Theorem cntzsubm 19579

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 19569	. . 3 ⊢ (𝑍‘𝑆) ⊆ 𝐵
4	3	a1i 11	. 2 ⊢ ((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) → (𝑍‘𝑆) ⊆ 𝐵)
5		eqid 2763	. . . . 5 ⊢ (0g‘𝑀) = (0g‘𝑀)
6	1, 5	mndidcl 18965	. . . 4 ⊢ (𝑀 ∈ Mnd → (0g‘𝑀) ∈ 𝐵)
7	6	adantr 486	. . 3 ⊢ ((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) → (0g‘𝑀) ∈ 𝐵)
8		simpll 779	. . . . . 6 ⊢ (((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ 𝑆) → 𝑀 ∈ Mnd)
9		simpr 490	. . . . . . 7 ⊢ ((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) → 𝑆 ⊆ 𝐵)
10	9	sselda 3913	. . . . . 6 ⊢ (((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝐵)
11		eqid 2763	. . . . . . 7 ⊢ (+g‘𝑀) = (+g‘𝑀)
12	1, 11, 5	mndlid 18970	. . . . . 6 ⊢ ((𝑀 ∈ Mnd ∧ 𝑥 ∈ 𝐵) → ((0g‘𝑀)(+g‘𝑀)𝑥) = 𝑥)
13	8, 10, 12	syl2anc 596	. . . . 5 ⊢ (((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ 𝑆) → ((0g‘𝑀)(+g‘𝑀)𝑥) = 𝑥)
14	1, 11, 5	mndrid 18971	. . . . . 6 ⊢ ((𝑀 ∈ Mnd ∧ 𝑥 ∈ 𝐵) → (𝑥(+g‘𝑀)(0g‘𝑀)) = 𝑥)
15	8, 10, 14	syl2anc 596	. . . . 5 ⊢ (((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ 𝑆) → (𝑥(+g‘𝑀)(0g‘𝑀)) = 𝑥)
16	13, 15	eqtr4d 2801	. . . 4 ⊢ (((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ 𝑆) → ((0g‘𝑀)(+g‘𝑀)𝑥) = (𝑥(+g‘𝑀)(0g‘𝑀)))
17	16	ralrimiva 3158	. . 3 ⊢ ((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) → ∀𝑥 ∈ 𝑆 ((0g‘𝑀)(+g‘𝑀)𝑥) = (𝑥(+g‘𝑀)(0g‘𝑀)))
18	1, 11, 2	elcntz 19563	. . . 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 487	. . 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 726	. 2 ⊢ ((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) → (0g‘𝑀) ∈ (𝑍‘𝑆))
21		simpll 779	. . . . 5 ⊢ (((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ (𝑍‘𝑆) ∧ 𝑧 ∈ (𝑍‘𝑆))) → 𝑀 ∈ Mnd)
22		simprl 783	. . . . . 6 ⊢ (((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ (𝑍‘𝑆) ∧ 𝑧 ∈ (𝑍‘𝑆))) → 𝑦 ∈ (𝑍‘𝑆))
23	3, 22	sselid 3911	. . . . 5 ⊢ (((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ (𝑍‘𝑆) ∧ 𝑧 ∈ (𝑍‘𝑆))) → 𝑦 ∈ 𝐵)
24		simprr 785	. . . . . 6 ⊢ (((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ (𝑍‘𝑆) ∧ 𝑧 ∈ (𝑍‘𝑆))) → 𝑧 ∈ (𝑍‘𝑆))
25	3, 24	sselid 3911	. . . . 5 ⊢ (((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ (𝑍‘𝑆) ∧ 𝑧 ∈ (𝑍‘𝑆))) → 𝑧 ∈ 𝐵)
26	1, 11	mndcl 18957	. . . . 5 ⊢ ((𝑀 ∈ Mnd ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → (𝑦(+g‘𝑀)𝑧) ∈ 𝐵)
27	21, 23, 25, 26	syl3anc 1399	. . . 4 ⊢ (((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ (𝑍‘𝑆) ∧ 𝑧 ∈ (𝑍‘𝑆))) → (𝑦(+g‘𝑀)𝑧) ∈ 𝐵)
28	21	adantr 486	. . . . . . 7 ⊢ ((((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ (𝑍‘𝑆) ∧ 𝑧 ∈ (𝑍‘𝑆))) ∧ 𝑥 ∈ 𝑆) → 𝑀 ∈ Mnd)
29	23	adantr 486	. . . . . . 7 ⊢ ((((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ (𝑍‘𝑆) ∧ 𝑧 ∈ (𝑍‘𝑆))) ∧ 𝑥 ∈ 𝑆) → 𝑦 ∈ 𝐵)
