Theorem cntrval2 33259

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 772	. . . . . . . 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 485	. . . . . . . . . 10 ⊢ (((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑝 ∈ 𝐵) → 𝑝 ∈ 𝐵)
6		simplr 774	. . . . . . . . . 10 ⊢ (((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑝 ∈ 𝐵) → 𝑧 ∈ 𝐵)
7	2, 4, 1, 5, 6	grpcld 18921	. . . . . . . . 9 ⊢ (((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑝 ∈ 𝐵) → (𝑝 + 𝑧) ∈ 𝐵)
8	2, 3, 1, 7, 5	grpsubcld 33128	. . . . . . . 8 ⊢ (((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑝 ∈ 𝐵) → ((𝑝 + 𝑧) − 𝑝) ∈ 𝐵)
9	2, 4	grprcan 18947	. . . . . . . 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 1380	. . . . . . 7 ⊢ (((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑝 ∈ 𝐵) → ((((𝑝 + 𝑧) − 𝑝) + 𝑝) = (𝑧 + 𝑝) ↔ ((𝑝 + 𝑧) − 𝑝) = 𝑧))
11	2, 4, 3	grpnpcan 19006	. . . . . . . . . 10 ⊢ ((𝑀 ∈ Grp ∧ (𝑝 + 𝑧) ∈ 𝐵 ∧ 𝑝 ∈ 𝐵) → (((𝑝 + 𝑧) − 𝑝) + 𝑝) = (𝑝 + 𝑧))
12	1, 7, 5, 11	syl3anc 1379	. . . . . . . . 9 ⊢ (((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑝 ∈ 𝐵) → (((𝑝 + 𝑧) − 𝑝) + 𝑝) = (𝑝 + 𝑧))
13	12	eqeq2d 2751	. . . . . . . 8 ⊢ (((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑝 ∈ 𝐵) → ((𝑧 + 𝑝) = (((𝑝 + 𝑧) − 𝑝) + 𝑝) ↔ (𝑧 + 𝑝) = (𝑝 + 𝑧)))
14		eqcom 2747	. . . . . . . 8 ⊢ ((𝑧 + 𝑝) = (((𝑝 + 𝑧) − 𝑝) + 𝑝) ↔ (((𝑝 + 𝑧) − 𝑝) + 𝑝) = (𝑧 + 𝑝))
15	13, 14	bitr3di 287	. . . . . . 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 776	. . . . . . . . . . 11 ⊢ ((((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑝 ∈ 𝐵) ∧ (𝑥 = 𝑝 ∧ 𝑦 = 𝑧)) → 𝑥 = 𝑝)
19		simprr 778	. . . . . . . . . . 11 ⊢ ((((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑝 ∈ 𝐵) ∧ (𝑥 = 𝑝 ∧ 𝑦 = 𝑧)) → 𝑦 = 𝑧)
20	18, 19	oveq12d 7381	. . . . . . . . . 10 ⊢ ((((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑝 ∈ 𝐵) ∧ (𝑥 = 𝑝 ∧ 𝑦 = 𝑧)) → (𝑥 + 𝑦) = (𝑝 + 𝑧))
21	20, 18	oveq12d 7381	. . . . . . . . 9 ⊢ ((((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑝 ∈ 𝐵) ∧ (𝑥 = 𝑝 ∧ 𝑦 = 𝑧)) → ((𝑥 + 𝑦) − 𝑥) = ((𝑝 + 𝑧) − 𝑝))
22		ovexd 7398	. . . . . . . . 9 ⊢ (((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑝 ∈ 𝐵) → ((𝑝 + 𝑧) − 𝑝) ∈ V)
23	17, 21, 5, 6, 22	ovmpod 7515	. . . . . . . 8 ⊢ (((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑝 ∈ 𝐵) → (𝑝 ⊕ 𝑧) = ((𝑝 + 𝑧) − 𝑝))
24	23	eqeq1d 2742	. . . . . . 7 ⊢ (((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑝 ∈ 𝐵) → ((𝑝 ⊕ 𝑧) = 𝑧 ↔ ((𝑝 + 𝑧) − 𝑝) = 𝑧))
25	10, 15, 24	3bitr4d 312	. . . . . 6 ⊢ (((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑝 ∈ 𝐵) → ((𝑧 + 𝑝) = (𝑝 + 𝑧) ↔ (𝑝 ⊕ 𝑧) = 𝑧))
26	25	ralbidva 3161	. . . . 5 ⊢ ((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) → (∀𝑝 ∈ 𝐵 (𝑧 + 𝑝) = (𝑝 + 𝑧) ↔ ∀𝑝 ∈ 𝐵 (𝑝 ⊕ 𝑧) = 𝑧))
27	26	pm5.32da 584	. . . 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 19303	. . . 4 ⊢ (𝑧 ∈ 𝑍 ↔ (𝑧 ∈ 𝐵 ∧ ∀𝑝 ∈ 𝐵 (𝑧 + 𝑝) = (𝑝 + 𝑧)))
30		rabid 3413	. . . 4 ⊢ (𝑧 ∈ {𝑧 ∈ 𝐵 ∣ ∀𝑝 ∈ 𝐵 (𝑝 ⊕ 𝑧) = 𝑧} ↔ (𝑧 ∈ 𝐵 ∧ ∀𝑝 ∈ 𝐵 (𝑝 ⊕ 𝑧) = 𝑧))
31	27, 29, 30	3bitr4g 315	. . 3 ⊢ (𝑀 ∈ Grp → (𝑧 ∈ 𝑍 ↔ 𝑧 ∈ {𝑧 ∈ 𝐵 ∣ ∀𝑝 ∈ 𝐵 (𝑝 ⊕ 𝑧) = 𝑧}))
32	2, 4, 3, 16	conjga 33258	. . . . 5 ⊢ (𝑀 ∈ Grp → ⊕ ∈ (𝑀 GrpAct 𝐵))
33	2, 32	fxpgaval 33255	. . . 4 ⊢ (𝑀 ∈ Grp → (𝐵FixPts ⊕ ) = {𝑧 ∈ 𝐵 ∣ ∀𝑝 ∈ 𝐵 (𝑝 ⊕ 𝑧) = 𝑧})
34	33	eleq2d 2826	. . 3 ⊢ (𝑀 ∈ Grp → (𝑧 ∈ (𝐵FixPts ⊕ ) ↔ 𝑧 ∈ {𝑧 ∈ 𝐵 ∣ ∀𝑝 ∈ 𝐵 (𝑝 ⊕ 𝑧) = 𝑧}))
35	31, 34	bitr4d 283	. 2 ⊢ (𝑀 ∈ Grp → (𝑧 ∈ 𝑍 ↔ 𝑧 ∈ (𝐵FixPts ⊕ )))
36	35	eqrdv 2738	1 ⊢ (𝑀 ∈ Grp → 𝑍 = (𝐵FixPts ⊕ ))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ∀wral 3054  {crab 3392  Vcvv 3432  ‘cfv 6492  (class class class)co 7363   ∈ cmpo 7365  Basecbs 17177  +_gcplusg 17218  Grpcgrp 18907  -_gcsg 18909  Cntrccntr 19289  FixPtscfxp 33251

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 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rmo 3345  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7320  df-ov 7366  df-oprab 7367  df-mpo 7368  df-1st 7938  df-2nd 7939  df-map 8772  df-0g 17402  df-mgm 18606  df-sgrp 18685  df-mnd 18701  df-grp 18910  df-minusg 18911  df-sbg 18912  df-ga 19263  df-cntz 19290  df-cntr 19291  df-fxp 33252

This theorem is referenced by: (None)