Theorem mgmhmco 18953

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)) → (𝐹 ∘ 𝐺) ∈ (𝑆 MgmHom 𝑈))

Proof of Theorem mgmhmco

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mgmhmrcl 18933	. . . 4 ⊢ (𝐹 ∈ (𝑇 MgmHom 𝑈) → (𝑇 ∈ Mgm ∧ 𝑈 ∈ Mgm))
2	1	simprd 501	. . 3 ⊢ (𝐹 ∈ (𝑇 MgmHom 𝑈) → 𝑈 ∈ Mgm)
3		mgmhmrcl 18933	. . . 4 ⊢ (𝐺 ∈ (𝑆 MgmHom 𝑇) → (𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm))
4	3	simpld 500	. . 3 ⊢ (𝐺 ∈ (𝑆 MgmHom 𝑇) → 𝑆 ∈ Mgm)
5	2, 4	anim12ci 626	. 2 ⊢ ((𝐹 ∈ (𝑇 MgmHom 𝑈) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) → (𝑆 ∈ Mgm ∧ 𝑈 ∈ Mgm))
6		eqid 2763	. . . . 5 ⊢ (Base‘𝑇) = (Base‘𝑇)
7		eqid 2763	. . . . 5 ⊢ (Base‘𝑈) = (Base‘𝑈)
8	6, 7	mgmhmf 18936	. . . 4 ⊢ (𝐹 ∈ (𝑇 MgmHom 𝑈) → 𝐹:(Base‘𝑇)⟶(Base‘𝑈))
9		eqid 2763	. . . . 5 ⊢ (Base‘𝑆) = (Base‘𝑆)
10	9, 6	mgmhmf 18936	. . . 4 ⊢ (𝐺 ∈ (𝑆 MgmHom 𝑇) → 𝐺:(Base‘𝑆)⟶(Base‘𝑇))
11		fco 6764	. . . 4 ⊢ ((𝐹:(Base‘𝑇)⟶(Base‘𝑈) ∧ 𝐺:(Base‘𝑆)⟶(Base‘𝑇)) → (𝐹 ∘ 𝐺):(Base‘𝑆)⟶(Base‘𝑈))
12	8, 10, 11	syl2an 608	. . 3 ⊢ ((𝐹 ∈ (𝑇 MgmHom 𝑈) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) → (𝐹 ∘ 𝐺):(Base‘𝑆)⟶(Base‘𝑈))
13		eqid 2763	. . . . . . . . . 10 ⊢ (+g‘𝑆) = (+g‘𝑆)
14		eqid 2763	. . . . . . . . . 10 ⊢ (+g‘𝑇) = (+g‘𝑇)
15	9, 13, 14	mgmhmlin 18938	. . . . . . . . 9 ⊢ ((𝐺 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐺‘(𝑥(+g‘𝑆)𝑦)) = ((𝐺‘𝑥)(+g‘𝑇)(𝐺‘𝑦)))
16	15	3expb 1138	. . . . . . . 8 ⊢ ((𝐺 ∈ (𝑆 MgmHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐺‘(𝑥(+g‘𝑆)𝑦)) = ((𝐺‘𝑥)(+g‘𝑇)(𝐺‘𝑦)))
17	16	adantll 727	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑇 MgmHom 𝑈) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐺‘(𝑥(+g‘𝑆)𝑦)) = ((𝐺‘𝑥)(+g‘𝑇)(𝐺‘𝑦)))
18	17	fveq2d 6920	. . . . . 6 ⊢ (((𝐹 ∈ (𝑇 MgmHom 𝑈) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐹‘(𝐺‘(𝑥(+g‘𝑆)𝑦))) = (𝐹‘((𝐺‘𝑥)(+g‘𝑇)(𝐺‘𝑦))))
19		simpll 779	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑇 MgmHom 𝑈) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → 𝐹 ∈ (𝑇 MgmHom 𝑈))
