Theorem cntzsubm 17591

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 ∧ 𝑆 ⊆ 𝐵) → (𝑍‘𝑆) ∈ (SubMnd‘𝑀))

Proof of Theorem cntzsubm

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cntzrec.b	. . . 4 ⊢ 𝐵 = (Base‘𝑀)
2		cntzrec.z	. . . 4 ⊢ 𝑍 = (Cntz‘𝑀)
3	1, 2	cntzssv 17584	. . 3 ⊢ (𝑍‘𝑆) ⊆ 𝐵
4	3	a1i 11	. 2 ⊢ ((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) → (𝑍‘𝑆) ⊆ 𝐵)
5		eqid 2610	. . . . 5 ⊢ (0g‘𝑀) = (0g‘𝑀)
6	1, 5	mndidcl 17131	. . . 4 ⊢ (𝑀 ∈ Mnd → (0g‘𝑀) ∈ 𝐵)
7	6	adantr 480	. . 3 ⊢ ((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) → (0g‘𝑀) ∈ 𝐵)
8		simpll 786	. . . . . 6 ⊢ (((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ 𝑆) → 𝑀 ∈ Mnd)
9		simpr 476	. . . . . . 7 ⊢ ((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) → 𝑆 ⊆ 𝐵)
10	9	sselda 3568	. . . . . 6 ⊢ (((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝐵)
11		eqid 2610	. . . . . . 7 ⊢ (+g‘𝑀) = (+g‘𝑀)
12	1, 11, 5	mndlid 17134	. . . . . 6 ⊢ ((𝑀 ∈ Mnd ∧ 𝑥 ∈ 𝐵) → ((0g‘𝑀)(+g‘𝑀)𝑥) = 𝑥)
13	8, 10, 12	syl2anc 691	. . . . 5 ⊢ (((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ 𝑆) → ((0g‘𝑀)(+g‘𝑀)𝑥) = 𝑥)
14	1, 11, 5	mndrid 17135	. . . . . 6 ⊢ ((𝑀 ∈ Mnd ∧ 𝑥 ∈ 𝐵) → (𝑥(+g‘𝑀)(0g‘𝑀)) = 𝑥)
15	8, 10, 14	syl2anc 691	. . . . 5 ⊢ (((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ 𝑆) → (𝑥(+g‘𝑀)(0g‘𝑀)) = 𝑥)
16	13, 15	eqtr4d 2647	. . . 4 ⊢ (((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ 𝑆) → ((0g‘𝑀)(+g‘𝑀)𝑥) = (𝑥(+g‘𝑀)(0g‘𝑀)))
17	16	ralrimiva 2949	. . 3 ⊢ ((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) → ∀𝑥 ∈ 𝑆 ((0g‘𝑀)(+g‘𝑀)𝑥) = (𝑥(+g‘𝑀)(0g‘𝑀)))
18	1, 11, 2	elcntz 17578	. . . 4 ⊢ (𝑆 ⊆ 𝐵 → ((0g‘𝑀) ∈ (𝑍‘𝑆) ↔ ((0g‘𝑀) ∈ 𝐵 ∧ ∀𝑥 ∈ 𝑆 ((0g‘𝑀)(+g‘𝑀)𝑥) = (𝑥(+g‘𝑀)(0g‘𝑀)))))
19	18	adantl 481	. . 3 ⊢ ((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) → ((0g‘𝑀) ∈ (𝑍‘𝑆) ↔ ((0g‘𝑀) ∈ 𝐵 ∧ ∀𝑥 ∈ 𝑆 ((0g‘𝑀)(+g‘𝑀)𝑥) = (𝑥(+g‘𝑀)(0g‘𝑀)))))
20	7, 17, 19	mpbir2and 959	. 2 ⊢ ((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) → (0g‘𝑀) ∈ (𝑍‘𝑆))
21		simpll 786	. . . . 5 ⊢ (((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ (𝑍‘𝑆) ∧ 𝑧 ∈ (𝑍‘𝑆))) → 𝑀 ∈ Mnd)
22		simprl 790	. . . . . 6 ⊢ (((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ (𝑍‘𝑆) ∧ 𝑧 ∈ (𝑍‘𝑆))) → 𝑦 ∈ (𝑍‘𝑆))
23	3, 22	sseldi 3566	. . . . 5 ⊢ (((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ (𝑍‘𝑆) ∧ 𝑧 ∈ (𝑍‘𝑆))) → 𝑦 ∈ 𝐵)
24		simprr 792	. . . . . 6 ⊢ (((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ (𝑍‘𝑆) ∧ 𝑧 ∈ (𝑍‘𝑆))) → 𝑧 ∈ (𝑍‘𝑆))
