Theorem mendassa 43635

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 43625	. . 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 43630	. . 3 ⊢ 𝑆 = (Scalar‘𝐴)
6	5	a1i 11	. 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → 𝑆 = (Scalar‘𝐴))
7		eqidd 2740	. 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → (Base‘𝑆) = (Base‘𝑆))
8		eqidd 2740	. 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → ( ·𝑠 ‘𝐴) = ( ·𝑠 ‘𝐴))
9		eqidd 2740	. 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → (.r‘𝐴) = (.r‘𝐴))
10	1, 4	mendlmod 43634	. 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → 𝐴 ∈ LMod)
11	1	mendring 43633	. . 3 ⊢ (𝑀 ∈ LMod → 𝐴 ∈ Ring)
12	11	adantr 481	. 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → 𝐴 ∈ Ring)
13		simpr3 1203	. . . . . . 7 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑧 ∈ (𝑀 LMHom 𝑀))
14		eqid 2739	. . . . . . . 8 ⊢ (Base‘𝑀) = (Base‘𝑀)
15	14, 14	lmhmf 21024	. . . . . . 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 7025	. . . . 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 6895	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑧 = (𝑣 ∈ (Base‘𝑀) ↦ (𝑧‘𝑣)))
19		simpr1 1201	. . . . . . 7 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑥 ∈ (Base‘𝑆))
20		simpr2 1202	. . . . . . 7 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑦 ∈ (𝑀 LMHom 𝑀))
21		eqid 2739	. . . . . . . 8 ⊢ ( ·𝑠 ‘𝑀) = ( ·𝑠 ‘𝑀)
22		eqid 2739	. . . . . . . 8 ⊢ (Base‘𝑆) = (Base‘𝑆)
23		eqid 2739	. . . . . . . 8 ⊢ ( ·𝑠 ‘𝐴) = ( ·𝑠 ‘𝐴)
24	1, 21, 2, 4, 22, 14, 23	mendvsca 43632	. . . . . . 7 ⊢ ((𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (𝑥( ·𝑠 ‘𝐴)𝑦) = (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠 ‘𝑀)𝑦))
25	19, 20, 24	syl2anc 590	. . . . . 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 6842	. . . . . . 7 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (Base‘𝑀) ∈ V)
27		simplr1 1222	. . . . . . 7 ⊢ ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) ∧ 𝑤 ∈ (Base‘𝑀)) → 𝑥 ∈ (Base‘𝑆))
28		fvexd 6842	. . . . . . 7 ⊢ ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) ∧ 𝑤 ∈ (Base‘𝑀)) → (𝑦‘𝑤) ∈ V)
29		fconstmpt 5680	. . . . . . . 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 21024	. . . . . . . . 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 6895	. . . . . . 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 7640	. . . . . 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 2774	. . . . 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 6827	. . . . . 6 ⊢ (𝑤 = (𝑧‘𝑣) → (𝑦‘𝑤) = (𝑦‘(𝑧‘𝑣)))
37	36	oveq2d 7372	. . . . 5 ⊢ (𝑤 = (𝑧‘𝑣) → (𝑥( ·𝑠 ‘𝑀)(𝑦‘𝑤)) = (𝑥( ·𝑠 ‘𝑀)(𝑦‘(𝑧‘𝑣))))
38	17, 18, 35, 37	fmptco 7071	. . . 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 1222	. . . . 5 ⊢ ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) ∧ 𝑣 ∈ (Base‘𝑀)) → 𝑥 ∈ (Base‘𝑆))
40		fvexd 6842	. . . . 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 5680	. . . . . 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 2739	. . . . . . . 8 ⊢ (.r‘𝐴) = (.r‘𝐴)
44	1, 2, 43	mendmulr 43629	. . . . . . 7 ⊢ ((𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑦(.r‘𝐴)𝑧) = (𝑦 ∘ 𝑧))
45	20, 13, 44	syl2anc 590	. . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑦(.r‘𝐴)𝑧) = (𝑦 ∘ 𝑧))
46		fcompt 7075	. . . . . . 7 ⊢ ((𝑦:(Base‘𝑀)⟶(Base‘𝑀) ∧ 𝑧:(Base‘𝑀)⟶(Base‘𝑀)) → (𝑦 ∘ 𝑧) = (𝑣 ∈ (Base‘𝑀) ↦ (𝑦‘(𝑧‘𝑣))))
47	32, 16, 46	syl2anc 590	. . . . . 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 2774	. . . . 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 7640	. . . 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 2777	. . 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 20868	. . . . 5 ⊢ ((𝐴 ∈ LMod ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (𝑥( ·𝑠 ‘𝐴)𝑦) ∈ (𝑀 LMHom 𝑀))
53	51, 19, 20, 52	syl3anc 1379	. . . 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 43629	. . . 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 590	. . 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 20222	. . . . 5 ⊢ ((𝐴 ∈ Ring ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑦(.r‘𝐴)𝑧) ∈ (𝑀 LMHom 𝑀))
58	56, 20, 13, 57	syl3anc 1379	. . . 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 43632	. . . 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 590	. . 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 2784	. 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 1223	. . . . . 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 21025	. . . . . 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 1379	. . . . 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 5165	. . . 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 780	. . . . . 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 20868	. . . . . 6 ⊢ ((𝑀 ∈ LMod ∧ 𝑥 ∈ (Base‘𝑆) ∧ (𝑧‘𝑣) ∈ (Base‘𝑀)) → (𝑥( ·𝑠 ‘𝑀)(𝑧‘𝑣)) ∈ (Base‘𝑀))
68	66, 39, 17, 67	syl3anc 1379	. . . . 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 43632	. . . . . . 7 ⊢ ((𝑥 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑥( ·𝑠 ‘𝐴)𝑧) = (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠 ‘𝑀)𝑧))
70	19, 13, 69	syl2anc 590	. . . . . 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 6842	. . . . . . 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 7640	. . . . . 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 2774	. . . . 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 6827	. . . . 5 ⊢ (𝑤 = (𝑥( ·𝑠 ‘𝑀)(𝑧‘𝑣)) → (𝑦‘𝑤) = (𝑦‘(𝑥( ·𝑠 ‘𝑀)(𝑧‘𝑣))))
75	68, 73, 33, 74	fmptco 7071	. . . 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 2784	. . 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 20868	. . . . 5 ⊢ ((𝐴 ∈ LMod ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑥( ·𝑠 ‘𝐴)𝑧) ∈ (𝑀 LMHom 𝑀))
78	51, 19, 13, 77	syl3anc 1379	. . . 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 43629	. . . 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 590	. . 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 2784	. 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 21840	1 ⊢ ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → 𝐴 ∈ AssAlg)

