Theorem mendassa 43202

Description: The module endomorphism algebra is an algebra. (Contributed by Mario Carneiro, 22-Sep-2015.)

Hypotheses

Ref	Expression
mendassa.a	⊢ 𝐴 = (MEndo‘𝑀)
mendassa.s	⊢ 𝑆 = (Scalar‘𝑀)

Assertion

Ref	Expression
mendassa	⊢ ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → 𝐴 ∈ AssAlg)

Proof of Theorem mendassa

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

Step	Hyp	Ref	Expression
1		mendassa.a	. . . 4 ⊢ 𝐴 = (MEndo‘𝑀)
2	1	mendbas 43192	. . 3 ⊢ (𝑀 LMHom 𝑀) = (Base‘𝐴)
3	2	a1i 11	. 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → (𝑀 LMHom 𝑀) = (Base‘𝐴))
4		mendassa.s	. . . 4 ⊢ 𝑆 = (Scalar‘𝑀)
5	1, 4	mendsca 43197	. . 3 ⊢ 𝑆 = (Scalar‘𝐴)
6	5	a1i 11	. 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → 𝑆 = (Scalar‘𝐴))
7		eqidd 2738	. 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → (Base‘𝑆) = (Base‘𝑆))
8		eqidd 2738	. 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → ( ·𝑠 ‘𝐴) = ( ·𝑠 ‘𝐴))
9		eqidd 2738	. 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → (.r‘𝐴) = (.r‘𝐴))
10	1, 4	mendlmod 43201	. 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → 𝐴 ∈ LMod)
11	1	mendring 43200	. . 3 ⊢ (𝑀 ∈ LMod → 𝐴 ∈ Ring)
12	11	adantr 480	. 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → 𝐴 ∈ Ring)
13		simpr3 1197	. . . . . . 7 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑧 ∈ (𝑀 LMHom 𝑀))
14		eqid 2737	. . . . . . . 8 ⊢ (Base‘𝑀) = (Base‘𝑀)
15	14, 14	lmhmf 21033	. . . . . . 7 ⊢ (𝑧 ∈ (𝑀 LMHom 𝑀) → 𝑧:(Base‘𝑀)⟶(Base‘𝑀))
16	13, 15	syl 17	. . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑧:(Base‘𝑀)⟶(Base‘𝑀))
17	16	ffvelcdmda 7104	. . . . 5 ⊢ ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) ∧ 𝑣 ∈ (Base‘𝑀)) → (𝑧‘𝑣) ∈ (Base‘𝑀))
18	16	feqmptd 6977	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑧 = (𝑣 ∈ (Base‘𝑀) ↦ (𝑧‘𝑣)))
19		simpr1 1195	. . . . . . 7 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑥 ∈ (Base‘𝑆))
20		simpr2 1196	. . . . . . 7 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑦 ∈ (𝑀 LMHom 𝑀))
21		eqid 2737	. . . . . . . 8 ⊢ ( ·𝑠 ‘𝑀) = ( ·𝑠 ‘𝑀)
22		eqid 2737	. . . . . . . 8 ⊢ (Base‘𝑆) = (Base‘𝑆)
23		eqid 2737	. . . . . . . 8 ⊢ ( ·𝑠 ‘𝐴) = ( ·𝑠 ‘𝐴)
24	1, 21, 2, 4, 22, 14, 23	mendvsca 43199	. . . . . . 7 ⊢ ((𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (𝑥( ·𝑠 ‘𝐴)𝑦) = (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠 ‘𝑀)𝑦))
25	19, 20, 24	syl2anc 584	. . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥( ·𝑠 ‘𝐴)𝑦) = (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠 ‘𝑀)𝑦))
26		fvexd 6921	. . . . . . 7 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (Base‘𝑀) ∈ V)