25	3, 24	sseldi 3566	. . . . 5 ⊢ (((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ (𝑍‘𝑆) ∧ 𝑧 ∈ (𝑍‘𝑆))) → 𝑧 ∈ 𝐵)
26	1, 11	mndcl 17124	. . . . 5 ⊢ ((𝑀 ∈ Mnd ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → (𝑦(+g‘𝑀)𝑧) ∈ 𝐵)
27	21, 23, 25, 26	syl3anc 1318	. . . 4 ⊢ (((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ (𝑍‘𝑆) ∧ 𝑧 ∈ (𝑍‘𝑆))) → (𝑦(+g‘𝑀)𝑧) ∈ 𝐵)
28	21	adantr 480	. . . . . . 7 ⊢ ((((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ (𝑍‘𝑆) ∧ 𝑧 ∈ (𝑍‘𝑆))) ∧ 𝑥 ∈ 𝑆) → 𝑀 ∈ Mnd)
29	23	adantr 480	. . . . . . 7 ⊢ ((((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ (𝑍‘𝑆) ∧ 𝑧 ∈ (𝑍‘𝑆))) ∧ 𝑥 ∈ 𝑆) → 𝑦 ∈ 𝐵)
30	25	adantr 480	. . . . . . 7 ⊢ ((((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ (𝑍‘𝑆) ∧ 𝑧 ∈ (𝑍‘𝑆))) ∧ 𝑥 ∈ 𝑆) → 𝑧 ∈ 𝐵)
31	10	adantlr 747	. . . . . . 7 ⊢ ((((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ (𝑍‘𝑆) ∧ 𝑧 ∈ (𝑍‘𝑆))) ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝐵)
32	1, 11	mndass 17125	. . . . . . 7 ⊢ ((𝑀 ∈ Mnd ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → ((𝑦(+g‘𝑀)𝑧)(+g‘𝑀)𝑥) = (𝑦(+g‘𝑀)(𝑧(+g‘𝑀)𝑥)))
33	28, 29, 30, 31, 32	syl13anc 1320	. . . . . 6 ⊢ ((((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ (𝑍‘𝑆) ∧ 𝑧 ∈ (𝑍‘𝑆))) ∧ 𝑥 ∈ 𝑆) → ((𝑦(+g‘𝑀)𝑧)(+g‘𝑀)𝑥) = (𝑦(+g‘𝑀)(𝑧(+g‘𝑀)𝑥)))
34	11, 2	cntzi 17585	. . . . . . . . 9 ⊢ ((𝑧 ∈ (𝑍‘𝑆) ∧ 𝑥 ∈ 𝑆) → (𝑧(+g‘𝑀)𝑥) = (𝑥(+g‘𝑀)𝑧))
35	24, 34	sylan 487	. . . . . . . 8 ⊢ ((((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ (𝑍‘𝑆) ∧ 𝑧 ∈ (𝑍‘𝑆))) ∧ 𝑥 ∈ 𝑆) → (𝑧(+g‘𝑀)𝑥) = (𝑥(+g‘𝑀)𝑧))
36	35	oveq2d 6565	. . . . . . 7 ⊢ ((((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ (𝑍‘𝑆) ∧ 𝑧 ∈ (𝑍‘𝑆))) ∧ 𝑥 ∈ 𝑆) → (𝑦(+g‘𝑀)(𝑧(+g‘𝑀)𝑥)) = (𝑦(+g‘𝑀)(𝑥(+g‘𝑀)𝑧)))
37	1, 11	mndass 17125	. . . . . . . 8 ⊢ ((𝑀 ∈ Mnd ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑦(+g‘𝑀)𝑥)(+g‘𝑀)𝑧) = (𝑦(+g‘𝑀)(𝑥(+g‘𝑀)𝑧)))
38	28, 29, 31, 30, 37	syl13anc 1320	. . . . . . 7 ⊢ ((((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ (𝑍‘𝑆) ∧ 𝑧 ∈ (𝑍‘𝑆))) ∧ 𝑥 ∈ 𝑆) → ((𝑦(+g‘𝑀)𝑥)(+g‘𝑀)𝑧) = (𝑦(+g‘𝑀)(𝑥(+g‘𝑀)𝑧)))
39	11, 2	cntzi 17585	. . . . . . . . 9 ⊢ ((𝑦 ∈ (𝑍‘𝑆) ∧ 𝑥 ∈ 𝑆) → (𝑦(+g‘𝑀)𝑥) = (𝑥(+g‘𝑀)𝑦))
40	22, 39	sylan 487	. . . . . . . 8 ⊢ ((((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ (𝑍‘𝑆) ∧ 𝑧 ∈ (𝑍‘𝑆))) ∧ 𝑥 ∈ 𝑆) → (𝑦(+g‘𝑀)𝑥) = (𝑥(+g‘𝑀)𝑦))
41	40	oveq1d 6564	. . . . . . 7 ⊢ ((((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ (𝑍‘𝑆) ∧ 𝑧 ∈ (𝑍‘𝑆))) ∧ 𝑥 ∈ 𝑆) → ((𝑦(+g‘𝑀)𝑥)(+g‘𝑀)𝑧) = ((𝑥(+g‘𝑀)𝑦)(+g‘𝑀)𝑧))
