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Theorem cntzsgrpcl 19352
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 = (𝑍𝑆)
Assertion
Ref Expression
cntzsgrpcl ((𝑀 ∈ Smgrp ∧ 𝑆𝐵) → ∀𝑦𝐶𝑧𝐶 (𝑦(+g𝑀)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.
StepHypRef Expression
1 simpll 767 . . . 4 (((𝑀 ∈ Smgrp ∧ 𝑆𝐵) ∧ (𝑦𝐶𝑧𝐶)) → 𝑀 ∈ Smgrp)
2 cntzsgrpcl.c . . . . . 6 𝐶 = (𝑍𝑆)
3 cntzsgrpcl.b . . . . . . 7 𝐵 = (Base‘𝑀)
4 cntzsgrpcl.z . . . . . . 7 𝑍 = (Cntz‘𝑀)
53, 4cntzssv 19346 . . . . . 6 (𝑍𝑆) ⊆ 𝐵
62, 5eqsstri 4030 . . . . 5 𝐶𝐵
7 simprl 771 . . . . 5 (((𝑀 ∈ Smgrp ∧ 𝑆𝐵) ∧ (𝑦𝐶𝑧𝐶)) → 𝑦𝐶)
86, 7sselid 3981 . . . 4 (((𝑀 ∈ Smgrp ∧ 𝑆𝐵) ∧ (𝑦𝐶𝑧𝐶)) → 𝑦𝐵)
9 simprr 773 . . . . 5 (((𝑀 ∈ Smgrp ∧ 𝑆𝐵) ∧ (𝑦𝐶𝑧𝐶)) → 𝑧𝐶)
106, 9sselid 3981 . . . 4 (((𝑀 ∈ Smgrp ∧ 𝑆𝐵) ∧ (𝑦𝐶𝑧𝐶)) → 𝑧𝐵)
11 eqid 2737 . . . . 5 (+g𝑀) = (+g𝑀)
123, 11sgrpcl 18739 . . . 4 ((𝑀 ∈ Smgrp ∧ 𝑦𝐵𝑧𝐵) → (𝑦(+g𝑀)𝑧) ∈ 𝐵)
131, 8, 10, 12syl3anc 1373 . . 3 (((𝑀 ∈ Smgrp ∧ 𝑆𝐵) ∧ (𝑦𝐶𝑧𝐶)) → (𝑦(+g𝑀)𝑧) ∈ 𝐵)
141adantr 480 . . . . . 6 ((((𝑀 ∈ Smgrp ∧ 𝑆𝐵) ∧ (𝑦𝐶𝑧𝐶)) ∧ 𝑥𝑆) → 𝑀 ∈ Smgrp)
158adantr 480 . . . . . 6 ((((𝑀 ∈ Smgrp ∧ 𝑆𝐵) ∧ (𝑦𝐶𝑧𝐶)) ∧ 𝑥𝑆) → 𝑦𝐵)
1610adantr 480 . . . . . 6 ((((𝑀 ∈ Smgrp ∧ 𝑆𝐵) ∧ (𝑦𝐶𝑧𝐶)) ∧ 𝑥𝑆) → 𝑧𝐵)
17 simpr 484 . . . . . . . 8 ((𝑀 ∈ Smgrp ∧ 𝑆𝐵) → 𝑆𝐵)
1817sselda 3983 . . . . . . 7 (((𝑀 ∈ Smgrp ∧ 𝑆𝐵) ∧ 𝑥𝑆) → 𝑥𝐵)
1918adantlr 715 . . . . . 6 ((((𝑀 ∈ Smgrp ∧ 𝑆𝐵) ∧ (𝑦𝐶𝑧𝐶)) ∧ 𝑥𝑆) → 𝑥𝐵)
