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Theorem cntzsubm 18404
Description: Centralizers in a monoid are submonoids. (Contributed by Stefan O'Rear, 6-Sep-2015.) (Revised by Mario Carneiro, 19-Apr-2016.)
Hypotheses
Ref Expression
cntzrec.b 𝐵 = (Base‘𝑀)
cntzrec.z 𝑍 = (Cntz‘𝑀)
Assertion
Ref Expression
cntzsubm ((𝑀 ∈ Mnd ∧ 𝑆𝐵) → (𝑍𝑆) ∈ (SubMnd‘𝑀))

Proof of Theorem cntzsubm
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cntzrec.b . . . 4 𝐵 = (Base‘𝑀)
2 cntzrec.z . . . 4 𝑍 = (Cntz‘𝑀)
31, 2cntzssv 18396 . . 3 (𝑍𝑆) ⊆ 𝐵
43a1i 11 . 2 ((𝑀 ∈ Mnd ∧ 𝑆𝐵) → (𝑍𝑆) ⊆ 𝐵)
5 eqid 2818 . . . . 5 (0g𝑀) = (0g𝑀)
61, 5mndidcl 17914 . . . 4 (𝑀 ∈ Mnd → (0g𝑀) ∈ 𝐵)
76adantr 481 . . 3 ((𝑀 ∈ Mnd ∧ 𝑆𝐵) → (0g𝑀) ∈ 𝐵)
8 simpll 763 . . . . . 6 (((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ 𝑥𝑆) → 𝑀 ∈ Mnd)
9 simpr 485 . . . . . . 7 ((𝑀 ∈ Mnd ∧ 𝑆𝐵) → 𝑆𝐵)
109sselda 3964 . . . . . 6 (((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ 𝑥𝑆) → 𝑥𝐵)
11 eqid 2818 . . . . . . 7 (+g𝑀) = (+g𝑀)
121, 11, 5mndlid 17919 . . . . . 6 ((𝑀 ∈ Mnd ∧ 𝑥𝐵) → ((0g𝑀)(+g𝑀)𝑥) = 𝑥)
138, 10, 12syl2anc 584 . . . . 5 (((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ 𝑥𝑆) → ((0g𝑀)(+g𝑀)𝑥) = 𝑥)
141, 11, 5mndrid 17920 . . . . . 6 ((𝑀 ∈ Mnd ∧ 𝑥𝐵) → (𝑥(+g𝑀)(0g𝑀)) = 𝑥)
158, 10, 14syl2anc 584 . . . . 5 (((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ 𝑥𝑆) → (𝑥(+g𝑀)(0g𝑀)) = 𝑥)
1613, 15eqtr4d 2856 . . . 4 (((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ 𝑥𝑆) → ((0g𝑀)(+g𝑀)𝑥) = (𝑥(+g𝑀)(0g𝑀)))
1716ralrimiva 3179 . . 3 ((𝑀 ∈ Mnd ∧ 𝑆𝐵) → ∀𝑥𝑆 ((0g𝑀)(+g𝑀)𝑥) = (𝑥(+g𝑀)(0g𝑀)))
181, 11, 2elcntz 18390 . . . 4 (𝑆𝐵 → ((0g𝑀) ∈ (𝑍𝑆) ↔ ((0g𝑀) ∈ 𝐵 ∧ ∀𝑥𝑆 ((0g𝑀)(+g𝑀)𝑥) = (𝑥(+g𝑀)(0g𝑀)))))
1918adantl 482 . . 3 ((𝑀 ∈ Mnd ∧ 𝑆𝐵) → ((0g𝑀) ∈ (𝑍𝑆) ↔ ((0g𝑀) ∈ 𝐵 ∧ ∀𝑥𝑆 ((0g𝑀)(+g𝑀)𝑥) = (𝑥(+g𝑀)(0g𝑀)))))
207, 17, 19mpbir2and 709 . 2 ((𝑀 ∈ Mnd ∧ 𝑆𝐵) → (0g𝑀) ∈ (𝑍𝑆))
21 simpll 763 . . . . 5 (((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ (𝑦 ∈ (𝑍𝑆) ∧ 𝑧 ∈ (𝑍𝑆))) → 𝑀 ∈ Mnd)
22 simprl 767 . . . . . 6 (((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ (𝑦 ∈ (𝑍𝑆) ∧ 𝑧 ∈ (𝑍𝑆))) → 𝑦 ∈ (𝑍𝑆))
233, 22sseldi 3962 . . . . 5 (((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ (𝑦 ∈ (𝑍𝑆) ∧ 𝑧 ∈ (𝑍𝑆))) → 𝑦𝐵)
