Theorem mhmco 13695

Description: The composition of monoid homomorphisms is a homomorphism. (Contributed by Mario Carneiro, 12-Jun-2015.)

Assertion

Ref	Expression
mhmco	⊢ ((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → (𝐹 ∘ 𝐺) ∈ (𝑆 MndHom 𝑈))

Proof of Theorem mhmco

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

Step	Hyp	Ref	Expression
1		mhmrcl2 13669	. . 3 ⊢ (𝐹 ∈ (𝑇 MndHom 𝑈) → 𝑈 ∈ Mnd)
2		mhmrcl1 13668	. . 3 ⊢ (𝐺 ∈ (𝑆 MndHom 𝑇) → 𝑆 ∈ Mnd)
3	1, 2	anim12ci 339	. 2 ⊢ ((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → (𝑆 ∈ Mnd ∧ 𝑈 ∈ Mnd))
4		eqid 2232	. . . . 5 ⊢ (Base‘𝑇) = (Base‘𝑇)
5		eqid 2232	. . . . 5 ⊢ (Base‘𝑈) = (Base‘𝑈)
6	4, 5	mhmf 13670	. . . 4 ⊢ (𝐹 ∈ (𝑇 MndHom 𝑈) → 𝐹:(Base‘𝑇)⟶(Base‘𝑈))
7		eqid 2232	. . . . 5 ⊢ (Base‘𝑆) = (Base‘𝑆)
8	7, 4	mhmf 13670	. . . 4 ⊢ (𝐺 ∈ (𝑆 MndHom 𝑇) → 𝐺:(Base‘𝑆)⟶(Base‘𝑇))
9		fco 5526	. . . 4 ⊢ ((𝐹:(Base‘𝑇)⟶(Base‘𝑈) ∧ 𝐺:(Base‘𝑆)⟶(Base‘𝑇)) → (𝐹 ∘ 𝐺):(Base‘𝑆)⟶(Base‘𝑈))
10	6, 8, 9	syl2an 289	. . 3 ⊢ ((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → (𝐹 ∘ 𝐺):(Base‘𝑆)⟶(Base‘𝑈))
11		eqid 2232	. . . . . . . . . 10 ⊢ (+g‘𝑆) = (+g‘𝑆)
12		eqid 2232	. . . . . . . . . 10 ⊢ (+g‘𝑇) = (+g‘𝑇)
13	7, 11, 12	mhmlin 13672	. . . . . . . . 9 ⊢ ((𝐺 ∈ (𝑆 MndHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐺‘(𝑥(+g‘𝑆)𝑦)) = ((𝐺‘𝑥)(+g‘𝑇)(𝐺‘𝑦)))
14	13	3expb 1231	. . . . . . . 8 ⊢ ((𝐺 ∈ (𝑆 MndHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐺‘(𝑥(+g‘𝑆)𝑦)) = ((𝐺‘𝑥)(+g‘𝑇)(𝐺‘𝑦)))
15	14	adantll 476	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐺‘(𝑥(+g‘𝑆)𝑦)) = ((𝐺‘𝑥)(+g‘𝑇)(𝐺‘𝑦)))
16	15	fveq2d 5673	. . . . . 6 ⊢ (((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐹‘(𝐺‘(𝑥(+g‘𝑆)𝑦))) = (𝐹‘((𝐺‘𝑥)(+g‘𝑇)(𝐺‘𝑦))))
17		simpll 527	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → 𝐹 ∈ (𝑇 MndHom 𝑈))
18	8	ad2antlr 489	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → 𝐺:(Base‘𝑆)⟶(Base‘𝑇))
19		simprl 531	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → 𝑥 ∈ (Base‘𝑆))
20	18, 19	ffvelcdmd 5812	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐺‘𝑥) ∈ (Base‘𝑇))
21		simprr 533	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → 𝑦 ∈ (Base‘𝑆))
22	18, 21	ffvelcdmd 5812	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐺‘𝑦) ∈ (Base‘𝑇))
23		eqid 2232	. . . . . . . 8 ⊢ (+g‘𝑈) = (+g‘𝑈)
