33802

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 779	. . . . . . . 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 490	. . . . . . . . . 10 ⊢ (((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑝 ∈ 𝐵) → 𝑝 ∈ 𝐵)
6		simplr 781	. . . . . . . . . 10 ⊢ (((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑝 ∈ 𝐵) → 𝑧 ∈ 𝐵)
7	2, 4, 1, 5, 6	grpcld 19187	. . . . . . . . 9 ⊢ (((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑝 ∈ 𝐵) → (𝑝 + 𝑧) ∈ 𝐵)
8	2, 3, 1, 7, 5	grpsubcld 33672	. . . . . . . 8 ⊢ (((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑝 ∈ 𝐵) → ((𝑝 + 𝑧) − 𝑝) ∈ 𝐵)
9	2, 4	grprcan 19213	. . . . . . . 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 1400	. . . . . . 7 ⊢ (((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑝 ∈ 𝐵) → ((((𝑝 + 𝑧) − 𝑝) + 𝑝) = (𝑧 + 𝑝) ↔ ((𝑝 + 𝑧) − 𝑝) = 𝑧))
11	2, 4, 3	grpnpcan 19271	. . . . . . . . . 10 ⊢ ((𝑀 ∈ Grp ∧ (𝑝 + 𝑧) ∈ 𝐵 ∧ 𝑝 ∈ 𝐵) → (((𝑝 + 𝑧) − 𝑝) + 𝑝) = (𝑝 + 𝑧))
12	1, 7, 5, 11	syl3anc 1399	. . . . . . . . 9 ⊢ (((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑝 ∈ 𝐵) → (((𝑝 + 𝑧) − 𝑝) + 𝑝) = (𝑝 + 𝑧))
13	12	eqeq2d 2774	. . . . . . . 8 ⊢ (((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑝 ∈ 𝐵) → ((𝑧 + 𝑝) = (((𝑝 + 𝑧) − 𝑝) + 𝑝) ↔ (𝑧 + 𝑝) = (𝑝 + 𝑧)))
14		eqcom 2770	. . . . . . . 8 ⊢ ((𝑧 + 𝑝) = (((𝑝 + 𝑧) − 𝑝) + 𝑝) ↔ (((𝑝 + 𝑧) − 𝑝) + 𝑝) = (𝑧 + 𝑝))
15	13, 14	bitr3di 289	. . . . . . 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 783	. . . . . . . . . . 11 ⊢ ((((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑝 ∈ 𝐵) ∧ (𝑥 = 𝑝 ∧ 𝑦 = 𝑧)) → 𝑥 = 𝑝)
19		simprr 785	. . . . . . . . . . 11 ⊢ ((((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑝 ∈ 𝐵) ∧ (𝑥 = 𝑝 ∧ 𝑦 = 𝑧)) → 𝑦 = 𝑧)
20	18, 19	oveq12d 7461	. . . . . . . . . 10 ⊢ ((((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑝 ∈ 𝐵) ∧ (𝑥 = 𝑝 ∧ 𝑦 = 𝑧)) → (𝑥 + 𝑦) = (𝑝 + 𝑧))
21	20, 18	oveq12d 7461	. . . . . . . . 9 ⊢ ((((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑝 ∈ 𝐵) ∧ (𝑥 = 𝑝 ∧ 𝑦 = 𝑧)) → ((𝑥 + 𝑦) − 𝑥) = ((𝑝 + 𝑧) − 𝑝))
22		ovexd 7478	. . . . . . . . 9 ⊢ (((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑝 ∈ 𝐵) → ((𝑝 + 𝑧) − 𝑝) ∈ V)
23	17, 21, 5, 6, 22	ovmpod 7596	. . . . . . . 8 ⊢ (((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑝 ∈ 𝐵) → (𝑝 ⊕ 𝑧) = ((𝑝 + 𝑧) − 𝑝))
24	23	eqeq1d 2765	. . . . . . 7 ⊢ (((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑝 ∈ 𝐵) → ((𝑝 ⊕ 𝑧) = 𝑧 ↔ ((𝑝 + 𝑧) − 𝑝) = 𝑧))
25	10, 15, 24	3bitr4d 314	. . . . . 6 ⊢ (((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑝 ∈ 𝐵) → ((𝑧 + 𝑝) = (𝑝 + 𝑧) ↔ (𝑝 ⊕ 𝑧) = 𝑧))
26	25	ralbidva 3187	. . . . 5 ⊢ ((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) → (∀𝑝 ∈ 𝐵 (𝑧 + 𝑝) = (𝑝 + 𝑧) ↔ ∀𝑝 ∈ 𝐵 (𝑝 ⊕ 𝑧) = 𝑧))
27	26	pm5.32da 590	. . . 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 19574	. . . 4 ⊢ (𝑧 ∈ 𝑍 ↔ (𝑧 ∈ 𝐵 ∧ ∀𝑝 ∈ 𝐵 (𝑧 + 𝑝) = (𝑝 + 𝑧)))
30		rabid 3436	. . . 4 ⊢ (𝑧 ∈ {𝑧 ∈ 𝐵 ∣ ∀𝑝 ∈ 𝐵 (𝑝 ⊕ 𝑧) = 𝑧} ↔ (𝑧 ∈ 𝐵 ∧ ∀𝑝 ∈ 𝐵 (𝑝 ⊕ 𝑧) = 𝑧))
31	27, 29, 30	3bitr4g 317	. . 3 ⊢ (𝑀 ∈ Grp → (𝑧 ∈ 𝑍 ↔ 𝑧 ∈ {𝑧 ∈ 𝐵 ∣ ∀𝑝 ∈ 𝐵 (𝑝 ⊕ 𝑧) = 𝑧}))
32	2, 4, 3, 16	conjga 33801	. . . . 5 ⊢ (𝑀 ∈ Grp → ⊕ ∈ (𝑀 GrpAct 𝐵))
33	2, 32	fxpgaval 33798	. . . 4 ⊢ (𝑀 ∈ Grp → (𝐵FixPts ⊕ ) = {𝑧 ∈ 𝐵 ∣ ∀𝑝 ∈ 𝐵 (𝑝 ⊕ 𝑧) = 𝑧})
34	33	eleq2d 2850	. . 3 ⊢ (𝑀 ∈ Grp → (𝑧 ∈ (𝐵FixPts ⊕ ) ↔ 𝑧 ∈ {𝑧 ∈ 𝐵 ∣ ∀𝑝 ∈ 𝐵 (𝑝 ⊕ 𝑧) = 𝑧}))
35	31, 34	bitr4d 285	. 2 ⊢ (𝑀 ∈ Grp → (𝑧 ∈ 𝑍 ↔ 𝑧 ∈ (𝐵FixPts ⊕ )))
36	35	eqrdv 2761	1 ⊢ (𝑀 ∈ Grp → 𝑍 = (𝐵FixPts ⊕ ))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∀wral 3080  {crab 3416  Vcvv 3454  ‘cfv 6558  (class class class)co 7443   ∈ cmpo 7445  Basecbs 17416  +_gcplusg 17457  Grpcgrp 19173  -_gcsg 19175  Cntrccntr 19560  FixPtscfxp 33794

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-riota 7400  df-ov 7446  df-oprab 7447  df-mpo 7448  df-1st 8025  df-2nd 8026  df-map 8877  df-0g 17641  df-mgm 18845  df-sgrp 18937  df-mnd 18953  df-grp 19176  df-minusg 19177  df-sbg 19178  df-ga 19534  df-cntz 19561  df-cntr 19562  df-fxp 33795

This theorem is used by: (None)