24 simprr 769 . . . . . 6 (((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ (𝑦 ∈ (𝑍𝑆) ∧ 𝑧 ∈ (𝑍𝑆))) → 𝑧 ∈ (𝑍𝑆))
253, 24sseldi 3962 . . . . 5 (((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ (𝑦 ∈ (𝑍𝑆) ∧ 𝑧 ∈ (𝑍𝑆))) → 𝑧𝐵)
261, 11mndcl 17907 . . . . 5 ((𝑀 ∈ Mnd ∧ 𝑦𝐵𝑧𝐵) → (𝑦(+g𝑀)𝑧) ∈ 𝐵)
2721, 23, 25, 26syl3anc 1363 . . . 4 (((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ (𝑦 ∈ (𝑍𝑆) ∧ 𝑧 ∈ (𝑍𝑆))) → (𝑦(+g𝑀)𝑧) ∈ 𝐵)
2821adantr 481 . . . . . . 7 ((((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ (𝑦 ∈ (𝑍𝑆) ∧ 𝑧 ∈ (𝑍𝑆))) ∧ 𝑥𝑆) → 𝑀 ∈ Mnd)
2923adantr 481 . . . . . . 7 ((((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ (𝑦 ∈ (𝑍𝑆) ∧ 𝑧 ∈ (𝑍𝑆))) ∧ 𝑥𝑆) → 𝑦𝐵)
3025adantr 481 . . . . . . 7 ((((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ (𝑦 ∈ (𝑍𝑆) ∧ 𝑧 ∈ (𝑍𝑆))) ∧ 𝑥𝑆) → 𝑧𝐵)
3110adantlr 711 . . . . . . 7 ((((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ (𝑦 ∈ (𝑍𝑆) ∧ 𝑧 ∈ (𝑍𝑆))) ∧ 𝑥𝑆) → 𝑥𝐵)
321, 11mndass 17908 . . . . . . 7 ((𝑀 ∈ Mnd ∧ (𝑦𝐵𝑧𝐵𝑥𝐵)) → ((𝑦(+g𝑀)𝑧)(+g𝑀)𝑥) = (𝑦(+g𝑀)(𝑧(+g𝑀)𝑥)))
3328, 29, 30, 31, 32syl13anc 1364 . . . . . 6 ((((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ (𝑦 ∈ (𝑍𝑆) ∧ 𝑧 ∈ (𝑍𝑆))) ∧ 𝑥𝑆) → ((𝑦(+g𝑀)𝑧)(+g𝑀)𝑥) = (𝑦(+g𝑀)(𝑧(+g𝑀)𝑥)))
3411, 2cntzi 18397 . . . . . . . . 9 ((𝑧 ∈ (𝑍𝑆) ∧ 𝑥𝑆) → (𝑧(+g𝑀)𝑥) = (𝑥(+g𝑀)𝑧))
3524, 34sylan 580 . . . . . . . 8 ((((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ (𝑦 ∈ (𝑍𝑆) ∧ 𝑧 ∈ (𝑍𝑆))) ∧ 𝑥𝑆) → (𝑧(+g𝑀)𝑥) = (𝑥(+g𝑀)𝑧))
3635oveq2d 7161 . . . . . . 7 ((((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ (𝑦 ∈ (𝑍𝑆) ∧ 𝑧 ∈ (𝑍𝑆))) ∧ 𝑥𝑆) → (𝑦(+g𝑀)(𝑧(+g𝑀)𝑥)) = (𝑦(+g𝑀)(𝑥(+g𝑀)𝑧)))
371, 11mndass 17908 . . . . . . . 8 ((𝑀 ∈ Mnd ∧ (𝑦𝐵𝑥𝐵𝑧𝐵)) → ((𝑦(+g𝑀)𝑥)(+g𝑀)𝑧) = (𝑦(+g𝑀)(𝑥(+g𝑀)𝑧)))
3828, 29, 31, 30, 37syl13anc 1364 . . . . . . 7 ((((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ (𝑦 ∈ (𝑍𝑆) ∧ 𝑧 ∈ (𝑍𝑆))) ∧ 𝑥𝑆) → ((𝑦(+g𝑀)𝑥)(+g𝑀)𝑧) = (𝑦(+g𝑀)(𝑥(+g𝑀)𝑧)))
3911, 2cntzi 18397 . . . . . . . . 9 ((𝑦 ∈ (𝑍𝑆) ∧ 𝑥𝑆) → (𝑦(+g𝑀)𝑥) = (𝑥(+g𝑀)𝑦))
4022, 39sylan 580 . . . . . . . 8 ((((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ (𝑦 ∈ (𝑍𝑆) ∧ 𝑧 ∈ (𝑍𝑆))) ∧ 𝑥𝑆) → (𝑦(+g𝑀)𝑥) = (𝑥(+g𝑀)𝑦))
4140oveq1d 7160 . . . . . . 7 ((((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ (𝑦 ∈ (𝑍𝑆) ∧ 𝑧 ∈ (𝑍𝑆))) ∧ 𝑥𝑆) → ((𝑦(+g𝑀)𝑥)(+g𝑀)𝑧) = ((𝑥(+g𝑀)𝑦)(+g𝑀)𝑧))
