Theorem cntrval2 33234

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 767	. . . . . . . 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 769	. . . . . . . . . 10 ⊢ (((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑝 ∈ 𝐵) → 𝑧 ∈ 𝐵)
7	2, 4, 1, 5, 6	grpcld 18881	. . . . . . . . 9 ⊢ (((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑝 ∈ 𝐵) → (𝑝 + 𝑧) ∈ 𝐵)
8	2, 3, 1, 7, 5	grpsubcld 33104	. . . . . . . 8 ⊢ (((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑝 ∈ 𝐵) → ((𝑝 + 𝑧) − 𝑝) ∈ 𝐵)
9	2, 4	grprcan 18907	. . . . . . . 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 1375	. . . . . . 7 ⊢ (((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑝 ∈ 𝐵) → ((((𝑝 + 𝑧) − 𝑝) + 𝑝) = (𝑧 + 𝑝) ↔ ((𝑝 + 𝑧) − 𝑝) = 𝑧))
11	2, 4, 3	grpnpcan 18966	. . . . . . . . . 10 ⊢ ((𝑀 ∈ Grp ∧ (𝑝 + 𝑧) ∈ 𝐵 ∧ 𝑝 ∈ 𝐵) → (((𝑝 + 𝑧) − 𝑝) + 𝑝) = (𝑝 + 𝑧))
12	1, 7, 5, 11	syl3anc 1374	. . . . . . . . 9 ⊢ (((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑝 ∈ 𝐵) → (((𝑝 + 𝑧) − 𝑝) + 𝑝) = (𝑝 + 𝑧))
13	12	eqeq2d 2748	. . . . . . . 8 ⊢ (((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑝 ∈ 𝐵) → ((𝑧 + 𝑝) = (((𝑝 + 𝑧) − 𝑝) + 𝑝) ↔ (𝑧 + 𝑝) = (𝑝 + 𝑧)))
14		eqcom 2744	. . . . . . . 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 771	. . . . . . . . . . 11 ⊢ ((((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑝 ∈ 𝐵) ∧ (𝑥 = 𝑝 ∧ 𝑦 = 𝑧)) → 𝑥 = 𝑝)
19		simprr 773	. . . . . . . . . . 11 ⊢ ((((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑝 ∈ 𝐵) ∧ (𝑥 = 𝑝 ∧ 𝑦 = 𝑧)) → 𝑦 = 𝑧)
20	18, 19	oveq12d 7378	. . . . . . . . . 10 ⊢ ((((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑝 ∈ 𝐵) ∧ (𝑥 = 𝑝 ∧ 𝑦 = 𝑧)) → (𝑥 + 𝑦) = (𝑝 + 𝑧))
21	20, 18	oveq12d 7378	. . . . . . . . 9 ⊢ ((((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑝 ∈ 𝐵) ∧ (𝑥 = 𝑝 ∧ 𝑦 = 𝑧)) → ((𝑥 + 𝑦) − 𝑥) = ((𝑝 + 𝑧) − 𝑝))
22		ovexd 7395	. . . . . . . . 9 ⊢ (((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑝 ∈ 𝐵) → ((𝑝 + 𝑧) − 𝑝) ∈ V)
23	17, 21, 5, 6, 22	ovmpod 7512	. . . . . . . 8 ⊢ (((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑝 ∈ 𝐵) → (𝑝 ⊕ 𝑧) = ((𝑝 + 𝑧) − 𝑝))
24	23	eqeq1d 2739	. . . . . . 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 3158	. . . . 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 19263	. . . 4 ⊢ (𝑧 ∈ 𝑍 ↔ (𝑧 ∈ 𝐵 ∧ ∀𝑝 ∈ 𝐵 (𝑧 + 𝑝) = (𝑝 + 𝑧)))
30		rabid 3421	. . . 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 33233	. . . . 5 ⊢ (𝑀 ∈ Grp → ⊕ ∈ (𝑀 GrpAct 𝐵))
33	2, 32	fxpgaval 33230	. . . 4 ⊢ (𝑀 ∈ Grp → (𝐵FixPts ⊕ ) = {𝑧 ∈ 𝐵 ∣ ∀𝑝 ∈ 𝐵 (𝑝 ⊕ 𝑧) = 𝑧})
34	33	eleq2d 2823	. . 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 2735	1 ⊢ (𝑀 ∈ Grp → 𝑍 = (𝐵FixPts ⊕ ))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  {crab 3400  Vcvv 3441  ‘cfv 6493  (class class class)co 7360   ∈ cmpo 7362  Basecbs 17140  +_gcplusg 17181  Grpcgrp 18867  -_gcsg 18869  Cntrccntr 19249  FixPtscfxp 33226

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5225  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rmo 3351  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  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 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-1st 7935  df-2nd 7936  df-map 8769  df-0g 17365  df-mgm 18569  df-sgrp 18648  df-mnd 18664  df-grp 18870  df-minusg 18871  df-sbg 18872  df-ga 19223  df-cntz 19250  df-cntr 19251  df-fxp 33227

This theorem is referenced by: (None)