27		simplr1 1216	. . . . . . 7 ⊢ ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) ∧ 𝑤 ∈ (Base‘𝑀)) → 𝑥 ∈ (Base‘𝑆))
28		fvexd 6921	. . . . . . 7 ⊢ ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) ∧ 𝑤 ∈ (Base‘𝑀)) → (𝑦‘𝑤) ∈ V)
29		fconstmpt 5747	. . . . . . . 8 ⊢ ((Base‘𝑀) × {𝑥}) = (𝑤 ∈ (Base‘𝑀) ↦ 𝑥)
30	29	a1i 11	. . . . . . 7 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((Base‘𝑀) × {𝑥}) = (𝑤 ∈ (Base‘𝑀) ↦ 𝑥))
31	14, 14	lmhmf 21033	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝑀 LMHom 𝑀) → 𝑦:(Base‘𝑀)⟶(Base‘𝑀))
32	20, 31	syl 17	. . . . . . . 8 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑦:(Base‘𝑀)⟶(Base‘𝑀))
33	32	feqmptd 6977	. . . . . . 7 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑦 = (𝑤 ∈ (Base‘𝑀) ↦ (𝑦‘𝑤)))
34	26, 27, 28, 30, 33	offval2 7717	. . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠 ‘𝑀)𝑦) = (𝑤 ∈ (Base‘𝑀) ↦ (𝑥( ·𝑠 ‘𝑀)(𝑦‘𝑤))))
35	25, 34	eqtrd 2777	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥( ·𝑠 ‘𝐴)𝑦) = (𝑤 ∈ (Base‘𝑀) ↦ (𝑥( ·𝑠 ‘𝑀)(𝑦‘𝑤))))
36		fveq2 6906	. . . . . 6 ⊢ (𝑤 = (𝑧‘𝑣) → (𝑦‘𝑤) = (𝑦‘(𝑧‘𝑣)))
37	36	oveq2d 7447	. . . . 5 ⊢ (𝑤 = (𝑧‘𝑣) → (𝑥( ·𝑠 ‘𝑀)(𝑦‘𝑤)) = (𝑥( ·𝑠 ‘𝑀)(𝑦‘(𝑧‘𝑣))))
38	17, 18, 35, 37	fmptco 7149	. . . 4 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((𝑥( ·𝑠 ‘𝐴)𝑦) ∘ 𝑧) = (𝑣 ∈ (Base‘𝑀) ↦ (𝑥( ·𝑠 ‘𝑀)(𝑦‘(𝑧‘𝑣)))))
39		simplr1 1216	. . . . 5 ⊢ ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) ∧ 𝑣 ∈ (Base‘𝑀)) → 𝑥 ∈ (Base‘𝑆))
40		fvexd 6921	. . . . 5 ⊢ ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) ∧ 𝑣 ∈ (Base‘𝑀)) → (𝑦‘(𝑧‘𝑣)) ∈ V)
41		fconstmpt 5747	. . . . . 6 ⊢ ((Base‘𝑀) × {𝑥}) = (𝑣 ∈ (Base‘𝑀) ↦ 𝑥)
42	41	a1i 11	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((Base‘𝑀) × {𝑥}) = (𝑣 ∈ (Base‘𝑀) ↦ 𝑥))
43		eqid 2737	. . . . . . . 8 ⊢ (.r‘𝐴) = (.r‘𝐴)
44	1, 2, 43	mendmulr 43196	. . . . . . 7 ⊢ ((𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑦(.r‘𝐴)𝑧) = (𝑦 ∘ 𝑧))
45	20, 13, 44	syl2anc 584	. . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑦(.r‘𝐴)𝑧) = (𝑦 ∘ 𝑧))
46		fcompt 7153	. . . . . . 7 ⊢ ((𝑦:(Base‘𝑀)⟶(Base‘𝑀) ∧ 𝑧:(Base‘𝑀)⟶(Base‘𝑀)) → (𝑦 ∘ 𝑧) = (𝑣 ∈ (Base‘𝑀) ↦ (𝑦‘(𝑧‘𝑣))))
47	32, 16, 46	syl2anc 584	. . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑦 ∘ 𝑧) = (𝑣 ∈ (Base‘𝑀) ↦ (𝑦‘(𝑧‘𝑣))))
48	45, 47	eqtrd 2777	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑦(.r‘𝐴)𝑧) = (𝑣 ∈ (Base‘𝑀) ↦ (𝑦‘(𝑧‘𝑣))))
49	26, 39, 40, 42, 48	offval2 7717	. . . 4 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠 ‘𝑀)(𝑦(.r‘𝐴)𝑧)) = (𝑣 ∈ (Base‘𝑀) ↦ (𝑥( ·𝑠 ‘𝑀)(𝑦‘(𝑧‘𝑣)))))
50	38, 49	eqtr4d 2780	. . 3 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((𝑥( ·𝑠 ‘𝐴)𝑦) ∘ 𝑧) = (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠 ‘𝑀)(𝑦(.r‘𝐴)𝑧)))