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  Vcvv 3431  {csn 4555   ↦ cmpt 5153   × cxp 5616   ∘ ccom 5622  ⟶wf 6481  ‘cfv 6485  (class class class)co 7356   ∘_f cof 7618  Basecbs 17170  ._rcmulr 17212  Scalarcsca 17214   ·_𝑠 cvsca 17215  Ringcrg 20205  CRingccrg 20206  LModclmod 20850   LMHom clmhm 21009  AssAlgcasa 21825  MEndocmend 43616

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 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-tp 4560  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-of 7620  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-er 8633  df-map 8765  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12166  df-2 12235  df-3 12236  df-4 12237  df-5 12238  df-6 12239  df-n0 12429  df-z 12516  df-uz 12780  df-fz 13453  df-struct 17108  df-sets 17125  df-slot 17143  df-ndx 17155  df-base 17171  df-plusg 17224  df-mulr 17225  df-sca 17227  df-vsca 17228  df-0g 17395  df-mgm 18599  df-sgrp 18678  df-mnd 18694  df-mhm 18742  df-grp 18903  df-minusg 18904  df-ghm 19179  df-cmn 19748  df-abl 19749  df-mgp 20113  df-rng 20125  df-ur 20154  df-ring 20207  df-cring 20208  df-lmod 20852  df-lmhm 21012  df-assa 21828  df-mend 43617

This theorem is referenced by: (None)