Theorem lmhmco 21007

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 2737	. 2 ⊢ (Base‘𝑀) = (Base‘𝑀)
2		eqid 2737	. 2 ⊢ ( ·𝑠 ‘𝑀) = ( ·𝑠 ‘𝑀)
3		eqid 2737	. 2 ⊢ ( ·𝑠 ‘𝑂) = ( ·𝑠 ‘𝑂)
4		eqid 2737	. 2 ⊢ (Scalar‘𝑀) = (Scalar‘𝑀)
5		eqid 2737	. 2 ⊢ (Scalar‘𝑂) = (Scalar‘𝑂)
6		eqid 2737	. 2 ⊢ (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀))
7		lmhmlmod1 20997	. . 3 ⊢ (𝐺 ∈ (𝑀 LMHom 𝑁) → 𝑀 ∈ LMod)
8	7	adantl 481	. 2 ⊢ ((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) → 𝑀 ∈ LMod)
9		lmhmlmod2 20996	. . 3 ⊢ (𝐹 ∈ (𝑁 LMHom 𝑂) → 𝑂 ∈ LMod)
10	9	adantr 480	. 2 ⊢ ((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) → 𝑂 ∈ LMod)
11		eqid 2737	. . . 4 ⊢ (Scalar‘𝑁) = (Scalar‘𝑁)
12	11, 5	lmhmsca 20994	. . 3 ⊢ (𝐹 ∈ (𝑁 LMHom 𝑂) → (Scalar‘𝑂) = (Scalar‘𝑁))
13	4, 11	lmhmsca 20994	. . 3 ⊢ (𝐺 ∈ (𝑀 LMHom 𝑁) → (Scalar‘𝑁) = (Scalar‘𝑀))
14	12, 13	sylan9eq 2792	. 2 ⊢ ((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) → (Scalar‘𝑂) = (Scalar‘𝑀))
15		lmghm 20995	. . 3 ⊢ (𝐹 ∈ (𝑁 LMHom 𝑂) → 𝐹 ∈ (𝑁 GrpHom 𝑂))
16		lmghm 20995	. . 3 ⊢ (𝐺 ∈ (𝑀 LMHom 𝑁) → 𝐺 ∈ (𝑀 GrpHom 𝑁))
17		ghmco 19177	. . 3 ⊢ ((𝐹 ∈ (𝑁 GrpHom 𝑂) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) → (𝐹 ∘ 𝐺) ∈ (𝑀 GrpHom 𝑂))
18	15, 16, 17	syl2an 597	. 2 ⊢ ((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) → (𝐹 ∘ 𝐺) ∈ (𝑀 GrpHom 𝑂))
19		simplr 769	. . . . . 6 ⊢ (((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝐺 ∈ (𝑀 LMHom 𝑁))
20		simprl 771	. . . . . 6 ⊢ (((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝑥 ∈ (Base‘(Scalar‘𝑀)))
21		simprr 773	. . . . . 6 ⊢ (((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝑦 ∈ (Base‘𝑀))
22		eqid 2737	. . . . . . 7 ⊢ ( ·𝑠 ‘𝑁) = ( ·𝑠 ‘𝑁)
23	4, 6, 1, 2, 22	lmhmlin 20999	. . . . . 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 1374	. . . . 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 6846	. . . 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 767	. . . . 5 ⊢ (((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝐹 ∈ (𝑁 LMHom 𝑂))
27	13	fveq2d 6846	. . . . . . 7 ⊢ (𝐺 ∈ (𝑀 LMHom 𝑁) → (Base‘(Scalar‘𝑁)) = (Base‘(Scalar‘𝑀)))
28	27	ad2antlr 728	. . . . . 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 2840	. . . . 5 ⊢ (((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝑥 ∈ (Base‘(Scalar‘𝑁)))
30		eqid 2737	. . . . . . . . 9 ⊢ (Base‘𝑁) = (Base‘𝑁)
31	1, 30	lmhmf 20998	. . . . . . . 8 ⊢ (𝐺 ∈ (𝑀 LMHom 𝑁) → 𝐺:(Base‘𝑀)⟶(Base‘𝑁))
32	31	adantl 481	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) → 𝐺:(Base‘𝑀)⟶(Base‘𝑁))
33	32	ffvelcdmda 7038	. . . . . 6 ⊢ (((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝐺‘𝑦) ∈ (Base‘𝑁))
34	33	adantrl 717	. . . . 5 ⊢ (((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝐺‘𝑦) ∈ (Base‘𝑁))
35		eqid 2737	. . . . . 6 ⊢ (Base‘(Scalar‘𝑁)) = (Base‘(Scalar‘𝑁))
36	11, 35, 30, 22, 3	lmhmlin 20999	. . . . 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 1374	. . . 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 2772	. . 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 6671	. . . 4 ⊢ ((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) → 𝐺 Fn (Base‘𝑀))
40	7	ad2antlr 728	. . . . 5 ⊢ (((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝑀 ∈ LMod)
41	1, 4, 2, 6	lmodvscl 20841	. . . . 5 ⊢ ((𝑀 ∈ LMod ∧ 𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝑥( ·𝑠 ‘𝑀)𝑦) ∈ (Base‘𝑀))
42	40, 20, 21, 41	syl3anc 1374	. . . 4 ⊢ (((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝑥( ·𝑠 ‘𝑀)𝑦) ∈ (Base‘𝑀))
43		fvco2 6939	. . . 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 686	. . 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 6939	. . . . 5 ⊢ ((𝐺 Fn (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) → ((𝐹 ∘ 𝐺)‘𝑦) = (𝐹‘(𝐺‘𝑦)))
46	39, 21, 45	syl2an2r 686	. . . 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 7384	. . 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 2782	. 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 21003	1 ⊢ ((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) → (𝐹 ∘ 𝐺) ∈ (𝑀 LMHom 𝑂))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ∘ ccom 5636   Fn wfn 6495  ⟶wf 6496  ‘cfv 6500  (class class class)co 7368  Basecbs 17148  Scalarcsca 17192   ·_𝑠 cvsca 17193   GrpHom cghm 19153  LModclmod 20823   LMHom clmhm 20983

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-fv 6508  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-1st 7943  df-2nd 7944  df-map 8777  df-0g 17373  df-mgm 18577  df-sgrp 18656  df-mnd 18672  df-mhm 18720  df-grp 18878  df-ghm 19154  df-lmod 20825  df-lmhm 20986

This theorem is referenced by:  lmimco  21811  nmhmco  24712  mendring  43539