Theorem mgmhmco 42816

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 42796	. . . 4 ⊢ (𝐹 ∈ (𝑇 MgmHom 𝑈) → (𝑇 ∈ Mgm ∧ 𝑈 ∈ Mgm))
2	1	simprd 491	. . 3 ⊢ (𝐹 ∈ (𝑇 MgmHom 𝑈) → 𝑈 ∈ Mgm)
3		mgmhmrcl 42796	. . . 4 ⊢ (𝐺 ∈ (𝑆 MgmHom 𝑇) → (𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm))
4	3	simpld 490	. . 3 ⊢ (𝐺 ∈ (𝑆 MgmHom 𝑇) → 𝑆 ∈ Mgm)
5	2, 4	anim12ci 607	. 2 ⊢ ((𝐹 ∈ (𝑇 MgmHom 𝑈) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) → (𝑆 ∈ Mgm ∧ 𝑈 ∈ Mgm))
6		eqid 2778	. . . . 5 ⊢ (Base‘𝑇) = (Base‘𝑇)
7		eqid 2778	. . . . 5 ⊢ (Base‘𝑈) = (Base‘𝑈)
8	6, 7	mgmhmf 42799	. . . 4 ⊢ (𝐹 ∈ (𝑇 MgmHom 𝑈) → 𝐹:(Base‘𝑇)⟶(Base‘𝑈))
9		eqid 2778	. . . . 5 ⊢ (Base‘𝑆) = (Base‘𝑆)
10	9, 6	mgmhmf 42799	. . . 4 ⊢ (𝐺 ∈ (𝑆 MgmHom 𝑇) → 𝐺:(Base‘𝑆)⟶(Base‘𝑇))
11		fco 6308	. . . 4 ⊢ ((𝐹:(Base‘𝑇)⟶(Base‘𝑈) ∧ 𝐺:(Base‘𝑆)⟶(Base‘𝑇)) → (𝐹 ∘ 𝐺):(Base‘𝑆)⟶(Base‘𝑈))
12	8, 10, 11	syl2an 589	. . 3 ⊢ ((𝐹 ∈ (𝑇 MgmHom 𝑈) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) → (𝐹 ∘ 𝐺):(Base‘𝑆)⟶(Base‘𝑈))
13		eqid 2778	. . . . . . . . . 10 ⊢ (+g‘𝑆) = (+g‘𝑆)
14		eqid 2778	. . . . . . . . . 10 ⊢ (+g‘𝑇) = (+g‘𝑇)
15	9, 13, 14	mgmhmlin 42801	. . . . . . . . 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 1110	. . . . . . . 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 704	. . . . . . 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 6450	. . . . . 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 757	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑇 MgmHom 𝑈) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → 𝐹 ∈ (𝑇 MgmHom 𝑈))
20	10	ad2antlr 717	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑇 MgmHom 𝑈) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → 𝐺:(Base‘𝑆)⟶(Base‘𝑇))
21		simprl 761	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑇 MgmHom 𝑈) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → 𝑥 ∈ (Base‘𝑆))
22	20, 21	ffvelrnd 6624	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑇 MgmHom 𝑈) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐺‘𝑥) ∈ (Base‘𝑇))
23		simprr 763	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑇 MgmHom 𝑈) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → 𝑦 ∈ (Base‘𝑆))
24	20, 23	ffvelrnd 6624	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑇 MgmHom 𝑈) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐺‘𝑦) ∈ (Base‘𝑇))
25		eqid 2778	. . . . . . . 8 ⊢ (+g‘𝑈) = (+g‘𝑈)
26	6, 14, 25	mgmhmlin 42801	. . . . . . 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 1439	. . . . . 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 2814	. . . . 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 475	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑇 MgmHom 𝑈) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) → 𝑆 ∈ Mgm)
30	9, 13	mgmcl 17631	. . . . . . . 8 ⊢ ((𝑆 ∈ Mgm ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑥(+g‘𝑆)𝑦) ∈ (Base‘𝑆))
31	30	3expb 1110	. . . . . . 7 ⊢ ((𝑆 ∈ Mgm ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝑥(+g‘𝑆)𝑦) ∈ (Base‘𝑆))
32	29, 31	sylan 575	. . . . . 6 ⊢ (((𝐹 ∈ (𝑇 MgmHom 𝑈) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝑥(+g‘𝑆)𝑦) ∈ (Base‘𝑆))
33		fvco3 6535	. . . . . 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 579	. . . . 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 6535	. . . . . . 7 ⊢ ((𝐺:(Base‘𝑆)⟶(Base‘𝑇) ∧ 𝑥 ∈ (Base‘𝑆)) → ((𝐹 ∘ 𝐺)‘𝑥) = (𝐹‘(𝐺‘𝑥)))
36	20, 21, 35	syl2anc 579	. . . . . 6 ⊢ (((𝐹 ∈ (𝑇 MgmHom 𝑈) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → ((𝐹 ∘ 𝐺)‘𝑥) = (𝐹‘(𝐺‘𝑥)))
37		fvco3 6535	. . . . . . 7 ⊢ ((𝐺:(Base‘𝑆)⟶(Base‘𝑇) ∧ 𝑦 ∈ (Base‘𝑆)) → ((𝐹 ∘ 𝐺)‘𝑦) = (𝐹‘(𝐺‘𝑦)))
38	20, 23, 37	syl2anc 579	. . . . . 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 6940	. . . . 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 2824	. . . 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 3153	. . 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 507	. 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 42798	. 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 578	1 ⊢ ((𝐹 ∈ (𝑇 MgmHom 𝑈) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) → (𝐹 ∘ 𝐺) ∈ (𝑆 MgmHom 𝑈))

Syntax hints:   → wi 4   ∧ wa 386   = wceq 1601   ∈ wcel 2107  ∀wral 3090   ∘ ccom 5359  ⟶wf 6131  ‘cfv 6135  (class class class)co 6922  Basecbs 16255  +_gcplusg 16338  Mgmcmgm 17626   MgmHom cmgmhm 42792

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-br 4887  df-opab 4949  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-fv 6143  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-map 8142  df-mgm 17628  df-mgmhm 42794

This theorem is referenced by:  rnghmco  42922