Theorem cntzmhm 19274

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 2737	. . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺)
2		eqid 2737	. . . 4 ⊢ (Base‘𝐻) = (Base‘𝐻)
3	1, 2	mhmf 18718	. . 3 ⊢ (𝐹 ∈ (𝐺 MndHom 𝐻) → 𝐹:(Base‘𝐺)⟶(Base‘𝐻))
4		cntzmhm.z	. . . . 5 ⊢ 𝑍 = (Cntz‘𝐺)
5	1, 4	cntzssv 19261	. . . 4 ⊢ (𝑍‘𝑆) ⊆ (Base‘𝐺)
6	5	sseli 3930	. . 3 ⊢ (𝐴 ∈ (𝑍‘𝑆) → 𝐴 ∈ (Base‘𝐺))
7		ffvelcdm 7028	. . 3 ⊢ ((𝐹:(Base‘𝐺)⟶(Base‘𝐻) ∧ 𝐴 ∈ (Base‘𝐺)) → (𝐹‘𝐴) ∈ (Base‘𝐻))
8	3, 6, 7	syl2an 597	. 2 ⊢ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍‘𝑆)) → (𝐹‘𝐴) ∈ (Base‘𝐻))
9		eqid 2737	. . . . . . . 8 ⊢ (+g‘𝐺) = (+g‘𝐺)
10	9, 4	cntzi 19262	. . . . . . 7 ⊢ ((𝐴 ∈ (𝑍‘𝑆) ∧ 𝑥 ∈ 𝑆) → (𝐴(+g‘𝐺)𝑥) = (𝑥(+g‘𝐺)𝐴))
11	10	adantll 715	. . . . . 6 ⊢ (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍‘𝑆)) ∧ 𝑥 ∈ 𝑆) → (𝐴(+g‘𝐺)𝑥) = (𝑥(+g‘𝐺)𝐴))
12	11	fveq2d 6839	. . . . 5 ⊢ (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍‘𝑆)) ∧ 𝑥 ∈ 𝑆) → (𝐹‘(𝐴(+g‘𝐺)𝑥)) = (𝐹‘(𝑥(+g‘𝐺)𝐴)))
13		simpll 767	. . . . . 6 ⊢ (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍‘𝑆)) ∧ 𝑥 ∈ 𝑆) → 𝐹 ∈ (𝐺 MndHom 𝐻))
14	6	ad2antlr 728	. . . . . 6 ⊢ (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍‘𝑆)) ∧ 𝑥 ∈ 𝑆) → 𝐴 ∈ (Base‘𝐺))
15	1, 4	cntzrcl 19260	. . . . . . . . 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 3934	. . . . . 6 ⊢ (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍‘𝑆)) ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ (Base‘𝐺))
19		eqid 2737	. . . . . . 7 ⊢ (+g‘𝐻) = (+g‘𝐻)
20	1, 9, 19	mhmlin 18722	. . . . . 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 1374	. . . . 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 18722	. . . . . 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 1374	. . . . 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 2780	. . . 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 3129	. . 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 6664	. . . 4 ⊢ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍‘𝑆)) → 𝐹 Fn (Base‘𝐺))
28		oveq2 7368	. . . . . 6 ⊢ (𝑦 = (𝐹‘𝑥) → ((𝐹‘𝐴)(+g‘𝐻)𝑦) = ((𝐹‘𝐴)(+g‘𝐻)(𝐹‘𝑥)))
29		oveq1 7367	. . . . . 6 ⊢ (𝑦 = (𝐹‘𝑥) → (𝑦(+g‘𝐻)(𝐹‘𝐴)) = ((𝐹‘𝑥)(+g‘𝐻)(𝐹‘𝐴)))
30	28, 29	eqeq12d 2753	. . . . 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 7185	. . . 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 585	. . 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 6031	. . . 4 ⊢ (𝐹 “ 𝑆) ⊆ ran 𝐹
35	26	frnd 6671	. . . 4 ⊢ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍‘𝑆)) → ran 𝐹 ⊆ (Base‘𝐻))
36	34, 35	sstrid 3946	. . 3 ⊢ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍‘𝑆)) → (𝐹 “ 𝑆) ⊆ (Base‘𝐻))
37		cntzmhm.y	. . . 4 ⊢ 𝑌 = (Cntz‘𝐻)
38	2, 19, 37	elcntz 19255	. . 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 714	1 ⊢ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍‘𝑆)) → (𝐹‘𝐴) ∈ (𝑌‘(𝐹 “ 𝑆)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  Vcvv 3441   ⊆ wss 3902  ran crn 5626   “ cima 5628   Fn wfn 6488  ⟶wf 6489  ‘cfv 6493  (class class class)co 7360  Basecbs 17140  +_gcplusg 17181   MndHom cmhm 18710  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  ax-un 7682

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-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-ov 7363  df-oprab 7364  df-mpo 7365  df-map 8769  df-mhm 18712  df-cntz 19250

This theorem is referenced by:  cntzmhm2  19275