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Theorem cntzsubm 18469
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 18461 . . 3 (𝑍𝑆) ⊆ 𝐵
43a1i 11 . 2 ((𝑀 ∈ Mnd ∧ 𝑆𝐵) → (𝑍𝑆) ⊆ 𝐵)
5 eqid 2824 . . . . 5 (0g𝑀) = (0g𝑀)
61, 5mndidcl 17929 . . . 4 (𝑀 ∈ Mnd → (0g𝑀) ∈ 𝐵)
76adantr 484 . . 3 ((𝑀 ∈ Mnd ∧ 𝑆𝐵) → (0g𝑀) ∈ 𝐵)
8 simpll 766 . . . . . 6 (((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ 𝑥𝑆) → 𝑀 ∈ Mnd)
9 simpr 488 . . . . . . 7 ((𝑀 ∈ Mnd ∧ 𝑆𝐵) → 𝑆𝐵)
109sselda 3954 . . . . . 6 (((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ 𝑥𝑆) → 𝑥𝐵)
11 eqid 2824 . . . . . . 7 (+g𝑀) = (+g𝑀)
121, 11, 5mndlid 17934 . . . . . 6 ((𝑀 ∈ Mnd ∧ 𝑥𝐵) → ((0g𝑀)(+g𝑀)𝑥) = 𝑥)
138, 10, 12syl2anc 587 . . . . 5 (((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ 𝑥𝑆) → ((0g𝑀)(+g𝑀)𝑥) = 𝑥)
141, 11, 5mndrid 17935 . . . . . 6 ((𝑀 ∈ Mnd ∧ 𝑥𝐵) → (𝑥(+g𝑀)(0g𝑀)) = 𝑥)
158, 10, 14syl2anc 587 . . . . 5 (((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ 𝑥𝑆) → (𝑥(+g𝑀)(0g𝑀)) = 𝑥)
1613, 15eqtr4d 2862 . . . 4 (((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ 𝑥𝑆) → ((0g𝑀)(+g𝑀)𝑥) = (𝑥(+g𝑀)(0g𝑀)))
1716ralrimiva 3177 . . 3 ((𝑀 ∈ Mnd ∧ 𝑆𝐵) → ∀𝑥𝑆 ((0g𝑀)(+g𝑀)𝑥) = (𝑥(+g𝑀)(0g𝑀)))
181, 11, 2elcntz 18455 . . . 4 (𝑆𝐵 → ((0g𝑀) ∈ (𝑍𝑆) ↔ ((0g𝑀) ∈ 𝐵 ∧ ∀𝑥𝑆 ((0g𝑀)(+g𝑀)𝑥) = (𝑥(+g𝑀)(0g𝑀)))))
1918adantl 485 . . 3 ((𝑀 ∈ Mnd ∧ 𝑆𝐵) → ((0g𝑀) ∈ (𝑍𝑆) ↔ ((0g𝑀) ∈ 𝐵 ∧ ∀𝑥𝑆 ((0g𝑀)(+g𝑀)𝑥) = (𝑥(+g𝑀)(0g𝑀)))))
207, 17, 19mpbir2and 712 . 2 ((𝑀 ∈ Mnd ∧ 𝑆𝐵) → (0g𝑀) ∈ (𝑍𝑆))
21 simpll 766 . . . . 5 (((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ (𝑦 ∈ (𝑍𝑆) ∧ 𝑧 ∈ (𝑍𝑆))) → 𝑀 ∈ Mnd)
22 simprl 770 . . . . . 6 (((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ (𝑦 ∈ (𝑍𝑆) ∧ 𝑧 ∈ (𝑍𝑆))) → 𝑦 ∈ (𝑍𝑆))
233, 22sseldi 3952 . . . . 5 (((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ (𝑦 ∈ (𝑍𝑆) ∧ 𝑧 ∈ (𝑍𝑆))) → 𝑦𝐵)
24 simprr 772 . . . . . 6 (((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ (𝑦 ∈ (𝑍𝑆) ∧ 𝑧 ∈ (𝑍𝑆))) → 𝑧 ∈ (𝑍𝑆))
253, 24sseldi 3952 . . . . 5 (((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ (𝑦 ∈ (𝑍𝑆) ∧ 𝑧 ∈ (𝑍𝑆))) → 𝑧𝐵)
261, 11mndcl 17922 . . . . 5 ((𝑀 ∈ Mnd ∧ 𝑦𝐵𝑧𝐵) → (𝑦(+g𝑀)𝑧) ∈ 𝐵)
