Theorem cntzsgrpcl 19287

Description: Centralizers are closed under the semigroup operation. (Contributed by AV, 17-Feb-2025.)

Hypotheses

Ref	Expression
cntzsgrpcl.b	⊢ 𝐵 = (Base‘𝑀)
cntzsgrpcl.z	⊢ 𝑍 = (Cntz‘𝑀)
cntzsgrpcl.c	⊢ 𝐶 = (𝑍‘𝑆)

Assertion

Ref	Expression
cntzsgrpcl	⊢ ((𝑀 ∈ Smgrp ∧ 𝑆 ⊆ 𝐵) → ∀𝑦 ∈ 𝐶 ∀𝑧 ∈ 𝐶 (𝑦(+g‘𝑀)𝑧) ∈ 𝐶)

Distinct variable groups:   𝑦,𝐵,𝑧   𝑦,𝐶,𝑧   𝑦,𝑀,𝑧   𝑦,𝑆,𝑧   𝑦,𝑍,𝑧

Proof of Theorem cntzsgrpcl

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpll 765	. . . 4 ⊢ (((𝑀 ∈ Smgrp ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → 𝑀 ∈ Smgrp)
2		cntzsgrpcl.c	. . . . . 6 ⊢ 𝐶 = (𝑍‘𝑆)
3		cntzsgrpcl.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑀)
4		cntzsgrpcl.z	. . . . . . 7 ⊢ 𝑍 = (Cntz‘𝑀)
5	3, 4	cntzssv 19281	. . . . . 6 ⊢ (𝑍‘𝑆) ⊆ 𝐵
6	2, 5	eqsstri 4007	. . . . 5 ⊢ 𝐶 ⊆ 𝐵
7		simprl 769	. . . . 5 ⊢ (((𝑀 ∈ Smgrp ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → 𝑦 ∈ 𝐶)
8	6, 7	sselid 3970	. . . 4 ⊢ (((𝑀 ∈ Smgrp ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → 𝑦 ∈ 𝐵)
9		simprr 771	. . . . 5 ⊢ (((𝑀 ∈ Smgrp ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → 𝑧 ∈ 𝐶)
10	6, 9	sselid 3970	. . . 4 ⊢ (((𝑀 ∈ Smgrp ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → 𝑧 ∈ 𝐵)
11		eqid 2725	. . . . 5 ⊢ (+g‘𝑀) = (+g‘𝑀)
12	3, 11	sgrpcl 18683	. . . 4 ⊢ ((𝑀 ∈ Smgrp ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → (𝑦(+g‘𝑀)𝑧) ∈ 𝐵)
13	1, 8, 10, 12	syl3anc 1368	. . 3 ⊢ (((𝑀 ∈ Smgrp ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → (𝑦(+g‘𝑀)𝑧) ∈ 𝐵)
14	1	adantr 479	. . . . . 6 ⊢ ((((𝑀 ∈ Smgrp ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) ∧ 𝑥 ∈ 𝑆) → 𝑀 ∈ Smgrp)
15	8	adantr 479	. . . . . 6 ⊢ ((((𝑀 ∈ Smgrp ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) ∧ 𝑥 ∈ 𝑆) → 𝑦 ∈ 𝐵)
16	10	adantr 479	. . . . . 6 ⊢ ((((𝑀 ∈ Smgrp ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) ∧ 𝑥 ∈ 𝑆) → 𝑧 ∈ 𝐵)
17		simpr 483	. . . . . . . 8 ⊢ ((𝑀 ∈ Smgrp ∧ 𝑆 ⊆ 𝐵) → 𝑆 ⊆ 𝐵)