4236, 38, 413eqtr2d 2859 . . . . . 6 ((((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ (𝑦 ∈ (𝑍𝑆) ∧ 𝑧 ∈ (𝑍𝑆))) ∧ 𝑥𝑆) → (𝑦(+g𝑀)(𝑧(+g𝑀)𝑥)) = ((𝑥(+g𝑀)𝑦)(+g𝑀)𝑧))
431, 11mndass 17908 . . . . . . 7 ((𝑀 ∈ Mnd ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ((𝑥(+g𝑀)𝑦)(+g𝑀)𝑧) = (𝑥(+g𝑀)(𝑦(+g𝑀)𝑧)))
4428, 31, 29, 30, 43syl13anc 1364 . . . . . 6 ((((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ (𝑦 ∈ (𝑍𝑆) ∧ 𝑧 ∈ (𝑍𝑆))) ∧ 𝑥𝑆) → ((𝑥(+g𝑀)𝑦)(+g𝑀)𝑧) = (𝑥(+g𝑀)(𝑦(+g𝑀)𝑧)))
4533, 42, 443eqtrd 2857 . . . . 5 ((((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ (𝑦 ∈ (𝑍𝑆) ∧ 𝑧 ∈ (𝑍𝑆))) ∧ 𝑥𝑆) → ((𝑦(+g𝑀)𝑧)(+g𝑀)𝑥) = (𝑥(+g𝑀)(𝑦(+g𝑀)𝑧)))
4645ralrimiva 3179 . . . 4 (((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ (𝑦 ∈ (𝑍𝑆) ∧ 𝑧 ∈ (𝑍𝑆))) → ∀𝑥𝑆 ((𝑦(+g𝑀)𝑧)(+g𝑀)𝑥) = (𝑥(+g𝑀)(𝑦(+g𝑀)𝑧)))
471, 11, 2elcntz 18390 . . . . 5 (𝑆𝐵 → ((𝑦(+g𝑀)𝑧) ∈ (𝑍𝑆) ↔ ((𝑦(+g𝑀)𝑧) ∈ 𝐵 ∧ ∀𝑥𝑆 ((𝑦(+g𝑀)𝑧)(+g𝑀)𝑥) = (𝑥(+g𝑀)(𝑦(+g𝑀)𝑧)))))
4847ad2antlr 723 . . . 4 (((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ (𝑦 ∈ (𝑍𝑆) ∧ 𝑧 ∈ (𝑍𝑆))) → ((𝑦(+g𝑀)𝑧) ∈ (𝑍𝑆) ↔ ((𝑦(+g𝑀)𝑧) ∈ 𝐵 ∧ ∀𝑥𝑆 ((𝑦(+g𝑀)𝑧)(+g𝑀)𝑥) = (𝑥(+g𝑀)(𝑦(+g𝑀)𝑧)))))
4927, 46, 48mpbir2and 709 . . 3 (((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ (𝑦 ∈ (𝑍𝑆) ∧ 𝑧 ∈ (𝑍𝑆))) → (𝑦(+g𝑀)𝑧) ∈ (𝑍𝑆))
5049ralrimivva 3188 . 2 ((𝑀 ∈ Mnd ∧ 𝑆𝐵) → ∀𝑦 ∈ (𝑍𝑆)∀𝑧 ∈ (𝑍𝑆)(𝑦(+g𝑀)𝑧) ∈ (𝑍𝑆))
511, 5, 11issubm 17956 . . 3 (𝑀 ∈ Mnd → ((𝑍𝑆) ∈ (SubMnd‘𝑀) ↔ ((𝑍𝑆) ⊆ 𝐵 ∧ (0g𝑀) ∈ (𝑍𝑆) ∧ ∀𝑦 ∈ (𝑍𝑆)∀𝑧 ∈ (𝑍𝑆)(𝑦(+g𝑀)𝑧) ∈ (𝑍𝑆))))
5251adantr 481 . 2 ((𝑀 ∈ Mnd ∧ 𝑆𝐵) → ((𝑍𝑆) ∈ (SubMnd‘𝑀) ↔ ((𝑍𝑆) ⊆ 𝐵 ∧ (0g𝑀) ∈ (𝑍𝑆) ∧ ∀𝑦 ∈ (𝑍𝑆)∀𝑧 ∈ (𝑍𝑆)(𝑦(+g𝑀)𝑧) ∈ (𝑍𝑆))))
534, 20, 50, 52mpbir3and 1334 1 ((𝑀 ∈ Mnd ∧ 𝑆𝐵) → (𝑍𝑆) ∈ (SubMnd‘𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  wral 3135  wss 3933  cfv 6348  (class class class)co 7145  Basecbs 16471  +gcplusg 16553  0gc0g 16701  Mndcmnd 17899  SubMndcsubmnd 17943  Cntzccntz 18383
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-0g 16703  df-mgm 17840  df-sgrp 17889  df-mnd 17900  df-submnd 17945  df-cntz 18385
This theorem is referenced by:  cntzsubg  18405  cntrcmnd  18891  cntzspan  18893  dprdfadd  19071  cntzsubr  19497
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