MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cntzmhm Structured version   Visualization version   GIF version

Theorem cntzmhm 17711
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 ∈ (𝑍𝑆)) → (𝐹𝐴) ∈ (𝑌‘(𝐹𝑆)))

Proof of Theorem cntzmhm
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2621 . . . 4 (Base‘𝐺) = (Base‘𝐺)
2 eqid 2621 . . . 4 (Base‘𝐻) = (Base‘𝐻)
31, 2mhmf 17280 . . 3 (𝐹 ∈ (𝐺 MndHom 𝐻) → 𝐹:(Base‘𝐺)⟶(Base‘𝐻))
4 cntzmhm.z . . . . 5 𝑍 = (Cntz‘𝐺)
51, 4cntzssv 17701 . . . 4 (𝑍𝑆) ⊆ (Base‘𝐺)
65sseli 3584 . . 3 (𝐴 ∈ (𝑍𝑆) → 𝐴 ∈ (Base‘𝐺))
7 ffvelrn 6323 . . 3 ((𝐹:(Base‘𝐺)⟶(Base‘𝐻) ∧ 𝐴 ∈ (Base‘𝐺)) → (𝐹𝐴) ∈ (Base‘𝐻))
83, 6, 7syl2an 494 . 2 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍𝑆)) → (𝐹𝐴) ∈ (Base‘𝐻))
9 eqid 2621 . . . . . . . 8 (+g𝐺) = (+g𝐺)
109, 4cntzi 17702 . . . . . . 7 ((𝐴 ∈ (𝑍𝑆) ∧ 𝑥𝑆) → (𝐴(+g𝐺)𝑥) = (𝑥(+g𝐺)𝐴))
1110adantll 749 . . . . . 6 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍𝑆)) ∧ 𝑥𝑆) → (𝐴(+g𝐺)𝑥) = (𝑥(+g𝐺)𝐴))
1211fveq2d 6162 . . . . 5 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍𝑆)) ∧ 𝑥𝑆) → (𝐹‘(𝐴(+g𝐺)𝑥)) = (𝐹‘(𝑥(+g𝐺)𝐴)))
13 simpll 789 . . . . . 6 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍𝑆)) ∧ 𝑥𝑆) → 𝐹 ∈ (𝐺 MndHom 𝐻))
146ad2antlr 762 . . . . . 6 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍𝑆)) ∧ 𝑥𝑆) → 𝐴 ∈ (Base‘𝐺))
151, 4cntzrcl 17700 . . . . . . . . 9 (𝐴 ∈ (𝑍𝑆) → (𝐺 ∈ V ∧ 𝑆 ⊆ (Base‘𝐺)))
1615adantl 482 . . . . . . . 8 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍𝑆)) → (𝐺 ∈ V ∧ 𝑆 ⊆ (Base‘𝐺)))
1716simprd 479 . . . . . . 7 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍𝑆)) → 𝑆 ⊆ (Base‘𝐺))
1817sselda 3588 . . . . . 6 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍𝑆)) ∧ 𝑥𝑆) → 𝑥 ∈ (Base‘𝐺))
19 eqid 2621 . . . . . . 7 (+g𝐻) = (+g𝐻)
201, 9, 19mhmlin 17282 . . . . . 6 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (Base‘𝐺) ∧ 𝑥 ∈ (Base‘𝐺)) → (𝐹‘(𝐴(+g𝐺)𝑥)) = ((𝐹𝐴)(+g𝐻)(𝐹𝑥)))
2113, 14, 18, 20syl3anc 1323 . . . . 5 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍𝑆)) ∧ 𝑥𝑆) → (𝐹‘(𝐴(+g𝐺)𝑥)) = ((𝐹𝐴)(+g𝐻)(𝐹𝑥)))
221, 9, 19mhmlin 17282 . . . . . 6 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝐴 ∈ (Base‘𝐺)) → (𝐹‘(𝑥(+g𝐺)𝐴)) = ((𝐹𝑥)(+g𝐻)(𝐹𝐴)))
2313, 18, 14, 22syl3anc 1323 . . . . 5 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍𝑆)) ∧ 𝑥𝑆) → (𝐹‘(𝑥(+g𝐺)𝐴)) = ((𝐹𝑥)(+g𝐻)(𝐹𝐴)))