20	10	ad2antlr 740	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑇 MgmHom 𝑈) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → 𝐺:(Base‘𝑆)⟶(Base‘𝑇))
21		simprl 783	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑇 MgmHom 𝑈) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → 𝑥 ∈ (Base‘𝑆))
22	20, 21	ffvelcdmd 7117	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑇 MgmHom 𝑈) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐺‘𝑥) ∈ (Base‘𝑇))
23		simprr 785	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑇 MgmHom 𝑈) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → 𝑦 ∈ (Base‘𝑆))
24	20, 23	ffvelcdmd 7117	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑇 MgmHom 𝑈) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐺‘𝑦) ∈ (Base‘𝑇))
25		eqid 2763	. . . . . . . 8 ⊢ (+g‘𝑈) = (+g‘𝑈)
26	6, 14, 25	mgmhmlin 18938	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑇 MgmHom 𝑈) ∧ (𝐺‘𝑥) ∈ (Base‘𝑇) ∧ (𝐺‘𝑦) ∈ (Base‘𝑇)) → (𝐹‘((𝐺‘𝑥)(+g‘𝑇)(𝐺‘𝑦))) = ((𝐹‘(𝐺‘𝑥))(+g‘𝑈)(𝐹‘(𝐺‘𝑦))))
27	19, 22, 24, 26	syl3anc 1399	. . . . . 6 ⊢ (((𝐹 ∈ (𝑇 MgmHom 𝑈) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐹‘((𝐺‘𝑥)(+g‘𝑇)(𝐺‘𝑦))) = ((𝐹‘(𝐺‘𝑥))(+g‘𝑈)(𝐹‘(𝐺‘𝑦))))
28	18, 27	eqtrd 2798	. . . . 5 ⊢ (((𝐹 ∈ (𝑇 MgmHom 𝑈) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐹‘(𝐺‘(𝑥(+g‘𝑆)𝑦))) = ((𝐹‘(𝐺‘𝑥))(+g‘𝑈)(𝐹‘(𝐺‘𝑦))))
29	4	adantl 487	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑇 MgmHom 𝑈) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) → 𝑆 ∈ Mgm)
30	9, 13	mgmcl 18869	. . . . . . . 8 ⊢ ((𝑆 ∈ Mgm ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑥(+g‘𝑆)𝑦) ∈ (Base‘𝑆))
31	30	3expb 1138	. . . . . . 7 ⊢ ((𝑆 ∈ Mgm ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝑥(+g‘𝑆)𝑦) ∈ (Base‘𝑆))
32	29, 31	sylan 592	. . . . . 6 ⊢ (((𝐹 ∈ (𝑇 MgmHom 𝑈) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝑥(+g‘𝑆)𝑦) ∈ (Base‘𝑆))
33		fvco3 7016	. . . . . 6 ⊢ ((𝐺:(Base‘𝑆)⟶(Base‘𝑇) ∧ (𝑥(+g‘𝑆)𝑦) ∈ (Base‘𝑆)) → ((𝐹 ∘ 𝐺)‘(𝑥(+g‘𝑆)𝑦)) = (𝐹‘(𝐺‘(𝑥(+g‘𝑆)𝑦))))
