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Theorem mgmhmco 44360
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 44340 . . . 4 (𝐹 ∈ (𝑇 MgmHom 𝑈) → (𝑇 ∈ Mgm ∧ 𝑈 ∈ Mgm))
21simprd 499 . . 3 (𝐹 ∈ (𝑇 MgmHom 𝑈) → 𝑈 ∈ Mgm)
3 mgmhmrcl 44340 . . . 4 (𝐺 ∈ (𝑆 MgmHom 𝑇) → (𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm))
43simpld 498 . . 3 (𝐺 ∈ (𝑆 MgmHom 𝑇) → 𝑆 ∈ Mgm)
52, 4anim12ci 616 . 2 ((𝐹 ∈ (𝑇 MgmHom 𝑈) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) → (𝑆 ∈ Mgm ∧ 𝑈 ∈ Mgm))
6 eqid 2822 . . . . 5 (Base‘𝑇) = (Base‘𝑇)
7 eqid 2822 . . . . 5 (Base‘𝑈) = (Base‘𝑈)
86, 7mgmhmf 44343 . . . 4 (𝐹 ∈ (𝑇 MgmHom 𝑈) → 𝐹:(Base‘𝑇)⟶(Base‘𝑈))
9 eqid 2822 . . . . 5 (Base‘𝑆) = (Base‘𝑆)
109, 6mgmhmf 44343 . . . 4 (𝐺 ∈ (𝑆 MgmHom 𝑇) → 𝐺:(Base‘𝑆)⟶(Base‘𝑇))
11 fco 6512 . . . 4 ((𝐹:(Base‘𝑇)⟶(Base‘𝑈) ∧ 𝐺:(Base‘𝑆)⟶(Base‘𝑇)) → (𝐹𝐺):(Base‘𝑆)⟶(Base‘𝑈))
128, 10, 11syl2an 598 . . 3 ((𝐹 ∈ (𝑇 MgmHom 𝑈) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) → (𝐹𝐺):(Base‘𝑆)⟶(Base‘𝑈))
13 eqid 2822 . . . . . . . . . 10 (+g𝑆) = (+g𝑆)
14 eqid 2822 . . . . . . . . . 10 (+g𝑇) = (+g𝑇)
159, 13, 14mgmhmlin 44345 . . . . . . . . 9 ((𝐺 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐺‘(𝑥(+g𝑆)𝑦)) = ((𝐺𝑥)(+g𝑇)(𝐺𝑦)))
16153expb 1117 . . . . . . . 8 ((𝐺 ∈ (𝑆 MgmHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐺‘(𝑥(+g𝑆)𝑦)) = ((𝐺𝑥)(+g𝑇)(𝐺𝑦)))
1716adantll 713 . . . . . . 7 (((𝐹 ∈ (𝑇 MgmHom 𝑈) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐺‘(𝑥(+g𝑆)𝑦)) = ((𝐺𝑥)(+g𝑇)(𝐺𝑦)))
1817fveq2d 6656 . . . . . 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 766 . . . . . . 7 (((𝐹 ∈ (𝑇 MgmHom 𝑈) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → 𝐹 ∈ (𝑇 MgmHom 𝑈))
2010ad2antlr 726 . . . . . . . 8 (((𝐹 ∈ (𝑇 MgmHom 𝑈) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → 𝐺:(Base‘𝑆)⟶(Base‘𝑇))
21 simprl 770 . . . . . . . 8 (((𝐹 ∈ (𝑇 MgmHom 𝑈) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → 𝑥 ∈ (Base‘𝑆))
2220, 21ffvelrnd 6834 . . . . . . 7 (((𝐹 ∈ (𝑇 MgmHom 𝑈) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐺𝑥) ∈ (Base‘𝑇))
23 simprr 772 . . . . . . . 8 (((𝐹 ∈ (𝑇 MgmHom 𝑈) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → 𝑦 ∈ (Base‘𝑆))
2420, 23ffvelrnd 6834 . . . . . . 7 (((𝐹 ∈ (𝑇 MgmHom 𝑈) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐺𝑦) ∈ (Base‘𝑇))
25 eqid 2822 . . . . . . . 8 (+g𝑈) = (+g𝑈)
266, 14, 25mgmhmlin 44345 . . . . . . 7 ((𝐹 ∈ (𝑇 MgmHom 𝑈) ∧ (𝐺𝑥) ∈ (Base‘𝑇) ∧ (𝐺𝑦) ∈ (Base‘𝑇)) → (𝐹‘((𝐺𝑥)(+g𝑇)(𝐺𝑦))) = ((𝐹‘(𝐺𝑥))(+g𝑈)(𝐹‘(𝐺𝑦))))
2719, 22, 24, 26syl3anc 1368 . . . . . 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 2857 . . . . 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 485 . . . . . . 7 ((𝐹 ∈ (𝑇 MgmHom 𝑈) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) → 𝑆 ∈ Mgm)
309, 13mgmcl 17846 . . . . . . . 8 ((𝑆 ∈ Mgm ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑥(+g𝑆)𝑦) ∈ (Base‘𝑆))
31303expb 1117 . . . . . . 7 ((𝑆 ∈ Mgm ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝑥(+g𝑆)𝑦) ∈ (Base‘𝑆))
3229, 31sylan 583 . . . . . 6 (((𝐹 ∈ (𝑇 MgmHom 𝑈) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝑥(+g𝑆)𝑦) ∈ (Base‘𝑆))
33 fvco3 6742 . . . . . 6 ((𝐺:(Base‘𝑆)⟶(Base‘𝑇) ∧ (𝑥(+g𝑆)𝑦) ∈ (Base‘𝑆)) → ((𝐹𝐺)‘(𝑥(+g𝑆)𝑦)) = (𝐹‘(𝐺‘(𝑥(+g𝑆)𝑦))))
3420, 32, 33syl2anc 587 . . . . 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 6742 . . . . . . 7 ((𝐺:(Base‘𝑆)⟶(Base‘𝑇) ∧ 𝑥 ∈ (Base‘𝑆)) → ((𝐹𝐺)‘𝑥) = (𝐹‘(𝐺𝑥)))
3620, 21, 35syl2anc 587 . . . . . 6 (((𝐹 ∈ (𝑇 MgmHom 𝑈) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → ((𝐹𝐺)‘𝑥) = (𝐹‘(𝐺𝑥)))
37 fvco3 6742 . . . . . . 7 ((𝐺:(Base‘𝑆)⟶(Base‘𝑇) ∧ 𝑦 ∈ (Base‘𝑆)) → ((𝐹𝐺)‘𝑦) = (𝐹‘(𝐺𝑦)))
3820, 23, 37syl2anc 587 . . . . . 6 (((𝐹 ∈ (𝑇 MgmHom 𝑈) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → ((𝐹𝐺)‘𝑦) = (𝐹‘(𝐺𝑦)))
3936, 38oveq12d 7158 . . . . 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 2867 . . . 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 3181 . . 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 515 . 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 44342 . 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 586 1 ((𝐹 ∈ (𝑇 MgmHom 𝑈) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) → (𝐹𝐺) ∈ (𝑆 MgmHom 𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2114  wral 3130  ccom 5536  wf 6330  cfv 6334  (class class class)co 7140  Basecbs 16474  +gcplusg 16556  Mgmcmgm 17841   MgmHom cmgmhm 44336
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 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-fv 6342  df-ov 7143  df-oprab 7144  df-mpo 7145  df-map 8395  df-mgm 17843  df-mgmhm 44338
This theorem is referenced by:  rnghmco  44470
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