18	17	sselda 3972	. . . . . . 7 ⊢ (((𝑀 ∈ Smgrp ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝐵)
19	18	adantlr 713	. . . . . 6 ⊢ ((((𝑀 ∈ Smgrp ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝐵)
20	3, 11	sgrpass 18682	. . . . . 6 ⊢ ((𝑀 ∈ Smgrp ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → ((𝑦(+g‘𝑀)𝑧)(+g‘𝑀)𝑥) = (𝑦(+g‘𝑀)(𝑧(+g‘𝑀)𝑥)))
21	14, 15, 16, 19, 20	syl13anc 1369	. . . . 5 ⊢ ((((𝑀 ∈ Smgrp ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) ∧ 𝑥 ∈ 𝑆) → ((𝑦(+g‘𝑀)𝑧)(+g‘𝑀)𝑥) = (𝑦(+g‘𝑀)(𝑧(+g‘𝑀)𝑥)))
22	2	eleq2i 2817	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝐶 ↔ 𝑧 ∈ (𝑍‘𝑆))
23	11, 4	cntzi 19282	. . . . . . . . 9 ⊢ ((𝑧 ∈ (𝑍‘𝑆) ∧ 𝑥 ∈ 𝑆) → (𝑧(+g‘𝑀)𝑥) = (𝑥(+g‘𝑀)𝑧))
24	22, 23	sylanb 579	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝐶 ∧ 𝑥 ∈ 𝑆) → (𝑧(+g‘𝑀)𝑥) = (𝑥(+g‘𝑀)𝑧))
25	9, 24	sylan 578	. . . . . . 7 ⊢ ((((𝑀 ∈ Smgrp ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) ∧ 𝑥 ∈ 𝑆) → (𝑧(+g‘𝑀)𝑥) = (𝑥(+g‘𝑀)𝑧))
26	25	oveq2d 7431	. . . . . 6 ⊢ ((((𝑀 ∈ Smgrp ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) ∧ 𝑥 ∈ 𝑆) → (𝑦(+g‘𝑀)(𝑧(+g‘𝑀)𝑥)) = (𝑦(+g‘𝑀)(𝑥(+g‘𝑀)𝑧)))
27	3, 11	sgrpass 18682	. . . . . . 7 ⊢ ((𝑀 ∈ Smgrp ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑦(+g‘𝑀)𝑥)(+g‘𝑀)𝑧) = (𝑦(+g‘𝑀)(𝑥(+g‘𝑀)𝑧)))
28	14, 15, 19, 16, 27	syl13anc 1369	. . . . . 6 ⊢ ((((𝑀 ∈ Smgrp ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) ∧ 𝑥 ∈ 𝑆) → ((𝑦(+g‘𝑀)𝑥)(+g‘𝑀)𝑧) = (𝑦(+g‘𝑀)(𝑥(+g‘𝑀)𝑧)))
29	2	eleq2i 2817	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐶 ↔ 𝑦 ∈ (𝑍‘𝑆))
30	11, 4	cntzi 19282	. . . . . . . . 9 ⊢ ((𝑦 ∈ (𝑍‘𝑆) ∧ 𝑥 ∈ 𝑆) → (𝑦(+g‘𝑀)𝑥) = (𝑥(+g‘𝑀)𝑦))
31	29, 30	sylanb 579	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝐶 ∧ 𝑥 ∈ 𝑆) → (𝑦(+g‘𝑀)𝑥) = (𝑥(+g‘𝑀)𝑦))
32	7, 31	sylan 578	. . . . . . 7 ⊢ ((((𝑀 ∈ Smgrp ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) ∧ 𝑥 ∈ 𝑆) → (𝑦(+g‘𝑀)𝑥) = (𝑥(+g‘𝑀)𝑦))
