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Theorem cntzmhm 19199
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 2732 . . . 4 (Base‘𝐺) = (Base‘𝐺)
2 eqid 2732 . . . 4 (Base‘𝐻) = (Base‘𝐻)
31, 2mhmf 18673 . . 3 (𝐹 ∈ (𝐺 MndHom 𝐻) → 𝐹:(Base‘𝐺)⟶(Base‘𝐻))
4 cntzmhm.z . . . . 5 𝑍 = (Cntz‘𝐺)
51, 4cntzssv 19186 . . . 4 (𝑍𝑆) ⊆ (Base‘𝐺)
65sseli 3977 . . 3 (𝐴 ∈ (𝑍𝑆) → 𝐴 ∈ (Base‘𝐺))
7 ffvelcdm 7080 . . 3 ((𝐹:(Base‘𝐺)⟶(Base‘𝐻) ∧ 𝐴 ∈ (Base‘𝐺)) → (𝐹𝐴) ∈ (Base‘𝐻))
83, 6, 7syl2an 596 . 2 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍𝑆)) → (𝐹𝐴) ∈ (Base‘𝐻))
9 eqid 2732 . . . . . . . 8 (+g𝐺) = (+g𝐺)
109, 4cntzi 19187 . . . . . . 7 ((𝐴 ∈ (𝑍𝑆) ∧ 𝑥𝑆) → (𝐴(+g𝐺)𝑥) = (𝑥(+g𝐺)𝐴))
1110adantll 712 . . . . . 6 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍𝑆)) ∧ 𝑥𝑆) → (𝐴(+g𝐺)𝑥) = (𝑥(+g𝐺)𝐴))
1211fveq2d 6892 . . . . 5 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍𝑆)) ∧ 𝑥𝑆) → (𝐹‘(𝐴(+g𝐺)𝑥)) = (𝐹‘(𝑥(+g𝐺)𝐴)))
13 simpll 765 . . . . . 6 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍𝑆)) ∧ 𝑥𝑆) → 𝐹 ∈ (𝐺 MndHom 𝐻))
146ad2antlr 725 . . . . . 6 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍𝑆)) ∧ 𝑥𝑆) → 𝐴 ∈ (Base‘𝐺))
151, 4cntzrcl 19185 . . . . . . . . 9 (𝐴 ∈ (𝑍𝑆) → (𝐺 ∈ V ∧ 𝑆 ⊆ (Base‘𝐺)))
1615adantl 482 . . . . . . . 8 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍𝑆)) → (𝐺 ∈ V ∧ 𝑆 ⊆ (Base‘𝐺)))
1716simprd 496 . . . . . . 7 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍𝑆)) → 𝑆 ⊆ (Base‘𝐺))
1817sselda 3981 . . . . . 6 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍𝑆)) ∧ 𝑥𝑆) → 𝑥 ∈ (Base‘𝐺))
19 eqid 2732 . . . . . . 7 (+g𝐻) = (+g𝐻)
201, 9, 19mhmlin 18675 . . . . . 6 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (Base‘𝐺) ∧ 𝑥 ∈ (Base‘𝐺)) → (𝐹‘(𝐴(+g𝐺)𝑥)) = ((𝐹𝐴)(+g𝐻)(𝐹𝑥)))
2113, 14, 18, 20syl3anc 1371 . . . . 5 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍𝑆)) ∧ 𝑥𝑆) → (𝐹‘(𝐴(+g𝐺)𝑥)) = ((𝐹𝐴)(+g𝐻)(𝐹𝑥)))
221, 9, 19mhmlin 18675 . . . . . 6 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝐴 ∈ (Base‘𝐺)) → (𝐹‘(𝑥(+g𝐺)𝐴)) = ((𝐹𝑥)(+g𝐻)(𝐹𝐴)))
2313, 18, 14, 22syl3anc 1371 . . . . 5 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍𝑆)) ∧ 𝑥𝑆) → (𝐹‘(𝑥(+g𝐺)𝐴)) = ((𝐹𝑥)(+g𝐻)(𝐹𝐴)))