2412, 21, 233eqtr3d 2663 . . . 4 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍𝑆)) ∧ 𝑥𝑆) → ((𝐹𝐴)(+g𝐻)(𝐹𝑥)) = ((𝐹𝑥)(+g𝐻)(𝐹𝐴)))
2524ralrimiva 2962 . . 3 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍𝑆)) → ∀𝑥𝑆 ((𝐹𝐴)(+g𝐻)(𝐹𝑥)) = ((𝐹𝑥)(+g𝐻)(𝐹𝐴)))
263adantr 481 . . . . 5 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍𝑆)) → 𝐹:(Base‘𝐺)⟶(Base‘𝐻))
27 ffn 6012 . . . . 5 (𝐹:(Base‘𝐺)⟶(Base‘𝐻) → 𝐹 Fn (Base‘𝐺))
2826, 27syl 17 . . . 4 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍𝑆)) → 𝐹 Fn (Base‘𝐺))
29 oveq2 6623 . . . . . 6 (𝑦 = (𝐹𝑥) → ((𝐹𝐴)(+g𝐻)𝑦) = ((𝐹𝐴)(+g𝐻)(𝐹𝑥)))
30 oveq1 6622 . . . . . 6 (𝑦 = (𝐹𝑥) → (𝑦(+g𝐻)(𝐹𝐴)) = ((𝐹𝑥)(+g𝐻)(𝐹𝐴)))
3129, 30eqeq12d 2636 . . . . 5 (𝑦 = (𝐹𝑥) → (((𝐹𝐴)(+g𝐻)𝑦) = (𝑦(+g𝐻)(𝐹𝐴)) ↔ ((𝐹𝐴)(+g𝐻)(𝐹𝑥)) = ((𝐹𝑥)(+g𝐻)(𝐹𝐴))))
3231ralima 6463 . . . 4 ((𝐹 Fn (Base‘𝐺) ∧ 𝑆 ⊆ (Base‘𝐺)) → (∀𝑦 ∈ (𝐹𝑆)((𝐹𝐴)(+g𝐻)𝑦) = (𝑦(+g𝐻)(𝐹𝐴)) ↔ ∀𝑥𝑆 ((𝐹𝐴)(+g𝐻)(𝐹𝑥)) = ((𝐹𝑥)(+g𝐻)(𝐹𝐴))))
3328, 17, 32syl2anc 692 . . 3 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍𝑆)) → (∀𝑦 ∈ (𝐹𝑆)((𝐹𝐴)(+g𝐻)𝑦) = (𝑦(+g𝐻)(𝐹𝐴)) ↔ ∀𝑥𝑆 ((𝐹𝐴)(+g𝐻)(𝐹𝑥)) = ((𝐹𝑥)(+g𝐻)(𝐹𝐴))))
3425, 33mpbird 247 . 2 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍𝑆)) → ∀𝑦 ∈ (𝐹𝑆)((𝐹𝐴)(+g𝐻)𝑦) = (𝑦(+g𝐻)(𝐹𝐴)))
35 imassrn 5446 . . . 4 (𝐹𝑆) ⊆ ran 𝐹
36 frn 6020 . . . . 5 (𝐹:(Base‘𝐺)⟶(Base‘𝐻) → ran 𝐹 ⊆ (Base‘𝐻))
3726, 36syl 17 . . . 4 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍𝑆)) → ran 𝐹 ⊆ (Base‘𝐻))
3835, 37syl5ss 3599 . . 3 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍𝑆)) → (𝐹𝑆) ⊆ (Base‘𝐻))
39 cntzmhm.y . . . 4 𝑌 = (Cntz‘𝐻)
402, 19, 39elcntz 17695 . . 3 ((𝐹𝑆) ⊆ (Base‘𝐻) → ((𝐹𝐴) ∈ (𝑌‘(𝐹𝑆)) ↔ ((𝐹𝐴) ∈ (Base‘𝐻) ∧ ∀𝑦 ∈ (𝐹𝑆)((𝐹𝐴)(+g𝐻)𝑦) = (𝑦(+g𝐻)(𝐹𝐴)))))
4138, 40syl 17 . 2 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍𝑆)) → ((𝐹𝐴) ∈ (𝑌‘(𝐹𝑆)) ↔ ((𝐹𝐴) ∈ (Base‘𝐻) ∧ ∀𝑦 ∈ (𝐹𝑆)((𝐹𝐴)(+g𝐻)𝑦) = (𝑦(+g𝐻)(𝐹𝐴)))))
428, 34, 41mpbir2and 956 1 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍𝑆)) → (𝐹𝐴) ∈ (𝑌‘(𝐹𝑆)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2908  Vcvv 3190  wss 3560  ran crn 5085  cima 5087   Fn wfn 5852  wf 5853  cfv 5857  (class class class)co 6615  Basecbs 15800  +gcplusg 15881   MndHom cmhm 17273  Cntzccntz 17688
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-map 7819  df-mhm 17275  df-cntz 17690
This theorem is referenced by:  cntzmhm2  17712
  Copyright terms: Public domain W3C validator