203, 11sgrpass 18738 . . . . . 6 ((𝑀 ∈ Smgrp ∧ (𝑦𝐵𝑧𝐵𝑥𝐵)) → ((𝑦(+g𝑀)𝑧)(+g𝑀)𝑥) = (𝑦(+g𝑀)(𝑧(+g𝑀)𝑥)))
2114, 15, 16, 19, 20syl13anc 1374 . . . . 5 ((((𝑀 ∈ Smgrp ∧ 𝑆𝐵) ∧ (𝑦𝐶𝑧𝐶)) ∧ 𝑥𝑆) → ((𝑦(+g𝑀)𝑧)(+g𝑀)𝑥) = (𝑦(+g𝑀)(𝑧(+g𝑀)𝑥)))
222eleq2i 2833 . . . . . . . . 9 (𝑧𝐶𝑧 ∈ (𝑍𝑆))
2311, 4cntzi 19347 . . . . . . . . 9 ((𝑧 ∈ (𝑍𝑆) ∧ 𝑥𝑆) → (𝑧(+g𝑀)𝑥) = (𝑥(+g𝑀)𝑧))
2422, 23sylanb 581 . . . . . . . 8 ((𝑧𝐶𝑥𝑆) → (𝑧(+g𝑀)𝑥) = (𝑥(+g𝑀)𝑧))
259, 24sylan 580 . . . . . . 7 ((((𝑀 ∈ Smgrp ∧ 𝑆𝐵) ∧ (𝑦𝐶𝑧𝐶)) ∧ 𝑥𝑆) → (𝑧(+g𝑀)𝑥) = (𝑥(+g𝑀)𝑧))
2625oveq2d 7447 . . . . . 6 ((((𝑀 ∈ Smgrp ∧ 𝑆𝐵) ∧ (𝑦𝐶𝑧𝐶)) ∧ 𝑥𝑆) → (𝑦(+g𝑀)(𝑧(+g𝑀)𝑥)) = (𝑦(+g𝑀)(𝑥(+g𝑀)𝑧)))
273, 11sgrpass 18738 . . . . . . 7 ((𝑀 ∈ Smgrp ∧ (𝑦𝐵𝑥𝐵𝑧𝐵)) → ((𝑦(+g𝑀)𝑥)(+g𝑀)𝑧) = (𝑦(+g𝑀)(𝑥(+g𝑀)𝑧)))
2814, 15, 19, 16, 27syl13anc 1374 . . . . . 6 ((((𝑀 ∈ Smgrp ∧ 𝑆𝐵) ∧ (𝑦𝐶𝑧𝐶)) ∧ 𝑥𝑆) → ((𝑦(+g𝑀)𝑥)(+g𝑀)𝑧) = (𝑦(+g𝑀)(𝑥(+g𝑀)𝑧)))
292eleq2i 2833 . . . . . . . . 9 (𝑦𝐶𝑦 ∈ (𝑍𝑆))
3011, 4cntzi 19347 . . . . . . . . 9 ((𝑦 ∈ (𝑍𝑆) ∧ 𝑥𝑆) → (𝑦(+g𝑀)𝑥) = (𝑥(+g𝑀)𝑦))
3129, 30sylanb 581 . . . . . . . 8 ((𝑦𝐶𝑥𝑆) → (𝑦(+g𝑀)𝑥) = (𝑥(+g𝑀)𝑦))
327, 31sylan 580 . . . . . . 7 ((((𝑀 ∈ Smgrp ∧ 𝑆𝐵) ∧ (𝑦𝐶𝑧𝐶)) ∧ 𝑥𝑆) → (𝑦(+g𝑀)𝑥) = (𝑥(+g𝑀)𝑦))
3332oveq1d 7446 . . . . . 6 ((((𝑀 ∈ Smgrp ∧ 𝑆𝐵) ∧ (𝑦𝐶𝑧𝐶)) ∧ 𝑥𝑆) → ((𝑦(+g𝑀)𝑥)(+g𝑀)𝑧) = ((𝑥(+g𝑀)𝑦)(+g𝑀)𝑧))
3426, 28, 333eqtr2d 2783 . . . . 5 ((((𝑀 ∈ Smgrp ∧ 𝑆𝐵) ∧ (𝑦𝐶𝑧𝐶)) ∧ 𝑥𝑆) → (𝑦(+g𝑀)(𝑧(+g𝑀)𝑥)) = ((𝑥(+g𝑀)𝑦)(+g𝑀)𝑧))