30	25	adantr 486	. . . . . . 7 ⊢ ((((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ (𝑍‘𝑆) ∧ 𝑧 ∈ (𝑍‘𝑆))) ∧ 𝑥 ∈ 𝑆) → 𝑧 ∈ 𝐵)
31	10	adantlr 728	. . . . . . 7 ⊢ ((((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ (𝑍‘𝑆) ∧ 𝑧 ∈ (𝑍‘𝑆))) ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝐵)
32	1, 11	mndass 18958	. . . . . . 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 1400	. . . . . 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 19570	. . . . . . . . 9 ⊢ ((𝑧 ∈ (𝑍‘𝑆) ∧ 𝑥 ∈ 𝑆) → (𝑧(+g‘𝑀)𝑥) = (𝑥(+g‘𝑀)𝑧))
35	24, 34	sylan 592	. . . . . . . 8 ⊢ ((((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ (𝑍‘𝑆) ∧ 𝑧 ∈ (𝑍‘𝑆))) ∧ 𝑥 ∈ 𝑆) → (𝑧(+g‘𝑀)𝑥) = (𝑥(+g‘𝑀)𝑧))
36	35	oveq2d 7456	. . . . . . 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 18958	. . . . . . . 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 1400	. . . . . . 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 19570	. . . . . . . . 9 ⊢ ((𝑦 ∈ (𝑍‘𝑆) ∧ 𝑥 ∈ 𝑆) → (𝑦(+g‘𝑀)𝑥) = (𝑥(+g‘𝑀)𝑦))
40	22, 39	sylan 592	. . . . . . . 8 ⊢ ((((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ (𝑍‘𝑆) ∧ 𝑧 ∈ (𝑍‘𝑆))) ∧ 𝑥 ∈ 𝑆) → (𝑦(+g‘𝑀)𝑥) = (𝑥(+g‘𝑀)𝑦))
41	40	oveq1d 7455	. . . . . . 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 2804	. . . . . 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 18958	. . . . . . 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 1400	. . . . . 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 2802	. . . . 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 3158	. . . 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 19563	. . . . 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 740	. . . 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 726	. . 3 ⊢ (((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ (𝑍‘𝑆) ∧ 𝑧 ∈ (𝑍‘𝑆))) → (𝑦(+g‘𝑀)𝑧) ∈ (𝑍‘𝑆))
50	49	ralrimivva 3209	. 2 ⊢ ((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) → ∀𝑦 ∈ (𝑍‘𝑆)∀𝑧 ∈ (𝑍‘𝑆)(𝑦(+g‘𝑀)𝑧) ∈ (𝑍‘𝑆))
51	1, 5, 11	issubm 19024	. . 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 486	. 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 1361	1 ⊢ ((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) → (𝑍‘𝑆) ∈ (SubMnd‘𝑀))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  ∀wral 3080   ⊆ wss 3900  ‘cfv 6555  (class class class)co 7440  Basecbs 17413  +_gcplusg 17454  0_gc0g 17636  Mndcmnd 18949  SubMndcsubmnd 19003  Cntzccntz 19556

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-riota 7397  df-ov 7443  df-0g 17638  df-mgm 18842  df-sgrp 18934  df-mnd 18950  df-submnd 19005  df-cntz 19558

This theorem is used by:  cntzsubg  19580  cntrcmnd  20083  cntzspan  20085  dprdfadd  20263  cntzsubr  20885