2412, 21, 233eqtr3d 2780 . . . 4 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍𝑆)) ∧ 𝑥𝑆) → ((𝐹𝐴)(+g𝐻)(𝐹𝑥)) = ((𝐹𝑥)(+g𝐻)(𝐹𝐴)))
2524ralrimiva 3146 . . 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))
2726ffnd 6715 . . . 4 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍𝑆)) → 𝐹 Fn (Base‘𝐺))
28 oveq2 7413 . . . . . 6 (𝑦 = (𝐹𝑥) → ((𝐹𝐴)(+g𝐻)𝑦) = ((𝐹𝐴)(+g𝐻)(𝐹𝑥)))
29 oveq1 7412 . . . . . 6 (𝑦 = (𝐹𝑥) → (𝑦(+g𝐻)(𝐹𝐴)) = ((𝐹𝑥)(+g𝐻)(𝐹𝐴)))
3028, 29eqeq12d 2748 . . . . 5 (𝑦 = (𝐹𝑥) → (((𝐹𝐴)(+g𝐻)𝑦) = (𝑦(+g𝐻)(𝐹𝐴)) ↔ ((𝐹𝐴)(+g𝐻)(𝐹𝑥)) = ((𝐹𝑥)(+g𝐻)(𝐹𝐴))))
3130ralima 7236 . . . 4 ((𝐹 Fn (Base‘𝐺) ∧ 𝑆 ⊆ (Base‘𝐺)) → (∀𝑦 ∈ (𝐹𝑆)((𝐹𝐴)(+g𝐻)𝑦) = (𝑦(+g𝐻)(𝐹𝐴)) ↔ ∀𝑥𝑆 ((𝐹𝐴)(+g𝐻)(𝐹𝑥)) = ((𝐹𝑥)(+g𝐻)(𝐹𝐴))))
3227, 17, 31syl2anc 584 . . 3 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍𝑆)) → (∀𝑦 ∈ (𝐹𝑆)((𝐹𝐴)(+g𝐻)𝑦) = (𝑦(+g𝐻)(𝐹𝐴)) ↔ ∀𝑥𝑆 ((𝐹𝐴)(+g𝐻)(𝐹𝑥)) = ((𝐹𝑥)(+g𝐻)(𝐹𝐴))))
3325, 32mpbird 256 . 2 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍𝑆)) → ∀𝑦 ∈ (𝐹𝑆)((𝐹𝐴)(+g𝐻)𝑦) = (𝑦(+g𝐻)(𝐹𝐴)))
34 imassrn 6068 . . . 4 (𝐹𝑆) ⊆ ran 𝐹
3526frnd 6722 . . . 4 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍𝑆)) → ran 𝐹 ⊆ (Base‘𝐻))
3634, 35sstrid 3992 . . 3 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍𝑆)) → (𝐹𝑆) ⊆ (Base‘𝐻))
37 cntzmhm.y . . . 4 𝑌 = (Cntz‘𝐻)
382, 19, 37elcntz 19180 . . 3 ((𝐹𝑆) ⊆ (Base‘𝐻) → ((𝐹𝐴) ∈ (𝑌‘(𝐹𝑆)) ↔ ((𝐹𝐴) ∈ (Base‘𝐻) ∧ ∀𝑦 ∈ (𝐹𝑆)((𝐹𝐴)(+g𝐻)𝑦) = (𝑦(+g𝐻)(𝐹𝐴)))))
3936, 38syl 17 . 2 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍𝑆)) → ((𝐹𝐴) ∈ (𝑌‘(𝐹𝑆)) ↔ ((𝐹𝐴) ∈ (Base‘𝐻) ∧ ∀𝑦 ∈ (𝐹𝑆)((𝐹𝐴)(+g𝐻)𝑦) = (𝑦(+g𝐻)(𝐹𝐴)))))
408, 33, 39mpbir2and 711 1 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍𝑆)) → (𝐹𝐴) ∈ (𝑌‘(𝐹𝑆)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wral 3061  Vcvv 3474  wss 3947  ran crn 5676  cima 5678   Fn wfn 6535  wf 6536  cfv 6540  (class class class)co 7405  Basecbs 17140  +gcplusg 17193   MndHom cmhm 18665  Cntzccntz 19173
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410  df-map 8818  df-mhm 18667  df-cntz 19175
This theorem is referenced by:  cntzmhm2  19200
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