Theorem cntzsubm 19271

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 19261	. . 3 ⊢ (𝑍‘𝑆) ⊆ 𝐵
4	3	a1i 11	. 2 ⊢ ((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) → (𝑍‘𝑆) ⊆ 𝐵)
5		eqid 2737	. . . . 5 ⊢ (0g‘𝑀) = (0g‘𝑀)
6	1, 5	mndidcl 18678	. . . 4 ⊢ (𝑀 ∈ Mnd → (0g‘𝑀) ∈ 𝐵)
7	6	adantr 480	. . 3 ⊢ ((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) → (0g‘𝑀) ∈ 𝐵)
8		simpll 767	. . . . . 6 ⊢ (((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ 𝑆) → 𝑀 ∈ Mnd)
9		simpr 484	. . . . . . 7 ⊢ ((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) → 𝑆 ⊆ 𝐵)
10	9	sselda 3934	. . . . . 6 ⊢ (((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝐵)
11		eqid 2737	. . . . . . 7 ⊢ (+g‘𝑀) = (+g‘𝑀)
12	1, 11, 5	mndlid 18683	. . . . . 6 ⊢ ((𝑀 ∈ Mnd ∧ 𝑥 ∈ 𝐵) → ((0g‘𝑀)(+g‘𝑀)𝑥) = 𝑥)
13	8, 10, 12	syl2anc 585	. . . . 5 ⊢ (((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ 𝑆) → ((0g‘𝑀)(+g‘𝑀)𝑥) = 𝑥)
14	1, 11, 5	mndrid 18684	. . . . . 6 ⊢ ((𝑀 ∈ Mnd ∧ 𝑥 ∈ 𝐵) → (𝑥(+g‘𝑀)(0g‘𝑀)) = 𝑥)
15	8, 10, 14	syl2anc 585	. . . . 5 ⊢ (((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ 𝑆) → (𝑥(+g‘𝑀)(0g‘𝑀)) = 𝑥)
16	13, 15	eqtr4d 2775	. . . 4 ⊢ (((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) ∧ 𝑥 ∈ 𝑆) → ((0g‘𝑀)(+g‘𝑀)𝑥) = (𝑥(+g‘𝑀)(0g‘𝑀)))
17	16	ralrimiva 3129	. . 3 ⊢ ((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) → ∀𝑥 ∈ 𝑆 ((0g‘𝑀)(+g‘𝑀)𝑥) = (𝑥(+g‘𝑀)(0g‘𝑀)))
18	1, 11, 2	elcntz 19255	. . . 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 714	. 2 ⊢ ((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) → (0g‘𝑀) ∈ (𝑍‘𝑆))
21		simpll 767	. . . . 5 ⊢ (((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ (𝑍‘𝑆) ∧ 𝑧 ∈ (𝑍‘𝑆))) → 𝑀 ∈ Mnd)
22		simprl 771	. . . . . 6 ⊢ (((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ (𝑍‘𝑆) ∧ 𝑧 ∈ (𝑍‘𝑆))) → 𝑦 ∈ (𝑍‘𝑆))
23	3, 22	sselid 3932	. . . . 5 ⊢ (((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ (𝑍‘𝑆) ∧ 𝑧 ∈ (𝑍‘𝑆))) → 𝑦 ∈ 𝐵)
24		simprr 773	. . . . . 6 ⊢ (((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ (𝑍‘𝑆) ∧ 𝑧 ∈ (𝑍‘𝑆))) → 𝑧 ∈ (𝑍‘𝑆))
25	3, 24	sselid 3932	. . . . 5 ⊢ (((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ (𝑍‘𝑆) ∧ 𝑧 ∈ (𝑍‘𝑆))) → 𝑧 ∈ 𝐵)
26	1, 11	mndcl 18671	. . . . 5 ⊢ ((𝑀 ∈ Mnd ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → (𝑦(+g‘𝑀)𝑧) ∈ 𝐵)
27	21, 23, 25, 26	syl3anc 1374	. . . 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 716	. . . . . . 7 ⊢ ((((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ (𝑍‘𝑆) ∧ 𝑧 ∈ (𝑍‘𝑆))) ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝐵)
32	1, 11	mndass 18672	. . . . . . 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 1375	. . . . . 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 19262	. . . . . . . . 9 ⊢ ((𝑧 ∈ (𝑍‘𝑆) ∧ 𝑥 ∈ 𝑆) → (𝑧(+g‘𝑀)𝑥) = (𝑥(+g‘𝑀)𝑧))
35	24, 34	sylan 581	. . . . . . . 8 ⊢ ((((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ (𝑍‘𝑆) ∧ 𝑧 ∈ (𝑍‘𝑆))) ∧ 𝑥 ∈ 𝑆) → (𝑧(+g‘𝑀)𝑥) = (𝑥(+g‘𝑀)𝑧))
36	35	oveq2d 7376	. . . . . . 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 18672	. . . . . . . 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 1375	. . . . . . 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 19262	. . . . . . . . 9 ⊢ ((𝑦 ∈ (𝑍‘𝑆) ∧ 𝑥 ∈ 𝑆) → (𝑦(+g‘𝑀)𝑥) = (𝑥(+g‘𝑀)𝑦))
40	22, 39	sylan 581	. . . . . . . 8 ⊢ ((((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ (𝑍‘𝑆) ∧ 𝑧 ∈ (𝑍‘𝑆))) ∧ 𝑥 ∈ 𝑆) → (𝑦(+g‘𝑀)𝑥) = (𝑥(+g‘𝑀)𝑦))
41	40	oveq1d 7375	. . . . . . 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 2778	. . . . . 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 18672	. . . . . . 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 1375	. . . . . 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 2776	. . . . 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 3129	. . . 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 19255	. . . . 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 728	. . . 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 714	. . 3 ⊢ (((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) ∧ (𝑦 ∈ (𝑍‘𝑆) ∧ 𝑧 ∈ (𝑍‘𝑆))) → (𝑦(+g‘𝑀)𝑧) ∈ (𝑍‘𝑆))
50	49	ralrimivva 3180	. 2 ⊢ ((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) → ∀𝑦 ∈ (𝑍‘𝑆)∀𝑧 ∈ (𝑍‘𝑆)(𝑦(+g‘𝑀)𝑧) ∈ (𝑍‘𝑆))
51	1, 5, 11	issubm 18732	. . 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 1344	1 ⊢ ((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) → (𝑍‘𝑆) ∈ (SubMnd‘𝑀))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052   ⊆ wss 3902  ‘cfv 6493  (class class class)co 7360  Basecbs 17140  +_gcplusg 17181  0_gc0g 17363  Mndcmnd 18663  SubMndcsubmnd 18711  Cntzccntz 19248

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5225  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rmo 3351  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7317  df-ov 7363  df-0g 17365  df-mgm 18569  df-sgrp 18648  df-mnd 18664  df-submnd 18713  df-cntz 19250

This theorem is referenced by:  cntzsubg  19272  cntrcmnd  19775  cntzspan  19777  dprdfadd  19955  cntzsubr  20543