353, 11sgrpass 18738 . . . . . 6 ((𝑀 ∈ Smgrp ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ((𝑥(+g𝑀)𝑦)(+g𝑀)𝑧) = (𝑥(+g𝑀)(𝑦(+g𝑀)𝑧)))
3614, 19, 15, 16, 35syl13anc 1374 . . . . 5 ((((𝑀 ∈ Smgrp ∧ 𝑆𝐵) ∧ (𝑦𝐶𝑧𝐶)) ∧ 𝑥𝑆) → ((𝑥(+g𝑀)𝑦)(+g𝑀)𝑧) = (𝑥(+g𝑀)(𝑦(+g𝑀)𝑧)))
3721, 34, 363eqtrd 2781 . . . 4 ((((𝑀 ∈ Smgrp ∧ 𝑆𝐵) ∧ (𝑦𝐶𝑧𝐶)) ∧ 𝑥𝑆) → ((𝑦(+g𝑀)𝑧)(+g𝑀)𝑥) = (𝑥(+g𝑀)(𝑦(+g𝑀)𝑧)))
3837ralrimiva 3146 . . 3 (((𝑀 ∈ Smgrp ∧ 𝑆𝐵) ∧ (𝑦𝐶𝑧𝐶)) → ∀𝑥𝑆 ((𝑦(+g𝑀)𝑧)(+g𝑀)𝑥) = (𝑥(+g𝑀)(𝑦(+g𝑀)𝑧)))
392eleq2i 2833 . . . . 5 ((𝑦(+g𝑀)𝑧) ∈ 𝐶 ↔ (𝑦(+g𝑀)𝑧) ∈ (𝑍𝑆))
403, 11, 4elcntz 19340 . . . . 5 (𝑆𝐵 → ((𝑦(+g𝑀)𝑧) ∈ (𝑍𝑆) ↔ ((𝑦(+g𝑀)𝑧) ∈ 𝐵 ∧ ∀𝑥𝑆 ((𝑦(+g𝑀)𝑧)(+g𝑀)𝑥) = (𝑥(+g𝑀)(𝑦(+g𝑀)𝑧)))))
4139, 40bitrid 283 . . . 4 (𝑆𝐵 → ((𝑦(+g𝑀)𝑧) ∈ 𝐶 ↔ ((𝑦(+g𝑀)𝑧) ∈ 𝐵 ∧ ∀𝑥𝑆 ((𝑦(+g𝑀)𝑧)(+g𝑀)𝑥) = (𝑥(+g𝑀)(𝑦(+g𝑀)𝑧)))))
4241ad2antlr 727 . . 3 (((𝑀 ∈ Smgrp ∧ 𝑆𝐵) ∧ (𝑦𝐶𝑧𝐶)) → ((𝑦(+g𝑀)𝑧) ∈ 𝐶 ↔ ((𝑦(+g𝑀)𝑧) ∈ 𝐵 ∧ ∀𝑥𝑆 ((𝑦(+g𝑀)𝑧)(+g𝑀)𝑥) = (𝑥(+g𝑀)(𝑦(+g𝑀)𝑧)))))
4313, 38, 42mpbir2and 713 . 2 (((𝑀 ∈ Smgrp ∧ 𝑆𝐵) ∧ (𝑦𝐶𝑧𝐶)) → (𝑦(+g𝑀)𝑧) ∈ 𝐶)
4443ralrimivva 3202 1 ((𝑀 ∈ Smgrp ∧ 𝑆𝐵) → ∀𝑦𝐶𝑧𝐶 (𝑦(+g𝑀)𝑧) ∈ 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wral 3061  wss 3951  cfv 6561  (class class class)co 7431  Basecbs 17247  +gcplusg 17297  Smgrpcsgrp 18731  Cntzccntz 19333
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-mgm 18653  df-sgrp 18732  df-cntz 19335
This theorem is referenced by:  cntzsubrng  20567
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