Theorem cntzmhm 19280

Description: Centralizers in a monoid are preserved by monoid homomorphisms. (Contributed by Mario Carneiro, 24-Apr-2016.)

Hypotheses

Ref	Expression
cntzmhm.z	⊢ 𝑍 = (Cntz‘𝐺)
cntzmhm.y	⊢ 𝑌 = (Cntz‘𝐻)

Assertion

Ref	Expression
cntzmhm	⊢ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍‘𝑆)) → (𝐹‘𝐴) ∈ (𝑌‘(𝐹 “ 𝑆)))

Proof of Theorem cntzmhm

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

Step	Hyp	Ref	Expression
1		eqid 2730	. . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺)
2		eqid 2730	. . . 4 ⊢ (Base‘𝐻) = (Base‘𝐻)
3	1, 2	mhmf 18723	. . 3 ⊢ (𝐹 ∈ (𝐺 MndHom 𝐻) → 𝐹:(Base‘𝐺)⟶(Base‘𝐻))
4		cntzmhm.z	. . . . 5 ⊢ 𝑍 = (Cntz‘𝐺)
5	1, 4	cntzssv 19267	. . . 4 ⊢ (𝑍‘𝑆) ⊆ (Base‘𝐺)
6	5	sseli 3945	. . 3 ⊢ (𝐴 ∈ (𝑍‘𝑆) → 𝐴 ∈ (Base‘𝐺))
7		ffvelcdm 7056	. . 3 ⊢ ((𝐹:(Base‘𝐺)⟶(Base‘𝐻) ∧ 𝐴 ∈ (Base‘𝐺)) → (𝐹‘𝐴) ∈ (Base‘𝐻))
8	3, 6, 7	syl2an 596	. 2 ⊢ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍‘𝑆)) → (𝐹‘𝐴) ∈ (Base‘𝐻))
9		eqid 2730	. . . . . . . 8 ⊢ (+g‘𝐺) = (+g‘𝐺)
10	9, 4	cntzi 19268	. . . . . . 7 ⊢ ((𝐴 ∈ (𝑍‘𝑆) ∧ 𝑥 ∈ 𝑆) → (𝐴(+g‘𝐺)𝑥) = (𝑥(+g‘𝐺)𝐴))
11	10	adantll 714	. . . . . 6 ⊢ (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍‘𝑆)) ∧ 𝑥 ∈ 𝑆) → (𝐴(+g‘𝐺)𝑥) = (𝑥(+g‘𝐺)𝐴))
12	11	fveq2d 6865	. . . . 5 ⊢ (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍‘𝑆)) ∧ 𝑥 ∈ 𝑆) → (𝐹‘(𝐴(+g‘𝐺)𝑥)) = (𝐹‘(𝑥(+g‘𝐺)𝐴)))
13		simpll 766	. . . . . 6 ⊢ (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍‘𝑆)) ∧ 𝑥 ∈ 𝑆) → 𝐹 ∈ (𝐺 MndHom 𝐻))
14	6	ad2antlr 727	. . . . . 6 ⊢ (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍‘𝑆)) ∧ 𝑥 ∈ 𝑆) → 𝐴 ∈ (Base‘𝐺))
15	1, 4	cntzrcl 19266	. . . . . . . . 9 ⊢ (𝐴 ∈ (𝑍‘𝑆) → (𝐺 ∈ V ∧ 𝑆 ⊆ (Base‘𝐺)))
16	15	adantl 481	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍‘𝑆)) → (𝐺 ∈ V ∧ 𝑆 ⊆ (Base‘𝐺)))
17	16	simprd 495	. . . . . . 7 ⊢ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍‘𝑆)) → 𝑆 ⊆ (Base‘𝐺))
18	17	sselda 3949	. . . . . 6 ⊢ (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍‘𝑆)) ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ (Base‘𝐺))
19		eqid 2730	. . . . . . 7 ⊢ (+g‘𝐻) = (+g‘𝐻)
20	1, 9, 19	mhmlin 18727	. . . . . 6 ⊢ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (Base‘𝐺) ∧ 𝑥 ∈ (Base‘𝐺)) → (𝐹‘(𝐴(+g‘𝐺)𝑥)) = ((𝐹‘𝐴)(+g‘𝐻)(𝐹‘𝑥)))
21	13, 14, 18, 20	syl3anc 1373	. . . . 5 ⊢ (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍‘𝑆)) ∧ 𝑥 ∈ 𝑆) → (𝐹‘(𝐴(+g‘𝐺)𝑥)) = ((𝐹‘𝐴)(+g‘𝐻)(𝐹‘𝑥)))
22	1, 9, 19	mhmlin 18727	. . . . . 6 ⊢ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝐴 ∈ (Base‘𝐺)) → (𝐹‘(𝑥(+g‘𝐺)𝐴)) = ((𝐹‘𝑥)(+g‘𝐻)(𝐹‘𝐴)))