24	4, 12, 23	mhmlin 13672	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ (𝐺‘𝑥) ∈ (Base‘𝑇) ∧ (𝐺‘𝑦) ∈ (Base‘𝑇)) → (𝐹‘((𝐺‘𝑥)(+g‘𝑇)(𝐺‘𝑦))) = ((𝐹‘(𝐺‘𝑥))(+g‘𝑈)(𝐹‘(𝐺‘𝑦))))
25	17, 20, 22, 24	syl3anc 1274	. . . . . 6 ⊢ (((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐹‘((𝐺‘𝑥)(+g‘𝑇)(𝐺‘𝑦))) = ((𝐹‘(𝐺‘𝑥))(+g‘𝑈)(𝐹‘(𝐺‘𝑦))))
26	16, 25	eqtrd 2265	. . . . 5 ⊢ (((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐹‘(𝐺‘(𝑥(+g‘𝑆)𝑦))) = ((𝐹‘(𝐺‘𝑥))(+g‘𝑈)(𝐹‘(𝐺‘𝑦))))
27	2	adantl 277	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → 𝑆 ∈ Mnd)
28	7, 11	mndcl 13628	. . . . . . . 8 ⊢ ((𝑆 ∈ Mnd ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑥(+g‘𝑆)𝑦) ∈ (Base‘𝑆))
29	28	3expb 1231	. . . . . . 7 ⊢ ((𝑆 ∈ Mnd ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝑥(+g‘𝑆)𝑦) ∈ (Base‘𝑆))
30	27, 29	sylan 283	. . . . . 6 ⊢ (((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝑥(+g‘𝑆)𝑦) ∈ (Base‘𝑆))
31		fvco3 5747	. . . . . 6 ⊢ ((𝐺:(Base‘𝑆)⟶(Base‘𝑇) ∧ (𝑥(+g‘𝑆)𝑦) ∈ (Base‘𝑆)) → ((𝐹 ∘ 𝐺)‘(𝑥(+g‘𝑆)𝑦)) = (𝐹‘(𝐺‘(𝑥(+g‘𝑆)𝑦))))
32	18, 30, 31	syl2anc 411	. . . . 5 ⊢ (((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → ((𝐹 ∘ 𝐺)‘(𝑥(+g‘𝑆)𝑦)) = (𝐹‘(𝐺‘(𝑥(+g‘𝑆)𝑦))))
33		fvco3 5747	. . . . . . 7 ⊢ ((𝐺:(Base‘𝑆)⟶(Base‘𝑇) ∧ 𝑥 ∈ (Base‘𝑆)) → ((𝐹 ∘ 𝐺)‘𝑥) = (𝐹‘(𝐺‘𝑥)))
34	18, 19, 33	syl2anc 411	. . . . . 6 ⊢ (((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → ((𝐹 ∘ 𝐺)‘𝑥) = (𝐹‘(𝐺‘𝑥)))
35		fvco3 5747	. . . . . . 7 ⊢ ((𝐺:(Base‘𝑆)⟶(Base‘𝑇) ∧ 𝑦 ∈ (Base‘𝑆)) → ((𝐹 ∘ 𝐺)‘𝑦) = (𝐹‘(𝐺‘𝑦)))
36	18, 21, 35	syl2anc 411	. . . . . 6 ⊢ (((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → ((𝐹 ∘ 𝐺)‘𝑦) = (𝐹‘(𝐺‘𝑦)))
37	34, 36	oveq12d 6067	. . . . 5 ⊢ (((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (((𝐹 ∘ 𝐺)‘𝑥)(+g‘𝑈)((𝐹 ∘ 𝐺)‘𝑦)) = ((𝐹‘(𝐺‘𝑥))(+g‘𝑈)(𝐹‘(𝐺‘𝑦))))
38	26, 32, 37	3eqtr4d 2275	. . . 4 ⊢ (((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → ((𝐹 ∘ 𝐺)‘(𝑥(+g‘𝑆)𝑦)) = (((𝐹 ∘ 𝐺)‘𝑥)(+g‘𝑈)((𝐹 ∘ 𝐺)‘𝑦)))
39	38	ralrimivva 2624	. . 3 ⊢ ((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → ∀𝑥 ∈ (Base‘𝑆)∀𝑦 ∈ (Base‘𝑆)((𝐹 ∘ 𝐺)‘(𝑥(+g‘𝑆)𝑦)) = (((𝐹 ∘ 𝐺)‘𝑥)(+g‘𝑈)((𝐹 ∘ 𝐺)‘𝑦)))
40	8	adantl 277	. . . . 5 ⊢ ((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → 𝐺:(Base‘𝑆)⟶(Base‘𝑇))
41		eqid 2232	. . . . . . 7 ⊢ (0g‘𝑆) = (0g‘𝑆)
