Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  mgmhmco Structured version   Visualization version   GIF version

Theorem mgmhmco 41591
 Description: The composition of magma homomorphisms is a homomorphism. (Contributed by AV, 27-Feb-2020.)
Assertion
Ref Expression
mgmhmco ((𝐹 ∈ (𝑇 MgmHom 𝑈) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) → (𝐹𝐺) ∈ (𝑆 MgmHom 𝑈))

Proof of Theorem mgmhmco
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mgmhmrcl 41571 . . . 4 (𝐹 ∈ (𝑇 MgmHom 𝑈) → (𝑇 ∈ Mgm ∧ 𝑈 ∈ Mgm))
21simprd 478 . . 3 (𝐹 ∈ (𝑇 MgmHom 𝑈) → 𝑈 ∈ Mgm)
3 mgmhmrcl 41571 . . . 4 (𝐺 ∈ (𝑆 MgmHom 𝑇) → (𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm))
43simpld 474 . . 3 (𝐺 ∈ (𝑆 MgmHom 𝑇) → 𝑆 ∈ Mgm)
52, 4anim12ci 589 . 2 ((𝐹 ∈ (𝑇 MgmHom 𝑈) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) → (𝑆 ∈ Mgm ∧ 𝑈 ∈ Mgm))
6 eqid 2610 . . . . 5 (Base‘𝑇) = (Base‘𝑇)
7 eqid 2610 . . . . 5 (Base‘𝑈) = (Base‘𝑈)
86, 7mgmhmf 41574 . . . 4 (𝐹 ∈ (𝑇 MgmHom 𝑈) → 𝐹:(Base‘𝑇)⟶(Base‘𝑈))
9 eqid 2610 . . . . 5 (Base‘𝑆) = (Base‘𝑆)
109, 6mgmhmf 41574 . . . 4 (𝐺 ∈ (𝑆 MgmHom 𝑇) → 𝐺:(Base‘𝑆)⟶(Base‘𝑇))
11 fco 5971 . . . 4 ((𝐹:(Base‘𝑇)⟶(Base‘𝑈) ∧ 𝐺:(Base‘𝑆)⟶(Base‘𝑇)) → (𝐹𝐺):(Base‘𝑆)⟶(Base‘𝑈))
128, 10, 11syl2an 493 . . 3 ((𝐹 ∈ (𝑇 MgmHom 𝑈) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) → (𝐹𝐺):(Base‘𝑆)⟶(Base‘𝑈))
13 eqid 2610 . . . . . . . . . 10 (+g𝑆) = (+g𝑆)
14 eqid 2610 . . . . . . . . . 10 (+g𝑇) = (+g𝑇)
159, 13, 14mgmhmlin 41576 . . . . . . . . 9 ((𝐺 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐺‘(𝑥(+g𝑆)𝑦)) = ((𝐺𝑥)(+g𝑇)(𝐺𝑦)))
16153expb 1258 . . . . . . . 8 ((𝐺 ∈ (𝑆 MgmHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐺‘(𝑥(+g𝑆)𝑦)) = ((𝐺𝑥)(+g𝑇)(𝐺𝑦)))
1716adantll 746 . . . . . . 7 (((𝐹 ∈ (𝑇 MgmHom 𝑈) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐺‘(𝑥(+g𝑆)𝑦)) = ((𝐺𝑥)(+g𝑇)(𝐺𝑦)))
1817fveq2d 6107 . . . . . 6 (((𝐹 ∈ (𝑇 MgmHom 𝑈) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐹‘(𝐺‘(𝑥(+g𝑆)𝑦))) = (𝐹‘((𝐺𝑥)(+g𝑇)(𝐺𝑦))))
19 simpll 786 . . . . . . 7 (((𝐹 ∈ (𝑇 MgmHom 𝑈) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → 𝐹 ∈ (𝑇 MgmHom 𝑈))
2010ad2antlr 759 . . . . . . . 8 (((𝐹 ∈ (𝑇 MgmHom 𝑈) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → 𝐺:(Base‘𝑆)⟶(Base‘𝑇))
21 simprl 790 . . . . . . . 8 (((𝐹 ∈ (𝑇 MgmHom 𝑈) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → 𝑥 ∈ (Base‘𝑆))
2220, 21ffvelrnd 6268 . . . . . . 7 (((𝐹 ∈ (𝑇 MgmHom 𝑈) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐺𝑥) ∈ (Base‘𝑇))
23 simprr 792 . . . . . . . 8 (((𝐹 ∈ (𝑇 MgmHom 𝑈) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → 𝑦 ∈ (Base‘𝑆))
2420, 23ffvelrnd 6268 . . . . . . 7 (((𝐹 ∈ (𝑇 MgmHom 𝑈) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐺𝑦) ∈ (Base‘𝑇))
25 eqid 2610 . . . . . . . 8 (+g𝑈) = (+g𝑈)
266, 14, 25mgmhmlin 41576 . . . . . . 7 ((𝐹 ∈ (𝑇 MgmHom 𝑈) ∧ (𝐺𝑥) ∈ (Base‘𝑇) ∧ (𝐺𝑦) ∈ (Base‘𝑇)) → (𝐹‘((𝐺𝑥)(+g𝑇)(𝐺𝑦))) = ((𝐹‘(𝐺𝑥))(+g𝑈)(𝐹‘(𝐺𝑦))))
2719, 22, 24, 26syl3anc 1318 . . . . . 6 (((𝐹 ∈ (𝑇 MgmHom 𝑈) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐹‘((𝐺𝑥)(+g𝑇)(𝐺𝑦))) = ((𝐹‘(𝐺𝑥))(+g𝑈)(𝐹‘(𝐺𝑦))))
