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Theorem cntzsubm 17974
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 17967 . . 3 (𝑍𝑆) ⊆ 𝐵
43a1i 11 . 2 ((𝑀 ∈ Mnd ∧ 𝑆𝐵) → (𝑍𝑆) ⊆ 𝐵)
5 eqid 2770 . . . . 5 (0g𝑀) = (0g𝑀)
61, 5mndidcl 17515 . . . 4 (𝑀 ∈ Mnd → (0g𝑀) ∈ 𝐵)
76adantr 466 . . 3 ((𝑀 ∈ Mnd ∧ 𝑆𝐵) → (0g𝑀) ∈ 𝐵)
8 simpll 742 . . . . . 6 (((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ 𝑥𝑆) → 𝑀 ∈ Mnd)
9 simpr 471 . . . . . . 7 ((𝑀 ∈ Mnd ∧ 𝑆𝐵) → 𝑆𝐵)
109sselda 3750 . . . . . 6 (((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ 𝑥𝑆) → 𝑥𝐵)
11 eqid 2770 . . . . . . 7 (+g𝑀) = (+g𝑀)
121, 11, 5mndlid 17518 . . . . . 6 ((𝑀 ∈ Mnd ∧ 𝑥𝐵) → ((0g𝑀)(+g𝑀)𝑥) = 𝑥)
138, 10, 12syl2anc 565 . . . . 5 (((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ 𝑥𝑆) → ((0g𝑀)(+g𝑀)𝑥) = 𝑥)
141, 11, 5mndrid 17519 . . . . . 6 ((𝑀 ∈ Mnd ∧ 𝑥𝐵) → (𝑥(+g𝑀)(0g𝑀)) = 𝑥)
158, 10, 14syl2anc 565 . . . . 5 (((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ 𝑥𝑆) → (𝑥(+g𝑀)(0g𝑀)) = 𝑥)
1613, 15eqtr4d 2807 . . . 4 (((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ 𝑥𝑆) → ((0g𝑀)(+g𝑀)𝑥) = (𝑥(+g𝑀)(0g𝑀)))
1716ralrimiva 3114 . . 3 ((𝑀 ∈ Mnd ∧ 𝑆𝐵) → ∀𝑥𝑆 ((0g𝑀)(+g𝑀)𝑥) = (𝑥(+g𝑀)(0g𝑀)))
181, 11, 2elcntz 17961 . . . 4 (𝑆𝐵 → ((0g𝑀) ∈ (𝑍𝑆) ↔ ((0g𝑀) ∈ 𝐵 ∧ ∀𝑥𝑆 ((0g𝑀)(+g𝑀)𝑥) = (𝑥(+g𝑀)(0g𝑀)))))
1918adantl 467 . . 3 ((𝑀 ∈ Mnd ∧ 𝑆𝐵) → ((0g𝑀) ∈ (𝑍𝑆) ↔ ((0g𝑀) ∈ 𝐵 ∧ ∀𝑥𝑆 ((0g𝑀)(+g𝑀)𝑥) = (𝑥(+g𝑀)(0g𝑀)))))
207, 17, 19mpbir2and 684 . 2 ((𝑀 ∈ Mnd ∧ 𝑆𝐵) → (0g𝑀) ∈ (𝑍𝑆))
21 simpll 742 . . . . 5 (((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ (𝑦 ∈ (𝑍𝑆) ∧ 𝑧 ∈ (𝑍𝑆))) → 𝑀 ∈ Mnd)
22 simprl 746 . . . . . 6 (((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ (𝑦 ∈ (𝑍𝑆) ∧ 𝑧 ∈ (𝑍𝑆))) → 𝑦 ∈ (𝑍𝑆))
233, 22sseldi 3748 . . . . 5 (((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ (𝑦 ∈ (𝑍𝑆) ∧ 𝑧 ∈ (𝑍𝑆))) → 𝑦𝐵)
24 simprr 748 . . . . . 6 (((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ (𝑦 ∈ (𝑍𝑆) ∧ 𝑧 ∈ (𝑍𝑆))) → 𝑧 ∈ (𝑍𝑆))
253, 24sseldi 3748 . . . . 5 (((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ (𝑦 ∈ (𝑍𝑆) ∧ 𝑧 ∈ (𝑍𝑆))) → 𝑧𝐵)