33	32	oveq1d 7430	. . . . . 6 ⊢ ((((𝑀 ∈ Smgrp ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) ∧ 𝑥 ∈ 𝑆) → ((𝑦(+g‘𝑀)𝑥)(+g‘𝑀)𝑧) = ((𝑥(+g‘𝑀)𝑦)(+g‘𝑀)𝑧))
34	26, 28, 33	3eqtr2d 2771	. . . . 5 ⊢ ((((𝑀 ∈ Smgrp ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) ∧ 𝑥 ∈ 𝑆) → (𝑦(+g‘𝑀)(𝑧(+g‘𝑀)𝑥)) = ((𝑥(+g‘𝑀)𝑦)(+g‘𝑀)𝑧))
35	3, 11	sgrpass 18682	. . . . . 6 ⊢ ((𝑀 ∈ Smgrp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥(+g‘𝑀)𝑦)(+g‘𝑀)𝑧) = (𝑥(+g‘𝑀)(𝑦(+g‘𝑀)𝑧)))
36	14, 19, 15, 16, 35	syl13anc 1369	. . . . 5 ⊢ ((((𝑀 ∈ Smgrp ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) ∧ 𝑥 ∈ 𝑆) → ((𝑥(+g‘𝑀)𝑦)(+g‘𝑀)𝑧) = (𝑥(+g‘𝑀)(𝑦(+g‘𝑀)𝑧)))
37	21, 34, 36	3eqtrd 2769	. . . 4 ⊢ ((((𝑀 ∈ Smgrp ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) ∧ 𝑥 ∈ 𝑆) → ((𝑦(+g‘𝑀)𝑧)(+g‘𝑀)𝑥) = (𝑥(+g‘𝑀)(𝑦(+g‘𝑀)𝑧)))
38	37	ralrimiva 3136	. . 3 ⊢ (((𝑀 ∈ Smgrp ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → ∀𝑥 ∈ 𝑆 ((𝑦(+g‘𝑀)𝑧)(+g‘𝑀)𝑥) = (𝑥(+g‘𝑀)(𝑦(+g‘𝑀)𝑧)))
39	2	eleq2i 2817	. . . . 5 ⊢ ((𝑦(+g‘𝑀)𝑧) ∈ 𝐶 ↔ (𝑦(+g‘𝑀)𝑧) ∈ (𝑍‘𝑆))
40	3, 11, 4	elcntz 19275	. . . . 5 ⊢ (𝑆 ⊆ 𝐵 → ((𝑦(+g‘𝑀)𝑧) ∈ (𝑍‘𝑆) ↔ ((𝑦(+g‘𝑀)𝑧) ∈ 𝐵 ∧ ∀𝑥 ∈ 𝑆 ((𝑦(+g‘𝑀)𝑧)(+g‘𝑀)𝑥) = (𝑥(+g‘𝑀)(𝑦(+g‘𝑀)𝑧)))))
41	39, 40	bitrid 282	. . . 4 ⊢ (𝑆 ⊆ 𝐵 → ((𝑦(+g‘𝑀)𝑧) ∈ 𝐶 ↔ ((𝑦(+g‘𝑀)𝑧) ∈ 𝐵 ∧ ∀𝑥 ∈ 𝑆 ((𝑦(+g‘𝑀)𝑧)(+g‘𝑀)𝑥) = (𝑥(+g‘𝑀)(𝑦(+g‘𝑀)𝑧)))))
42	41	ad2antlr 725	. . 3 ⊢ (((𝑀 ∈ Smgrp ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → ((𝑦(+g‘𝑀)𝑧) ∈ 𝐶 ↔ ((𝑦(+g‘𝑀)𝑧) ∈ 𝐵 ∧ ∀𝑥 ∈ 𝑆 ((𝑦(+g‘𝑀)𝑧)(+g‘𝑀)𝑥) = (𝑥(+g‘𝑀)(𝑦(+g‘𝑀)𝑧)))))
43	13, 38, 42	mpbir2and 711	. 2 ⊢ (((𝑀 ∈ Smgrp ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → (𝑦(+g‘𝑀)𝑧) ∈ 𝐶)
44	43	ralrimivva 3191	1 ⊢ ((𝑀 ∈ Smgrp ∧ 𝑆 ⊆ 𝐵) → ∀𝑦 ∈ 𝐶 ∀𝑧 ∈ 𝐶 (𝑦(+g‘𝑀)𝑧) ∈ 𝐶)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098  ∀wral 3051   ⊆ wss 3940  ‘cfv 6542  (class class class)co 7415  Basecbs 17177  +_gcplusg 17230  Smgrpcsgrp 18675  Cntzccntz 19268

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7418  df-mgm 18597  df-sgrp 18676  df-cntz 19270

This theorem is referenced by:  cntzsubrng  20506