2818, 27eqtrd 2644 . . . . 5 (((𝐹 ∈ (𝑇 MgmHom 𝑈) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐹‘(𝐺‘(𝑥(+g𝑆)𝑦))) = ((𝐹‘(𝐺𝑥))(+g𝑈)(𝐹‘(𝐺𝑦))))
294adantl 481 . . . . . . 7 ((𝐹 ∈ (𝑇 MgmHom 𝑈) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) → 𝑆 ∈ Mgm)
309, 13mgmcl 17068 . . . . . . . 8 ((𝑆 ∈ Mgm ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑥(+g𝑆)𝑦) ∈ (Base‘𝑆))
31303expb 1258 . . . . . . 7 ((𝑆 ∈ Mgm ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝑥(+g𝑆)𝑦) ∈ (Base‘𝑆))
3229, 31sylan 487 . . . . . 6 (((𝐹 ∈ (𝑇 MgmHom 𝑈) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝑥(+g𝑆)𝑦) ∈ (Base‘𝑆))
33 fvco3 6185 . . . . . 6 ((𝐺:(Base‘𝑆)⟶(Base‘𝑇) ∧ (𝑥(+g𝑆)𝑦) ∈ (Base‘𝑆)) → ((𝐹𝐺)‘(𝑥(+g𝑆)𝑦)) = (𝐹‘(𝐺‘(𝑥(+g𝑆)𝑦))))
3420, 32, 33syl2anc 691 . . . . 5 (((𝐹 ∈ (𝑇 MgmHom 𝑈) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → ((𝐹𝐺)‘(𝑥(+g𝑆)𝑦)) = (𝐹‘(𝐺‘(𝑥(+g𝑆)𝑦))))
35 fvco3 6185 . . . . . . 7 ((𝐺:(Base‘𝑆)⟶(Base‘𝑇) ∧ 𝑥 ∈ (Base‘𝑆)) → ((𝐹𝐺)‘𝑥) = (𝐹‘(𝐺𝑥)))
3620, 21, 35syl2anc 691 . . . . . 6 (((𝐹 ∈ (𝑇 MgmHom 𝑈) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → ((𝐹𝐺)‘𝑥) = (𝐹‘(𝐺𝑥)))
37 fvco3 6185 . . . . . . 7 ((𝐺:(Base‘𝑆)⟶(Base‘𝑇) ∧ 𝑦 ∈ (Base‘𝑆)) → ((𝐹𝐺)‘𝑦) = (𝐹‘(𝐺𝑦)))
3820, 23, 37syl2anc 691 . . . . . 6 (((𝐹 ∈ (𝑇 MgmHom 𝑈) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → ((𝐹𝐺)‘𝑦) = (𝐹‘(𝐺𝑦)))
3936, 38oveq12d 6567 . . . . 5 (((𝐹 ∈ (𝑇 MgmHom 𝑈) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (((𝐹𝐺)‘𝑥)(+g𝑈)((𝐹𝐺)‘𝑦)) = ((𝐹‘(𝐺𝑥))(+g𝑈)(𝐹‘(𝐺𝑦))))
4028, 34, 393eqtr4d 2654 . . . 4 (((𝐹 ∈ (𝑇 MgmHom 𝑈) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → ((𝐹𝐺)‘(𝑥(+g𝑆)𝑦)) = (((𝐹𝐺)‘𝑥)(+g𝑈)((𝐹𝐺)‘𝑦)))
4140ralrimivva 2954 . . 3 ((𝐹 ∈ (𝑇 MgmHom 𝑈) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) → ∀𝑥 ∈ (Base‘𝑆)∀𝑦 ∈ (Base‘𝑆)((𝐹𝐺)‘(𝑥(+g𝑆)𝑦)) = (((𝐹𝐺)‘𝑥)(+g𝑈)((𝐹𝐺)‘𝑦)))
4212, 41jca 553 . 2 ((𝐹 ∈ (𝑇 MgmHom 𝑈) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) → ((𝐹𝐺):(Base‘𝑆)⟶(Base‘𝑈) ∧ ∀𝑥 ∈ (Base‘𝑆)∀𝑦 ∈ (Base‘𝑆)((𝐹𝐺)‘(𝑥(+g𝑆)𝑦)) = (((𝐹𝐺)‘𝑥)(+g𝑈)((𝐹𝐺)‘𝑦))))
439, 7, 13, 25ismgmhm 41573 . 2 ((𝐹𝐺) ∈ (𝑆 MgmHom 𝑈) ↔ ((𝑆 ∈ Mgm ∧ 𝑈 ∈ Mgm) ∧ ((𝐹𝐺):(Base‘𝑆)⟶(Base‘𝑈) ∧ ∀𝑥 ∈ (Base‘𝑆)∀𝑦 ∈ (Base‘𝑆)((𝐹𝐺)‘(𝑥(+g𝑆)𝑦)) = (((𝐹𝐺)‘𝑥)(+g𝑈)((𝐹𝐺)‘𝑦)))))
445, 42, 43sylanbrc 695 1 ((𝐹 ∈ (𝑇 MgmHom 𝑈) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) → (𝐹𝐺) ∈ (𝑆 MgmHom 𝑈))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896   ∘ ccom 5042  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  +gcplusg 15768  Mgmcmgm 17063   MgmHom cmgmhm 41567 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-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-rab 2905  df-v 3175  df-sbc 3403  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-br 4584  df-opab 4644  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-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-map 7746  df-mgm 17065  df-mgmhm 41569 This theorem is referenced by:  rnghmco  41697
 Copyright terms: Public domain W3C validator