261, 11mndcl 17508 . . . . 5 ((𝑀 ∈ Mnd ∧ 𝑦𝐵𝑧𝐵) → (𝑦(+g𝑀)𝑧) ∈ 𝐵)
2721, 23, 25, 26syl3anc 1475 . . . 4 (((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ (𝑦 ∈ (𝑍𝑆) ∧ 𝑧 ∈ (𝑍𝑆))) → (𝑦(+g𝑀)𝑧) ∈ 𝐵)
2821adantr 466 . . . . . . 7 ((((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ (𝑦 ∈ (𝑍𝑆) ∧ 𝑧 ∈ (𝑍𝑆))) ∧ 𝑥𝑆) → 𝑀 ∈ Mnd)
2923adantr 466 . . . . . . 7 ((((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ (𝑦 ∈ (𝑍𝑆) ∧ 𝑧 ∈ (𝑍𝑆))) ∧ 𝑥𝑆) → 𝑦𝐵)
3025adantr 466 . . . . . . 7 ((((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ (𝑦 ∈ (𝑍𝑆) ∧ 𝑧 ∈ (𝑍𝑆))) ∧ 𝑥𝑆) → 𝑧𝐵)
3110adantlr 686 . . . . . . 7 ((((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ (𝑦 ∈ (𝑍𝑆) ∧ 𝑧 ∈ (𝑍𝑆))) ∧ 𝑥𝑆) → 𝑥𝐵)
321, 11mndass 17509 . . . . . . 7 ((𝑀 ∈ Mnd ∧ (𝑦𝐵𝑧𝐵𝑥𝐵)) → ((𝑦(+g𝑀)𝑧)(+g𝑀)𝑥) = (𝑦(+g𝑀)(𝑧(+g𝑀)𝑥)))
3328, 29, 30, 31, 32syl13anc 1477 . . . . . 6 ((((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ (𝑦 ∈ (𝑍𝑆) ∧ 𝑧 ∈ (𝑍𝑆))) ∧ 𝑥𝑆) → ((𝑦(+g𝑀)𝑧)(+g𝑀)𝑥) = (𝑦(+g𝑀)(𝑧(+g𝑀)𝑥)))
3411, 2cntzi 17968 . . . . . . . . 9 ((𝑧 ∈ (𝑍𝑆) ∧ 𝑥𝑆) → (𝑧(+g𝑀)𝑥) = (𝑥(+g𝑀)𝑧))
3524, 34sylan 561 . . . . . . . 8 ((((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ (𝑦 ∈ (𝑍𝑆) ∧ 𝑧 ∈ (𝑍𝑆))) ∧ 𝑥𝑆) → (𝑧(+g𝑀)𝑥) = (𝑥(+g𝑀)𝑧))
3635oveq2d 6808 . . . . . . 7 ((((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ (𝑦 ∈ (𝑍𝑆) ∧ 𝑧 ∈ (𝑍𝑆))) ∧ 𝑥𝑆) → (𝑦(+g𝑀)(𝑧(+g𝑀)𝑥)) = (𝑦(+g𝑀)(𝑥(+g𝑀)𝑧)))
371, 11mndass 17509 . . . . . . . 8 ((𝑀 ∈ Mnd ∧ (𝑦𝐵𝑥𝐵𝑧𝐵)) → ((𝑦(+g𝑀)𝑥)(+g𝑀)𝑧) = (𝑦(+g𝑀)(𝑥(+g𝑀)𝑧)))
3828, 29, 31, 30, 37syl13anc 1477 . . . . . . 7 ((((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ (𝑦 ∈ (𝑍𝑆) ∧ 𝑧 ∈ (𝑍𝑆))) ∧ 𝑥𝑆) → ((𝑦(+g𝑀)𝑥)(+g𝑀)𝑧) = (𝑦(+g𝑀)(𝑥(+g𝑀)𝑧)))
3911, 2cntzi 17968 . . . . . . . . 9 ((𝑦 ∈ (𝑍𝑆) ∧ 𝑥𝑆) → (𝑦(+g𝑀)𝑥) = (𝑥(+g𝑀)𝑦))
4022, 39sylan 561 . . . . . . . 8 ((((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ (𝑦 ∈ (𝑍𝑆) ∧ 𝑧 ∈ (𝑍𝑆))) ∧ 𝑥𝑆) → (𝑦(+g𝑀)𝑥) = (𝑥(+g𝑀)𝑦))
4140oveq1d 6807 . . . . . . 7 ((((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ (𝑦 ∈ (𝑍𝑆) ∧ 𝑧 ∈ (𝑍𝑆))) ∧ 𝑥𝑆) → ((𝑦(+g𝑀)𝑥)(+g𝑀)𝑧) = ((𝑥(+g𝑀)𝑦)(+g𝑀)𝑧))