2721, 23, 25, 26syl3anc 1368 . . . 4 (((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ (𝑦 ∈ (𝑍𝑆) ∧ 𝑧 ∈ (𝑍𝑆))) → (𝑦(+g𝑀)𝑧) ∈ 𝐵)
2821adantr 484 . . . . . . 7 ((((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ (𝑦 ∈ (𝑍𝑆) ∧ 𝑧 ∈ (𝑍𝑆))) ∧ 𝑥𝑆) → 𝑀 ∈ Mnd)
2923adantr 484 . . . . . . 7 ((((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ (𝑦 ∈ (𝑍𝑆) ∧ 𝑧 ∈ (𝑍𝑆))) ∧ 𝑥𝑆) → 𝑦𝐵)
3025adantr 484 . . . . . . 7 ((((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ (𝑦 ∈ (𝑍𝑆) ∧ 𝑧 ∈ (𝑍𝑆))) ∧ 𝑥𝑆) → 𝑧𝐵)
3110adantlr 714 . . . . . . 7 ((((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ (𝑦 ∈ (𝑍𝑆) ∧ 𝑧 ∈ (𝑍𝑆))) ∧ 𝑥𝑆) → 𝑥𝐵)
321, 11mndass 17923 . . . . . . 7 ((𝑀 ∈ Mnd ∧ (𝑦𝐵𝑧𝐵𝑥𝐵)) → ((𝑦(+g𝑀)𝑧)(+g𝑀)𝑥) = (𝑦(+g𝑀)(𝑧(+g𝑀)𝑥)))
3328, 29, 30, 31, 32syl13anc 1369 . . . . . 6 ((((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ (𝑦 ∈ (𝑍𝑆) ∧ 𝑧 ∈ (𝑍𝑆))) ∧ 𝑥𝑆) → ((𝑦(+g𝑀)𝑧)(+g𝑀)𝑥) = (𝑦(+g𝑀)(𝑧(+g𝑀)𝑥)))
3411, 2cntzi 18462 . . . . . . . . 9 ((𝑧 ∈ (𝑍𝑆) ∧ 𝑥𝑆) → (𝑧(+g𝑀)𝑥) = (𝑥(+g𝑀)𝑧))
3524, 34sylan 583 . . . . . . . 8 ((((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ (𝑦 ∈ (𝑍𝑆) ∧ 𝑧 ∈ (𝑍𝑆))) ∧ 𝑥𝑆) → (𝑧(+g𝑀)𝑥) = (𝑥(+g𝑀)𝑧))
3635oveq2d 7166 . . . . . . 7 ((((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ (𝑦 ∈ (𝑍𝑆) ∧ 𝑧 ∈ (𝑍𝑆))) ∧ 𝑥𝑆) → (𝑦(+g𝑀)(𝑧(+g𝑀)𝑥)) = (𝑦(+g𝑀)(𝑥(+g𝑀)𝑧)))
371, 11mndass 17923 . . . . . . . 8 ((𝑀 ∈ Mnd ∧ (𝑦𝐵𝑥𝐵𝑧𝐵)) → ((𝑦(+g𝑀)𝑥)(+g𝑀)𝑧) = (𝑦(+g𝑀)(𝑥(+g𝑀)𝑧)))
3828, 29, 31, 30, 37syl13anc 1369 . . . . . . 7 ((((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ (𝑦 ∈ (𝑍𝑆) ∧ 𝑧 ∈ (𝑍𝑆))) ∧ 𝑥𝑆) → ((𝑦(+g𝑀)𝑥)(+g𝑀)𝑧) = (𝑦(+g𝑀)(𝑥(+g𝑀)𝑧)))
3911, 2cntzi 18462 . . . . . . . . 9 ((𝑦 ∈ (𝑍𝑆) ∧ 𝑥𝑆) → (𝑦(+g𝑀)𝑥) = (𝑥(+g𝑀)𝑦))
4022, 39sylan 583 . . . . . . . 8 ((((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ (𝑦 ∈ (𝑍𝑆) ∧ 𝑧 ∈ (𝑍𝑆))) ∧ 𝑥𝑆) → (𝑦(+g𝑀)𝑥) = (𝑥(+g𝑀)𝑦))
4140oveq1d 7165 . . . . . . 7 ((((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ (𝑦 ∈ (𝑍𝑆) ∧ 𝑧 ∈ (𝑍𝑆))) ∧ 𝑥𝑆) → ((𝑦(+g𝑀)𝑥)(+g𝑀)𝑧) = ((𝑥(+g𝑀)𝑦)(+g𝑀)𝑧))
4236, 38, 413eqtr2d 2865 . . . . . 6 ((((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ (𝑦 ∈ (𝑍𝑆) ∧ 𝑧 ∈ (𝑍𝑆))) ∧ 𝑥𝑆) → (𝑦(+g𝑀)(𝑧(+g𝑀)𝑥)) = ((𝑥(+g𝑀)𝑦)(+g𝑀)𝑧))
431, 11mndass 17923 . . . . . . 7 ((𝑀 ∈ Mnd ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ((𝑥(+g𝑀)𝑦)(+g𝑀)𝑧) = (𝑥(+g𝑀)(𝑦(+g𝑀)𝑧)))
