MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cntzsubm Structured version   Visualization version   GIF version

Theorem cntzsubm 17749
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 17742 . . 3 (𝑍𝑆) ⊆ 𝐵
43a1i 11 . 2 ((𝑀 ∈ Mnd ∧ 𝑆𝐵) → (𝑍𝑆) ⊆ 𝐵)
5 eqid 2620 . . . . 5 (0g𝑀) = (0g𝑀)
61, 5mndidcl 17289 . . . 4 (𝑀 ∈ Mnd → (0g𝑀) ∈ 𝐵)
76adantr 481 . . 3 ((𝑀 ∈ Mnd ∧ 𝑆𝐵) → (0g𝑀) ∈ 𝐵)
8 simpll 789 . . . . . 6 (((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ 𝑥𝑆) → 𝑀 ∈ Mnd)
9 simpr 477 . . . . . . 7 ((𝑀 ∈ Mnd ∧ 𝑆𝐵) → 𝑆𝐵)
109sselda 3595 . . . . . 6 (((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ 𝑥𝑆) → 𝑥𝐵)
11 eqid 2620 . . . . . . 7 (+g𝑀) = (+g𝑀)
121, 11, 5mndlid 17292 . . . . . 6 ((𝑀 ∈ Mnd ∧ 𝑥𝐵) → ((0g𝑀)(+g𝑀)𝑥) = 𝑥)
138, 10, 12syl2anc 692 . . . . 5 (((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ 𝑥𝑆) → ((0g𝑀)(+g𝑀)𝑥) = 𝑥)
141, 11, 5mndrid 17293 . . . . . 6 ((𝑀 ∈ Mnd ∧ 𝑥𝐵) → (𝑥(+g𝑀)(0g𝑀)) = 𝑥)
158, 10, 14syl2anc 692 . . . . 5 (((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ 𝑥𝑆) → (𝑥(+g𝑀)(0g𝑀)) = 𝑥)
1613, 15eqtr4d 2657 . . . 4 (((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ 𝑥𝑆) → ((0g𝑀)(+g𝑀)𝑥) = (𝑥(+g𝑀)(0g𝑀)))
1716ralrimiva 2963 . . 3 ((𝑀 ∈ Mnd ∧ 𝑆𝐵) → ∀𝑥𝑆 ((0g𝑀)(+g𝑀)𝑥) = (𝑥(+g𝑀)(0g𝑀)))
181, 11, 2elcntz 17736 . . . 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 956 . 2 ((𝑀 ∈ Mnd ∧ 𝑆𝐵) → (0g𝑀) ∈ (𝑍𝑆))
21 simpll 789 . . . . 5 (((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ (𝑦 ∈ (𝑍𝑆) ∧ 𝑧 ∈ (𝑍𝑆))) → 𝑀 ∈ Mnd)
22 simprl 793 . . . . . 6 (((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ (𝑦 ∈ (𝑍𝑆) ∧ 𝑧 ∈ (𝑍𝑆))) → 𝑦 ∈ (𝑍𝑆))
233, 22sseldi 3593 . . . . 5 (((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ (𝑦 ∈ (𝑍𝑆) ∧ 𝑧 ∈ (𝑍𝑆))) → 𝑦𝐵)
24 simprr 795 . . . . . 6 (((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ (𝑦 ∈ (𝑍𝑆) ∧ 𝑧 ∈ (𝑍𝑆))) → 𝑧 ∈ (𝑍𝑆))
253, 24sseldi 3593 . . . . 5 (((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ (𝑦 ∈ (𝑍𝑆) ∧ 𝑧 ∈ (𝑍𝑆))) → 𝑧𝐵)
261, 11mndcl 17282 . . . . 5 ((𝑀 ∈ Mnd ∧ 𝑦𝐵𝑧𝐵) → (𝑦(+g𝑀)𝑧) ∈ 𝐵)
2721, 23, 25, 26syl3anc 1324 . . . 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 750 . . . . . . 7 ((((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ (𝑦 ∈ (𝑍𝑆) ∧ 𝑧 ∈ (𝑍𝑆))) ∧ 𝑥𝑆) → 𝑥𝐵)
321, 11mndass 17283 . . . . . . 7 ((𝑀 ∈ Mnd ∧ (𝑦𝐵𝑧𝐵𝑥𝐵)) → ((𝑦(+g𝑀)𝑧)(+g𝑀)𝑥) = (𝑦(+g𝑀)(𝑧(+g𝑀)𝑥)))
3328, 29, 30, 31, 32syl13anc 1326 . . . . . 6 ((((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ (𝑦 ∈ (𝑍𝑆) ∧ 𝑧 ∈ (𝑍𝑆))) ∧ 𝑥𝑆) → ((𝑦(+g𝑀)𝑧)(+g𝑀)𝑥) = (𝑦(+g𝑀)(𝑧(+g𝑀)𝑥)))
3411, 2cntzi 17743 . . . . . . . . 9 ((𝑧 ∈ (𝑍𝑆) ∧ 𝑥𝑆) → (𝑧(+g𝑀)𝑥) = (𝑥(+g𝑀)𝑧))
3524, 34sylan 488 . . . . . . . 8 ((((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ (𝑦 ∈ (𝑍𝑆) ∧ 𝑧 ∈ (𝑍𝑆))) ∧ 𝑥𝑆) → (𝑧(+g𝑀)𝑥) = (𝑥(+g𝑀)𝑧))
