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Theorem mendassa 41507
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.
StepHypRef Expression
1 mendassa.a . . . 4 𝐴 = (MEndo‘𝑀)
21mendbas 41497 . . 3 (𝑀 LMHom 𝑀) = (Base‘𝐴)
32a1i 11 . 2 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → (𝑀 LMHom 𝑀) = (Base‘𝐴))
4 mendassa.s . . . 4 𝑆 = (Scalar‘𝑀)
51, 4mendsca 41502 . . 3 𝑆 = (Scalar‘𝐴)
65a1i 11 . 2 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → 𝑆 = (Scalar‘𝐴))
7 eqidd 2737 . 2 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → (Base‘𝑆) = (Base‘𝑆))
8 eqidd 2737 . 2 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → ( ·𝑠𝐴) = ( ·𝑠𝐴))
9 eqidd 2737 . 2 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → (.r𝐴) = (.r𝐴))
101, 4mendlmod 41506 . 2 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → 𝐴 ∈ LMod)
111mendring 41505 . . 3 (𝑀 ∈ LMod → 𝐴 ∈ Ring)
1211adantr 481 . 2 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → 𝐴 ∈ Ring)
13 simpr 485 . 2 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → 𝑆 ∈ CRing)
14 simpr3 1196 . . . . . . 7 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑧 ∈ (𝑀 LMHom 𝑀))
15 eqid 2736 . . . . . . . 8 (Base‘𝑀) = (Base‘𝑀)
1615, 15lmhmf 20495 . . . . . . 7 (𝑧 ∈ (𝑀 LMHom 𝑀) → 𝑧:(Base‘𝑀)⟶(Base‘𝑀))
1714, 16syl 17 . . . . . 6 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑧:(Base‘𝑀)⟶(Base‘𝑀))
1817ffvelcdmda 7035 . . . . 5 ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) ∧ 𝑣 ∈ (Base‘𝑀)) → (𝑧𝑣) ∈ (Base‘𝑀))
1917feqmptd 6910 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑧 = (𝑣 ∈ (Base‘𝑀) ↦ (𝑧𝑣)))
20 simpr1 1194 . . . . . . 7 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑥 ∈ (Base‘𝑆))
21 simpr2 1195 . . . . . . 7 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑦 ∈ (𝑀 LMHom 𝑀))
22 eqid 2736 . . . . . . . 8 ( ·𝑠𝑀) = ( ·𝑠𝑀)
23 eqid 2736 . . . . . . . 8 (Base‘𝑆) = (Base‘𝑆)
24 eqid 2736 . . . . . . . 8 ( ·𝑠𝐴) = ( ·𝑠𝐴)
251, 22, 2, 4, 23, 15, 24mendvsca 41504 . . . . . . 7 ((𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (𝑥( ·𝑠𝐴)𝑦) = (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠𝑀)𝑦))
2620, 21, 25syl2anc 584 . . . . . 6 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥( ·𝑠𝐴)𝑦) = (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠𝑀)𝑦))
27 fvexd 6857 . . . . . . 7 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (Base‘𝑀) ∈ V)
28 simplr1 1215 . . . . . . 7 ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) ∧ 𝑤 ∈ (Base‘𝑀)) → 𝑥 ∈ (Base‘𝑆))
