Theorem cntzmhm 18945

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 2738	. . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺)
2		eqid 2738	. . . 4 ⊢ (Base‘𝐻) = (Base‘𝐻)
3	1, 2	mhmf 18435	. . 3 ⊢ (𝐹 ∈ (𝐺 MndHom 𝐻) → 𝐹:(Base‘𝐺)⟶(Base‘𝐻))
4		cntzmhm.z	. . . . 5 ⊢ 𝑍 = (Cntz‘𝐺)
5	1, 4	cntzssv 18934	. . . 4 ⊢ (𝑍‘𝑆) ⊆ (Base‘𝐺)
6	5	sseli 3917	. . 3 ⊢ (𝐴 ∈ (𝑍‘𝑆) → 𝐴 ∈ (Base‘𝐺))
7		ffvelrn 6959	. . 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 2738	. . . . . . . 8 ⊢ (+g‘𝐺) = (+g‘𝐺)
10	9, 4	cntzi 18935	. . . . . . 7 ⊢ ((𝐴 ∈ (𝑍‘𝑆) ∧ 𝑥 ∈ 𝑆) → (𝐴(+g‘𝐺)𝑥) = (𝑥(+g‘𝐺)𝐴))
11	10	adantll 711	. . . . . 6 ⊢ (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍‘𝑆)) ∧ 𝑥 ∈ 𝑆) → (𝐴(+g‘𝐺)𝑥) = (𝑥(+g‘𝐺)𝐴))
12	11	fveq2d 6778	. . . . 5 ⊢ (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍‘𝑆)) ∧ 𝑥 ∈ 𝑆) → (𝐹‘(𝐴(+g‘𝐺)𝑥)) = (𝐹‘(𝑥(+g‘𝐺)𝐴)))
13		simpll 764	. . . . . 6 ⊢ (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍‘𝑆)) ∧ 𝑥 ∈ 𝑆) → 𝐹 ∈ (𝐺 MndHom 𝐻))
14	6	ad2antlr 724	. . . . . 6 ⊢ (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍‘𝑆)) ∧ 𝑥 ∈ 𝑆) → 𝐴 ∈ (Base‘𝐺))
15	1, 4	cntzrcl 18933	. . . . . . . . 9 ⊢ (𝐴 ∈ (𝑍‘𝑆) → (𝐺 ∈ V ∧ 𝑆 ⊆ (Base‘𝐺)))
16	15	adantl 482	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍‘𝑆)) → (𝐺 ∈ V ∧ 𝑆 ⊆ (Base‘𝐺)))
17	16	simprd 496	. . . . . . 7 ⊢ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍‘𝑆)) → 𝑆 ⊆ (Base‘𝐺))
18	17	sselda 3921	. . . . . 6 ⊢ (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍‘𝑆)) ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ (Base‘𝐺))
19		eqid 2738	. . . . . . 7 ⊢ (+g‘𝐻) = (+g‘𝐻)
20	1, 9, 19	mhmlin 18437	. . . . . 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 1370	. . . . 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 18437	. . . . . 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 1370	. . . . 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 2786	. . . 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 3103	. . 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 481	. . . . 5 ⊢ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍‘𝑆)) → 𝐹:(Base‘𝐺)⟶(Base‘𝐻))
27	26	ffnd 6601	. . . 4 ⊢ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍‘𝑆)) → 𝐹 Fn (Base‘𝐺))
28		oveq2 7283	. . . . . 6 ⊢ (𝑦 = (𝐹‘𝑥) → ((𝐹‘𝐴)(+g‘𝐻)𝑦) = ((𝐹‘𝐴)(+g‘𝐻)(𝐹‘𝑥)))
29		oveq1 7282	. . . . . 6 ⊢ (𝑦 = (𝐹‘𝑥) → (𝑦(+g‘𝐻)(𝐹‘𝐴)) = ((𝐹‘𝑥)(+g‘𝐻)(𝐹‘𝐴)))
30	28, 29	eqeq12d 2754	. . . . 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 7114	. . . 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 256	. 2 ⊢ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍‘𝑆)) → ∀𝑦 ∈ (𝐹 “ 𝑆)((𝐹‘𝐴)(+g‘𝐻)𝑦) = (𝑦(+g‘𝐻)(𝐹‘𝐴)))
34		imassrn 5980	. . . 4 ⊢ (𝐹 “ 𝑆) ⊆ ran 𝐹
35	26	frnd 6608	. . . 4 ⊢ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍‘𝑆)) → ran 𝐹 ⊆ (Base‘𝐻))
36	34, 35	sstrid 3932	. . 3 ⊢ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍‘𝑆)) → (𝐹 “ 𝑆) ⊆ (Base‘𝐻))
37		cntzmhm.y	. . . 4 ⊢ 𝑌 = (Cntz‘𝐻)
38	2, 19, 37	elcntz 18928	. . 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 710	1 ⊢ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍‘𝑆)) → (𝐹‘𝐴) ∈ (𝑌‘(𝐹 “ 𝑆)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∀wral 3064  Vcvv 3432   ⊆ wss 3887  ran crn 5590   “ cima 5592   Fn wfn 6428  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275  Basecbs 16912  +_gcplusg 16962   MndHom cmhm 18428  Cntzccntz 18921

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-map 8617  df-mhm 18430  df-cntz 18923

This theorem is referenced by:  cntzmhm2  18946