Theorem lmhmco 19934

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 2738	. 2 ⊢ (Base‘𝑀) = (Base‘𝑀)
2		eqid 2738	. 2 ⊢ ( ·𝑠 ‘𝑀) = ( ·𝑠 ‘𝑀)
3		eqid 2738	. 2 ⊢ ( ·𝑠 ‘𝑂) = ( ·𝑠 ‘𝑂)
4		eqid 2738	. 2 ⊢ (Scalar‘𝑀) = (Scalar‘𝑀)
5		eqid 2738	. 2 ⊢ (Scalar‘𝑂) = (Scalar‘𝑂)
6		eqid 2738	. 2 ⊢ (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀))
7		lmhmlmod1 19924	. . 3 ⊢ (𝐺 ∈ (𝑀 LMHom 𝑁) → 𝑀 ∈ LMod)
8	7	adantl 485	. 2 ⊢ ((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) → 𝑀 ∈ LMod)
9		lmhmlmod2 19923	. . 3 ⊢ (𝐹 ∈ (𝑁 LMHom 𝑂) → 𝑂 ∈ LMod)
10	9	adantr 484	. 2 ⊢ ((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) → 𝑂 ∈ LMod)
11		eqid 2738	. . . 4 ⊢ (Scalar‘𝑁) = (Scalar‘𝑁)
12	11, 5	lmhmsca 19921	. . 3 ⊢ (𝐹 ∈ (𝑁 LMHom 𝑂) → (Scalar‘𝑂) = (Scalar‘𝑁))
13	4, 11	lmhmsca 19921	. . 3 ⊢ (𝐺 ∈ (𝑀 LMHom 𝑁) → (Scalar‘𝑁) = (Scalar‘𝑀))
14	12, 13	sylan9eq 2793	. 2 ⊢ ((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) → (Scalar‘𝑂) = (Scalar‘𝑀))
15		lmghm 19922	. . 3 ⊢ (𝐹 ∈ (𝑁 LMHom 𝑂) → 𝐹 ∈ (𝑁 GrpHom 𝑂))
16		lmghm 19922	. . 3 ⊢ (𝐺 ∈ (𝑀 LMHom 𝑁) → 𝐺 ∈ (𝑀 GrpHom 𝑁))
17		ghmco 18496	. . 3 ⊢ ((𝐹 ∈ (𝑁 GrpHom 𝑂) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) → (𝐹 ∘ 𝐺) ∈ (𝑀 GrpHom 𝑂))
18	15, 16, 17	syl2an 599	. 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 2738	. . . . . . 7 ⊢ ( ·𝑠 ‘𝑁) = ( ·𝑠 ‘𝑁)
23	4, 6, 1, 2, 22	lmhmlin 19926	. . . . . 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 1372	. . . . 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 6678	. . . 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 6678	. . . . . . 7 ⊢ (𝐺 ∈ (𝑀 LMHom 𝑁) → (Base‘(Scalar‘𝑁)) = (Base‘(Scalar‘𝑀)))
28	27	ad2antlr 727	. . . . . 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 2836	. . . . 5 ⊢ (((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝑥 ∈ (Base‘(Scalar‘𝑁)))
30		eqid 2738	. . . . . . . . 9 ⊢ (Base‘𝑁) = (Base‘𝑁)
31	1, 30	lmhmf 19925	. . . . . . . 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	ffvelrnda 6861	. . . . . 6 ⊢ (((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝐺‘𝑦) ∈ (Base‘𝑁))
34	33	adantrl 716	. . . . 5 ⊢ (((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝐺‘𝑦) ∈ (Base‘𝑁))
35		eqid 2738	. . . . . 6 ⊢ (Base‘(Scalar‘𝑁)) = (Base‘(Scalar‘𝑁))
36	11, 35, 30, 22, 3	lmhmlin 19926	. . . . 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 1372	. . . 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 2773	. . 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 6505	. . . 4 ⊢ ((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) → 𝐺 Fn (Base‘𝑀))
40	7	ad2antlr 727	. . . . 5 ⊢ (((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝑀 ∈ LMod)
41	1, 4, 2, 6	lmodvscl 19770	. . . . 5 ⊢ ((𝑀 ∈ LMod ∧ 𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝑥( ·𝑠 ‘𝑀)𝑦) ∈ (Base‘𝑀))
42	40, 20, 21, 41	syl3anc 1372	. . . 4 ⊢ (((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝑥( ·𝑠 ‘𝑀)𝑦) ∈ (Base‘𝑀))
43		fvco2 6765	. . . 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 685	. . 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 6765	. . . . 5 ⊢ ((𝐺 Fn (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) → ((𝐹 ∘ 𝐺)‘𝑦) = (𝐹‘(𝐺‘𝑦)))
46	39, 21, 45	syl2an2r 685	. . . 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 7186	. . 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 2783	. 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 19930	1 ⊢ ((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) → (𝐹 ∘ 𝐺) ∈ (𝑀 LMHom 𝑂))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1542   ∈ wcel 2114   ∘ ccom 5529   Fn wfn 6334  ⟶wf 6335  ‘cfv 6339  (class class class)co 7170  Basecbs 16586  Scalarcsca 16671   ·_𝑠 cvsca 16672   GrpHom cghm 18473  LModclmod 19753   LMHom clmhm 19910

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 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rmo 3061  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-riota 7127  df-ov 7173  df-oprab 7174  df-mpo 7175  df-map 8439  df-0g 16818  df-mgm 17968  df-sgrp 18017  df-mnd 18028  df-mhm 18072  df-grp 18222  df-ghm 18474  df-lmod 19755  df-lmhm 19913

This theorem is referenced by:  lmimco  20660  nmhmco  23509  mendring  40589