Theorem mhmco 18789

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 18754	. . 3 ⊢ (𝐹 ∈ (𝑇 MndHom 𝑈) → 𝑈 ∈ Mnd)
2		mhmrcl1 18753	. . 3 ⊢ (𝐺 ∈ (𝑆 MndHom 𝑇) → 𝑆 ∈ Mnd)
3	1, 2	anim12ci 620	. 2 ⊢ ((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → (𝑆 ∈ Mnd ∧ 𝑈 ∈ Mnd))
4		eqid 2740	. . . . 5 ⊢ (Base‘𝑇) = (Base‘𝑇)
5		eqid 2740	. . . . 5 ⊢ (Base‘𝑈) = (Base‘𝑈)
6	4, 5	mhmf 18755	. . . 4 ⊢ (𝐹 ∈ (𝑇 MndHom 𝑈) → 𝐹:(Base‘𝑇)⟶(Base‘𝑈))
7		eqid 2740	. . . . 5 ⊢ (Base‘𝑆) = (Base‘𝑆)
8	7, 4	mhmf 18755	. . . 4 ⊢ (𝐺 ∈ (𝑆 MndHom 𝑇) → 𝐺:(Base‘𝑆)⟶(Base‘𝑇))
9		fco 6686	. . . 4 ⊢ ((𝐹:(Base‘𝑇)⟶(Base‘𝑈) ∧ 𝐺:(Base‘𝑆)⟶(Base‘𝑇)) → (𝐹 ∘ 𝐺):(Base‘𝑆)⟶(Base‘𝑈))
10	6, 8, 9	syl2an 602	. . 3 ⊢ ((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → (𝐹 ∘ 𝐺):(Base‘𝑆)⟶(Base‘𝑈))
11		eqid 2740	. . . . . . . . . 10 ⊢ (+g‘𝑆) = (+g‘𝑆)
12		eqid 2740	. . . . . . . . . 10 ⊢ (+g‘𝑇) = (+g‘𝑇)
13	7, 11, 12	mhmlin 18759	. . . . . . . . 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 1126	. . . . . . . 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 720	. . . . . . 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 6838	. . . . . 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 772	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → 𝐹 ∈ (𝑇 MndHom 𝑈))
18	8	ad2antlr 733	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → 𝐺:(Base‘𝑆)⟶(Base‘𝑇))
19		simprl 776	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → 𝑥 ∈ (Base‘𝑆))
20	18, 19	ffvelcdmd 7033	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐺‘𝑥) ∈ (Base‘𝑇))
21		simprr 778	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → 𝑦 ∈ (Base‘𝑆))
22	18, 21	ffvelcdmd 7033	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐺‘𝑦) ∈ (Base‘𝑇))
23		eqid 2740	. . . . . . . 8 ⊢ (+g‘𝑈) = (+g‘𝑈)
24	4, 12, 23	mhmlin 18759	. . . . . . 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 1379	. . . . . 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 2775	. . . . 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 482	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → 𝑆 ∈ Mnd)
28	7, 11	mndcl 18708	. . . . . . . 8 ⊢ ((𝑆 ∈ Mnd ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑥(+g‘𝑆)𝑦) ∈ (Base‘𝑆))
29	28	3expb 1126	. . . . . . 7 ⊢ ((𝑆 ∈ Mnd ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝑥(+g‘𝑆)𝑦) ∈ (Base‘𝑆))
30	27, 29	sylan 586	. . . . . 6 ⊢ (((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝑥(+g‘𝑆)𝑦) ∈ (Base‘𝑆))
31		fvco3 6934	. . . . . 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 590	. . . . 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 6934	. . . . . . 7 ⊢ ((𝐺:(Base‘𝑆)⟶(Base‘𝑇) ∧ 𝑥 ∈ (Base‘𝑆)) → ((𝐹 ∘ 𝐺)‘𝑥) = (𝐹‘(𝐺‘𝑥)))