4428, 31, 29, 30, 43syl13anc 1369 . . . . . 6 ((((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ (𝑦 ∈ (𝑍𝑆) ∧ 𝑧 ∈ (𝑍𝑆))) ∧ 𝑥𝑆) → ((𝑥(+g𝑀)𝑦)(+g𝑀)𝑧) = (𝑥(+g𝑀)(𝑦(+g𝑀)𝑧)))
4533, 42, 443eqtrd 2863 . . . . 5 ((((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ (𝑦 ∈ (𝑍𝑆) ∧ 𝑧 ∈ (𝑍𝑆))) ∧ 𝑥𝑆) → ((𝑦(+g𝑀)𝑧)(+g𝑀)𝑥) = (𝑥(+g𝑀)(𝑦(+g𝑀)𝑧)))
4645ralrimiva 3177 . . . 4 (((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ (𝑦 ∈ (𝑍𝑆) ∧ 𝑧 ∈ (𝑍𝑆))) → ∀𝑥𝑆 ((𝑦(+g𝑀)𝑧)(+g𝑀)𝑥) = (𝑥(+g𝑀)(𝑦(+g𝑀)𝑧)))
471, 11, 2elcntz 18455 . . . . 5 (𝑆𝐵 → ((𝑦(+g𝑀)𝑧) ∈ (𝑍𝑆) ↔ ((𝑦(+g𝑀)𝑧) ∈ 𝐵 ∧ ∀𝑥𝑆 ((𝑦(+g𝑀)𝑧)(+g𝑀)𝑥) = (𝑥(+g𝑀)(𝑦(+g𝑀)𝑧)))))
4847ad2antlr 726 . . . 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 712 . . 3 (((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ (𝑦 ∈ (𝑍𝑆) ∧ 𝑧 ∈ (𝑍𝑆))) → (𝑦(+g𝑀)𝑧) ∈ (𝑍𝑆))
5049ralrimivva 3186 . 2 ((𝑀 ∈ Mnd ∧ 𝑆𝐵) → ∀𝑦 ∈ (𝑍𝑆)∀𝑧 ∈ (𝑍𝑆)(𝑦(+g𝑀)𝑧) ∈ (𝑍𝑆))
511, 5, 11issubm 17971 . . 3 (𝑀 ∈ Mnd → ((𝑍𝑆) ∈ (SubMnd‘𝑀) ↔ ((𝑍𝑆) ⊆ 𝐵 ∧ (0g𝑀) ∈ (𝑍𝑆) ∧ ∀𝑦 ∈ (𝑍𝑆)∀𝑧 ∈ (𝑍𝑆)(𝑦(+g𝑀)𝑧) ∈ (𝑍𝑆))))
5251adantr 484 . 2 ((𝑀 ∈ Mnd ∧ 𝑆𝐵) → ((𝑍𝑆) ∈ (SubMnd‘𝑀) ↔ ((𝑍𝑆) ⊆ 𝐵 ∧ (0g𝑀) ∈ (𝑍𝑆) ∧ ∀𝑦 ∈ (𝑍𝑆)∀𝑧 ∈ (𝑍𝑆)(𝑦(+g𝑀)𝑧) ∈ (𝑍𝑆))))
534, 20, 50, 52mpbir3and 1339 1 ((𝑀 ∈ Mnd ∧ 𝑆𝐵) → (𝑍𝑆) ∈ (SubMnd‘𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2115  wral 3133  wss 3920  cfv 6344  (class class class)co 7150  Basecbs 16486  +gcplusg 16568  0gc0g 16716  Mndcmnd 17914  SubMndcsubmnd 17958  Cntzccntz 18448
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5177  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3483  df-sbc 3760  df-csb 3868  df-dif 3923  df-un 3925  df-in 3927  df-ss 3937  df-nul 4278  df-if 4452  df-pw 4525  df-sn 4552  df-pr 4554  df-op 4558  df-uni 4826  df-iun 4908  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-iota 6303  df-fun 6346  df-fn 6347  df-f 6348  df-f1 6349  df-fo 6350  df-f1o 6351  df-fv 6352  df-riota 7108  df-ov 7153  df-0g 16718  df-mgm 17855  df-sgrp 17904  df-mnd 17915  df-submnd 17960  df-cntz 18450
This theorem is referenced by:  cntzsubg  18470  cntrcmnd  18965  cntzspan  18967  dprdfadd  19145  cntzsubr  19571
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