Theorem cntzsgrpcl 19404

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 778	. . . 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 19398	. . . . . 6 ⊢ (𝑍‘𝑆) ⊆ 𝐵
6	2, 5	eqsstri 3991	. . . . 5 ⊢ 𝐶 ⊆ 𝐵
7		simprl 782	. . . . 5 ⊢ (((𝑀 ∈ Smgrp ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → 𝑦 ∈ 𝐶)
8	6, 7	sselid 3943	. . . 4 ⊢ (((𝑀 ∈ Smgrp ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → 𝑦 ∈ 𝐵)
9		simprr 784	. . . . 5 ⊢ (((𝑀 ∈ Smgrp ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → 𝑧 ∈ 𝐶)
10	6, 9	sselid 3943	. . . 4 ⊢ (((𝑀 ∈ Smgrp ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → 𝑧 ∈ 𝐵)
11		eqid 2769	. . . . 5 ⊢ (+g‘𝑀) = (+g‘𝑀)
12	3, 11	sgrpcl 18784	. . . 4 ⊢ ((𝑀 ∈ Smgrp ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → (𝑦(+g‘𝑀)𝑧) ∈ 𝐵)
13	1, 8, 10, 12	syl3anc 1396	. . 3 ⊢ (((𝑀 ∈ Smgrp ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → (𝑦(+g‘𝑀)𝑧) ∈ 𝐵)
14	1	adantr 485	. . . . . 6 ⊢ ((((𝑀 ∈ Smgrp ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) ∧ 𝑥 ∈ 𝑆) → 𝑀 ∈ Smgrp)
15	8	adantr 485	. . . . . 6 ⊢ ((((𝑀 ∈ Smgrp ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) ∧ 𝑥 ∈ 𝑆) → 𝑦 ∈ 𝐵)
16	10	adantr 485	. . . . . 6 ⊢ ((((𝑀 ∈ Smgrp ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) ∧ 𝑥 ∈ 𝑆) → 𝑧 ∈ 𝐵)
17		simpr 489	. . . . . . . 8 ⊢ ((𝑀 ∈ Smgrp ∧ 𝑆 ⊆ 𝐵) → 𝑆 ⊆ 𝐵)
18	17	sselda 3945	. . . . . . 7 ⊢ (((𝑀 ∈ Smgrp ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝐵)
19	18	adantlr 727	. . . . . 6 ⊢ ((((𝑀 ∈ Smgrp ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝐵)
20	3, 11	sgrpass 18783	. . . . . 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 1397	. . . . 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 2861	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝐶 ↔ 𝑧 ∈ (𝑍‘𝑆))
23	11, 4	cntzi 19399	. . . . . . . . 9 ⊢ ((𝑧 ∈ (𝑍‘𝑆) ∧ 𝑥 ∈ 𝑆) → (𝑧(+g‘𝑀)𝑥) = (𝑥(+g‘𝑀)𝑧))
24	22, 23	sylanb 592	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝐶 ∧ 𝑥 ∈ 𝑆) → (𝑧(+g‘𝑀)𝑥) = (𝑥(+g‘𝑀)𝑧))
25	9, 24	sylan 591	. . . . . . 7 ⊢ ((((𝑀 ∈ Smgrp ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) ∧ 𝑥 ∈ 𝑆) → (𝑧(+g‘𝑀)𝑥) = (𝑥(+g‘𝑀)𝑧))
26	25	oveq2d 7427	. . . . . 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 18783	. . . . . . 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 1397	. . . . . 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 2861	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐶 ↔ 𝑦 ∈ (𝑍‘𝑆))
30	11, 4	cntzi 19399	. . . . . . . . 9 ⊢ ((𝑦 ∈ (𝑍‘𝑆) ∧ 𝑥 ∈ 𝑆) → (𝑦(+g‘𝑀)𝑥) = (𝑥(+g‘𝑀)𝑦))
31	29, 30	sylanb 592	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝐶 ∧ 𝑥 ∈ 𝑆) → (𝑦(+g‘𝑀)𝑥) = (𝑥(+g‘𝑀)𝑦))
32	7, 31	sylan 591	. . . . . . 7 ⊢ ((((𝑀 ∈ Smgrp ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) ∧ 𝑥 ∈ 𝑆) → (𝑦(+g‘𝑀)𝑥) = (𝑥(+g‘𝑀)𝑦))
33	32	oveq1d 7426	. . . . . 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 2810	. . . . 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 18783	. . . . . 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 1397	. . . . 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 2808	. . . 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 3163	. . 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 2861	. . . . 5 ⊢ ((𝑦(+g‘𝑀)𝑧) ∈ 𝐶 ↔ (𝑦(+g‘𝑀)𝑧) ∈ (𝑍‘𝑆))
40	3, 11, 4	elcntz 19392	. . . . 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 739	. . 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 725	. 2 ⊢ (((𝑀 ∈ Smgrp ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐶)) → (𝑦(+g‘𝑀)𝑧) ∈ 𝐶)
44	43	ralrimivva 3214	1 ⊢ ((𝑀 ∈ Smgrp ∧ 𝑆 ⊆ 𝐵) → ∀𝑦 ∈ 𝐶 ∀𝑧 ∈ 𝐶 (𝑦(+g‘𝑀)𝑧) ∈ 𝐶)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ∀wral 3085   ⊆ wss 3913  ‘cfv 6537  (class class class)co 7411  Basecbs 17269  +_gcplusg 17310  Smgrpcsgrp 18776  Cntzccntz 19385

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7414  df-mgm 18698  df-sgrp 18777  df-cntz 19387

This theorem is referenced by:  cntzsubrng  20652