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Theorem cntzmhm 17594
 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 2610 . . . 4 (Base‘𝐺) = (Base‘𝐺)
2 eqid 2610 . . . 4 (Base‘𝐻) = (Base‘𝐻)
31, 2mhmf 17163 . . 3 (𝐹 ∈ (𝐺 MndHom 𝐻) → 𝐹:(Base‘𝐺)⟶(Base‘𝐻))
4 cntzmhm.z . . . . 5 𝑍 = (Cntz‘𝐺)
51, 4cntzssv 17584 . . . 4 (𝑍𝑆) ⊆ (Base‘𝐺)
65sseli 3564 . . 3 (𝐴 ∈ (𝑍𝑆) → 𝐴 ∈ (Base‘𝐺))
7 ffvelrn 6265 . . 3 ((𝐹:(Base‘𝐺)⟶(Base‘𝐻) ∧ 𝐴 ∈ (Base‘𝐺)) → (𝐹𝐴) ∈ (Base‘𝐻))
83, 6, 7syl2an 493 . 2 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍𝑆)) → (𝐹𝐴) ∈ (Base‘𝐻))
9 eqid 2610 . . . . . . . 8 (+g𝐺) = (+g𝐺)
109, 4cntzi 17585 . . . . . . 7 ((𝐴 ∈ (𝑍𝑆) ∧ 𝑥𝑆) → (𝐴(+g𝐺)𝑥) = (𝑥(+g𝐺)𝐴))
1110adantll 746 . . . . . 6 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍𝑆)) ∧ 𝑥𝑆) → (𝐴(+g𝐺)𝑥) = (𝑥(+g𝐺)𝐴))
1211fveq2d 6107 . . . . 5 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍𝑆)) ∧ 𝑥𝑆) → (𝐹‘(𝐴(+g𝐺)𝑥)) = (𝐹‘(𝑥(+g𝐺)𝐴)))
13 simpll 786 . . . . . 6 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍𝑆)) ∧ 𝑥𝑆) → 𝐹 ∈ (𝐺 MndHom 𝐻))
146ad2antlr 759 . . . . . 6 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍𝑆)) ∧ 𝑥𝑆) → 𝐴 ∈ (Base‘𝐺))
151, 4cntzrcl 17583 . . . . . . . . 9 (𝐴 ∈ (𝑍𝑆) → (𝐺 ∈ V ∧ 𝑆 ⊆ (Base‘𝐺)))
1615adantl 481 . . . . . . . 8 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍𝑆)) → (𝐺 ∈ V ∧ 𝑆 ⊆ (Base‘𝐺)))
1716simprd 478 . . . . . . 7 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍𝑆)) → 𝑆 ⊆ (Base‘𝐺))
1817sselda 3568 . . . . . 6 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍𝑆)) ∧ 𝑥𝑆) → 𝑥 ∈ (Base‘𝐺))
19 eqid 2610 . . . . . . 7 (+g𝐻) = (+g𝐻)
201, 9, 19mhmlin 17165 . . . . . 6 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (Base‘𝐺) ∧ 𝑥 ∈ (Base‘𝐺)) → (𝐹‘(𝐴(+g𝐺)𝑥)) = ((𝐹𝐴)(+g𝐻)(𝐹𝑥)))
2113, 14, 18, 20syl3anc 1318 . . . . 5 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍𝑆)) ∧ 𝑥𝑆) → (𝐹‘(𝐴(+g𝐺)𝑥)) = ((𝐹𝐴)(+g𝐻)(𝐹𝑥)))
221, 9, 19mhmlin 17165 . . . . . 6 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝐴 ∈ (Base‘𝐺)) → (𝐹‘(𝑥(+g𝐺)𝐴)) = ((𝐹𝑥)(+g𝐻)(𝐹𝐴)))
2313, 18, 14, 22syl3anc 1318 . . . . 5 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍𝑆)) ∧ 𝑥𝑆) → (𝐹‘(𝑥(+g𝐺)𝐴)) = ((𝐹𝑥)(+g𝐻)(𝐹𝐴)))
2412, 21, 233eqtr3d 2652 . . . 4 (((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍𝑆)) ∧ 𝑥𝑆) → ((𝐹𝐴)(+g𝐻)(𝐹𝑥)) = ((𝐹𝑥)(+g𝐻)(𝐹𝐴)))
2524ralrimiva 2949 . . 3 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍𝑆)) → ∀𝑥𝑆 ((𝐹𝐴)(+g𝐻)(𝐹𝑥)) = ((𝐹𝑥)(+g𝐻)(𝐹𝐴)))
263adantr 480 . . . . 5 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍𝑆)) → 𝐹:(Base‘𝐺)⟶(Base‘𝐻))
27 ffn 5958 . . . . 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 6557 . . . . . 6 (𝑦 = (𝐹𝑥) → ((𝐹𝐴)(+g𝐻)𝑦) = ((𝐹𝐴)(+g𝐻)(𝐹𝑥)))
30 oveq1 6556 . . . . . 6 (𝑦 = (𝐹𝑥) → (𝑦(+g𝐻)(𝐹𝐴)) = ((𝐹𝑥)(+g𝐻)(𝐹𝐴)))
3129, 30eqeq12d 2625 . . . . 5 (𝑦 = (𝐹𝑥) → (((𝐹𝐴)(+g𝐻)𝑦) = (𝑦(+g𝐻)(𝐹𝐴)) ↔ ((𝐹𝐴)(+g𝐻)(𝐹𝑥)) = ((𝐹𝑥)(+g𝐻)(𝐹𝐴))))
3231ralima 6402 . . . 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 691 . . 3 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍𝑆)) → (∀𝑦 ∈ (𝐹𝑆)((𝐹𝐴)(+g𝐻)𝑦) = (𝑦(+g𝐻)(𝐹𝐴)) ↔ ∀𝑥𝑆 ((𝐹𝐴)(+g𝐻)(𝐹𝑥)) = ((𝐹𝑥)(+g𝐻)(𝐹𝐴))))
3425, 33mpbird 246 . 2 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍𝑆)) → ∀𝑦 ∈ (𝐹𝑆)((𝐹𝐴)(+g𝐻)𝑦) = (𝑦(+g𝐻)(𝐹𝐴)))
35 imassrn 5396 . . . 4 (𝐹𝑆) ⊆ ran 𝐹
36 frn 5966 . . . . 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 3579 . . 3 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍𝑆)) → (𝐹𝑆) ⊆ (Base‘𝐻))
39 cntzmhm.y . . . 4 𝑌 = (Cntz‘𝐻)
402, 19, 39elcntz 17578 . . 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 959 1 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍𝑆)) → (𝐹𝐴) ∈ (𝑌‘(𝐹𝑆)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173   ⊆ wss 3540  ran crn 5039   “ cima 5041   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  +gcplusg 15768   MndHom cmhm 17156  Cntzccntz 17571 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-map 7746  df-mhm 17158  df-cntz 17573 This theorem is referenced by:  cntzmhm2  17595
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