Theorem cntzsubm 19408

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 19398	. . 3 ⊢ (𝑍‘𝑆) ⊆ 𝐵
4	3	a1i 11	. 2 ⊢ ((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) → (𝑍‘𝑆) ⊆ 𝐵)
5		eqid 2769	. . . . 5 ⊢ (0g‘𝑀) = (0g‘𝑀)
6	1, 5	mndidcl 18807	. . . 4 ⊢ (𝑀 ∈ Mnd → (0g‘𝑀) ∈ 𝐵)
7	6	adantr 485	. . 3 ⊢ ((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) → (0g‘𝑀) ∈ 𝐵)
8		simpll 778	. . . . . 6 ⊢ (((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ 𝑆) → 𝑀 ∈ Mnd)
9		simpr 489	. . . . . . 7 ⊢ ((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) → 𝑆 ⊆ 𝐵)
10	9	sselda 3943	. . . . . 6 ⊢ (((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝐵)
11		eqid 2769	. . . . . . 7 ⊢ (+g‘𝑀) = (+g‘𝑀)
12	1, 11, 5	mndlid 18812	. . . . . 6 ⊢ ((𝑀 ∈ Mnd ∧ 𝑥 ∈ 𝐵) → ((0g‘𝑀)(+g‘𝑀)𝑥) = 𝑥)
13	8, 10, 12	syl2anc 595	. . . . 5 ⊢ (((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ 𝑆) → ((0g‘𝑀)(+g‘𝑀)𝑥) = 𝑥)
14	1, 11, 5	mndrid 18813	. . . . . 6 ⊢ ((𝑀 ∈ Mnd ∧ 𝑥 ∈ 𝐵) → (𝑥(+g‘𝑀)(0g‘𝑀)) = 𝑥)
15	8, 10, 14	syl2anc 595	. . . . 5 ⊢ (((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ 𝑆) → (𝑥(+g‘𝑀)(0g‘𝑀)) = 𝑥)
16	13, 15	eqtr4d 2807	. . . 4 ⊢ (((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ 𝑆) → ((0g‘𝑀)(+g‘𝑀)𝑥) = (𝑥(+g‘𝑀)(0g‘𝑀)))
17	16	ralrimiva 3163	. . 3 ⊢ ((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) → ∀𝑥 ∈ 𝑆 ((0g‘𝑀)(+g‘𝑀)𝑥) = (𝑥(+g‘𝑀)(0g‘𝑀)))
18	1, 11, 2	elcntz 19392	. . . 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 486	. . 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 725	. 2 ⊢ ((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) → (0g‘𝑀) ∈ (𝑍‘𝑆))
21		simpll 778	. . . . 5 ⊢ (((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ (𝑍‘𝑆) ∧ 𝑧 ∈ (𝑍‘𝑆))) → 𝑀 ∈ Mnd)
22		simprl 782	. . . . . 6 ⊢ (((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ (𝑍‘𝑆) ∧ 𝑧 ∈ (𝑍‘𝑆))) → 𝑦 ∈ (𝑍‘𝑆))
23	3, 22	sselid 3941	. . . . 5 ⊢ (((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ (𝑍‘𝑆) ∧ 𝑧 ∈ (𝑍‘𝑆))) → 𝑦 ∈ 𝐵)
24		simprr 784	. . . . . 6 ⊢ (((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ (𝑍‘𝑆) ∧ 𝑧 ∈ (𝑍‘𝑆))) → 𝑧 ∈ (𝑍‘𝑆))
25	3, 24	sselid 3941	. . . . 5 ⊢ (((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ (𝑍‘𝑆) ∧ 𝑧 ∈ (𝑍‘𝑆))) → 𝑧 ∈ 𝐵)
26	1, 11	mndcl 18800	. . . . 5 ⊢ ((𝑀 ∈ Mnd ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → (𝑦(+g‘𝑀)𝑧) ∈ 𝐵)
27	21, 23, 25, 26	syl3anc 1396	. . . 4 ⊢ (((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ (𝑍‘𝑆) ∧ 𝑧 ∈ (𝑍‘𝑆))) → (𝑦(+g‘𝑀)𝑧) ∈ 𝐵)
28	21	adantr 485	. . . . . . 7 ⊢ ((((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ (𝑍‘𝑆) ∧ 𝑧 ∈ (𝑍‘𝑆))) ∧ 𝑥 ∈ 𝑆) → 𝑀 ∈ Mnd)
29	23	adantr 485	. . . . . . 7 ⊢ ((((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ (𝑍‘𝑆) ∧ 𝑧 ∈ (𝑍‘𝑆))) ∧ 𝑥 ∈ 𝑆) → 𝑦 ∈ 𝐵)
30	25	adantr 485	. . . . . . 7 ⊢ ((((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ (𝑍‘𝑆) ∧ 𝑧 ∈ (𝑍‘𝑆))) ∧ 𝑥 ∈ 𝑆) → 𝑧 ∈ 𝐵)