42	7, 41	mndidcl 13635	. . . . . 6 ⊢ (𝑆 ∈ Mnd → (0g‘𝑆) ∈ (Base‘𝑆))
43	27, 42	syl 14	. . . . 5 ⊢ ((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → (0g‘𝑆) ∈ (Base‘𝑆))
44		fvco3 5747	. . . . 5 ⊢ ((𝐺:(Base‘𝑆)⟶(Base‘𝑇) ∧ (0g‘𝑆) ∈ (Base‘𝑆)) → ((𝐹 ∘ 𝐺)‘(0g‘𝑆)) = (𝐹‘(𝐺‘(0g‘𝑆))))
45	40, 43, 44	syl2anc 411	. . . 4 ⊢ ((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → ((𝐹 ∘ 𝐺)‘(0g‘𝑆)) = (𝐹‘(𝐺‘(0g‘𝑆))))
46		eqid 2232	. . . . . . 7 ⊢ (0g‘𝑇) = (0g‘𝑇)
47	41, 46	mhm0 13673	. . . . . 6 ⊢ (𝐺 ∈ (𝑆 MndHom 𝑇) → (𝐺‘(0g‘𝑆)) = (0g‘𝑇))
48	47	adantl 277	. . . . 5 ⊢ ((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → (𝐺‘(0g‘𝑆)) = (0g‘𝑇))
49	48	fveq2d 5673	. . . 4 ⊢ ((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → (𝐹‘(𝐺‘(0g‘𝑆))) = (𝐹‘(0g‘𝑇)))
50		eqid 2232	. . . . . 6 ⊢ (0g‘𝑈) = (0g‘𝑈)
51	46, 50	mhm0 13673	. . . . 5 ⊢ (𝐹 ∈ (𝑇 MndHom 𝑈) → (𝐹‘(0g‘𝑇)) = (0g‘𝑈))
52	51	adantr 276	. . . 4 ⊢ ((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → (𝐹‘(0g‘𝑇)) = (0g‘𝑈))
53	45, 49, 52	3eqtrd 2269	. . 3 ⊢ ((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → ((𝐹 ∘ 𝐺)‘(0g‘𝑆)) = (0g‘𝑈))
54	10, 39, 53	3jca 1204	. 2 ⊢ ((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 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)) ∧ ((𝐹 ∘ 𝐺)‘(0g‘𝑆)) = (0g‘𝑈)))
55	7, 5, 11, 23, 41, 50	ismhm 13666	. 2 ⊢ ((𝐹 ∘ 𝐺) ∈ (𝑆 MndHom 𝑈) ↔ ((𝑆 ∈ Mnd ∧ 𝑈 ∈ Mnd) ∧ ((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)) ∧ ((𝐹 ∘ 𝐺)‘(0g‘𝑆)) = (0g‘𝑈))))
56	3, 54, 55	sylanbrc 417	1 ⊢ ((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → (𝐹 ∘ 𝐺) ∈ (𝑆 MndHom 𝑈))

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 1005   = wceq 1398   ∈ wcel 2203  ∀wral 2520   ∘ ccom 4752  ⟶wf 5347  ‘cfv 5351  (class class class)co 6049  Basecbs 13204  +_gcplusg 13282  0_gc0g 13461  Mndcmnd 13621   MndHom cmhm 13662

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-cnex 8217  ax-resscn 8218  ax-1re 8220  ax-addrcl 8223

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-fv 5359  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-map 6883  df-inn 9237  df-2 9295  df-ndx 13207  df-slot 13208  df-base 13210  df-plusg 13295  df-0g 13463  df-mgm 13561  df-sgrp 13607  df-mnd 13622  df-mhm 13664

This theorem is referenced by:  ghmco  13973  rhmco  14311  lgseisenlem4  15938