Theorem lmhmco 21090

Description: The composition of two module-linear functions is module-linear. (Contributed by Stefan O'Rear, 4-Sep-2015.)

Assertion

Ref	Expression
lmhmco	⊢ ((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) → (𝐹 ∘ 𝐺) ∈ (𝑀 LMHom 𝑂))

Proof of Theorem lmhmco

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

Step	Hyp	Ref	Expression
1		eqid 2761	. 2 ⊢ (Base‘𝑀) = (Base‘𝑀)
2		eqid 2761	. 2 ⊢ ( ·𝑠 ‘𝑀) = ( ·𝑠 ‘𝑀)
3		eqid 2761	. 2 ⊢ ( ·𝑠 ‘𝑂) = ( ·𝑠 ‘𝑂)
4		eqid 2761	. 2 ⊢ (Scalar‘𝑀) = (Scalar‘𝑀)
5		eqid 2761	. 2 ⊢ (Scalar‘𝑂) = (Scalar‘𝑂)
6		eqid 2761	. 2 ⊢ (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀))
7		lmhmlmod1 21080	. . 3 ⊢ (𝐺 ∈ (𝑀 LMHom 𝑁) → 𝑀 ∈ LMod)
8	7	adantl 485	. 2 ⊢ ((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) → 𝑀 ∈ LMod)
9		lmhmlmod2 21079	. . 3 ⊢ (𝐹 ∈ (𝑁 LMHom 𝑂) → 𝑂 ∈ LMod)
10	9	adantr 484	. 2 ⊢ ((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) → 𝑂 ∈ LMod)
11		eqid 2761	. . . 4 ⊢ (Scalar‘𝑁) = (Scalar‘𝑁)
12	11, 5	lmhmsca 21077	. . 3 ⊢ (𝐹 ∈ (𝑁 LMHom 𝑂) → (Scalar‘𝑂) = (Scalar‘𝑁))
13	4, 11	lmhmsca 21077	. . 3 ⊢ (𝐺 ∈ (𝑀 LMHom 𝑁) → (Scalar‘𝑁) = (Scalar‘𝑀))
14	12, 13	sylan9eq 2816	. 2 ⊢ ((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) → (Scalar‘𝑂) = (Scalar‘𝑀))
15		lmghm 21078	. . 3 ⊢ (𝐹 ∈ (𝑁 LMHom 𝑂) → 𝐹 ∈ (𝑁 GrpHom 𝑂))
16		lmghm 21078	. . 3 ⊢ (𝐺 ∈ (𝑀 LMHom 𝑁) → 𝐺 ∈ (𝑀 GrpHom 𝑁))
17		ghmco 19259	. . 3 ⊢ ((𝐹 ∈ (𝑁 GrpHom 𝑂) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) → (𝐹 ∘ 𝐺) ∈ (𝑀 GrpHom 𝑂))
18	15, 16, 17	syl2an 605	. 2 ⊢ ((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) → (𝐹 ∘ 𝐺) ∈ (𝑀 GrpHom 𝑂))
19		simplr 778	. . . . . 6 ⊢ (((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝐺 ∈ (𝑀 LMHom 𝑁))
20		simprl 780	. . . . . 6 ⊢ (((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝑥 ∈ (Base‘(Scalar‘𝑀)))
21		simprr 782	. . . . . 6 ⊢ (((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝑦 ∈ (Base‘𝑀))
22		eqid 2761	. . . . . . 7 ⊢ ( ·𝑠 ‘𝑁) = ( ·𝑠 ‘𝑁)
23	4, 6, 1, 2, 22	lmhmlin 21082	. . . . . 6 ⊢ ((𝐺 ∈ (𝑀 LMHom 𝑁) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝐺‘(𝑥( ·𝑠 ‘𝑀)𝑦)) = (𝑥( ·𝑠 ‘𝑁)(𝐺‘𝑦)))
24	19, 20, 21, 23	syl3anc 1389	. . . . 5 ⊢ (((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝐺‘(𝑥( ·𝑠 ‘𝑀)𝑦)) = (𝑥( ·𝑠 ‘𝑁)(𝐺‘𝑦)))
25	24	fveq2d 6867	. . . 4 ⊢ (((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝐹‘(𝐺‘(𝑥( ·𝑠 ‘𝑀)𝑦))) = (𝐹‘(𝑥( ·𝑠 ‘𝑁)(𝐺‘𝑦))))
26		simpll 776	. . . . 5 ⊢ (((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝐹 ∈ (𝑁 LMHom 𝑂))
27	13	fveq2d 6867	. . . . . . 7 ⊢ (𝐺 ∈ (𝑀 LMHom 𝑁) → (Base‘(Scalar‘𝑁)) = (Base‘(Scalar‘𝑀)))
28	27	ad2antlr 737	. . . . . 6 ⊢ (((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → (Base‘(Scalar‘𝑁)) = (Base‘(Scalar‘𝑀)))
29	20, 28	eleqtrrd 2864	. . . . 5 ⊢ (((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝑥 ∈ (Base‘(Scalar‘𝑁)))
30		eqid 2761	. . . . . . . . 9 ⊢ (Base‘𝑁) = (Base‘𝑁)