29 fvexd 6857 . . . . . . 7 ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) ∧ 𝑤 ∈ (Base‘𝑀)) → (𝑦𝑤) ∈ V)
30 fconstmpt 5694 . . . . . . . 8 ((Base‘𝑀) × {𝑥}) = (𝑤 ∈ (Base‘𝑀) ↦ 𝑥)
3130a1i 11 . . . . . . 7 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((Base‘𝑀) × {𝑥}) = (𝑤 ∈ (Base‘𝑀) ↦ 𝑥))
3215, 15lmhmf 20495 . . . . . . . . 9 (𝑦 ∈ (𝑀 LMHom 𝑀) → 𝑦:(Base‘𝑀)⟶(Base‘𝑀))
3321, 32syl 17 . . . . . . . 8 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑦:(Base‘𝑀)⟶(Base‘𝑀))
3433feqmptd 6910 . . . . . . 7 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑦 = (𝑤 ∈ (Base‘𝑀) ↦ (𝑦𝑤)))
3527, 28, 29, 31, 34offval2 7637 . . . . . 6 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠𝑀)𝑦) = (𝑤 ∈ (Base‘𝑀) ↦ (𝑥( ·𝑠𝑀)(𝑦𝑤))))
3626, 35eqtrd 2776 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥( ·𝑠𝐴)𝑦) = (𝑤 ∈ (Base‘𝑀) ↦ (𝑥( ·𝑠𝑀)(𝑦𝑤))))
37 fveq2 6842 . . . . . 6 (𝑤 = (𝑧𝑣) → (𝑦𝑤) = (𝑦‘(𝑧𝑣)))
3837oveq2d 7373 . . . . 5 (𝑤 = (𝑧𝑣) → (𝑥( ·𝑠𝑀)(𝑦𝑤)) = (𝑥( ·𝑠𝑀)(𝑦‘(𝑧𝑣))))
3918, 19, 36, 38fmptco 7075 . . . 4 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((𝑥( ·𝑠𝐴)𝑦) ∘ 𝑧) = (𝑣 ∈ (Base‘𝑀) ↦ (𝑥( ·𝑠𝑀)(𝑦‘(𝑧𝑣)))))
40 simplr1 1215 . . . . 5 ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) ∧ 𝑣 ∈ (Base‘𝑀)) → 𝑥 ∈ (Base‘𝑆))
41 fvexd 6857 . . . . 5 ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) ∧ 𝑣 ∈ (Base‘𝑀)) → (𝑦‘(𝑧𝑣)) ∈ V)
42 fconstmpt 5694 . . . . . 6 ((Base‘𝑀) × {𝑥}) = (𝑣 ∈ (Base‘𝑀) ↦ 𝑥)
4342a1i 11 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((Base‘𝑀) × {𝑥}) = (𝑣 ∈ (Base‘𝑀) ↦ 𝑥))
44 eqid 2736 . . . . . . . 8 (.r𝐴) = (.r𝐴)
451, 2, 44mendmulr 41501 . . . . . . 7 ((𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑦(.r𝐴)𝑧) = (𝑦𝑧))
4621, 14, 45syl2anc 584 . . . . . 6 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑦(.r𝐴)𝑧) = (𝑦𝑧))
47 fcompt 7079 . . . . . . 7 ((𝑦:(Base‘𝑀)⟶(Base‘𝑀) ∧ 𝑧:(Base‘𝑀)⟶(Base‘𝑀)) → (𝑦𝑧) = (𝑣 ∈ (Base‘𝑀) ↦ (𝑦‘(𝑧𝑣))))
4833, 17, 47syl2anc 584 . . . . . 6 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑦𝑧) = (𝑣 ∈ (Base‘𝑀) ↦ (𝑦‘(𝑧𝑣))))
4946, 48eqtrd 2776 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑦(.r𝐴)𝑧) = (𝑣 ∈ (Base‘𝑀) ↦ (𝑦‘(𝑧𝑣))))
5027, 40, 41, 43, 49offval2 7637 . . . 4 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠𝑀)(𝑦(.r𝐴)𝑧)) = (𝑣 ∈ (Base‘𝑀) ↦ (𝑥( ·𝑠𝑀)(𝑦‘(𝑧𝑣)))))
5139, 50eqtr4d 2779 . . 3 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((𝑥( ·𝑠𝐴)𝑦) ∘ 𝑧) = (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠𝑀)(𝑦(.r𝐴)𝑧)))
5210adantr 481 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝐴 ∈ LMod)
532, 5, 24, 23lmodvscl 20339 . . . . 5 ((𝐴 ∈ LMod ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (𝑥( ·𝑠𝐴)𝑦) ∈ (𝑀 LMHom 𝑀))
5452, 20, 21, 53syl3anc 1371 . . . 4 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥( ·𝑠𝐴)𝑦) ∈ (𝑀 LMHom 𝑀))
551, 2, 44mendmulr 41501 . . . 4 (((𝑥( ·𝑠𝐴)𝑦) ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → ((𝑥( ·𝑠𝐴)𝑦)(.r𝐴)𝑧) = ((𝑥( ·𝑠𝐴)𝑦) ∘ 𝑧))
5654, 14, 55syl2anc 584 . . 3 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((𝑥( ·𝑠𝐴)𝑦)(.r𝐴)𝑧) = ((𝑥( ·𝑠𝐴)𝑦) ∘ 𝑧))
5712adantr 481 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝐴 ∈ Ring)
582, 44ringcl 19981 . . . . 5 ((𝐴 ∈ Ring ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑦(.r𝐴)𝑧) ∈ (𝑀 LMHom 𝑀))
5957, 21, 14, 58syl3anc 1371 . . . 4 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑦(.r𝐴)𝑧) ∈ (𝑀 LMHom 𝑀))
601, 22, 2, 4, 23, 15, 24mendvsca 41504 . . . 4 ((𝑥 ∈ (Base‘𝑆) ∧ (𝑦(.r𝐴)𝑧) ∈ (𝑀 LMHom 𝑀)) → (𝑥( ·𝑠𝐴)(𝑦(.r𝐴)𝑧)) = (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠𝑀)(𝑦(.r𝐴)𝑧)))
6120, 59, 60syl2anc 584 . . 3 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥( ·𝑠𝐴)(𝑦(.r𝐴)𝑧)) = (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠𝑀)(𝑦(.r𝐴)𝑧)))
6251, 56, 613eqtr4d 2786 . 2 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((𝑥( ·𝑠𝐴)𝑦)(.r𝐴)𝑧) = (𝑥( ·𝑠𝐴)(𝑦(.r𝐴)𝑧)))
63 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))
644, 23, 15, 22, 22lmhmlin 20496 . . . . . 6 ((𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑥 ∈ (Base‘𝑆) ∧ (𝑧𝑣) ∈ (Base‘𝑀)) → (𝑦‘(𝑥( ·𝑠𝑀)(𝑧𝑣))) = (𝑥( ·𝑠𝑀)(𝑦‘(𝑧𝑣))))
6563, 40, 18, 64syl3anc 1371 . . . . 5 ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) ∧ 𝑣 ∈ (Base‘𝑀)) → (𝑦‘(𝑥( ·𝑠𝑀)(𝑧𝑣))) = (𝑥( ·𝑠𝑀)(𝑦‘(𝑧𝑣))))
6665mpteq2dva 5205 . . . 4 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑣 ∈ (Base‘𝑀) ↦ (𝑦‘(𝑥( ·𝑠𝑀)(𝑧𝑣)))) = (𝑣 ∈ (Base‘𝑀) ↦ (𝑥( ·𝑠𝑀)(𝑦‘(𝑧𝑣)))))
67 simplll 773 . . . . . 6 ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) ∧ 𝑣 ∈ (Base‘𝑀)) → 𝑀 ∈ LMod)
6815, 4, 22, 23lmodvscl 20339 . . . . . 6 ((𝑀 ∈ LMod ∧ 𝑥 ∈ (Base‘𝑆) ∧ (𝑧𝑣) ∈ (Base‘𝑀)) → (𝑥( ·𝑠𝑀)(𝑧𝑣)) ∈ (Base‘𝑀))
6967, 40, 18, 68syl3anc 1371 . . . . 5 ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) ∧ 𝑣 ∈ (Base‘𝑀)) → (𝑥( ·𝑠𝑀)(𝑧𝑣)) ∈ (Base‘𝑀))
701, 22, 2, 4, 23, 15, 24mendvsca 41504 . . . . . . 7 ((𝑥 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑥( ·𝑠𝐴)𝑧) = (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠𝑀)𝑧))
7120, 14, 70syl2anc 584 . . . . . 6 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥( ·𝑠𝐴)𝑧) = (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠𝑀)𝑧))