31	10	adantlr 727	. . . . . . 7 ⊢ ((((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ (𝑍‘𝑆) ∧ 𝑧 ∈ (𝑍‘𝑆))) ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝐵)
32	1, 11	mndass 18801	. . . . . . 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 1397	. . . . . 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 19399	. . . . . . . . 9 ⊢ ((𝑧 ∈ (𝑍‘𝑆) ∧ 𝑥 ∈ 𝑆) → (𝑧(+g‘𝑀)𝑥) = (𝑥(+g‘𝑀)𝑧))
35	24, 34	sylan 591	. . . . . . . 8 ⊢ ((((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ (𝑍‘𝑆) ∧ 𝑧 ∈ (𝑍‘𝑆))) ∧ 𝑥 ∈ 𝑆) → (𝑧(+g‘𝑀)𝑥) = (𝑥(+g‘𝑀)𝑧))
36	35	oveq2d 7427	. . . . . . 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 18801	. . . . . . . 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 1397	. . . . . . 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 19399	. . . . . . . . 9 ⊢ ((𝑦 ∈ (𝑍‘𝑆) ∧ 𝑥 ∈ 𝑆) → (𝑦(+g‘𝑀)𝑥) = (𝑥(+g‘𝑀)𝑦))
40	22, 39	sylan 591	. . . . . . . 8 ⊢ ((((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ (𝑍‘𝑆) ∧ 𝑧 ∈ (𝑍‘𝑆))) ∧ 𝑥 ∈ 𝑆) → (𝑦(+g‘𝑀)𝑥) = (𝑥(+g‘𝑀)𝑦))
41	40	oveq1d 7426	. . . . . . 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 2810	. . . . . 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 18801	. . . . . . 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 1397	. . . . . 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 2808	. . . . 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 3163	. . . 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 19392	. . . . 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 739	. . . 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 725	. . 3 ⊢ (((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ (𝑍‘𝑆) ∧ 𝑧 ∈ (𝑍‘𝑆))) → (𝑦(+g‘𝑀)𝑧) ∈ (𝑍‘𝑆))
50	49	ralrimivva 3214	. 2 ⊢ ((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) → ∀𝑦 ∈ (𝑍‘𝑆)∀𝑧 ∈ (𝑍‘𝑆)(𝑦(+g‘𝑀)𝑧) ∈ (𝑍‘𝑆))
51	1, 5, 11	issubm 18861	. . 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 485	. 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 1359	1 ⊢ ((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) → (𝑍‘𝑆) ∈ (SubMnd‘𝑀))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149  ∀wral 3085   ⊆ wss 3911  ‘cfv 6537  (class class class)co 7411  Basecbs 17269  +_gcplusg 17310  0_gc0g 17492  Mndcmnd 18792  SubMndcsubmnd 18840  Cntzccntz 19385

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5337  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rmo 3375  df-reu 3376  df-rab 3423  df-v 3463  df-sbc 3752  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4491  df-pw 4567  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7368  df-ov 7414  df-0g 17494  df-mgm 18698  df-sgrp 18777  df-mnd 18793  df-submnd 18842  df-cntz 19387

This theorem is referenced by:  cntzsubg  19409  cntrcmnd  19912  cntzspan  19914  dprdfadd  20092  cntzsubr  20691