31	1, 30	lmhmf 21081	. . . . . . . 8 ⊢ (𝐺 ∈ (𝑀 LMHom 𝑁) → 𝐺:(Base‘𝑀)⟶(Base‘𝑁))
32	31	adantl 485	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) → 𝐺:(Base‘𝑀)⟶(Base‘𝑁))
33	32	ffvelcdmda 7061	. . . . . 6 ⊢ (((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝐺‘𝑦) ∈ (Base‘𝑁))
34	33	adantrl 726	. . . . 5 ⊢ (((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝐺‘𝑦) ∈ (Base‘𝑁))
35		eqid 2761	. . . . . 6 ⊢ (Base‘(Scalar‘𝑁)) = (Base‘(Scalar‘𝑁))
36	11, 35, 30, 22, 3	lmhmlin 21082	. . . . 5 ⊢ ((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑁)) ∧ (𝐺‘𝑦) ∈ (Base‘𝑁)) → (𝐹‘(𝑥( ·𝑠 ‘𝑁)(𝐺‘𝑦))) = (𝑥( ·𝑠 ‘𝑂)(𝐹‘(𝐺‘𝑦))))
37	26, 29, 34, 36	syl3anc 1389	. . . 4 ⊢ (((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝐹‘(𝑥( ·𝑠 ‘𝑁)(𝐺‘𝑦))) = (𝑥( ·𝑠 ‘𝑂)(𝐹‘(𝐺‘𝑦))))
38	25, 37	eqtrd 2796	. . 3 ⊢ (((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝐹‘(𝐺‘(𝑥( ·𝑠 ‘𝑀)𝑦))) = (𝑥( ·𝑠 ‘𝑂)(𝐹‘(𝐺‘𝑦))))
39	32	ffnd 6688	. . . 4 ⊢ ((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) → 𝐺 Fn (Base‘𝑀))
40	7	ad2antlr 737	. . . . 5 ⊢ (((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝑀 ∈ LMod)
41	1, 4, 2, 6	lmodvscl 20925	. . . . 5 ⊢ ((𝑀 ∈ LMod ∧ 𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝑥( ·𝑠 ‘𝑀)𝑦) ∈ (Base‘𝑀))
42	40, 20, 21, 41	syl3anc 1389	. . . 4 ⊢ (((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝑥( ·𝑠 ‘𝑀)𝑦) ∈ (Base‘𝑀))
43		fvco2 6960	. . . 4 ⊢ ((𝐺 Fn (Base‘𝑀) ∧ (𝑥( ·𝑠 ‘𝑀)𝑦) ∈ (Base‘𝑀)) → ((𝐹 ∘ 𝐺)‘(𝑥( ·𝑠 ‘𝑀)𝑦)) = (𝐹‘(𝐺‘(𝑥( ·𝑠 ‘𝑀)𝑦))))
44	39, 42, 43	syl2an2r 695	. . 3 ⊢ (((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → ((𝐹 ∘ 𝐺)‘(𝑥( ·𝑠 ‘𝑀)𝑦)) = (𝐹‘(𝐺‘(𝑥( ·𝑠 ‘𝑀)𝑦))))
45		fvco2 6960	. . . . 5 ⊢ ((𝐺 Fn (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) → ((𝐹 ∘ 𝐺)‘𝑦) = (𝐹‘(𝐺‘𝑦)))
46	39, 21, 45	syl2an2r 695	. . . 4 ⊢ (((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → ((𝐹 ∘ 𝐺)‘𝑦) = (𝐹‘(𝐺‘𝑦)))
47	46	oveq2d 7408	. . 3 ⊢ (((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝑥( ·𝑠 ‘𝑂)((𝐹 ∘ 𝐺)‘𝑦)) = (𝑥( ·𝑠 ‘𝑂)(𝐹‘(𝐺‘𝑦))))
48	38, 44, 47	3eqtr4d 2806	. 2 ⊢ (((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → ((𝐹 ∘ 𝐺)‘(𝑥( ·𝑠 ‘𝑀)𝑦)) = (𝑥( ·𝑠 ‘𝑂)((𝐹 ∘ 𝐺)‘𝑦)))
49	1, 2, 3, 4, 5, 6, 8, 10, 14, 18, 48	islmhmd 21086	1 ⊢ ((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) → (𝐹 ∘ 𝐺) ∈ (𝑀 LMHom 𝑂))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1559   ∈ wcel 2141   ∘ ccom 5649   Fn wfn 6512  ⟶wf 6513  ‘cfv 6517  (class class class)co 7392  Basecbs 17228  Scalarcsca 17272   ·_𝑠 cvsca 17273   GrpHom cghm 19236  LModclmod 20907   LMHom clmhm 21066

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-fv 6525  df-riota 7349  df-ov 7395  df-oprab 7396  df-mpo 7397  df-1st 7966  df-2nd 7967  df-map 8805  df-0g 17453  df-mgm 18657  df-sgrp 18736  df-mnd 18752  df-mhm 18800  df-grp 18961  df-ghm 19237  df-lmod 20909  df-lmhm 21069

This theorem is referenced by:  lmimco  21876  nmhmco  24796  mendring  43729