3635oveq2d 6651 . . . . . . 7 ((((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ (𝑦 ∈ (𝑍𝑆) ∧ 𝑧 ∈ (𝑍𝑆))) ∧ 𝑥𝑆) → (𝑦(+g𝑀)(𝑧(+g𝑀)𝑥)) = (𝑦(+g𝑀)(𝑥(+g𝑀)𝑧)))
371, 11mndass 17283 . . . . . . . 8 ((𝑀 ∈ Mnd ∧ (𝑦𝐵𝑥𝐵𝑧𝐵)) → ((𝑦(+g𝑀)𝑥)(+g𝑀)𝑧) = (𝑦(+g𝑀)(𝑥(+g𝑀)𝑧)))
3828, 29, 31, 30, 37syl13anc 1326 . . . . . . 7 ((((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ (𝑦 ∈ (𝑍𝑆) ∧ 𝑧 ∈ (𝑍𝑆))) ∧ 𝑥𝑆) → ((𝑦(+g𝑀)𝑥)(+g𝑀)𝑧) = (𝑦(+g𝑀)(𝑥(+g𝑀)𝑧)))
3911, 2cntzi 17743 . . . . . . . . 9 ((𝑦 ∈ (𝑍𝑆) ∧ 𝑥𝑆) → (𝑦(+g𝑀)𝑥) = (𝑥(+g𝑀)𝑦))
4022, 39sylan 488 . . . . . . . 8 ((((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ (𝑦 ∈ (𝑍𝑆) ∧ 𝑧 ∈ (𝑍𝑆))) ∧ 𝑥𝑆) → (𝑦(+g𝑀)𝑥) = (𝑥(+g𝑀)𝑦))
4140oveq1d 6650 . . . . . . 7 ((((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ (𝑦 ∈ (𝑍𝑆) ∧ 𝑧 ∈ (𝑍𝑆))) ∧ 𝑥𝑆) → ((𝑦(+g𝑀)𝑥)(+g𝑀)𝑧) = ((𝑥(+g𝑀)𝑦)(+g𝑀)𝑧))
4236, 38, 413eqtr2d 2660 . . . . . 6 ((((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ (𝑦 ∈ (𝑍𝑆) ∧ 𝑧 ∈ (𝑍𝑆))) ∧ 𝑥𝑆) → (𝑦(+g𝑀)(𝑧(+g𝑀)𝑥)) = ((𝑥(+g𝑀)𝑦)(+g𝑀)𝑧))
431, 11mndass 17283 . . . . . . 7 ((𝑀 ∈ Mnd ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ((𝑥(+g𝑀)𝑦)(+g𝑀)𝑧) = (𝑥(+g𝑀)(𝑦(+g𝑀)𝑧)))
4428, 31, 29, 30, 43syl13anc 1326 . . . . . 6 ((((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ (𝑦 ∈ (𝑍𝑆) ∧ 𝑧 ∈ (𝑍𝑆))) ∧ 𝑥𝑆) → ((𝑥(+g𝑀)𝑦)(+g𝑀)𝑧) = (𝑥(+g𝑀)(𝑦(+g𝑀)𝑧)))
4533, 42, 443eqtrd 2658 . . . . 5 ((((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ (𝑦 ∈ (𝑍𝑆) ∧ 𝑧 ∈ (𝑍𝑆))) ∧ 𝑥𝑆) → ((𝑦(+g𝑀)𝑧)(+g𝑀)𝑥) = (𝑥(+g𝑀)(𝑦(+g𝑀)𝑧)))
4645ralrimiva 2963 . . . 4 (((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ (𝑦 ∈ (𝑍𝑆) ∧ 𝑧 ∈ (𝑍𝑆))) → ∀𝑥𝑆 ((𝑦(+g𝑀)𝑧)(+g𝑀)𝑥) = (𝑥(+g𝑀)(𝑦(+g𝑀)𝑧)))
471, 11, 2elcntz 17736 . . . . 5 (𝑆𝐵 → ((𝑦(+g𝑀)𝑧) ∈ (𝑍𝑆) ↔ ((𝑦(+g𝑀)𝑧) ∈ 𝐵 ∧ ∀𝑥𝑆 ((𝑦(+g𝑀)𝑧)(+g𝑀)𝑥) = (𝑥(+g𝑀)(𝑦(+g𝑀)𝑧)))))
4847ad2antlr 762 . . . 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 956 . . 3 (((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ (𝑦 ∈ (𝑍𝑆) ∧ 𝑧 ∈ (𝑍𝑆))) → (𝑦(+g𝑀)𝑧) ∈ (𝑍𝑆))
5049ralrimivva 2968 . 2 ((𝑀 ∈ Mnd ∧ 𝑆𝐵) → ∀𝑦 ∈ (𝑍𝑆)∀𝑧 ∈ (𝑍𝑆)(𝑦(+g𝑀)𝑧) ∈ (𝑍𝑆))
511, 5, 11issubm 17328 . . 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 1243 1 ((𝑀 ∈ Mnd ∧ 𝑆𝐵) → (𝑍𝑆) ∈ (SubMnd‘𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1481  wcel 1988  wral 2909  wss 3567  cfv 5876  (class class class)co 6635  Basecbs 15838  +gcplusg 15922  0gc0g 16081  Mndcmnd 17275  SubMndcsubmnd 17315  Cntzccntz 17729
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-riota 6596  df-ov 6638  df-0g 16083  df-mgm 17223  df-sgrp 17265  df-mnd 17276  df-submnd 17317  df-cntz 17731
This theorem is referenced by:  cntzsubg  17750  cntzspan  18228  dprdfadd  18400  cntzsubr  18793
  Copyright terms: Public domain W3C validator