Theorem cntzmhm 19417

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 2762	. . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺)
2		eqid 2762	. . . 4 ⊢ (Base‘𝐻) = (Base‘𝐻)
3	1, 2	mhmf 18853	. . 3 ⊢ (𝐹 ∈ (𝐺 MndHom 𝐻) → 𝐹:(Base‘𝐺)⟶(Base‘𝐻))
4		cntzmhm.z	. . . . 5 ⊢ 𝑍 = (Cntz‘𝐺)
5	1, 4	cntzssv 19404	. . . 4 ⊢ (𝑍‘𝑆) ⊆ (Base‘𝐺)
6	5	sseli 3932	. . 3 ⊢ (𝐴 ∈ (𝑍‘𝑆) → 𝐴 ∈ (Base‘𝐺))
7		ffvelcdm 7076	. . 3 ⊢ ((𝐹:(Base‘𝐺)⟶(Base‘𝐻) ∧ 𝐴 ∈ (Base‘𝐺)) → (𝐹‘𝐴) ∈ (Base‘𝐻))
8	3, 6, 7	syl2an 607	. 2 ⊢ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍‘𝑆)) → (𝐹‘𝐴) ∈ (Base‘𝐻))
9		eqid 2762	. . . . . . . 8 ⊢ (+g‘𝐺) = (+g‘𝐺)
10	9, 4	cntzi 19405	. . . . . . 7 ⊢ ((𝐴 ∈ (𝑍‘𝑆) ∧ 𝑥 ∈ 𝑆) → (𝐴(+g‘𝐺)𝑥) = (𝑥(+g‘𝐺)𝐴))
11	10	adantll 726	. . . . . 6 ⊢ (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍‘𝑆)) ∧ 𝑥 ∈ 𝑆) → (𝐴(+g‘𝐺)𝑥) = (𝑥(+g‘𝐺)𝐴))
12	11	fveq2d 6885	. . . . 5 ⊢ (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍‘𝑆)) ∧ 𝑥 ∈ 𝑆) → (𝐹‘(𝐴(+g‘𝐺)𝑥)) = (𝐹‘(𝑥(+g‘𝐺)𝐴)))
13		simpll 778	. . . . . 6 ⊢ (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍‘𝑆)) ∧ 𝑥 ∈ 𝑆) → 𝐹 ∈ (𝐺 MndHom 𝐻))
14	6	ad2antlr 739	. . . . . 6 ⊢ (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍‘𝑆)) ∧ 𝑥 ∈ 𝑆) → 𝐴 ∈ (Base‘𝐺))
15	1, 4	cntzrcl 19403	. . . . . . . . 9 ⊢ (𝐴 ∈ (𝑍‘𝑆) → (𝐺 ∈ V ∧ 𝑆 ⊆ (Base‘𝐺)))
16	15	adantl 486	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍‘𝑆)) → (𝐺 ∈ V ∧ 𝑆 ⊆ (Base‘𝐺)))
17	16	simprd 500	. . . . . . 7 ⊢ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍‘𝑆)) → 𝑆 ⊆ (Base‘𝐺))
18	17	sselda 3936	. . . . . 6 ⊢ (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍‘𝑆)) ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ (Base‘𝐺))
19		eqid 2762	. . . . . . 7 ⊢ (+g‘𝐻) = (+g‘𝐻)
20	1, 9, 19	mhmlin 18857	. . . . . 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 1397	. . . . 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 18857	. . . . . 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 1397	. . . . 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 2805	. . . 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 3156	. . 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 485	. . . . 5 ⊢ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍‘𝑆)) → 𝐹:(Base‘𝐺)⟶(Base‘𝐻))
27	26	ffnd 6706	. . . 4 ⊢ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍‘𝑆)) → 𝐹 Fn (Base‘𝐺))
28		oveq2 7420	. . . . . 6 ⊢ (𝑦 = (𝐹‘𝑥) → ((𝐹‘𝐴)(+g‘𝐻)𝑦) = ((𝐹‘𝐴)(+g‘𝐻)(𝐹‘𝑥)))
29		oveq1 7419	. . . . . 6 ⊢ (𝑦 = (𝐹‘𝑥) → (𝑦(+g‘𝐻)(𝐹‘𝐴)) = ((𝐹‘𝑥)(+g‘𝐻)(𝐹‘𝐴)))
30	28, 29	eqeq12d 2778	. . . . 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 7235	. . . 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 595	. . 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 260	. 2 ⊢ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍‘𝑆)) → ∀𝑦 ∈ (𝐹 “ 𝑆)((𝐹‘𝐴)(+g‘𝐻)𝑦) = (𝑦(+g‘𝐻)(𝐹‘𝐴)))
34		imassrn 6072	. . . 4 ⊢ (𝐹 “ 𝑆) ⊆ ran 𝐹
35	26	frnd 6714	. . . 4 ⊢ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍‘𝑆)) → ran 𝐹 ⊆ (Base‘𝐻))
36	34, 35	sstrid 3947	. . 3 ⊢ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍‘𝑆)) → (𝐹 “ 𝑆) ⊆ (Base‘𝐻))
37		cntzmhm.y	. . . 4 ⊢ 𝑌 = (Cntz‘𝐻)
38	2, 19, 37	elcntz 19398	. . 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 18	. 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 725	1 ⊢ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍‘𝑆)) → (𝐹‘𝐴) ∈ (𝑌‘(𝐹 “ 𝑆)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1569   ∈ wcel 2142  ∀wral 3078  Vcvv 3454   ⊆ wss 3904  ran crn 5661   “ cima 5663   Fn wfn 6531  ⟶wf 6532  ‘cfv 6536  (class class class)co 7412  Basecbs 17275  +_gcplusg 17316   MndHom cmhm 18845  Cntzccntz 19391

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5555  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7415  df-oprab 7416  df-mpo 7417  df-map 8824  df-mhm 18847  df-cntz 19393

This theorem is used by:  cntzmhm2  19418