34	20, 32, 33	syl2anc 596	. . . . 5 ⊢ (((𝐹 ∈ (𝑇 MgmHom 𝑈) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → ((𝐹 ∘ 𝐺)‘(𝑥(+g‘𝑆)𝑦)) = (𝐹‘(𝐺‘(𝑥(+g‘𝑆)𝑦))))
35		fvco3 7016	. . . . . . 7 ⊢ ((𝐺:(Base‘𝑆)⟶(Base‘𝑇) ∧ 𝑥 ∈ (Base‘𝑆)) → ((𝐹 ∘ 𝐺)‘𝑥) = (𝐹‘(𝐺‘𝑥)))
36	20, 21, 35	syl2anc 596	. . . . . 6 ⊢ (((𝐹 ∈ (𝑇 MgmHom 𝑈) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → ((𝐹 ∘ 𝐺)‘𝑥) = (𝐹‘(𝐺‘𝑥)))
37		fvco3 7016	. . . . . . 7 ⊢ ((𝐺:(Base‘𝑆)⟶(Base‘𝑇) ∧ 𝑦 ∈ (Base‘𝑆)) → ((𝐹 ∘ 𝐺)‘𝑦) = (𝐹‘(𝐺‘𝑦)))
38	20, 23, 37	syl2anc 596	. . . . . 6 ⊢ (((𝐹 ∈ (𝑇 MgmHom 𝑈) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → ((𝐹 ∘ 𝐺)‘𝑦) = (𝐹‘(𝐺‘𝑦)))
39	36, 38	oveq12d 7469	. . . . 5 ⊢ (((𝐹 ∈ (𝑇 MgmHom 𝑈) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (((𝐹 ∘ 𝐺)‘𝑥)(+g‘𝑈)((𝐹 ∘ 𝐺)‘𝑦)) = ((𝐹‘(𝐺‘𝑥))(+g‘𝑈)(𝐹‘(𝐺‘𝑦))))
40	28, 34, 39	3eqtr4d 2808	. . . 4 ⊢ (((𝐹 ∈ (𝑇 MgmHom 𝑈) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → ((𝐹 ∘ 𝐺)‘(𝑥(+g‘𝑆)𝑦)) = (((𝐹 ∘ 𝐺)‘𝑥)(+g‘𝑈)((𝐹 ∘ 𝐺)‘𝑦)))
41	40	ralrimivva 3209	. . 3 ⊢ ((𝐹 ∈ (𝑇 MgmHom 𝑈) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) → ∀𝑥 ∈ (Base‘𝑆)∀𝑦 ∈ (Base‘𝑆)((𝐹 ∘ 𝐺)‘(𝑥(+g‘𝑆)𝑦)) = (((𝐹 ∘ 𝐺)‘𝑥)(+g‘𝑈)((𝐹 ∘ 𝐺)‘𝑦)))
42	12, 41	jca 521	. 2 ⊢ ((𝐹 ∈ (𝑇 MgmHom 𝑈) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) → ((𝐹 ∘ 𝐺):(Base‘𝑆)⟶(Base‘𝑈) ∧ ∀𝑥 ∈ (Base‘𝑆)∀𝑦 ∈ (Base‘𝑆)((𝐹 ∘ 𝐺)‘(𝑥(+g‘𝑆)𝑦)) = (((𝐹 ∘ 𝐺)‘𝑥)(+g‘𝑈)((𝐹 ∘ 𝐺)‘𝑦))))
43	9, 7, 13, 25	ismgmhm 18935	. 2 ⊢ ((𝐹 ∘ 𝐺) ∈ (𝑆 MgmHom 𝑈) ↔ ((𝑆 ∈ Mgm ∧ 𝑈 ∈ Mgm) ∧ ((𝐹 ∘ 𝐺):(Base‘𝑆)⟶(Base‘𝑈) ∧ ∀𝑥 ∈ (Base‘𝑆)∀𝑦 ∈ (Base‘𝑆)((𝐹 ∘ 𝐺)‘(𝑥(+g‘𝑆)𝑦)) = (((𝐹 ∘ 𝐺)‘𝑥)(+g‘𝑈)((𝐹 ∘ 𝐺)‘𝑦)))))
44	5, 42, 43	sylanbrc 595	1 ⊢ ((𝐹 ∈ (𝑇 MgmHom 𝑈) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) → (𝐹 ∘ 𝐺) ∈ (𝑆 MgmHom 𝑈))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∀wral 3080   ∘ ccom 5662  ⟶wf 6562  ‘cfv 6566  (class class class)co 7451  Basecbs 17431  +_gcplusg 17472  Mgmcmgm 18864   MgmHom cmgmhm 18929

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-fv 6574  df-ov 7454  df-oprab 7455  df-mpo 7456  df-map 8887  df-mgm 18866  df-mgmhm 18931

This theorem is used by:  rnghmco  20738