51	10	adantr 480	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝐴 ∈ LMod)
52	2, 5, 23, 22	lmodvscl 20876	. . . . 5 ⊢ ((𝐴 ∈ LMod ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (𝑥( ·𝑠 ‘𝐴)𝑦) ∈ (𝑀 LMHom 𝑀))
53	51, 19, 20, 52	syl3anc 1373	. . . 4 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥( ·𝑠 ‘𝐴)𝑦) ∈ (𝑀 LMHom 𝑀))
54	1, 2, 43	mendmulr 43196	. . . 4 ⊢ (((𝑥( ·𝑠 ‘𝐴)𝑦) ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → ((𝑥( ·𝑠 ‘𝐴)𝑦)(.r‘𝐴)𝑧) = ((𝑥( ·𝑠 ‘𝐴)𝑦) ∘ 𝑧))
55	53, 13, 54	syl2anc 584	. . 3 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((𝑥( ·𝑠 ‘𝐴)𝑦)(.r‘𝐴)𝑧) = ((𝑥( ·𝑠 ‘𝐴)𝑦) ∘ 𝑧))
56	12	adantr 480	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝐴 ∈ Ring)
57	2, 43	ringcl 20247	. . . . 5 ⊢ ((𝐴 ∈ Ring ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑦(.r‘𝐴)𝑧) ∈ (𝑀 LMHom 𝑀))
58	56, 20, 13, 57	syl3anc 1373	. . . 4 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑦(.r‘𝐴)𝑧) ∈ (𝑀 LMHom 𝑀))
59	1, 21, 2, 4, 22, 14, 23	mendvsca 43199	. . . 4 ⊢ ((𝑥 ∈ (Base‘𝑆) ∧ (𝑦(.r‘𝐴)𝑧) ∈ (𝑀 LMHom 𝑀)) → (𝑥( ·𝑠 ‘𝐴)(𝑦(.r‘𝐴)𝑧)) = (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠 ‘𝑀)(𝑦(.r‘𝐴)𝑧)))
60	19, 58, 59	syl2anc 584	. . 3 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥( ·𝑠 ‘𝐴)(𝑦(.r‘𝐴)𝑧)) = (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠 ‘𝑀)(𝑦(.r‘𝐴)𝑧)))
61	50, 55, 60	3eqtr4d 2787	. 2 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((𝑥( ·𝑠 ‘𝐴)𝑦)(.r‘𝐴)𝑧) = (𝑥( ·𝑠 ‘𝐴)(𝑦(.r‘𝐴)𝑧)))
62		simplr2 1217	. . . . . 6 ⊢ ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) ∧ 𝑣 ∈ (Base‘𝑀)) → 𝑦 ∈ (𝑀 LMHom 𝑀))
63	4, 22, 14, 21, 21	lmhmlin 21034	. . . . . 6 ⊢ ((𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑥 ∈ (Base‘𝑆) ∧ (𝑧‘𝑣) ∈ (Base‘𝑀)) → (𝑦‘(𝑥( ·𝑠 ‘𝑀)(𝑧‘𝑣))) = (𝑥( ·𝑠 ‘𝑀)(𝑦‘(𝑧‘𝑣))))
64	62, 39, 17, 63	syl3anc 1373	. . . . 5 ⊢ ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) ∧ 𝑣 ∈ (Base‘𝑀)) → (𝑦‘(𝑥( ·𝑠 ‘𝑀)(𝑧‘𝑣))) = (𝑥( ·𝑠 ‘𝑀)(𝑦‘(𝑧‘𝑣))))
65	64	mpteq2dva 5242	. . . 4 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑣 ∈ (Base‘𝑀) ↦ (𝑦‘(𝑥( ·𝑠 ‘𝑀)(𝑧‘𝑣)))) = (𝑣 ∈ (Base‘𝑀) ↦ (𝑥( ·𝑠 ‘𝑀)(𝑦‘(𝑧‘𝑣)))))
66		simplll 775	. . . . . 6 ⊢ ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) ∧ 𝑣 ∈ (Base‘𝑀)) → 𝑀 ∈ LMod)
67	14, 4, 21, 22	lmodvscl 20876	. . . . . 6 ⊢ ((𝑀 ∈ LMod ∧ 𝑥 ∈ (Base‘𝑆) ∧ (𝑧‘𝑣) ∈ (Base‘𝑀)) → (𝑥( ·𝑠 ‘𝑀)(𝑧‘𝑣)) ∈ (Base‘𝑀))
68	66, 39, 17, 67	syl3anc 1373	. . . . 5 ⊢ ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) ∧ 𝑣 ∈ (Base‘𝑀)) → (𝑥( ·𝑠 ‘𝑀)(𝑧‘𝑣)) ∈ (Base‘𝑀))
69	1, 21, 2, 4, 22, 14, 23	mendvsca 43199	. . . . . . 7 ⊢ ((𝑥 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑥( ·𝑠 ‘𝐴)𝑧) = (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠 ‘𝑀)𝑧))