34	18, 19, 33	syl2anc 590	. . . . . 6 ⊢ (((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → ((𝐹 ∘ 𝐺)‘𝑥) = (𝐹‘(𝐺‘𝑥)))
35		fvco3 6934	. . . . . . 7 ⊢ ((𝐺:(Base‘𝑆)⟶(Base‘𝑇) ∧ 𝑦 ∈ (Base‘𝑆)) → ((𝐹 ∘ 𝐺)‘𝑦) = (𝐹‘(𝐺‘𝑦)))
36	18, 21, 35	syl2anc 590	. . . . . 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 7381	. . . . 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 2785	. . . 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 3183	. . 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 482	. . . . 5 ⊢ ((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → 𝐺:(Base‘𝑆)⟶(Base‘𝑇))
41		eqid 2740	. . . . . . 7 ⊢ (0g‘𝑆) = (0g‘𝑆)
42	7, 41	mndidcl 18715	. . . . . 6 ⊢ (𝑆 ∈ Mnd → (0g‘𝑆) ∈ (Base‘𝑆))
43	27, 42	syl 17	. . . . 5 ⊢ ((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → (0g‘𝑆) ∈ (Base‘𝑆))
44		fvco3 6934	. . . . 5 ⊢ ((𝐺:(Base‘𝑆)⟶(Base‘𝑇) ∧ (0g‘𝑆) ∈ (Base‘𝑆)) → ((𝐹 ∘ 𝐺)‘(0g‘𝑆)) = (𝐹‘(𝐺‘(0g‘𝑆))))
45	40, 43, 44	syl2anc 590	. . . 4 ⊢ ((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → ((𝐹 ∘ 𝐺)‘(0g‘𝑆)) = (𝐹‘(𝐺‘(0g‘𝑆))))
46		eqid 2740	. . . . . . 7 ⊢ (0g‘𝑇) = (0g‘𝑇)
47	41, 46	mhm0 18760	. . . . . 6 ⊢ (𝐺 ∈ (𝑆 MndHom 𝑇) → (𝐺‘(0g‘𝑆)) = (0g‘𝑇))
48	47	adantl 482	. . . . 5 ⊢ ((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → (𝐺‘(0g‘𝑆)) = (0g‘𝑇))
49	48	fveq2d 6838	. . . 4 ⊢ ((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → (𝐹‘(𝐺‘(0g‘𝑆))) = (𝐹‘(0g‘𝑇)))
50		eqid 2740	. . . . . 6 ⊢ (0g‘𝑈) = (0g‘𝑈)
51	46, 50	mhm0 18760	. . . . 5 ⊢ (𝐹 ∈ (𝑇 MndHom 𝑈) → (𝐹‘(0g‘𝑇)) = (0g‘𝑈))
52	51	adantr 481	. . . 4 ⊢ ((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → (𝐹‘(0g‘𝑇)) = (0g‘𝑈))
53	45, 49, 52	3eqtrd 2779	. . 3 ⊢ ((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → ((𝐹 ∘ 𝐺)‘(0g‘𝑆)) = (0g‘𝑈))
54	10, 39, 53	3jca 1134	. 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 18751	. 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 589	1 ⊢ ((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → (𝐹 ∘ 𝐺) ∈ (𝑆 MndHom 𝑈))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  ∀wral 3054   ∘ ccom 5629  ⟶wf 6488  ‘cfv 6492  (class class class)co 7363  Basecbs 17177  +_gcplusg 17218  0_gc0g 17400  Mndcmnd 18700   MndHom cmhm 18747

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rmo 3345  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500  df-riota 7320  df-ov 7366  df-oprab 7367  df-mpo 7368  df-map 8772  df-0g 17402  df-mgm 18606  df-sgrp 18685  df-mnd 18701  df-mhm 18749

This theorem is referenced by:  ghmco  19209  rhmco  20479  zrhpsgnmhm  21566  lgseisenlem4  27366