Theorem mendassa 41921

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 41911	. . 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 41916	. . 3 ⊢ 𝑆 = (Scalar‘𝐴)
6	5	a1i 11	. 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → 𝑆 = (Scalar‘𝐴))
7		eqidd 2733	. 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → (Base‘𝑆) = (Base‘𝑆))
8		eqidd 2733	. 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → ( ·𝑠 ‘𝐴) = ( ·𝑠 ‘𝐴))
9		eqidd 2733	. 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → (.r‘𝐴) = (.r‘𝐴))
10	1, 4	mendlmod 41920	. 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → 𝐴 ∈ LMod)
11	1	mendring 41919	. . 3 ⊢ (𝑀 ∈ LMod → 𝐴 ∈ Ring)
12	11	adantr 481	. 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → 𝐴 ∈ Ring)
13		simpr3 1196	. . . . . . 7 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑧 ∈ (𝑀 LMHom 𝑀))
14		eqid 2732	. . . . . . . 8 ⊢ (Base‘𝑀) = (Base‘𝑀)
15	14, 14	lmhmf 20637	. . . . . . 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 7083	. . . . 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 6957	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑧 = (𝑣 ∈ (Base‘𝑀) ↦ (𝑧‘𝑣)))
19		simpr1 1194	. . . . . . 7 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑥 ∈ (Base‘𝑆))
20		simpr2 1195	. . . . . . 7 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑦 ∈ (𝑀 LMHom 𝑀))
21		eqid 2732	. . . . . . . 8 ⊢ ( ·𝑠 ‘𝑀) = ( ·𝑠 ‘𝑀)
22		eqid 2732	. . . . . . . 8 ⊢ (Base‘𝑆) = (Base‘𝑆)
23		eqid 2732	. . . . . . . 8 ⊢ ( ·𝑠 ‘𝐴) = ( ·𝑠 ‘𝐴)
24	1, 21, 2, 4, 22, 14, 23	mendvsca 41918	. . . . . . 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 6903	. . . . . . 7 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (Base‘𝑀) ∈ V)
27		simplr1 1215	. . . . . . 7 ⊢ ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) ∧ 𝑤 ∈ (Base‘𝑀)) → 𝑥 ∈ (Base‘𝑆))
28		fvexd 6903	. . . . . . 7 ⊢ ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) ∧ 𝑤 ∈ (Base‘𝑀)) → (𝑦‘𝑤) ∈ V)
29		fconstmpt 5736	. . . . . . . 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 20637	. . . . . . . . 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 6957	. . . . . . 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 7686	. . . . . 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 2772	. . . . 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 6888	. . . . . 6 ⊢ (𝑤 = (𝑧‘𝑣) → (𝑦‘𝑤) = (𝑦‘(𝑧‘𝑣)))
37	36	oveq2d 7421	. . . . 5 ⊢ (𝑤 = (𝑧‘𝑣) → (𝑥( ·𝑠 ‘𝑀)(𝑦‘𝑤)) = (𝑥( ·𝑠 ‘𝑀)(𝑦‘(𝑧‘𝑣))))
38	17, 18, 35, 37	fmptco 7123	. . . 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 1215	. . . . 5 ⊢ ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) ∧ 𝑣 ∈ (Base‘𝑀)) → 𝑥 ∈ (Base‘𝑆))
40		fvexd 6903	. . . . 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 5736	. . . . . 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 2732	. . . . . . . 8 ⊢ (.r‘𝐴) = (.r‘𝐴)
44	1, 2, 43	mendmulr 41915	. . . . . . 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 7127	. . . . . . 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 2772	. . . . 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 7686	. . . 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 2775	. . 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 481	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝐴 ∈ LMod)
52	2, 5, 23, 22	lmodvscl 20481	. . . . 5 ⊢ ((𝐴 ∈ LMod ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (𝑥( ·𝑠 ‘𝐴)𝑦) ∈ (𝑀 LMHom 𝑀))
53	51, 19, 20, 52	syl3anc 1371	. . . 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 41915	. . . 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 481	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝐴 ∈ Ring)
57	2, 43	ringcl 20066	. . . . 5 ⊢ ((𝐴 ∈ Ring ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑦(.r‘𝐴)𝑧) ∈ (𝑀 LMHom 𝑀))
58	56, 20, 13, 57	syl3anc 1371	. . . 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 41918	. . . 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 2782	. 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 1216	. . . . . 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 20638	. . . . . 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 1371	. . . . 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 5247	. . . 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 773	. . . . . 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 20481	. . . . . 6 ⊢ ((𝑀 ∈ LMod ∧ 𝑥 ∈ (Base‘𝑆) ∧ (𝑧‘𝑣) ∈ (Base‘𝑀)) → (𝑥( ·𝑠 ‘𝑀)(𝑧‘𝑣)) ∈ (Base‘𝑀))
68	66, 39, 17, 67	syl3anc 1371	. . . . 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 41918	. . . . . . 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 6903	. . . . . . 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 7686	. . . . . 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 2772	. . . . 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 6888	. . . . 5 ⊢ (𝑤 = (𝑥( ·𝑠 ‘𝑀)(𝑧‘𝑣)) → (𝑦‘𝑤) = (𝑦‘(𝑥( ·𝑠 ‘𝑀)(𝑧‘𝑣))))
75	68, 73, 33, 74	fmptco 7123	. . . 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 2782	. . 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 20481	. . . . 5 ⊢ ((𝐴 ∈ LMod ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑥( ·𝑠 ‘𝐴)𝑧) ∈ (𝑀 LMHom 𝑀))
78	51, 19, 13, 77	syl3anc 1371	. . . 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 41915	. . . 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 2782	. 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 21410	1 ⊢ ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → 𝐴 ∈ AssAlg)

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  Vcvv 3474  {csn 4627   ↦ cmpt 5230   × cxp 5673   ∘ ccom 5679  ⟶wf 6536  ‘cfv 6540  (class class class)co 7405   ∘_f cof 7664  Basecbs 17140  ._rcmulr 17194  Scalarcsca 17196   ·_𝑠 cvsca 17197  Ringcrg 20049  CRingccrg 20050  LModclmod 20463   LMHom clmhm 20622  AssAlgcasa 21396  MEndocmend 41902

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-tp 4632  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-of 7666  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-er 8699  df-map 8818  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-nn 12209  df-2 12271  df-3 12272  df-4 12273  df-5 12274  df-6 12275  df-n0 12469  df-z 12555  df-uz 12819  df-fz 13481  df-struct 17076  df-sets 17093  df-slot 17111  df-ndx 17123  df-base 17141  df-plusg 17206  df-mulr 17207  df-sca 17209  df-vsca 17210  df-0g 17383  df-mgm 18557  df-sgrp 18606  df-mnd 18622  df-mhm 18667  df-grp 18818  df-minusg 18819  df-ghm 19084  df-cmn 19644  df-abl 19645  df-mgp 19982  df-ur 19999  df-ring 20051  df-cring 20052  df-lmod 20465  df-lmhm 20625  df-assa 21399  df-mend 41903

This theorem is referenced by: (None)