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Theorem cntzsgrpcl 19284
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 766 . . . 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 19278 . . . . . 6 (𝑍𝑆) ⊆ 𝐵
62, 5eqsstri 4014 . . . . 5 𝐶𝐵
7 simprl 770 . . . . 5 (((𝑀 ∈ Smgrp ∧ 𝑆𝐵) ∧ (𝑦𝐶𝑧𝐶)) → 𝑦𝐶)
86, 7sselid 3978 . . . 4 (((𝑀 ∈ Smgrp ∧ 𝑆𝐵) ∧ (𝑦𝐶𝑧𝐶)) → 𝑦𝐵)
9 simprr 772 . . . . 5 (((𝑀 ∈ Smgrp ∧ 𝑆𝐵) ∧ (𝑦𝐶𝑧𝐶)) → 𝑧𝐶)
106, 9sselid 3978 . . . 4 (((𝑀 ∈ Smgrp ∧ 𝑆𝐵) ∧ (𝑦𝐶𝑧𝐶)) → 𝑧𝐵)
11 eqid 2728 . . . . 5 (+g𝑀) = (+g𝑀)
123, 11sgrpcl 18685 . . . 4 ((𝑀 ∈ Smgrp ∧ 𝑦𝐵𝑧𝐵) → (𝑦(+g𝑀)𝑧) ∈ 𝐵)
131, 8, 10, 12syl3anc 1369 . . 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 3980 . . . . . . 7 (((𝑀 ∈ Smgrp ∧ 𝑆𝐵) ∧ 𝑥𝑆) → 𝑥𝐵)
1918adantlr 714 . . . . . 6 ((((𝑀 ∈ Smgrp ∧ 𝑆𝐵) ∧ (𝑦𝐶𝑧𝐶)) ∧ 𝑥𝑆) → 𝑥𝐵)
203, 11sgrpass 18684 . . . . . 6 ((𝑀 ∈ Smgrp ∧ (𝑦𝐵𝑧𝐵𝑥𝐵)) → ((𝑦(+g𝑀)𝑧)(+g𝑀)𝑥) = (𝑦(+g𝑀)(𝑧(+g𝑀)𝑥)))
2114, 15, 16, 19, 20syl13anc 1370 . . . . 5 ((((𝑀 ∈ Smgrp ∧ 𝑆𝐵) ∧ (𝑦𝐶𝑧𝐶)) ∧ 𝑥𝑆) → ((𝑦(+g𝑀)𝑧)(+g𝑀)𝑥) = (𝑦(+g𝑀)(𝑧(+g𝑀)𝑥)))
222eleq2i 2821 . . . . . . . . 9 (𝑧𝐶𝑧 ∈ (𝑍𝑆))
2311, 4cntzi 19279 . . . . . . . . 9 ((𝑧 ∈ (𝑍𝑆) ∧ 𝑥𝑆) → (𝑧(+g𝑀)𝑥) = (𝑥(+g𝑀)𝑧))
2422, 23sylanb 580 . . . . . . . 8 ((𝑧𝐶𝑥𝑆) → (𝑧(+g𝑀)𝑥) = (𝑥(+g𝑀)𝑧))
259, 24sylan 579 . . . . . . 7 ((((𝑀 ∈ Smgrp ∧ 𝑆𝐵) ∧ (𝑦𝐶𝑧𝐶)) ∧ 𝑥𝑆) → (𝑧(+g𝑀)𝑥) = (𝑥(+g𝑀)𝑧))
2625oveq2d 7436 . . . . . 6 ((((𝑀 ∈ Smgrp ∧ 𝑆𝐵) ∧ (𝑦𝐶𝑧𝐶)) ∧ 𝑥𝑆) → (𝑦(+g𝑀)(𝑧(+g𝑀)𝑥)) = (𝑦(+g𝑀)(𝑥(+g𝑀)𝑧)))
273, 11sgrpass 18684 . . . . . . 7 ((𝑀 ∈ Smgrp ∧ (𝑦𝐵𝑥𝐵𝑧𝐵)) → ((𝑦(+g𝑀)𝑥)(+g𝑀)𝑧) = (𝑦(+g𝑀)(𝑥(+g𝑀)𝑧)))
2814, 15, 19, 16, 27syl13anc 1370 . . . . . 6 ((((𝑀 ∈ Smgrp ∧ 𝑆𝐵) ∧ (𝑦𝐶𝑧𝐶)) ∧ 𝑥𝑆) → ((𝑦(+g𝑀)𝑥)(+g𝑀)𝑧) = (𝑦(+g𝑀)(𝑥(+g𝑀)𝑧)))
292eleq2i 2821 . . . . . . . . 9 (𝑦𝐶𝑦 ∈ (𝑍𝑆))