70	19, 13, 69	syl2anc 584	. . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥( ·𝑠 ‘𝐴)𝑧) = (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠 ‘𝑀)𝑧))
71		fvexd 6921	. . . . . . 7 ⊢ ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) ∧ 𝑣 ∈ (Base‘𝑀)) → (𝑧‘𝑣) ∈ V)
72	26, 39, 71, 42, 18	offval2 7717	. . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠 ‘𝑀)𝑧) = (𝑣 ∈ (Base‘𝑀) ↦ (𝑥( ·𝑠 ‘𝑀)(𝑧‘𝑣))))
73	70, 72	eqtrd 2777	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥( ·𝑠 ‘𝐴)𝑧) = (𝑣 ∈ (Base‘𝑀) ↦ (𝑥( ·𝑠 ‘𝑀)(𝑧‘𝑣))))
74		fveq2 6906	. . . . 5 ⊢ (𝑤 = (𝑥( ·𝑠 ‘𝑀)(𝑧‘𝑣)) → (𝑦‘𝑤) = (𝑦‘(𝑥( ·𝑠 ‘𝑀)(𝑧‘𝑣))))
75	68, 73, 33, 74	fmptco 7149	. . . 4 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑦 ∘ (𝑥( ·𝑠 ‘𝐴)𝑧)) = (𝑣 ∈ (Base‘𝑀) ↦ (𝑦‘(𝑥( ·𝑠 ‘𝑀)(𝑧‘𝑣)))))
76	65, 75, 49	3eqtr4d 2787	. . 3 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑦 ∘ (𝑥( ·𝑠 ‘𝐴)𝑧)) = (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠 ‘𝑀)(𝑦(.r‘𝐴)𝑧)))
77	2, 5, 23, 22	lmodvscl 20876	. . . . 5 ⊢ ((𝐴 ∈ LMod ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑥( ·𝑠 ‘𝐴)𝑧) ∈ (𝑀 LMHom 𝑀))
78	51, 19, 13, 77	syl3anc 1373	. . . 4 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥( ·𝑠 ‘𝐴)𝑧) ∈ (𝑀 LMHom 𝑀))
79	1, 2, 43	mendmulr 43196	. . . 4 ⊢ ((𝑦 ∈ (𝑀 LMHom 𝑀) ∧ (𝑥( ·𝑠 ‘𝐴)𝑧) ∈ (𝑀 LMHom 𝑀)) → (𝑦(.r‘𝐴)(𝑥( ·𝑠 ‘𝐴)𝑧)) = (𝑦 ∘ (𝑥( ·𝑠 ‘𝐴)𝑧)))
80	20, 78, 79	syl2anc 584	. . 3 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑦(.r‘𝐴)(𝑥( ·𝑠 ‘𝐴)𝑧)) = (𝑦 ∘ (𝑥( ·𝑠 ‘𝐴)𝑧)))
81	76, 80, 60	3eqtr4d 2787	. 2 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑦(.r‘𝐴)(𝑥( ·𝑠 ‘𝐴)𝑧)) = (𝑥( ·𝑠 ‘𝐴)(𝑦(.r‘𝐴)𝑧)))
82	3, 6, 7, 8, 9, 10, 12, 61, 81	isassad 21885	1 ⊢ ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → 𝐴 ∈ AssAlg)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108  Vcvv 3480  {csn 4626   ↦ cmpt 5225   × cxp 5683   ∘ ccom 5689  ⟶wf 6557  ‘cfv 6561  (class class class)co 7431   ∘_f cof 7695  Basecbs 17247  ._rcmulr 17298  Scalarcsca 17300   ·_𝑠 cvsca 17301  Ringcrg 20230  CRingccrg 20231  LModclmod 20858   LMHom clmhm 21018  AssAlgcasa 21870  MEndocmend 43183

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-n0 12527  df-z 12614  df-uz 12879  df-fz 13548  df-struct 17184  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-plusg 17310  df-mulr 17311  df-sca 17313  df-vsca 17314  df-0g 17486  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-mhm 18796  df-grp 18954  df-minusg 18955  df-ghm 19231  df-cmn 19800  df-abl 19801  df-mgp 20138  df-rng 20150  df-ur 20179  df-ring 20232  df-cring 20233  df-lmod 20860  df-lmhm 21021  df-assa 21873  df-mend 43184

This theorem is referenced by: (None)