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Theorem cntzmhm 18536
 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 2758 . . . 4 (Base‘𝐺) = (Base‘𝐺)
2 eqid 2758 . . . 4 (Base‘𝐻) = (Base‘𝐻)
31, 2mhmf 18027 . . 3 (𝐹 ∈ (𝐺 MndHom 𝐻) → 𝐹:(Base‘𝐺)⟶(Base‘𝐻))
4 cntzmhm.z . . . . 5 𝑍 = (Cntz‘𝐺)
51, 4cntzssv 18525 . . . 4 (𝑍𝑆) ⊆ (Base‘𝐺)
65sseli 3888 . . 3 (𝐴 ∈ (𝑍𝑆) → 𝐴 ∈ (Base‘𝐺))
7 ffvelrn 6840 . . 3 ((𝐹:(Base‘𝐺)⟶(Base‘𝐻) ∧ 𝐴 ∈ (Base‘𝐺)) → (𝐹𝐴) ∈ (Base‘𝐻))
83, 6, 7syl2an 598 . 2 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍𝑆)) → (𝐹𝐴) ∈ (Base‘𝐻))
9 eqid 2758 . . . . . . . 8 (+g𝐺) = (+g𝐺)
109, 4cntzi 18526 . . . . . . 7 ((𝐴 ∈ (𝑍𝑆) ∧ 𝑥𝑆) → (𝐴(+g𝐺)𝑥) = (𝑥(+g𝐺)𝐴))
1110adantll 713 . . . . . 6 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍𝑆)) ∧ 𝑥𝑆) → (𝐴(+g𝐺)𝑥) = (𝑥(+g𝐺)𝐴))
1211fveq2d 6662 . . . . 5 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍𝑆)) ∧ 𝑥𝑆) → (𝐹‘(𝐴(+g𝐺)𝑥)) = (𝐹‘(𝑥(+g𝐺)𝐴)))
13 simpll 766 . . . . . 6 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍𝑆)) ∧ 𝑥𝑆) → 𝐹 ∈ (𝐺 MndHom 𝐻))
146ad2antlr 726 . . . . . 6 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍𝑆)) ∧ 𝑥𝑆) → 𝐴 ∈ (Base‘𝐺))
151, 4cntzrcl 18524 . . . . . . . . 9 (𝐴 ∈ (𝑍𝑆) → (𝐺 ∈ V ∧ 𝑆 ⊆ (Base‘𝐺)))
1615adantl 485 . . . . . . . 8 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍𝑆)) → (𝐺 ∈ V ∧ 𝑆 ⊆ (Base‘𝐺)))
1716simprd 499 . . . . . . 7 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍𝑆)) → 𝑆 ⊆ (Base‘𝐺))
1817sselda 3892 . . . . . 6 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍𝑆)) ∧ 𝑥𝑆) → 𝑥 ∈ (Base‘𝐺))
19 eqid 2758 . . . . . . 7 (+g𝐻) = (+g𝐻)
201, 9, 19mhmlin 18029 . . . . . 6 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (Base‘𝐺) ∧ 𝑥 ∈ (Base‘𝐺)) → (𝐹‘(𝐴(+g𝐺)𝑥)) = ((𝐹𝐴)(+g𝐻)(𝐹𝑥)))
2113, 14, 18, 20syl3anc 1368 . . . . 5 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍𝑆)) ∧ 𝑥𝑆) → (𝐹‘(𝐴(+g𝐺)𝑥)) = ((𝐹𝐴)(+g𝐻)(𝐹𝑥)))
221, 9, 19mhmlin 18029 . . . . . 6 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝐴 ∈ (Base‘𝐺)) → (𝐹‘(𝑥(+g𝐺)𝐴)) = ((𝐹𝑥)(+g𝐻)(𝐹𝐴)))
2313, 18, 14, 22syl3anc 1368 . . . . 5 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍𝑆)) ∧ 𝑥𝑆) → (𝐹‘(𝑥(+g𝐺)𝐴)) = ((𝐹𝑥)(+g𝐻)(𝐹𝐴)))
2412, 21, 233eqtr3d 2801 . . . 4 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍𝑆)) ∧ 𝑥𝑆) → ((𝐹𝐴)(+g𝐻)(𝐹𝑥)) = ((𝐹𝑥)(+g𝐻)(𝐹𝐴)))
2524ralrimiva 3113 . . 3 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍𝑆)) → ∀𝑥𝑆 ((𝐹𝐴)(+g𝐻)(𝐹𝑥)) = ((𝐹𝑥)(+g𝐻)(𝐹𝐴)))
263adantr 484 . . . . 5 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍𝑆)) → 𝐹:(Base‘𝐺)⟶(Base‘𝐻))
2726ffnd 6499 . . . 4 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍𝑆)) → 𝐹 Fn (Base‘𝐺))
28 oveq2 7158 . . . . . 6 (𝑦 = (𝐹𝑥) → ((𝐹𝐴)(+g𝐻)𝑦) = ((𝐹𝐴)(+g𝐻)(𝐹𝑥)))
29 oveq1 7157 . . . . . 6 (𝑦 = (𝐹𝑥) → (𝑦(+g𝐻)(𝐹𝐴)) = ((𝐹𝑥)(+g𝐻)(𝐹𝐴)))
3028, 29eqeq12d 2774 . . . . 5 (𝑦 = (𝐹𝑥) → (((𝐹𝐴)(+g𝐻)𝑦) = (𝑦(+g𝐻)(𝐹𝐴)) ↔ ((𝐹𝐴)(+g𝐻)(𝐹𝑥)) = ((𝐹𝑥)(+g𝐻)(𝐹𝐴))))
3130ralima 6992 . . . 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 587 . . 3 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍𝑆)) → (∀𝑦 ∈ (𝐹𝑆)((𝐹𝐴)(+g𝐻)𝑦) = (𝑦(+g𝐻)(𝐹𝐴)) ↔ ∀𝑥𝑆 ((𝐹𝐴)(+g𝐻)(𝐹𝑥)) = ((𝐹𝑥)(+g𝐻)(𝐹𝐴))))
3325, 32mpbird 260 . 2 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍𝑆)) → ∀𝑦 ∈ (𝐹𝑆)((𝐹𝐴)(+g𝐻)𝑦) = (𝑦(+g𝐻)(𝐹𝐴)))
34 imassrn 5912 . . . 4 (𝐹𝑆) ⊆ ran 𝐹
3526frnd 6505 . . . 4 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍𝑆)) → ran 𝐹 ⊆ (Base‘𝐻))
3634, 35sstrid 3903 . . 3 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍𝑆)) → (𝐹𝑆) ⊆ (Base‘𝐻))
37 cntzmhm.y . . . 4 𝑌 = (Cntz‘𝐻)
382, 19, 37elcntz 18519 . . 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 712 1 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍𝑆)) → (𝐹𝐴) ∈ (𝑌‘(𝐹𝑆)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3070  Vcvv 3409   ⊆ wss 3858  ran crn 5525   “ cima 5527   Fn wfn 6330  ⟶wf 6331  ‘cfv 6335  (class class class)co 7150  Basecbs 16541  +gcplusg 16623   MndHom cmhm 18020  Cntzccntz 18512 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5156  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-ov 7153  df-oprab 7154  df-mpo 7155  df-map 8418  df-mhm 18022  df-cntz 18514 This theorem is referenced by:  cntzmhm2  18537
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