3011, 4cntzi 19279 . . . . . . . . 9 ((𝑦 ∈ (𝑍𝑆) ∧ 𝑥𝑆) → (𝑦(+g𝑀)𝑥) = (𝑥(+g𝑀)𝑦))
3129, 30sylanb 580 . . . . . . . 8 ((𝑦𝐶𝑥𝑆) → (𝑦(+g𝑀)𝑥) = (𝑥(+g𝑀)𝑦))
327, 31sylan 579 . . . . . . 7 ((((𝑀 ∈ Smgrp ∧ 𝑆𝐵) ∧ (𝑦𝐶𝑧𝐶)) ∧ 𝑥𝑆) → (𝑦(+g𝑀)𝑥) = (𝑥(+g𝑀)𝑦))
3332oveq1d 7435 . . . . . 6 ((((𝑀 ∈ Smgrp ∧ 𝑆𝐵) ∧ (𝑦𝐶𝑧𝐶)) ∧ 𝑥𝑆) → ((𝑦(+g𝑀)𝑥)(+g𝑀)𝑧) = ((𝑥(+g𝑀)𝑦)(+g𝑀)𝑧))
3426, 28, 333eqtr2d 2774 . . . . 5 ((((𝑀 ∈ Smgrp ∧ 𝑆𝐵) ∧ (𝑦𝐶𝑧𝐶)) ∧ 𝑥𝑆) → (𝑦(+g𝑀)(𝑧(+g𝑀)𝑥)) = ((𝑥(+g𝑀)𝑦)(+g𝑀)𝑧))
353, 11sgrpass 18684 . . . . . 6 ((𝑀 ∈ Smgrp ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ((𝑥(+g𝑀)𝑦)(+g𝑀)𝑧) = (𝑥(+g𝑀)(𝑦(+g𝑀)𝑧)))
3614, 19, 15, 16, 35syl13anc 1370 . . . . 5 ((((𝑀 ∈ Smgrp ∧ 𝑆𝐵) ∧ (𝑦𝐶𝑧𝐶)) ∧ 𝑥𝑆) → ((𝑥(+g𝑀)𝑦)(+g𝑀)𝑧) = (𝑥(+g𝑀)(𝑦(+g𝑀)𝑧)))
3721, 34, 363eqtrd 2772 . . . 4 ((((𝑀 ∈ Smgrp ∧ 𝑆𝐵) ∧ (𝑦𝐶𝑧𝐶)) ∧ 𝑥𝑆) → ((𝑦(+g𝑀)𝑧)(+g𝑀)𝑥) = (𝑥(+g𝑀)(𝑦(+g𝑀)𝑧)))
3837ralrimiva 3143 . . 3 (((𝑀 ∈ Smgrp ∧ 𝑆𝐵) ∧ (𝑦𝐶𝑧𝐶)) → ∀𝑥𝑆 ((𝑦(+g𝑀)𝑧)(+g𝑀)𝑥) = (𝑥(+g𝑀)(𝑦(+g𝑀)𝑧)))
392eleq2i 2821 . . . . 5 ((𝑦(+g𝑀)𝑧) ∈ 𝐶 ↔ (𝑦(+g𝑀)𝑧) ∈ (𝑍𝑆))
403, 11, 4elcntz 19272 . . . . 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 726 . . 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 712 . 2 (((𝑀 ∈ Smgrp ∧ 𝑆𝐵) ∧ (𝑦𝐶𝑧𝐶)) → (𝑦(+g𝑀)𝑧) ∈ 𝐶)
4443ralrimivva 3197 1 ((𝑀 ∈ Smgrp ∧ 𝑆𝐵) → ∀𝑦𝐶𝑧𝐶 (𝑦(+g𝑀)𝑧) ∈ 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1534  wcel 2099  wral 3058  wss 3947  cfv 6548  (class class class)co 7420  Basecbs 17179  +gcplusg 17232  Smgrpcsgrp 18677  Cntzccntz 19265
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-ov 7423  df-mgm 18599  df-sgrp 18678  df-cntz 19267
This theorem is referenced by:  cntzsubrng  20503
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