42	36, 38, 41	3eqtr2d 2650	. . . . . 6 ⊢ ((((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ (𝑍‘𝑆) ∧ 𝑧 ∈ (𝑍‘𝑆))) ∧ 𝑥 ∈ 𝑆) → (𝑦(+g‘𝑀)(𝑧(+g‘𝑀)𝑥)) = ((𝑥(+g‘𝑀)𝑦)(+g‘𝑀)𝑧))
43	1, 11	mndass 17125	. . . . . . 7 ⊢ ((𝑀 ∈ Mnd ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥(+g‘𝑀)𝑦)(+g‘𝑀)𝑧) = (𝑥(+g‘𝑀)(𝑦(+g‘𝑀)𝑧)))
44	28, 31, 29, 30, 43	syl13anc 1320	. . . . . 6 ⊢ ((((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ (𝑍‘𝑆) ∧ 𝑧 ∈ (𝑍‘𝑆))) ∧ 𝑥 ∈ 𝑆) → ((𝑥(+g‘𝑀)𝑦)(+g‘𝑀)𝑧) = (𝑥(+g‘𝑀)(𝑦(+g‘𝑀)𝑧)))
45	33, 42, 44	3eqtrd 2648	. . . . 5 ⊢ ((((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ (𝑍‘𝑆) ∧ 𝑧 ∈ (𝑍‘𝑆))) ∧ 𝑥 ∈ 𝑆) → ((𝑦(+g‘𝑀)𝑧)(+g‘𝑀)𝑥) = (𝑥(+g‘𝑀)(𝑦(+g‘𝑀)𝑧)))
46	45	ralrimiva 2949	. . . 4 ⊢ (((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ (𝑍‘𝑆) ∧ 𝑧 ∈ (𝑍‘𝑆))) → ∀𝑥 ∈ 𝑆 ((𝑦(+g‘𝑀)𝑧)(+g‘𝑀)𝑥) = (𝑥(+g‘𝑀)(𝑦(+g‘𝑀)𝑧)))
47	1, 11, 2	elcntz 17578	. . . . 5 ⊢ (𝑆 ⊆ 𝐵 → ((𝑦(+g‘𝑀)𝑧) ∈ (𝑍‘𝑆) ↔ ((𝑦(+g‘𝑀)𝑧) ∈ 𝐵 ∧ ∀𝑥 ∈ 𝑆 ((𝑦(+g‘𝑀)𝑧)(+g‘𝑀)𝑥) = (𝑥(+g‘𝑀)(𝑦(+g‘𝑀)𝑧)))))
48	47	ad2antlr 759	. . . 4 ⊢ (((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ (𝑍‘𝑆) ∧ 𝑧 ∈ (𝑍‘𝑆))) → ((𝑦(+g‘𝑀)𝑧) ∈ (𝑍‘𝑆) ↔ ((𝑦(+g‘𝑀)𝑧) ∈ 𝐵 ∧ ∀𝑥 ∈ 𝑆 ((𝑦(+g‘𝑀)𝑧)(+g‘𝑀)𝑥) = (𝑥(+g‘𝑀)(𝑦(+g‘𝑀)𝑧)))))
49	27, 46, 48	mpbir2and 959	. . 3 ⊢ (((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ (𝑍‘𝑆) ∧ 𝑧 ∈ (𝑍‘𝑆))) → (𝑦(+g‘𝑀)𝑧) ∈ (𝑍‘𝑆))
50	49	ralrimivva 2954	. 2 ⊢ ((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) → ∀𝑦 ∈ (𝑍‘𝑆)∀𝑧 ∈ (𝑍‘𝑆)(𝑦(+g‘𝑀)𝑧) ∈ (𝑍‘𝑆))
51	1, 5, 11	issubm 17170	. . 3 ⊢ (𝑀 ∈ Mnd → ((𝑍‘𝑆) ∈ (SubMnd‘𝑀) ↔ ((𝑍‘𝑆) ⊆ 𝐵 ∧ (0g‘𝑀) ∈ (𝑍‘𝑆) ∧ ∀𝑦 ∈ (𝑍‘𝑆)∀𝑧 ∈ (𝑍‘𝑆)(𝑦(+g‘𝑀)𝑧) ∈ (𝑍‘𝑆))))
52	51	adantr 480	. 2 ⊢ ((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) → ((𝑍‘𝑆) ∈ (SubMnd‘𝑀) ↔ ((𝑍‘𝑆) ⊆ 𝐵 ∧ (0g‘𝑀) ∈ (𝑍‘𝑆) ∧ ∀𝑦 ∈ (𝑍‘𝑆)∀𝑧 ∈ (𝑍‘𝑆)(𝑦(+g‘𝑀)𝑧) ∈ (𝑍‘𝑆))))
53	4, 20, 50, 52	mpbir3and 1238	1 ⊢ ((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) → (𝑍‘𝑆) ∈ (SubMnd‘𝑀))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896   ⊆ wss 3540  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  +_gcplusg 15768  0_gc0g 15923  Mndcmnd 17117  SubMndcsubmnd 17157  Cntzccntz 17571

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-0g 15925  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-submnd 17159  df-cntz 17573

This theorem is referenced by:  cntzsubg  17592  cntzspan  18070  dprdfadd  18242  cntzsubr  18635