4236, 38, 413eqtr2d 2810 . . . . . 6 ((((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ (𝑦 ∈ (𝑍𝑆) ∧ 𝑧 ∈ (𝑍𝑆))) ∧ 𝑥𝑆) → (𝑦(+g𝑀)(𝑧(+g𝑀)𝑥)) = ((𝑥(+g𝑀)𝑦)(+g𝑀)𝑧))
431, 11mndass 17509 . . . . . . 7 ((𝑀 ∈ Mnd ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ((𝑥(+g𝑀)𝑦)(+g𝑀)𝑧) = (𝑥(+g𝑀)(𝑦(+g𝑀)𝑧)))
4428, 31, 29, 30, 43syl13anc 1477 . . . . . 6 ((((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ (𝑦 ∈ (𝑍𝑆) ∧ 𝑧 ∈ (𝑍𝑆))) ∧ 𝑥𝑆) → ((𝑥(+g𝑀)𝑦)(+g𝑀)𝑧) = (𝑥(+g𝑀)(𝑦(+g𝑀)𝑧)))
4533, 42, 443eqtrd 2808 . . . . 5 ((((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ (𝑦 ∈ (𝑍𝑆) ∧ 𝑧 ∈ (𝑍𝑆))) ∧ 𝑥𝑆) → ((𝑦(+g𝑀)𝑧)(+g𝑀)𝑥) = (𝑥(+g𝑀)(𝑦(+g𝑀)𝑧)))
4645ralrimiva 3114 . . . 4 (((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ (𝑦 ∈ (𝑍𝑆) ∧ 𝑧 ∈ (𝑍𝑆))) → ∀𝑥𝑆 ((𝑦(+g𝑀)𝑧)(+g𝑀)𝑥) = (𝑥(+g𝑀)(𝑦(+g𝑀)𝑧)))
471, 11, 2elcntz 17961 . . . . 5 (𝑆𝐵 → ((𝑦(+g𝑀)𝑧) ∈ (𝑍𝑆) ↔ ((𝑦(+g𝑀)𝑧) ∈ 𝐵 ∧ ∀𝑥𝑆 ((𝑦(+g𝑀)𝑧)(+g𝑀)𝑥) = (𝑥(+g𝑀)(𝑦(+g𝑀)𝑧)))))
4847ad2antlr 698 . . . 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 684 . . 3 (((𝑀 ∈ Mnd ∧ 𝑆𝐵) ∧ (𝑦 ∈ (𝑍𝑆) ∧ 𝑧 ∈ (𝑍𝑆))) → (𝑦(+g𝑀)𝑧) ∈ (𝑍𝑆))
5049ralrimivva 3119 . 2 ((𝑀 ∈ Mnd ∧ 𝑆𝐵) → ∀𝑦 ∈ (𝑍𝑆)∀𝑧 ∈ (𝑍𝑆)(𝑦(+g𝑀)𝑧) ∈ (𝑍𝑆))
511, 5, 11issubm 17554 . . 3 (𝑀 ∈ Mnd → ((𝑍𝑆) ∈ (SubMnd‘𝑀) ↔ ((𝑍𝑆) ⊆ 𝐵 ∧ (0g𝑀) ∈ (𝑍𝑆) ∧ ∀𝑦 ∈ (𝑍𝑆)∀𝑧 ∈ (𝑍𝑆)(𝑦(+g𝑀)𝑧) ∈ (𝑍𝑆))))
5251adantr 466 . 2 ((𝑀 ∈ Mnd ∧ 𝑆𝐵) → ((𝑍𝑆) ∈ (SubMnd‘𝑀) ↔ ((𝑍𝑆) ⊆ 𝐵 ∧ (0g𝑀) ∈ (𝑍𝑆) ∧ ∀𝑦 ∈ (𝑍𝑆)∀𝑧 ∈ (𝑍𝑆)(𝑦(+g𝑀)𝑧) ∈ (𝑍𝑆))))
534, 20, 50, 52mpbir3and 1426 1 ((𝑀 ∈ Mnd ∧ 𝑆𝐵) → (𝑍𝑆) ∈ (SubMnd‘𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  w3a 1070   = wceq 1630  wcel 2144  wral 3060  wss 3721  cfv 6031  (class class class)co 6792  Basecbs 16063  +gcplusg 16148  0gc0g 16307  Mndcmnd 17501  SubMndcsubmnd 17541  Cntzccntz 17954
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rmo 3068  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6753  df-ov 6795  df-0g 16309  df-mgm 17449  df-sgrp 17491  df-mnd 17502  df-submnd 17543  df-cntz 17956
This theorem is referenced by:  cntzsubg  17975  cntzspan  18453  dprdfadd  18626  cntzsubr  19021
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