Theorem cntzsgrpcl 19575

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 779	. . . 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 19569	. . . . . 6 ⊢ (𝑍‘𝑆) ⊆ 𝐵
6	2, 5	eqsstri 3959	. . . . 5 ⊢ 𝐶 ⊆ 𝐵
7		simprl 783	. . . . 5 ⊢ (((𝑀 ∈ Smgrp ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → 𝑦 ∈ 𝐶)
8	6, 7	sselid 3911	. . . 4 ⊢ (((𝑀 ∈ Smgrp ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → 𝑦 ∈ 𝐵)
9		simprr 785	. . . . 5 ⊢ (((𝑀 ∈ Smgrp ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → 𝑧 ∈ 𝐶)
10	6, 9	sselid 3911	. . . 4 ⊢ (((𝑀 ∈ Smgrp ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → 𝑧 ∈ 𝐵)
11		eqid 2763	. . . . 5 ⊢ (+g‘𝑀) = (+g‘𝑀)
12	3, 11	sgrpcl 18941	. . . 4 ⊢ ((𝑀 ∈ Smgrp ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → (𝑦(+g‘𝑀)𝑧) ∈ 𝐵)
13	1, 8, 10, 12	syl3anc 1399	. . 3 ⊢ (((𝑀 ∈ Smgrp ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → (𝑦(+g‘𝑀)𝑧) ∈ 𝐵)
14	1	adantr 486	. . . . . 6 ⊢ ((((𝑀 ∈ Smgrp ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) ∧ 𝑥 ∈ 𝑆) → 𝑀 ∈ Smgrp)
15	8	adantr 486	. . . . . 6 ⊢ ((((𝑀 ∈ Smgrp ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) ∧ 𝑥 ∈ 𝑆) → 𝑦 ∈ 𝐵)
16	10	adantr 486	. . . . . 6 ⊢ ((((𝑀 ∈ Smgrp ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) ∧ 𝑥 ∈ 𝑆) → 𝑧 ∈ 𝐵)
17		simpr 490	. . . . . . . 8 ⊢ ((𝑀 ∈ Smgrp ∧ 𝑆 ⊆ 𝐵) → 𝑆 ⊆ 𝐵)
18	17	sselda 3913	. . . . . . 7 ⊢ (((𝑀 ∈ Smgrp ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝐵)
19	18	adantlr 728	. . . . . 6 ⊢ ((((𝑀 ∈ Smgrp ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝐵)
20	3, 11	sgrpass 18940	. . . . . 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 1400	. . . . 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 2856	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝐶 ↔ 𝑧 ∈ (𝑍‘𝑆))
23	11, 4	cntzi 19570	. . . . . . . . 9 ⊢ ((𝑧 ∈ (𝑍‘𝑆) ∧ 𝑥 ∈ 𝑆) → (𝑧(+g‘𝑀)𝑥) = (𝑥(+g‘𝑀)𝑧))
24	22, 23	sylanb 593	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝐶 ∧ 𝑥 ∈ 𝑆) → (𝑧(+g‘𝑀)𝑥) = (𝑥(+g‘𝑀)𝑧))
25	9, 24	sylan 592	. . . . . . 7 ⊢ ((((𝑀 ∈ Smgrp ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) ∧ 𝑥 ∈ 𝑆) → (𝑧(+g‘𝑀)𝑥) = (𝑥(+g‘𝑀)𝑧))
26	25	oveq2d 7456	. . . . . 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 18940	. . . . . . 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 1400	. . . . . 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 2856	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐶 ↔ 𝑦 ∈ (𝑍‘𝑆))
30	11, 4	cntzi 19570	. . . . . . . . 9 ⊢ ((𝑦 ∈ (𝑍‘𝑆) ∧ 𝑥 ∈ 𝑆) → (𝑦(+g‘𝑀)𝑥) = (𝑥(+g‘𝑀)𝑦))
31	29, 30	sylanb 593	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝐶 ∧ 𝑥 ∈ 𝑆) → (𝑦(+g‘𝑀)𝑥) = (𝑥(+g‘𝑀)𝑦))
32	7, 31	sylan 592	. . . . . . 7 ⊢ ((((𝑀 ∈ Smgrp ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) ∧ 𝑥 ∈ 𝑆) → (𝑦(+g‘𝑀)𝑥) = (𝑥(+g‘𝑀)𝑦))
33	32	oveq1d 7455	. . . . . 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 2804	. . . . 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 18940	. . . . . 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 1400	. . . . 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 2802	. . . 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 3158	. . 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 2856	. . . . 5 ⊢ ((𝑦(+g‘𝑀)𝑧) ∈ 𝐶 ↔ (𝑦(+g‘𝑀)𝑧) ∈ (𝑍‘𝑆))
40	3, 11, 4	elcntz 19563	. . . . 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 286	. . . 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 740	. . 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 726	. 2 ⊢ (((𝑀 ∈ Smgrp ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → (𝑦(+g‘𝑀)𝑧) ∈ 𝐶)
44	43	ralrimivva 3209	1 ⊢ ((𝑀 ∈ Smgrp ∧ 𝑆 ⊆ 𝐵) → ∀𝑦 ∈ 𝐶 ∀𝑧 ∈ 𝐶 (𝑦(+g‘𝑀)𝑧) ∈ 𝐶)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∀wral 3080   ⊆ wss 3900  ‘cfv 6555  (class class class)co 7440  Basecbs 17413  +_gcplusg 17454  Smgrpcsgrp 18933  Cntzccntz 19556

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

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-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 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-ov 7443  df-mgm 18842  df-sgrp 18934  df-cntz 19558

This theorem is used by:  cntzsubrng  20846