72 fvexd 6857 . . . . . . 7 ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) ∧ 𝑣 ∈ (Base‘𝑀)) → (𝑧𝑣) ∈ V)
7327, 40, 72, 43, 19offval2 7637 . . . . . 6 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠𝑀)𝑧) = (𝑣 ∈ (Base‘𝑀) ↦ (𝑥( ·𝑠𝑀)(𝑧𝑣))))
7471, 73eqtrd 2776 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥( ·𝑠𝐴)𝑧) = (𝑣 ∈ (Base‘𝑀) ↦ (𝑥( ·𝑠𝑀)(𝑧𝑣))))
75 fveq2 6842 . . . . 5 (𝑤 = (𝑥( ·𝑠𝑀)(𝑧𝑣)) → (𝑦𝑤) = (𝑦‘(𝑥( ·𝑠𝑀)(𝑧𝑣))))
7669, 74, 34, 75fmptco 7075 . . . 4 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑦 ∘ (𝑥( ·𝑠𝐴)𝑧)) = (𝑣 ∈ (Base‘𝑀) ↦ (𝑦‘(𝑥( ·𝑠𝑀)(𝑧𝑣)))))
7766, 76, 503eqtr4d 2786 . . 3 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑦 ∘ (𝑥( ·𝑠𝐴)𝑧)) = (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠𝑀)(𝑦(.r𝐴)𝑧)))
782, 5, 24, 23lmodvscl 20339 . . . . 5 ((𝐴 ∈ LMod ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑥( ·𝑠𝐴)𝑧) ∈ (𝑀 LMHom 𝑀))
7952, 20, 14, 78syl3anc 1371 . . . 4 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥( ·𝑠𝐴)𝑧) ∈ (𝑀 LMHom 𝑀))
801, 2, 44mendmulr 41501 . . . 4 ((𝑦 ∈ (𝑀 LMHom 𝑀) ∧ (𝑥( ·𝑠𝐴)𝑧) ∈ (𝑀 LMHom 𝑀)) → (𝑦(.r𝐴)(𝑥( ·𝑠𝐴)𝑧)) = (𝑦 ∘ (𝑥( ·𝑠𝐴)𝑧)))
8121, 79, 80syl2anc 584 . . 3 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑦(.r𝐴)(𝑥( ·𝑠𝐴)𝑧)) = (𝑦 ∘ (𝑥( ·𝑠𝐴)𝑧)))
8277, 81, 613eqtr4d 2786 . 2 (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑦(.r𝐴)(𝑥( ·𝑠𝐴)𝑧)) = (𝑥( ·𝑠𝐴)(𝑦(.r𝐴)𝑧)))
833, 6, 7, 8, 9, 10, 12, 13, 62, 82isassad 21270 1 ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → 𝐴 ∈ AssAlg)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1087   = wceq 1541  wcel 2106  Vcvv 3445  {csn 4586  cmpt 5188   × cxp 5631  ccom 5637  wf 6492  cfv 6496  (class class class)co 7357  f cof 7615  Basecbs 17083  .rcmulr 17134  Scalarcsca 17136   ·𝑠 cvsca 17137  Ringcrg 19964  CRingccrg 19965  LModclmod 20322   LMHom clmhm 20480  AssAlgcasa 21256  MEndocmend 41488
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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-n0 12414  df-z 12500  df-uz 12764  df-fz 13425  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-plusg 17146  df-mulr 17147  df-sca 17149  df-vsca 17150  df-0g 17323  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-mhm 18601  df-grp 18751  df-minusg 18752  df-ghm 19006  df-cmn 19564  df-abl 19565  df-mgp 19897  df-ur 19914  df-ring 19966  df-cring 19967  df-lmod 20324  df-lmhm 20483  df-assa 21259  df-mend 41489
This theorem is referenced by: (None)
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