23	13, 18, 14, 22	syl3anc 1373	. . . . 5 ⊢ (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍‘𝑆)) ∧ 𝑥 ∈ 𝑆) → (𝐹‘(𝑥(+g‘𝐺)𝐴)) = ((𝐹‘𝑥)(+g‘𝐻)(𝐹‘𝐴)))
24	12, 21, 23	3eqtr3d 2773	. . . 4 ⊢ (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍‘𝑆)) ∧ 𝑥 ∈ 𝑆) → ((𝐹‘𝐴)(+g‘𝐻)(𝐹‘𝑥)) = ((𝐹‘𝑥)(+g‘𝐻)(𝐹‘𝐴)))
25	24	ralrimiva 3126	. . 3 ⊢ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍‘𝑆)) → ∀𝑥 ∈ 𝑆 ((𝐹‘𝐴)(+g‘𝐻)(𝐹‘𝑥)) = ((𝐹‘𝑥)(+g‘𝐻)(𝐹‘𝐴)))
26	3	adantr 480	. . . . 5 ⊢ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍‘𝑆)) → 𝐹:(Base‘𝐺)⟶(Base‘𝐻))
27	26	ffnd 6692	. . . 4 ⊢ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍‘𝑆)) → 𝐹 Fn (Base‘𝐺))
28		oveq2 7398	. . . . . 6 ⊢ (𝑦 = (𝐹‘𝑥) → ((𝐹‘𝐴)(+g‘𝐻)𝑦) = ((𝐹‘𝐴)(+g‘𝐻)(𝐹‘𝑥)))
29		oveq1 7397	. . . . . 6 ⊢ (𝑦 = (𝐹‘𝑥) → (𝑦(+g‘𝐻)(𝐹‘𝐴)) = ((𝐹‘𝑥)(+g‘𝐻)(𝐹‘𝐴)))
30	28, 29	eqeq12d 2746	. . . . 5 ⊢ (𝑦 = (𝐹‘𝑥) → (((𝐹‘𝐴)(+g‘𝐻)𝑦) = (𝑦(+g‘𝐻)(𝐹‘𝐴)) ↔ ((𝐹‘𝐴)(+g‘𝐻)(𝐹‘𝑥)) = ((𝐹‘𝑥)(+g‘𝐻)(𝐹‘𝐴))))
31	30	ralima 7214	. . . 4 ⊢ ((𝐹 Fn (Base‘𝐺) ∧ 𝑆 ⊆ (Base‘𝐺)) → (∀𝑦 ∈ (𝐹 “ 𝑆)((𝐹‘𝐴)(+g‘𝐻)𝑦) = (𝑦(+g‘𝐻)(𝐹‘𝐴)) ↔ ∀𝑥 ∈ 𝑆 ((𝐹‘𝐴)(+g‘𝐻)(𝐹‘𝑥)) = ((𝐹‘𝑥)(+g‘𝐻)(𝐹‘𝐴))))
32	27, 17, 31	syl2anc 584	. . 3 ⊢ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍‘𝑆)) → (∀𝑦 ∈ (𝐹 “ 𝑆)((𝐹‘𝐴)(+g‘𝐻)𝑦) = (𝑦(+g‘𝐻)(𝐹‘𝐴)) ↔ ∀𝑥 ∈ 𝑆 ((𝐹‘𝐴)(+g‘𝐻)(𝐹‘𝑥)) = ((𝐹‘𝑥)(+g‘𝐻)(𝐹‘𝐴))))
33	25, 32	mpbird 257	. 2 ⊢ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍‘𝑆)) → ∀𝑦 ∈ (𝐹 “ 𝑆)((𝐹‘𝐴)(+g‘𝐻)𝑦) = (𝑦(+g‘𝐻)(𝐹‘𝐴)))
34		imassrn 6045	. . . 4 ⊢ (𝐹 “ 𝑆) ⊆ ran 𝐹
35	26	frnd 6699	. . . 4 ⊢ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍‘𝑆)) → ran 𝐹 ⊆ (Base‘𝐻))
36	34, 35	sstrid 3961	. . 3 ⊢ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍‘𝑆)) → (𝐹 “ 𝑆) ⊆ (Base‘𝐻))
37		cntzmhm.y	. . . 4 ⊢ 𝑌 = (Cntz‘𝐻)
38	2, 19, 37	elcntz 19261	. . 3 ⊢ ((𝐹 “ 𝑆) ⊆ (Base‘𝐻) → ((𝐹‘𝐴) ∈ (𝑌‘(𝐹 “ 𝑆)) ↔ ((𝐹‘𝐴) ∈ (Base‘𝐻) ∧ ∀𝑦 ∈ (𝐹 “ 𝑆)((𝐹‘𝐴)(+g‘𝐻)𝑦) = (𝑦(+g‘𝐻)(𝐹‘𝐴)))))
39	36, 38	syl 17	. 2 ⊢ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍‘𝑆)) → ((𝐹‘𝐴) ∈ (𝑌‘(𝐹 “ 𝑆)) ↔ ((𝐹‘𝐴) ∈ (Base‘𝐻) ∧ ∀𝑦 ∈ (𝐹 “ 𝑆)((𝐹‘𝐴)(+g‘𝐻)𝑦) = (𝑦(+g‘𝐻)(𝐹‘𝐴)))))
40	8, 33, 39	mpbir2and 713	1 ⊢ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍‘𝑆)) → (𝐹‘𝐴) ∈ (𝑌‘(𝐹 “ 𝑆)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3045  Vcvv 3450   ⊆ wss 3917  ran crn 5642   “ cima 5644   Fn wfn 6509  ⟶wf 6510  ‘cfv 6514  (class class class)co 7390  Basecbs 17186  +_gcplusg 17227   MndHom cmhm 18715  Cntzccntz 19254

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-map 8804  df-mhm 18717  df-cntz 19256

This theorem is referenced by:  cntzmhm2  19281